Action Algebra[1]

A Model for Math and a Handbook for Arithmetic to Algebra

Royal Lyon Publications Klamath Falls, Oregon

Action Algebra A Model for Math and a Handbook for Arithmetic to Algebra

Copyright ©2010 by Ed Lyons All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author. Printed in the United States of America ISBN: 1453612122 First printing, June 2010 Second printing, September 2010 Corrections and additions are posted at our website: ActionAlgebra.com

Table of Contents Introduction Arithmetic to Algebra In Just Ten Minutes! . . . 8

Basic Principles Equality . . . . . . . . . . . . . . . . . . . 11 Value Change Needs Counterchange . . . . 12 Priority . . . . . . . . . . . . . . . . . . . . 13 First Calculate the Complicated . . . . . . . 13 Insight . . . . . . . . . . . . . . . . . . . . 15 Changing Looks does not Change Value . . . 15

Action Chart Zoom Levels Arithmetic: Numbers 1) Benefits of a Number System Line . . . . 20 2) Translating Numbers and Words . . . . . 21 3) Numbers are arrows . . . . . . . . . . . 21 4) Comparing numbers . . . . . . . . . . . 23 5) Kinds of numbers . . . . . . . . . . . . . 25 6) Parts of compound numbers . . . . . . . 26

Arithmetic: Combine 7) Adding on a Number Line . . . . . . . . 30 8) Subtracting on a Number Line . . . . . . 31 9) Adding Stacked Numbers . . . . . . . . . 32 10) Subtracting Stacked Numbers . . . . . . 33 11) Combining Stacked Integers . . . . . . . 34 12) Series of Signs . . . . . . . . . . . . . . 37 13) Combining Series of Signs . . . . . . . . 37 14) Combining Big Integers . . . . . . . . . 38 15) Combining Decimals . . . . . . . . . . . 38 16) Combining Tags . . . . . . . . . . . . . 39

Arithmetic: Multiply 17) Multiplying on a Grid . . . . . . . . . . 42 18) Learning the Times Table . . . . . . . . 45 19) Learning Multiples . . . . . . . . . . . . 47 20) Negative Numbers on a Grid . . . . . . 47 21) Multiplying Big Numbers . . . . . . . . . 51 22) Multiplying Bigger-Smaller . . . . . . . 53 23) Multiplying Decimals . . . . . . . . . . . 53 24) Multiplying Fractions . . . . . . . . . . . 54 25) Multiplying Tags by Merging . . . . . . . 55 26) Finding Common Multiples . . . . . . . 56

Arithmetic: Divide 27) Dividing on a Grid . . . . . . . . . . . . 58 28) Learning How to Shift . . . . . . . . . . 59 29) Dividing and Bigger-Smaller . . . . . . . 62 30) Speed Division . . . . . . . . . . . . . . 62 31) Long Division . . . . . . . . . . . . . . 63 32) Long Division with Decimals . . . . . . . 64 33) Factoring . . . . . . . . . . . . . . . . . 65 34) Prime Factoring . . . . . . . . . . . . . 66 35) Finding Common Factors . . . . . . . . 67 36) Reducing Fractions . . . . . . . . . . . . 68 37) Dividing Fractions Using Reciprocals . . . 70 38) Making Like Fractions . . . . . . . . . . 71 39) Combining Fractions . . . . . . . . . . . 72 40) Canceling Tags . . . . . . . . . . . . . . 73

Pre-Algebra: Exponents 41) Basics . . . . . . . . . . . . . . . . . . 76 42) Negative Exponents . . . . . . . . . . . 78 43) Multiplying Bases . . . . . . . . . . . . 80 44) Dividing Bases . . . . . . . . . . . . . . 81 45) Zoom Levels . . . . . . . . . . . . . . . 82 46) Groups with exponents . . . . . . . . . 84 47) Exponents in Fractions . . . . . . . . . . 84 48) Scientific Numbers . . . . . . . . . . . . 86 49) Adjust Scientifics . . . . . . . . . . . . . 87 50) Multiply Scientifics . . . . . . . . . . . . 88 51) Powers of Scientifics . . . . . . . . . . . 89 52) Dividing Scientifics . . . . . . . . . . . . 90 53) Combining Scientifics . . . . . . . . . . 90

Pre-Algebra: Morphs 54) Fractions and Mixed Numbers . . . . . . 92 55) Rounding . . . . . . . . . . . . . . . . 93 56) Fractions and Decimals . . . . . . . . . 94 57) Fractions and Percents . . . . . . . . . . 95 58) Decimals and Percents . . . . . . . . . . 96 59) Units . . . . . . . . . . . . . . . . . . . 97 60) Metric Units . . . . . . . . . . . . . . . 100

Pre-Algebra: Calculate 61) MUD before COLT . . . . . . . . . . . 62) FUN before MUD . . . . . . . . . . . . 63) IN before FUN . . . . . . . . . . . . . 64) Order with fractions . . . . . . . . . .

104 105 106 107

65) Nesting . . . . . . . . . . . . . . . . . 66) Absolute Value . . . . . . . . . . . . . 67) Formulas . . . . . . . . . . . . . . . . 68) Units in Formulas . . . . . . . . . . . . 69) 2D Shapes . . . . . . . . . . . . . . . 70) 3D Shapes . . . . . . . . . . . . . . . 71) Averages . . . . . . . . . . . . . . . . 72) Rates . . . . . . . . . . . . . . . . . . 73) Ratios . . . . . . . . . . . . . . . . .

108 108 109 110 112 113 114 114 116

Pre-Algebra: Roots 74) What is a Root? . . . . . . . . . . . . . 75) Reducing Roots . . . . . . . . . . . . . 76) Combining Roots . . . . . . . . . . . . 77) Multiplying Roots . . . . . . . . . . . . 78) Fractional Exponents . . . . . . . . . . 79) Rationalize Roots . . . . . . . . . . . . 80) Roots with Same Base . . . . . . . . .

118 119 120 121 122 123 124

Algebra: Polynomials Thinking in Algebra . . . . . . . . . . . . . As Few Variables as Possible . . . . . . . . 81) Distribution . . . . . . . . . . . . . . . 82) FOIL . . . . . . . . . . . . . . . . . . 83) Rationalize with conjugates . . . . . . . 84) Common Factoring . . . . . . . . . . . 85) Bifactoring . . . . . . . . . . . . . . . 86) Bifactor when a>1 . . . . . . . . . . . 87) Bifactor- other . . . . . . . . . . . . . 88) Squares . . . . . . . . . . . . . . . . . 89) Double factoring . . . . . . . . . . . . 90) Quadratic Formula . . . . . . . . . . . 91) Reduce fractions . . . . . . . . . . . . 92) Multiply fractions . . . . . . . . . . . . 93) Combine fractions . . . . . . . . . . . 94) Long Division . . . . . . . . . . . . .

126 129 130 131 132 133 133 135 136 137 137 138 140 141 141 142

Algebra: Linear Equations 95) Recognize equation types . . . . . . . . 96) Answer! . . . . . . . . . . . . . . . . . 97) Break variable term . . . . . . . . . . . 98) Combine like terms . . . . . . . . . . . 99) Decouple like terms . . . . . . . . . . . 100) Eliminate decimals . . . . . . . . . . 101) Eliminate fractions . . . . . . . . . . 102) Fill Parentheses . . . . . . . . . . . . 103) Flip complex fractions . . . . . . . . .

144 146 147 148 149 150 151 152 153

104) Figure functions . . . . . . . . . . . . 154 105) Proportions . . . . . . . . . . . . . . 155

Algebra: Quadratic Equations 106) Fill, Flip, or Figure . . . . . . . . . . . 107) Eliminate fractions or decimals . . . . 108) Descending order = 0 . . . . . . . . . 109) Common factor . . . . . . . . . . . . 110) Bifactor . . . . . . . . . . . . . . . . 111) Answer formula . . . . . . . . . . . .

156 157 158 159 160 161

Algebra: Other Equations 112) Linear . . . . . . . . . . . . . . . . . 113) Rational . . . . . . . . . . . . . . . . 114) Multi-variable . . . . . . . . . . . . . 115) Exponential . . . . . . . . . . . . . . 116) Inequalities . . . . . . . . . . . . . . 117) Radical . . . . . . . . . . . . . . . . System of Equations . . . . . . . . . . . . 118) Systems by Substitution . . . . . . . . 119) Systems by Elimination . . . . . . . . 120) Systems of Three . . . . . . . . . . .

162 163 165 166 167 168 169 169 172 175

Actions Explained Rule Sheet Goals & Methods Encrypted Education . . . . . . . . . . . . What Is Understanding? . . . . . . . . . . Readiness . . . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . My Student Is Stuck! . . . . . . . . . . . . Pre-Formal Math . . . . . . . . . . . . . . Activities . . . . . . . . . . . . . . . . . . Whiteboards and Vinyl . . . . . . . . . . .

Grade Sheets

194 195 196 197 199 200 200 202

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Action Algebra

Introduction This book is written for teachers and parents who want to understand the big picture of arithmetic to algebra so they can intelligently explain it to students in a connected framework. At the end of the book I show that a connected framework is understanding. The only assumptions I bring to this book is that you have a desire to see math in a new way and you have the ability to reason. I also assume that you took standard arithmetic and algebra courses sometime in the distant, hazy past and you may or may not have passed those classes. Some of you are teaching math whether you like it or not. Action Algebra covers the foundation or core of math from beginning numbers to advanced equations. I proceed in a logical, step-by-step manner in the same order of lessons 1-120 as with the students. Therefore, I leave some topics incomplete at their first presentation and finish them later after the additional principles are introduced. For example, in the second chapter on combining, only fractions with common denominators are used. Later, in the divison chapter, fractions with different denominators are covered. Some of you are in a position to teach your students from the very beginning, such as homeschoolers with young children. Others of you have some flexibility, but your student(s) already have years of habits (for better or for worse) ingrained in them. Still others of you are in a classroom with many students and a fixed curriculum. Understanding the common thinking processes connecting the huge variety of problems in the textbooks will be of help to any teacher in any situation. For example, many parents teaching Saxon are lost when trying to explain previous concepts more than a few lessons back. This handbook covers arithmetic, pre-algebra, and algebra in 40 lessons each. The focus is on the math, but with the worksheets and videos, many word problems are also covered. Topics that are applications or electives of math, such as statistics, geometry, trigonometry, and science are planned to be covered in future classes. Action Algebra lays a solid, complete system of understanding that will fully prepare a student for all their other courses. If possible, have your student(s) master these lessons before any other math. Believe it or not, every essential topic from arithmetic to algebra is covered in this book. The only thing "lacking" is the duplication of topics that the teach-reteach textbooks have made popular by their grossly inefficient methods. Years ago I figuratively started with grade 2 math and worked my way up to Algebra 2. I kept every new topic, but ripped out the pages dealing with a repeat or slight revision of the topic. At the end I had enough pages left to make two textbooks. That discovery spawned the development of this curriculum.

Introduction

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The Action Algebra worksheets can be used as a supplement to any curriculum, but the full power and time-savings will be realized when they are used as the curriculum itself. Because of their almost limitless variety and ability to be customized, students can study a topic in an organized, focused context, then practice it until it becomes automatic, then they can return to it for review as needed and at scheduled review points. There is a lot of time wasted in "gear changing" and the mind loses focus. (See the section at the end of the book on how to use the grade sheets.) So the full Action Algebra approach combines the best of both worlds. Repetitive drillwork is combined with a constructivist approach that results in students really knowing why and what they are doing. Students are not left to randomly discover principles, but neither are they engaged in almost mindless drill. They are guided to understand concepts and procedures in connection with each other. If American high school math students are ever to regain the top spot in the world, we must combine both approaches that are fighting with each other in the education arena. American ingenuity and American hard work are compatible, resulting in American excellence, quality, and performance. The 40 lessons (roughly one per week) in each class are not magic numbers. They could easily increase or decrease as time goes on. The point I am making with them now is that it is possible to cut the usual seat time in half or in third. For a student to accomplish this seemingly miraculous feat only means that they understand and review as they proceed. The consequences of this is that there will be more time to apply math both in math and science classes. It also means that schools will not only raise their graduation rates, but their average levels of achievement will raise much higher. Homeschoolers, of course, will cut down their time even further. But now the present lies nearer than the glorious future. For lower grade teachers I recommend reading the arithmetic and pre-algebra chapters. This will give you an understanding of the next level for which you are preparing your students. Just like with them, understanding the next level “seals in” the current level. For you, understanding the process of combining with negative numbers is crucial. If you feel you need more examples, please look at the worksheets and videos. For middle and high school teachers, the whole book is necessary, especially understanding how the Shift Action is involved in so many problems and steps. This is the single biggest concept that students need so they can tie together so many seemingly random steps. Also, the FA method of solving equations is very simple to teach as a unit outside of any textbook, then your curriculum can proceed with much greater ease. As with the elementary teachers, you may need more examples, so look at the worksheets (many of which have step by step solutions) and videos. One last note before launching into the math. You may want to look at the last chapters

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of the book on Goals and Methods, and on Pre-Formal education. Math is the most abstract of all classes and we must realize who the young students are that we teach as much as knowing how to teach a topic. Also, if you are thinking of using Action Algebra as your main curriculum, the Grade Sheet section will give you a good introduction. Now that we have addressed some technicalities, I hope you will find many Aha! moments as you begin studying this book. To help us get started with the big picture, Einstein will semi-seriously take us from arithmetic to algebra in ten minutes.

Arithmetic to Algebra In Just Ten Minutes! Once upon a time little Einstein stuck his finger in an olive and then in another olive and another and another until he had an olive on each finger. “Hmmm,” he thought, “There is something similar between the olives and my fingers. I have the same (what shall I call it?) number in both groups. As I was sticking my fingers in them I was counting.” Then he thought again. “What if I want to count more olives than I have fingers? I guess I should invent a symbol for each number and a way of re-using those symbols when I run out of fingers.” So little Einstein invented the number system with ten digits and place value. He was pretty clever about it, because his first digit, 0, represented having no olives on his fingers. Ten, 10, represented having his fingers completely full without any extra and ready to start putting olives on his mother’s fingers. The budding scientist was too smart to put them on his toes because he knew he would get them dirty and squish them sooner or later. Some days later, little Einstein started to get bored with his number system. He had counted all the olives in his father’s orchard, his neighbor’s orchard, and his uncle’s orchard across town. In fact, Einstein knew the number of olives in all the orchards around town. He also knew the numbers of cats, dogs, and horses. Yet little Einstein wanted to something more, something new. He sat down in the dirt road and thought and thought. Then it came to him! What if he could find out the number of all the olives in all the orchards together! Why not do something with the numbers he had already collected so that he could figure the answer without recounting?! Of course, that was a brilliant idea. He came up with a process of putting numbers together that he called adding. In no time flat, little Einstein knew the total of all the olives, animals, houses, and people in the town. Not long after that, he invented something called subtraction so he could accurately change his total when olives were eaten or exported. Sometimes, an animal died and he needed to take that into account as well.

Introduction

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Little Einstein was starting to catch on to the power of numbers and so it wasn’t long before he discovered he could multiply and divide the rows and columns of trees in the orchards to quickly find out how many were in each. As he shared his knowledge with the townspeople, they soon began to ask him questions and wanted to learn what he was doing. This forced little Einstein to invent symbols for each of his ideas so it could be written down and made permanent. So, in addition to his ten digits, he made symbols for his four operations that he could do with those numbers. One day at supper time he was struck with a puzzle. One of the olives he put on his finger split into pieces. He could not count them as 1, 2, 3 because they were not whole olives like the others. That’s when he realized he needed a way to keep track of partial things. Thus, fractions were born. He used a slash or a horizontal line because it reminded him of a cut. The top number represented how many pieces he ate and the bottom number represented the total number of pieces the olive had broken into. The number on top was usually smaller than the number on the bottom because some pieces fell on the floor. To save himself some time, little Einstein put a decimal at the end of the whole number of olives, then started counting tenths and tenths of tenths on the right side. That way he did not have draw a slash and put a bunch of zeroes below it. It was a special, convenient fraction that always meant tenths. Then, because he had whole numbers and tenths in a decimal number, he put a whole number in front of a fraction and called it a mixed number. Then, because people used dollars so much and were always figuring prices as some part of 100 pennies, he invented the percent symbol to make everybody’s life just a little bit easier. But it was his friend, Sherlock, who prompted Einstein to make some of his bigger discoveries. One day, Sherlock asked Einstein if he had any idea how many olives there might be in the whole world. Einstein replied that his friend’s question was not elementary. He would need to invent another kind of number to handle the enormous task of multiplying all the olives on all the trees in all the orchards of all the towns of all the countries of the world. So he made scientific numbers with a handy little device called an exponent, which compressed the multiplying of many numbers down into one. After all their research and calculations, Einstein and Sherlock discovered that some pieces of their data never changed and other data changed a lot. Einstein called the data pieces that stayed constant--get this-constants. Sherlock thought that bit of naming was too elementary, but could not argue with the logic. One of Einstein’s first constants was something he called “pie.” Actually, he spelled it “pi” because he did not want to offend any of his Greek neighbors. Pi was the ratio of the diameter of an olive to its circumference, which was always just about 3.14. Curious, eh?

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Right after eating pie and discovering pi, Einstein discovered “c.” This was the speed of light that he and Sherlock measured every time they took a flash picture of olives at night. Shortly thereafter, Einstein muttered E=mc2 as he tried to think of creative ways to destroy all the olives in a country in a very short time. But Einstein’s greatest number was still waiting to be discovered. In the laboratory, Sherlock was deep in calculations and very frustrated when Einstein walked in. “What’s the matter, old boy?” Einstein asked. Sherlock replied, “I go through the same long process over and over again as new numbers come in from the orchards. There must be a better way of solving these mysteries that are really just a repeat of the same kind of problem.” “Well now,” Einstein exclaimed, “Isn’t solving mysteries your cup of tea? Why should some unknown numbers stop you--.” Just at that moment, Einstein had an incredible insight. “Unknown numbers!” he cried.“They are not totally unknown. After all, we know they are numbers, we just don’t know exactly which one. The numbers just vary from time to time. Let’s call them variables and learn how to do adding, subtracting, multiplying, and dividing with variables!” Sherlock looked at the scientist with his mouth agape and jaw dropped. After a bit, he raised his index finger like he was checking the wind, and declared, “I think you’ve got an idea!” “If you could do that, we could make formulas and equations that hold the spots for our numbers before we get them from the orchards. We could do some of the calculations only once and never have to do them again! Instead of re-inventing the wheel and figuring out what to do with each number every time, we would have a template we could use over and over again. That’s just as good as recycling all the olive boxes!” Einstein raised his hand in the air as if he was posing on the steps of the Acropolis and pronounced, “We shall call doing math with variables, Algebra.”

Basic Principles

13

Basic Principles Three basic principles upon which math is founded are equality, priority, and insight. Applying them to math gives us: value change needs counterchange; first calculate the complicated; and, changing looks does not change value.

Equality Life is a constant balancing act. We have to balance work and play with rest and sleep. We need to get enough time alone to think for ourselves and do our own things, but we also need time with family and friends. We can’t be alone and with a group at the same time, so we have to balance our time between the two. Sometimes we might split our time between two different activities, like watching TV and doing homework. Yet, one still affects the other. We can’t do whatever we want whenever we want. Humans require balance or else we get sick or get a hangover. One way or another our lives demand, and get, balance. Balance is necessary because of limits. Unless the parents are infinitely wealthy, if sister gets more allowance, then brother gets less. If there are more eagles above the river, then there are less fish in the river. If you have driven more miles down the road, then there is less gas in the tank. These are like the Law of Conservation of Energy: Energy can neither be created nor destroyed, it just changes form. Cause matches effect. Action equals reaction. Input equals output. In other words, the pot of soup never grows, it just gets stirred. A math problem is the same. The answer must equal the problem. It is no different in value, just in format. For example 5 truckloads of 10

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crates of 200 boxes of cereal is a problem, not an answer to my question. I want to know how many boxes of cereal are coming to my store. 10,000 is an acceptable answer. 5 ^ 10 ^ 200 is accurate, but not acceptably simple enough. Now in my attempts to solve the problem, I cannot arbitrarily inflate a number or remove a number. I can do many things, but one thing I can never do is create or destroy value. Before must equal after at every step from beginning to end. Imagine a math problem taking place on a balance scale. (A very simple one can be a hanger with clothes pins holding different items in balance.) You can do whatever you want to items on one or both sides as long as your actions leave the hanger in balance. Folding a hanging sock doesn’t upset the balance so it is fine. However, removing a sock on one side requires the same kind of sock to be removed from the other side. Balance before = balance after. This principle of equality and balance seems to be telling us what we can’t do and therefore limits our options. However, it actually opens the door to two powerful Actions.

Value Change Needs Counterchange Because I cannot create or destroy but must maintain equality, I must make a counterchange for every change of value that I introduce. (Notice that I said, “I introduce.” I am not talking about the calculations that the problem tells me I must do.) For example, if I subtract 3 from one side, then I must subtract 3 from the other side. If I multiply the left by 28, then I need to multiply the right by 28. These are examples of the Sync Action. My change tips the scale out of balance, but my counterchange brings it back into balance, so that is perfectly “legal.” In other words, it really works. Another Action based on the principle of balance is Shift. It is used when I am dealing with only one side of an equation, or with an expression, which is a problem that is only one side of an equation. For example, if I have two water balloons hanging from the left side of my hanger and I want to take 6 oz. of water from one, then I must add 6 oz. of water to the other. In math this looks like 10 + 15 becoming 4 + 21. If I squeeze a balloon so that the top has less water the bottom automatically has more water. (Popping balloons not allowed!) In math this happens when we reduce fractions. 6/8 becomes 3/4. There are less pieces of the pie on top, but the size of the pieces got bigger on the bottom. This may not be readily apparent to you, but we will look at this Action many times with fractions and other examples. It is used a lot! Another example of Shift happens with units. When I change a 1 dollar bill I get 10 dimes in return. I have more things, but each item has less value. Another example from real life is air conditioning. To cool off my house I must heat

Basic Principles

15

up the outside. Think of this Action as the principle of the Up and Down. My house goes down in temperature, while the outside goes up in temperature. If you don’t believe this, go stick your hand over the exchanger!

Priority Years ago I saw a simple demonstration that I have never forgotten. A lady had a jar with several large rocks beside it. There was also a pile of gravel and sand. She put the sand in the jar, followed by the gravel, but only one rock would fit. Then she emptied the jar and started over from scratch. This time she put the rocks in first, and poured the gravel around them. Then she poured the sand in and shook the jar until it all fit. She succeeded by starting with the biggest stuff. Likewise, life is filled with order. You build a house from the foundation up and then from the outside in. Order matters or else the house will be ruined by the weather or collapse under a load. Life is filled with priorities. Starting the day with a good breakfast makes us healthier. To eat a good breakfast we have to wake up early, which means we need to go to bed on time. Getting our homework done on time gives us privileges like going outside to play and getting good grades. This gives us feelings of accomplishment and happiness. That makes us better, kinder people. Paying our bills before blowing our money on extras is another priority that wise people adhere to. Keeping one priority often helps us keep other priorities straight. As we figure out our priorities and follow them, that helps us achieve a balanced life.

First Calculate the Complicated Math also has its priorities. The important things must be calculated first, and that which is most complicated is most important. Why? You cannot count that which you do not know. Very loosely speaking, the simple goal of much of math is to count. We want a number, a value, which is a count of miles, hours, dollars, items, or other things. Before I can count, I must add, because adding is counting two or more groups of things. Before I can add, I must multiply, because multiplying is repetitive adding. Before I can multiply, I must calculate functions, because then I will know the final number to multiply. And in the midst of all that, I must pay attention to parentheses, because they can override anything at anytime. A pretty good rule of thumb is to first calculate the things you learned last. In other

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words, solve a problem in reverse order of when you learned the parts of it. For example, everyone learns adding before multiplying, but we should multiply before adding in a problem. Next comes exponents and other functions like roots and trigonometry. The order of learning these things will vary a little depending on the textbook, but all of these are on the same level of importance which is above multiplication. A simple real-life situation will illustrate the meaning. Let’s say it is your task to count the total production of toy blocks on a certain day. There is a pile of blocks waiting to be packaged. There are boxes of blocks stacked on pallets, and there is a machine cranking out blocks constantly. You can count the blocks in the pile easy enough, but to count the blocks in the boxes you must first count how many are in one box, then multiply by the number of boxes, then multiply by the number of pallets. You must go inside a box and count because you cannot count that which you do not know. Now you have a choice. You can wait for the machine to stop making blocks and let them get boxed up to do your count, or you can count what is available and keep them completely separate from the output of the machine while you wait. Either way you are giving priority to the machine before calculating your total. This simple illustration shows that functions (machines that, in professorial terms, map a set of numbers to another set) must be considered before boxes before loose items or you must have a way of separating them. Likewise, roots and logs come before multiplication which comes before addition, or you must have some way of completely separating them. Four Actions- In, Fun, MuD, COLTin that order- help us to calculate correctly. (IN FUNny MUD is a COLT) In, or Inbox, means I should work inside parentheses first. Parentheses ( ) and brackets [ ] and braces { } all act like mathematical boxes to group what is inside them. We must find a single value for the whole group before we can add or times it by what else is there. Fun is short for function. Exponents, logarithms, and trig functions are the

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common functions to be encountered in pre-algebra and algebra. They are little mysteries that must be unraveled before we know the final value we have to work with. MuD is short for Multiply and Divide. They are of equal importance because division is basically multiplication in reverse. So almost all rules that apply to multiplication apply to division, also. When talking about multiplication, keep division in mind. The NOPE trick of figuring negative and positive signs applies to both. COLT is short for Combine Only Like Terms (or Tags or Things). Combining is the all-in-one method of adding and subtracting that is covered in the second chapter. It is the one way that works for all of arithmetic and all of algebra. SSADDL is the how-to principle that goes with COLT, because every colt needs a saddle!

Insight We hope to raise our children with enough insight to know that changing costumes does not really change the actor. It is still the same person behind the mask. Similarly, we try to teach them that beauty is more than skin deep and that the value of persons does not depend on the color of their skin. Also, an old dollar is worth as much as a new dollar.

Changing Looks does not Change Value Most math steps depend on the Balance and Priority Actions, but the Insight Actions are nice helper tools. They are easy to use, but not needed as often. (So they tend to get forgotten.) This group of Actions are called Insight because it takes looking at the problem and your options in a slightly different way to figure out that if you used one of these, you could make things easier. None of these Actions changes the value of numbers, so counterchanges are not needed. The Show Action hides or unhides what is already there. Sort re-arranges what is there. Morph changes the form of a number into another equal form. Sub trades one value with another equal value. All of these Actions change only the looks of a number, but do not change the value of the number. It is like putting a new paint job on a car without changing the car itself. So that is a real quick introduction and overview of the 3 basic principles and the 10 Actions. As we proceed through the lessons I will amplify the Actions at appropriate points, then use them to explain the current problem.

Actions Action Chart First calculate the complicated

   

IN

You must first work inside the boxes ( ) [ ] { } and fraction bars to figure the answer you need to work with. Think of unwrapping a present from the inside out.

FUN

Then you must feed raw numbers into the mouth of the function (funnel, get it?!) to figure the answer you need to work with. Function processes input, you use only the output.

MUD

Then you can MUltiply and Divide all kinds by merging tags. Figure the sign by using NOPE- Negative Odd Positive Even. (MUD can get on all things, but do we like it? NOPE!)

COLT

Then Combine Only Like Things (Terms, Tags) by using SSADDL- Same Signs Add, Differents Destroy, Largest sign is answer sign. (SSADDL your COLT)

Value change must have counterchange

     

SYNC

You may do the same thing once to each whole side of an equation. 1 effect, 2 opposite sides.

SHIFT

You may change the value of an object at any time if you counter it with an equal, opposite change within that object. 2 opposite effects, 1 side.

SORT

Changing looks does not change values

You may re-arrange the objects in a level at any time, but never change a division part.

SHOW

You may show or hide invisible objects at any time.

MORPH

to another at any time.

SUB

You may convert an object from one format

You may replace object A with object B at any time, if A=B.

20

Action Algebra

Zoom Levels Zoom levels are not needed to teach or learn arithmetic, but these next two pages are inserted as an overview for teachers and they show what I meant by “objects” on the Action pages. Even teachers of basic math will benefit from this because they can see where their topics fit into the big picture.

equation

expression = expression term + term factor ^ factor

factor ^ factor

term + term factor ^ factor factor ^ factor

Zoom levels is a phrase I coined to describe the varying levels of focus in a math problem. Sometimes we are working on the factors within a term, while at other times we are working with the terms in an expression. As we advance through math it becomes increasingly important to be aware on what level we are on. Any given step of any problem takes place on only one level. The level can change from step to step, but it will never change within a step. For example, we don’t do something on the term level, then try to balance it on the factor level. As you can see in the diagram, factors multiply (or divide) to make a term (called “compound number” in arithmetic). Terms add (or subtract) to make up an expression. Expressions are linked with an equal sign to make an equation. So we have four levels where Actions happen. Groups, ( ) and [ ] and {} and fraction bars, can be used at any level. They can group factors into “superfactors” and terms into “superterms.” That is when our abstracting abilities really get tested. It happens a little bit in pre-algebra, but mostly in algebra. (Arithmetic teachers, you can breathe a sigh of relief!)

Zoom Levels

21

6^9 8xy -2(5+6x)

factor

factor

factor factor factor

factor

f a c t o r

Now let’s see what these things look like in real life. Any two things that multiply each other are factors. (Division is included, because division is reverse multiplication.) So all the different kinds of numbers and groups of numbers can be factors. It all depends on how they are connected. In the above examples you can see how the numbers and letters have a dual role. Not only are they numbers or variables, but they are also factors because they multiply each other. Also, as factors, they “bond” themselves into packages called, terms. The example on the right is interesting because of the grouping. The whole example is one big term made of -2 ^ ( ). However, inside the ( ) factor are two “subterms” of 5 and 6x. The 6x term has its own factors of 6 and x. Do you see why I call this “zoom levels?” You zoom in from the problem as a whole until you get down to the individual parts. One more note about the above examples: Each one is an expression, because an expression can be made of 1 term, just as a term can be made of only 1 factor. This means that a single number can be a single factor (times an invisible 1) making a single term making an expression. It all depends from which zoom level you choose to look at it.

3*8-5*7 3*8¤-5*7

6x+0-24y 6x¤+0¤-24y

9-7(4+9y) 9¤-7(4+9y)



4 ¤ + 9 y

A useful, and often challenging, exercise I do with algebra students is to give them random algebra expressions to be split into terms. I use a double slash or squiggly line so that it is not confused with some other symbol. The point is that students must “see” algebra. What helps me visualize this is I think of expressions like railroad trains. The terms are the cars coupled together with + and - signs. Inside the cars are boxes of stuff called factors. We can split trains apart at the couplings between the cars, but we never split the cars themselves because that would make a mess on the tracks. So the summary of the matter is that there are four zoom levels we need to be aware of as we progress through math. We will work with the objects in only one level at a time. That means factors are the objects in terms. Terms are the objects in expressions. Expressions are the objects in an equation.

22

Action Algebra

Arithmetic: Numbers This chapter covers numbers and the number system. It shows how numbers are arrows from the number line, which is an infinite arrow. We then compare numbers to each other. Finally, we identify the types of numbers and the parts of compound numbers. All of this is approached from a concrete, rather than abstract, perspective to make it clear on a child’s level.

1) Benefits of a Number System Line Of course, we just call it “number line,” but I am trying to capture the complete wisdom of the idea by saying, number system line. Without a number system, which requires the idea of place value, numbers would just be an endless invention of names. Not too helpful. If we did not organize numbers into an orderly sequence on a line, we would think of numbers in random order and places. Like counting the pennies in a jar, it would be too hard to precisely compare piles (sets) of things. Lining things up and comparing the lengths of the lines at a glance is the easiest way. If we are comparing two numbers and the number line is horizontal, then the number farthest to the right is greater. If the number line is turned vertically, then the highest number is greatest. The worksheet about locating numbers is a good time to point out that the number line starts at 0 and goes endlessly in both directions. It should also be pointed out that all counting begins at 0 and we count steps. Adults sometimes make the mistake of counting the starting point. Instead, we start at 0 and count the first step to 1, the second step to 2, and so on. This is like counting on our fingers and we already have 3 fingers up. We don’t start at 3 and count 1. We start at 3, move over

...,-3,-2,-1,0,1,2,3,... 0

Arithmetic: Numbers

23

to the fourth finger, then count 1. We are counting steps, not marks on the line. If the number is negative, we go down or to the left. This is just like a thermometer getting colder, or going down the stairs to the basement. (If you have anxieties about negative numbers, don’t show them. Children have not seen them before and so they have no hang-ups about them. Six year old children can easily do this even if they don’t have our abstract understanding of them. More details will be covered in lesson 3.)

2) Translating Numbers and Words From pre-formal activities, the student should already be familiar with both the idea and the wording of place value. Orally s/he should be able to count to 100, but now the transition to the written form needs to take place. This is one lesson where writing large may make a critical difference.

3) Numbers are arrows What was implied in lesson 1 is now clearly stated. A number is an arrow. It has both size and direction. It is a piece of a number line. To exactly describe the size of an arrow we use the digits 0 through 9 to “spell” numbers. The digit part of a number tells us the size of the number, which is the length of the arrow. Like an arrow, a number always has direction. In math, there are two basic directions: positive or negative. Arrows always have direction so numbers always have signs. Sign + digits = number. If you do not see a sign that means there is an invisible positive. You will never be wrong if you want to write it yourself. 8=+8 3=+3 +6=6 Numbers are not just bars. They are arrows. Think of their size AND direction. Think, I have $5 or I owe $5. I walk up 8 stairs or I walk down 8 stairs. The temperature is 15 degrees above zero or 15 degrees below zero. Some things may not be negative, but the numbers used to describe them can be. For example, can you eat -3 pieces of cake?! Can you be -5 feet tall? Of course not. My height can’t be negative. But I am always 5 feet tall whether I am climbing up a cliff or hanging upside down. Numbers always have size and direction.

24

Action Algebra

6 feet tall

6 feet above the floor=+6

floor = 0

6 feet tall

6 feet below the floor=-6

What is an arrow? An arrow is a line with a pointer on one end to tell us what direction it is going. The pointer end is called the head. The end without the pointer is the tail. This is where the arrow begins. Every arrow starts at 0 length and stretches out to its end where the pointer is. This is exactly what numbers do. They start at 0 on the number line and stretch a certain distance. The digits part of the number tells us the size, and the sign part of the number tells us the direction. + is up or to the right, and - is down or to the left. Just like an arrow is a line with a pointer, a number is a digit part with a sign part. We often leave out the sign which gives a number direction. Forgetting about it causes us to misunderstand adding and subtracting, which then causes us more problems when it comes time for pre-algebra. Even when we don’t see a sign in front of the digits, that just means there is an invisible + sign there. All “plain” numbers are positive. A number is negative only if there is a - sign in front. Negatives are normal Let’s look at the numbers that strike fear in the hearts of many. A negative number (or, a minus number, as some call them) is just a regular number that goes left on the number line instead of right. If the number line is in the vertical position then negatives go down. This is just like a thermometer that is minus 5 degrees below 0 when it is

Arithmetic: Numbers

25

really cold outside. It is also like being 1200 feet below sea level in Death Valley. There is nothing bad or different about negative numbers. In fact, they are really good when you are keeping score in golf! The real source of our anxiety about negative numbers comes from trying to add and subtract them. The typical way adding and subtracting is explained breaks down when it comes time to introduce negative numbers and it is this breakdown that is the real cause of confusion. The next chapter on combining will fix this problem.

4) Comparing numbers This lesson is not mathematically hard, but the language can be a little subtle. In normal life we use bigger, larger, and greater in similar ways to mean the same thing. However, in math we make a definite distinction between them. Bigger and larger mean the same thing: the size of the number, which is the length of the arrow. Greater, however, means position on the number line. Let me explain. Bigger, larger, smaller only want to know the size of the number, not the sign. Bigger and larger want to know which of two arrows stretches furthest from 0. The direction in which they stretch does not matter. Smaller does not care about direction either. It just want to know which arrow stretches the least from 0. These distinctions are useful when we subtract, because we want to make sure the bigger number is on top. We don’t care if it is positive or negative, only the size of the number determines if it is bigger, or larger.

26

Action Algebra

If you are familiar with absolute value, bigger/smaller is exactly the same idea. When we wonder if a number is greater than another, we are wondering if it is higher on the number line. As in the picture below, the greater number may actually be smaller, but because it ends at a higher spot, it is greater. So greater than and less than mean higher and lower. When you see these symbols < and > they are referring to greater/less than, not bigger/smaller.

+4

-8

+4 is greater than -8, because it is higher, but -8 is bigger than +4 because its arrow is longer Bigger, smaller, larger only see the length of the arrow, not its direction Greater than and less than include size and direction which gives a final position on the number line

When doing the worksheets it may help to put your finger over the signs when using bigger, larger, smaller. Now it is like both numbers are positive and pointing upward. That is exactly what absolute value will do later on, so this is not wrong or a “get by” trick. When working with greater/less than use a number line in the vertical position. Now it is easier to see that any positive number, no matter how small, is greater than any negative number, no matter how large. Also, any negative number is less than (lower) than any positive number. You should find students readily grasping the concepts separately, but may get mixed up when all the words are used on the all comparisons sheet. I wish I knew of an easy, obvious memory device here, but I don’t.

Arithmetic: Numbers

27

5) Kinds of numbers This is another memory lesson. All the students need to do is recognize the types of numbers. They do not need to do any comparison or math with them. This is like bird identification. The student does not need to know how they fly, just recognize what they look like. There are 8 basic kinds of numbers we use in arithmetic and algebra: 1) Integer. I use the more technical word instead of “whole number.” Whole number is not used consistently. Integers are simply positive and negative whole numbers, including 0. Integers are a subset of decimals.

+5

-2

16

-39

+8

2) Decimal. Anytime a number has a visible decimal point, I consider it a decimal. Technically, a number like 3.0 could also be considered an integer, so you can override the answer keys and give credit for that answer as well. For further study you could look up rational and irrational numbers in Wikipedia in case you come across it on standardized tests or textbooks, but the distinction is not central to arithmetic and algebra.

7.25

1.33...

0.09

.1

3) Percent. Usually it is the integers and decimals that have a % tacked on them, but any number type can be made into a percent. So I consider the % symbol trumps all else.

8.9%

6%

9 1/5%

30.18%

4) Fraction. Any kind of number can appear in a fraction on either the top or the bottom, or before or after the slash. However, we try to convert (Morph) the fraction into having only integers as soon as possible.

1/2

4/

13

-3/7

1 23

5) Mixed number, or simply, mixed. A visible integer next a fraction with only integers is a mixed. I rarely work with mixed numbers. Instead, I turn them into fractions, solve the problem, then re-convert back to a mixed.

5 1/2

-94/13

-11 3/7

1

62

6) Scientific. A decimal times 10 to a power is the basic form of a scientific number.

28

Action Algebra

Technically, the decimal must be between 1.0 and 9.99999... so that there is only a ones digit followed by decimal places. However, at this point, if any decimal followed by ^10N is called scientific, that is good enough.

2.7^104

9.003^10-15

7) Variable. Just an introduction is necessary here. Variables are letters that wrap mystery numbers within them. Algebra will tell us how to solve mystery numbers, but for arithmetic all we need to know is that they are shorthand numbers for things like apples, boxes, and miles. We need to know a little bit about the things so we know whether to add or not.

x

y

apple

a

8) Constant. There are just two special decimals that we abbreviate to letters in standard elementary math. They are & and e. A calculator will give you the long decimal values if you want them, but for now all the student needs to know is that & and e are constantly the same value in every problem.

&

e

i

6) Parts of compound numbers As I am sure you have already noticed, this first chapter on numbers has not been standard. While I have not relied on a young child’s inability to comprehend deep concepts, nevertheless a complete foundation has been laid for all the rest of arithmetic and algebra. There will be no need to teach, unteach, and then reteach. Starting with the very next chapter you will see the advantages of introducing all the details of numbers right up front. One continuous system and framework can be built that cuts out a tremendous amount of duplication and work arounds. Using the endless supply of Action Algebra worksheets, the student can progress at his or her own pace in a simple, straightforward fashion and still finish algebra years early. This leaves plenty of time for side topics, applications, and other investigations! I needed to say that to prepare you for wording new to you. Just like we have compound words, we have compound numbers. Fractions are good examples because they are numbers within numbers. The top (numerator) and the bottom

Arithmetic: Numbers

29

(denominator) are individual numbers, but stepping back and looking at the numbers with a bar in between we see a fraction. Thus, we have a compound number. A compound number is made up of a simple number (integer, decimal) followed by a tag. Tag “Tag” is not a regular math word. It is a word I made up to help you see the parts of a number and what they do. The tag always comes after the regular number and tells us what kind of compound number we are looking at. This is important because we must have matching tags before we can add or subtract two numbers, and we must know what is in the tag so we know what to multiply.

-1 8

xy

5 players 2 /3&

.3 7

/9

Regular numbers like 2 or 7.4 have blank tags. Sometimes the tag can be a variable, like x, or it can be an item from daily life, like shoes or books. Since fractions are compound numbers, they have tags you can see. The bottom number (or the right hand number if written sideways) is the tag. The fraction bar is included. (Algebra teachers: A tag is all factors in a term except the coefficient and/or numerator.) Compound Number Very simply put, a compound number, like a compound word, is made up of more than one part. Be mindful that one of the parts might be invisible. All compound numbers have at least two parts called the Front Number (frontnum, for short) and the Tag. An optional third part is attached to some numbers called the Unit (miles, feet, meters, pounds, etc...), but it is really part of the tag, also.

30

Action Algebra

Compound Number

Frontnum Tag Unit The algebra word for compound number is TERM. In short, the front number is the first part of every compound number and is the quantity part. It tells us how many tags we have. The frontnum is always an integer, decimal, or top of a fraction. Once in a while it is invisible, but we will talk about that later. The tag is everything after the frontnum that is attached by multiplication or division. This includes other numbers and all letters. Multiplication and division signs are included. These labels, “compound number” “frontnum” and “tag,” should seem new and strange to you because they are not standard vocabulary. However, they are labels for standard math items that you learned when you were in school that were left unnamed in the lower grades or not named at all. Young children can easily identify the parts of a compound number, even if they don’t yet understand everything those parts do. Rather than use a strange word like “coefficient” that still makes no real sense to me (a math teacher), they can easily and visually relate to “front number.” Term vs. compound number is a toss-up. If you want to skip the baby word and go right to “term” that would make sense to me, also. Tag labels the unlabeled so we have nothing to lose there. The main point is that children learning math for the first time will accept whatever words you use. What might feel strange to you will be accepted as normal by them.

Arithmetic: Numbers

31

32

Action Algebra

Arithmetic: Combine In this chapter we will learn how to add and subtract compound numbers. Adding and subtracting are just pieces of an overall process that is called “combining.” It is easier to learn how to combine positive and negative numbers right from the beginning. This chapter introduces our first Action, which is called COLT- Combine Only Like Things (like things have like tags). As you learn about adding and subtracting, you will begin to see that they are just like walking up and down the stairs of the icon. Above the water line is positive and below the water line is negative. We will start with a proper understanding of adding and subtracting, but quickly move to the all-in-one method of combining. This method will not only work for arithmetic, but it is also the far better way of adding and subtracting terms in algebra.



7) Adding on a Number Line

We usually think of adding this way: Put two numbers together to get a bigger number. If you use “bigger” the same way we used it in the previous chapter, then you are right. However, most people don’t use it that way or teach it that way. They fear negative numbers, so they think of only the up direction for bigger. They don’t realize that numbers also get bigger as you go farther from 0 in the down direction. But the previous chapter showed us they do!

-8 -6

-2

+5 0

-8 -4

-4

+3

+6 +11 +4 +7

Arrows lined up in the same direction, head to tail, is adding Let’s use arrows to help us make a better definition. Adding is lining up two arrows together in the same direction (head to tail). On the number line this means that adding

Arithmetic: Combine

33

is putting two numbers together so that the answer is farther away from zero. Therefore, two positive numbers add up to a bigger positive, while two negatives add up to a bigger negative number. So you see, it is not because numbers go up that they add, it is because they go in the same direction, even if that direction is down. Once again, if you understand the difference between “bigger” and “greater than” you are farther ahead than many. They mistakenly think that every time they add the answer must be higher on the number line, but really, the answer must be farther from zero, up or down. Think of this in practical terms it will make sense. If you owe someone $4 and someone else $2, how much do you owe altogether? Of course, you owe $6 total. In your head you knew that owing $4 was bad and so was owing $2. So putting them together meant that things were going to get worse. You added, not subtracted, the debts. Your answer got farther from, not closer to, 0. Again, let’s say you dig a hole 3 feet deep, then you dig another 2 feet. How deep is the hole? It only makes sense that if you go down, then down some more, you end up with a deeper hole, which means you must add the 3 and the 2 to get 5. Of course, it is a negative 5, because it is below 0, which is ground level. Teach adding this way to prepare the student to understanding subtraction correctly.

8) Subtracting on a Number Line



Adding lines up arrows in the same direction, so subtraction puts them together in opposite directions. Subtraction is not always “taking away,” but “taking away” is always subtraction. “Taking away” only deals with size, but of course, numbers have size and direction. Therefore, subtraction must take into account size and direction, which is both the number and its sign.

+6 +4

-2 0

-8 -12

+4

Subtracting lines up arrows in opposite directions, head to tail

34

Action Algebra

This is critical for us as adults to understand before teaching our students. We have been conditioned to think that subtraction is only taking away, but this leads to a mental road block. For example, if I have a stack of 3 books on the table, how can I take away 5? If I think that subtractions is only taking away, then this problem is impossible. When a student believing this enters pre-algebra and negative numbers, all sorts of mental difficulties and confusion arise. Some never get over it. Many take months to expand their thinking. To subtract 5 books from 3, I need to see that the original stack goes up 3 from 0, which is the tabletop. Then I need to see that I must go down 5, which of course will land me in negative territory below the tabletop at -2. I owe the table 2 books. The only correct way to tell if you need to subtract two numbers is by looking at BOTH of their signs. If the signs are different, subtract, but if they are the same, add. The answer might be positive or it might be negative, but it will always be closer to zero than the biggest number. Therefore, the usual advice to put the bigger number on top when setting up a subtraction problem is always correct. (At this point, you might want to take a peek at lesson 11 so you can see where this is all going, which is to the all-in-one method of combining.) Now look at the examples. -2+6 means you go left 2 then go right 6 to end at +4 for the answer. When using pencil and paper with just the numbers, notice that -2 and +6 have different signs. +4-12 means you right 4 and then go left 12 to end at -8 for the final answer. Again, notice the different signs and so the arrows go different directions.

9) Adding Stacked Numbers



This lesson is the standard lesson which teaches students to add numbers vertically. This will be a real test of a child’s abstract abilities. Some may need to wait, some may take many weeks to master it to the level of being automatic. Again I advise not to push. Challenge, but not push. There is a significant jump from concrete, pre-formal thinking to juggling the abstract idea of numbers in the head.

Arithmetic: Combine

+8 +6 +14

35 1 0

+37 +13 +50

1 0

+79 +76 +155

+72 +29 11 90 +101

+858 +245 13 90 1000 +1103

To prepare the way for combining, the worksheets put the biggest number on top and all the signs are written. The process of adding and carrying the spillover is the same as what you are familiar with. However, I have seen a variation that could be helpful to some students. Instead of writing the carry above the column to the left, the answer is written in one place below. It is a little bit more writing, but the place values are made plain all the way through. Notice also that the problem can be done either right to left or left to right.

10) Subtracting Stacked Numbers



With the exception of showing all the signs, this is a standard lesson on subtraction. The big number is on the top, so even the standard “take away” explanation will work here. (Take away is not wrong, it is just incomplete.) As you can see in the examples, the standard way of subtracting, with all the borrowing and slashing can be quite messy. If we as adults don’t like it, we can imagine the trouble this mess causes young children. So in the beginning you might want to break down the steps for them more clearly to aid their understanding. The cause for borrowing (as well as for carrying in addition) is place value. Because we cannot always store enough value in the top digit, we must borrow 10 times that place from the place to the left and temporarily squeeze it in. What we are really doing when we squeeze in extra value is making a new problem within the main problem. Look at the 24-19 example. After borrowing 10 from the 20 we can look at it as two problems, 14-9 and 10-10. Each of those problems only have 1 digit for an answer, which fit fine in one place. So we want to make sure not to borrow when we don’t need to, nor to borrow more than 10. In either case we will make a problem that results in two digits. And as Hardy use to say to Laurel, “That’s another fine mess you’ve gotten me into.” Let’s look at the 93-27 example in the middle and sort of dialog our way through it.

36

+15 -7 +8

Action Algebra 1

+24 / -19 +5 1

8 0 1

/ +93 -27 +66

6

/ +72 -59 +13 1

7 14 1

/ / +858 -269 +589

I am going to work right to left, because along the way the top digit might be smaller than the bottom digit. To solve that problem I need to have a big, rich neighbor on my left that has not spent all her money yet. So the first digit I will work with is the 3 and I see it must subtract a 7. For a final answer I can go into debt, but not in the middle of a problem, so I must borrow. The 9 is the big, rich neighbor and she is happy to loan me 1. But guess what?!! The 9 is in the tens place so it is really a 90 and the 1 she will loan me is really a 10. That will make my subtraction work! 10+3 is 13, so I now have 13 squeezed into the ones place that can easily have 7 taken away from it. 13-7 is 6, so I write a 6 in the ones place of my answer. Now I move to the second column and the 8 that remains from the 9 can subtract the 2 beneath it. 8-2 is 6, so I write a 6 in the tens place of my answer. I now have a final answer of 66. When demonstrating that to students you have two options. Slash and write the borrow real tiny, or make separate problems. (Here is where a big whiteboard can come in handy. Next to the main problem, you can write the two (or three) smaller problems, then put their answers back in place under the main problem.) Either way you do it, be sure to note to yourself and the students that they are using their place value skill and bigger/smaller skill from the Numbers chapter. Nothing a student learns is extra or useless. It all leads to something in a later chapter, or even in the very next lesson, which is about to happen!

11) Combining Stacked Integers



Starting with this lesson and before we complete the chapter, we will roll all the previous lessons into one. Being able to combine tags at the end of the chapter means the student is able to do all the skills to that point. It will be a good review spot. However, we must first start with combining plain integers (blank tags).

Arithmetic: Combine

37

Before beginning let me clear up some terminology. I have seen some books use combining the same way I do to mean either adding or subtracting. I have also seen some books use the word “adding” in the same way I use “combining.” I am comfortable with both usages, but in this book, I will use adding in only the way I have already described it. Adding is two arrows in the same direction head to tail. This translates to numbers on paper as I showed two lessons previous. Combining I will use only to describe the process I am about to show you, which will combine (no pun intended) adding and subtracting. Mastering this method, a student is set to conquer arithmetic, word problems, and algebra. Combining clears confusion Now lets put adding and subtracting together into one new process called, “combining.” Combining will tell you when to do the old-style adding and when to do the old-style subtraction and what the sign of the answer will be. You don’t need to memorize special cases and what to do in case a number is negative. Everything is all wrapped up into one overall process.

+7 +7 -7 -7 +2 -2 -2 +2 +9 +5 -9 -5

+15 +03 +18

+15 -15 -15 -03 +03 -03 +12 -12 -18

1) Always write largest number on top 2) Answer sign is Largest sign (top)

3) Same Signs Add, Differents Destroy Before I explain, just study the examples to see if you can find a pattern. Did you notice that the biggest number is always on top? Did you notice that the answer always has the same sign as the biggest number? In other words, the top sign is always the answer sign. Did you notice that when the signs are the same, the numbers add to get a bigger number farther from zero? Did you notice that when the signs are different, the numbers subtract to get a number closer to zero? This is the process you should have been taught starting in first grade. With combining, there is no need to learn, unlearn, and then re-learn separate processes with positive and negative numbers. Merge the two processes with correct ideas of “bigger/larger” into the

38

Action Algebra

one process of combining that ALWAYS works, even in algebra. (Remind the student that bigger and larger mean the same thing, just as they learned in the Numbers chapter. I use the word Larger here because it fits into a mnemonic.) SSADDL your COLT Most of the time numbers are not stacked vertically and you don’t want to take the time to re-write them that way. Here are the similar steps to combine positive and negative numbers when they are in normal, sideways format. Same Signs Add. -8-5 means combine by adding. +9+2 means combine by adding. 18+7 means combine by adding. Don’t forget the invisible + in front of 18! Differents Destroy. This is a shorthand way of saying different signs destroy each other. This goes back to the old Pacman game. The + are like cherries and the - are like Pacmans who eat cherries. Put a - and + together and they destroy each other like matter and antimatter. Poof! This is the same as a hole and a pile of dirt. If you fill the hole with the pile, then both the hole and the pile are gone. Poof! Positives and negatives destroy each other, when combined. Another little tip to fix this in the memory is that “different” is basically the same word as “difference.” The word, difference, is used in word problems as a clue to subtract. So you could say Different Difference to keep things straight, but people may look at you a little funny as if you are a verbal photocopier! (but you won’t forget!)

-6-4= 2 -6-4=3 -6-4=-10 1

-7+9= 2 -7+9=+ 3 -7+9=+2 1

1) Find sign of largest number 2) Copy it to answer sign 3) Same Signs Add, Differents Destroy

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39



12) Series of Signs

This lesson really belongs in the chapter on multiplication, but I need to insert it here because some students will be confused by their textbook. Because they teach adding and subtracting separately, some books insert an extra + or - sign intending to be helpful. This is not necessary once you know combining and you will later have to unlearn the crutch of depending on an extra sign. (I have seen more students confused by this device than helped.) So here is what to do. Count all the signs that look like - that are in front of the number. If the count is odd, the number is -. If the count is even, the number is +. It does not matter if you call the - sign a negative sign, a minus sign, or subtraction. Count all the - signs. This is the NOPE rule that goes with the MuD Action. I’ll explain why this works in the next chapter.



13) Combining Series of Signs

Now that the student knows how to condense a series of signs into one, he will be able to combine any number of numbers with any number of signs. (No need to go crazy here. Every problem can be broken down into combining two numbers at a time until the total is reached.)

4--7 = +4+7 -8+-4 = -8-4 --++-2-+-6=-2+6 9-(+7)=+9-7 -3+(-5)=-3-5 11+(+2)=+11+2

-=-

--=+

---=-

----=+

-----=-

------=+

-------=-

NOPE-Negative Odd Positive Even

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Action Algebra

If you think this looks like standard pre-algebra, you are right. If you think it is too early to introduce it to students, just remember that we have arrived here in a smooth progression. If the student has successfully handled the previous lessons, there is no reason to assume she will not handle this one successfully. Don’t let your fears and biases get in the way of the student’s blank slate! Also, the earlier something is learned, the more it is reviewed to the point of becoming automatic.

14) Combining Big Integers



This lesson introduces no new concepts. Rather, it consolidates previous learning and extends it to large numbers written sideways. It is up to the student to find the largest/ biggest number, put it on top, then add or subtract according to the signs.

4--7 = +4+7 = +11 -8+-4 = -8-4 = -12 --++-2-+-6 = -2+6 = +4 9-(+7) = +9-7 = +2 -3+(-5) = -3-5 = -8 11+(+2) = +11+2 = +13

15) Combining Decimals



Combining decimals is no different than combining integers, except that a decimal point is visible. Sure the decimals must be lined up, but we lined up the invisible decimal

Arithmetic: Combine

41

points in integers when we lined up the ones, tens, and so on. Why must we always line things up this way? Because of COLT, Combine Only Like Things. We add pennies with pennies and dimes with dimes, so we also add ones with ones and tenths with tenths. There is nothing magical about the decimal point. It is the place values that must be lined up. The icon looks like it is lined up on the right side. This is a small visual reminder to line up numbers correctly to the right side. If you fill in the invisible zeros past the decimal, then all numbers line up to the right, but the key is lining up the decimal so that place values match above and below. Notice that all the numbers above have blank tags. Because they are blank we don’t even need to draw or label the tags. If you put them in, you won’t be wrong, but combining blank tags gives you a blank tag.



16) Combining Tags

COLT says Combine Only Like Things. We know two things are alike if they have the same tag. So you could also say Combine Only Like Tags. Also, when you combine identical tags, you get the same tag for an answer. Now the question arises, What do I do if I need to combine things with different tags? Don’t! You can’t! Just stop and do nothing. You are done! (Multiplying can work with different tags, but combining cannot. Sometimes multiplying can change the tags so that combining can work.)

-400. - 75. -475.

-28. + 5. -23.

+5.10 + .26 +5.36

-75.30 - 2.04 -77.34

Think about it. You have 3 apples in your left hand and 5 oranges in your right hand. How many do you have altogether? Did you say 8? You should have asked me, How many what? Do you have 8 apples? No. Do you have 8 oranges? No. You have 8 fruits, but was that what I was asking for? You don’t know. Therefore, you can’t answer. Math is exact. The question and the numbers you have, must ALL match exactly. Otherwise, don’t answer the question, because you can’t. This may sound nitpicky and

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hard for students to understand, but actually it is easy. The rule is simple: tags match exactly or do nothing. No exceptions. Not a lot of thinking. Combine ONLY Like Tags. Frontnum Tag

+3 apples +5 apples +8 apples

+3 apples +5 oranges STOP

Frontnum Tag

+3 miles +5 miles +8 miles

+3 miles +5 books STOP

Frontnum Tag

Frontnum Tag

Frontnum Tag

Frontnum Tag

Frontnum Tag

Frontnum Tag

9.2 xy +3.7 xy -5.5 xy 9.2 % +3.7 % -5.5 %

+7 /4 -1 /4 +6 /4

+17 a +24 a +41 a

-8 /3 +3 /3 -5 /3

+17 & +24 & +41 &

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43

Fractions As we saw earlier, fractions are compound numbers with visible tags. Everything from the fraction bar down (or to the right) is part of the tag. Notice that the answer tag is the same as the problem tags. The frontnums get combined as regular numbers, because they are regular numbers. Variables Variables are always part of the tag. Look at the examples and you will see that when you combine x’s you get an x. When you combine y’s you get a y. Constants Combining with constants is no different than combining with variables or anything else in the tag. Use COLT. Percents Anything after the front number is part of the tag, even fancy symbols like the percent sign. And guess what?! When you combine percents you get a percent. When you are at the store you see there is a 10% sale taking place. Then you see little signs that say prices have been lowered another 5%. 10% + 5% = 15% discount. % + % = %.

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Arithmetic: Multiply In this chapter we will learn how to multiply compound numbers. We will see that the tags do not have to be the same, and in fact, multiplying makes them different. The icon gives hints that multiplying and dividing go together. Dividing is just multiplication in reverse. That is why we call this Action, MuD, for MUltiply and Divide. However, we will look at dividing in its own chapter. The icon looks like a small grid and reminds us that multiplying is based on a grid, while the stairs of the COLT icon were like a single number line. The icon also suggests what do with the + and - signs.

17) Multiplying on a Grid



Multiplying is putting two arrows together, tail to tail, at right angles. The answer is the area of the rectangle that they make. Here’s why. We have already seen that combining is counting numbers one after another without restarting the count. This is visually represented by arrows lined up head to tail on a single number line. What we want to do now is repetitive counting. We want to copy a number some number of times and get that total. If you think about that last sentence closely you will realize that we are introducing a new idea that we don’t yet have words for. The words I am about to use are not “official” as if they must be memorized. I am simply trying to describe an idea using whatever words are available. Here is the new idea that multiplying introduces: number role. Each of the two numbers in a multiplying problem have different roles. One number is the “original number” and the other number is the “copy number.” The original number gets duplicated according to the copy number. This is like putting the original number on a piece of paper on a copy machine. Then we punch in the copy number and we get that many pieces of paper on the output tray that each have the original number. Then we add up all those pages and get the total.

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45

length=6

copies

Do you see how this is different from combining where the numbers just sit there waiting for us to count them? The numbers in a combining problem have no roles. They are inert and lifeless. In a multiplying problem, however, they have different roles to perform. They have different meanings. total Even though we can interchange the original number and make it the copy number (which makes the copy number into the original number) and still get the same answer, whichever way we solve it, we give the numbers different roles. This subtle idea leads to new ideas. original Notice that length ^ length = area. Length ^ length does not equal another length. Multiplying makes a new thing, a different thing. Now compare this to combining. Length + length = length. Combining keeps everything the same. Multiplying makes things different. This is easy to forget when we work with plain numbers for so long and forget about the real things that they count. Numbers don’t exist to count themselves. They exist to count real things. When you combine real things you get the same kind of thing as an answer. When you multiply real things, you get a different thing as an answer. This common sense pattern of life can be used when solving word problems. For example, you are told that your room is 10 feet long by 12 feet wide. Then you are asked to find the area of the room. In the information and the question you have feet, feet, and area. You have more than one kind of thing. You can know automatically that combining the numbers will not give you the answer, because feet + feet = feet. So, you must multiply. Combining works on Multiplying literally adds another a number line, while dimension that combining does not multiplying works on a grid know exists. Two number lines put together on a grid lets us multiply two different things at once. If we know that different things are involved, then we know multiplication was used. If we use multiplication, then we know different things will result. area=54 Apply that idea to this question. If 5 trucks each have 1000 cookies, how many total cookies are there? We have trucks and cookies and must find cookies. What I am not sure length=9

9*6=54

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Action Algebra

crates each truck=6

about is whether I should multiply or combine. My question is answered as soon as I realize that trucks and cookies are different things. I cannot combine them, so therefore I multiply. So the lesson is that different things (different tags) and multiplying (which includes dividing) go together, while same things and combining go together.

Multiplying lets us work with more than one kind of thing at a time, because it works with more than one number line at a time. One number line represents trucks, while the other represents crates in each truck. The area of the rectangle tells us how many total crates are in all the trucks. One number line can work with trucks OR crates, not both.

9*6=54 total crates=54

trucks=9

What dividing really is We will do division problems in the next chapter, but it might be helpful at this point to contrast division with multiplication. Because dividing is just multiplication in reverse, it also uses more than one thing at a time. Division “unpacks” the answer that multiplication put together. What got packed was a rectangle on the grid. What did the packing were the two numbers multiplying each other on the two axes. Multiplying is length ^ length = area, so division is area _ length = length. It does not matter in what order you multiply the lengths, but you can see that order matters when it comes to dividing. Area _ length gives you length, but length _ area is nonsense!

length

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47

area length

Simply put, division is reverse multiplication where the order matters. Never change the order of a division problem or the division part of a main problem!

18) Learning the Times Table



Before getting too deep into multiplication, the students will need to know the times table. Sure a calculator can do the raw calculation, but then the students will never gain automaticity, which means they will mentally stumble every time they face a concept based on multiplication. (This applies the same to combining and dividing.) They should be able to work mentally with at least single digits and small double digits in all four of the basic operations. Memorizing the times table need not appear as a giant task if we cut out the duplication. We can further trim it by cutting out the 2’s and 9’s because there are better ways of dealing with them. So what is left is a smaller, easier to manage group of numbers. (A side benefit is that the squares become obvious along the diagonal.) This times table follows the good study habit of not studying the same thing twice. Students do not need to memorize 3^4 and 4^3, they understand that is the same problem. By making them study the same thing twice, or learning what they already know, they begin to doubt if they know it and so start slipping backwards. Review is good, but intensive study should focus only on what is not yet understood or memorized. So now here is why 2 and 9 are left off the table.

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Action Algebra

You can figure 2 ^ a number by doubling the number in your head. Just add number + number. 2^8=8+8=16 5^2=5+5=10 You can figure 9’s by either of two neat little tricks. Let’s solve the problem 4^9 Method A) Subtract 1 from 4 to get 3. That is your 10’s digit. Subtract the 3 from the 9. That is your 1’s digit. Your answer is 36. Always subtract 1, then subtract from 9.

^ 3 4 5 6 7 8 3 9 12 15 18 21 24 16 20 24 28 32 4 25 30 35 40 5 36 42 48 6 49 56 7 64 8

Method B) Hold your hands in front of you and curl your 4th finger (count left to right). Now count the fingers to the left of your curled finger, 3. This is your 10’s digit. Count the fingers to the right of your curled finger, 6. That is the 1’s digit. Your answer is 36. Try both of these tips with all the 9’s to prove it to yourself. Before leaving this lesson, look at the times table above one more time. Doesn’t that look like a less daunting task than memorizing the full table? If it looks easier to us, then it will also look easier to children!

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49

19) Learning Multiples



Multiples are very much related to the times table and they can help a student learn their multiplication facts. A list of multiples is generated by starting with any number then repeatedly adding that number to itself. Sometimes this is called skip-counting.

2, 4, 6, 8, 10, 12, 14, ... 5, 10, 15, 20, 25, 30, ... 7, 14, 21, 28, 35, 42, ... 11, 22, 33, 44, 55, 66, ... You can also start multiplying the number by 1, then by 2, then by 3, and so forth: 4^1=4; 4^2=8; 4^3=12; 4^4=16; 4^5=20; 4^6=24; etc... This is where the word “multiple” comes from. The reason to find multiples is to find common multiples, which will then be used by fractions and other problems later on.

20) Negative Numbers on a Grid



It is time to complete our understanding of negative numbers. We have a system that works for all adding and subtracting called, combining. Now we need a system that works for multiplying, and along with it, dividing. We also need to know how to tell the two systems apart.

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Action Algebra

2

-

-16

-4

3

+

+16

1

+

+16

+4

-4

-

+4*+4=+16 -4*+4=-16 -4*-4=+16 +4*-4=-16

+4

-16

4

Let’s return to our copier illustration, but this time we will copy money instead of plain paper. I want to solve the problem 8^10. In other words, I want to make 8 copies of a $10 bill. I put the bill on the glass, punch in 8, and out comes 8 bills. I add them up and get $80. Sweeeet!! Translating this problem to a grid I can put 10 on the horizontal axis (the X axis) and 8 on the vertical axis (Y axis). Since the 10 is positive it goes to the right. Since the 8 is positive it goes up. Therefore, the rectangle that they make is in the upper right quadrant (quadrant 1). Since the answer is positive (I have $80 in cash, not debt) we say that any answer in quadrant 1 is positive. Now let’s modify the problem. Instead of a $10 bill, I now have a $10 IOU. When I make 8 copies of the IOU, I will be in debt $80. That is -80. Going to the grid, my 8 is still positive and up, but the 10 is negative, so it goes left, not right. In which quadrant is my answer rectangle? In the upper left, quadrant 2. So any answers in the “northwest” quadrant are negative. Notice that so far the answer sign is following the NOPE rule that was introduced in

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51

the Combining chapter. Negative Odd Positive Even, and the first problem had 0 negative numbers, and 0 is even, and the answer was +. The second problem had 1 negative number, which is odd, and the answer was -. Now things get interesting because we must imagine a negative copier. Let’s pretend that we have a copier destroys existing copies instead of creating new ones. When I ask for 3 negative copies of my original, this strange machine destroys 3 originals. Negative copies, get it?! So now I put my $10 bill on the glass and punch in -8 copies. Rather than spitting out $80 for my spending pleasure, it reaches into my wallet and shreds 8 $10 bills into oblivion. Obviously, I am not happy, but the copier did what I told it to do! So now I am $80 poorer. The answer to -8^10 is -80. I show this on the grid with a +10 to the right and a -8 down. Therefore, my answer rectangle is in the lower right (southeast), which is quadrant 4. Thus, we can conclude that any answers in quadrant 4 are negative. And here comes the final mindbender. I put an IOU of $10 on the copier and punch in -8 copies. The copier goes into my wallet, pulls out 8 IOUS of $10 each and destroys them! Weird, but quite nice! Better than a bailout! I am $80 less in debt. A positive thing just happened to me because something bad got destroyed. A negative got negated. A double negative is positive. We will overlook the moral implications of arguing that two wrongs make a right and instead focus on a reversed reverse is forward. Does all that make sense?! If my debt (-) is destroyed (-) then that is a + result for me. On the grid that problem looks like this: the -8 goes down and the -10 goes left. Therefore, the answer rectangle is in the third quadrant (southwest), and so quadrant 4 is positive. In summary, multiplying is two arrows making a rectangle in one of four quadrants. Which quadrant the rectangle is in determines the sign. Telling the difference How do we tell the difference between a multiplying problem with + and -, and a combining problem with + and -? Multiplying always has * or ^ or ( ). If you see nothing that says to multiply, then combine.

-6+(-4) becomes -6-4

-6^+(-4) becomes -6^-4 8-(+5) becomes 8-5

8*-(+5) becomes 8*-5

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Action Algebra

Sometimes a sign touches the ( ) instead of a number. In that case, only the sign multiplies the sign inside. This is part of the general math notation that says that things that touch multiply each other. For example, 5x means 5 times x, and 7b means 7 times b. A couple of exceptions to this general way of writing are digits and mixed numbers. If two digits are next to each other, then they are spelling a number using place value. 25 means twenty five, not 2^5. Also, mixed numbers put a whole number next to a fraction. In that case, it means add, not times.

-2+5 = -7 but -2*+5 = -10 +1-8 = -7 but +1^-8 = -8 -3-6 = -9 but -3(-6) = -18 -4+9 = +5 but (-4)(+9) = -36 2

3

-

+

+

-

1

4

Arithmetic: Multiply

21) Multiplying Big Numbers

53



Now that we have a system in place for multiplying numbers with signs, we can now attack 2 and 3 digit numbers. This will introduce the distributive property and carrying. (Note: This explanation of distributive property is for you as a teacher. I have not found it necessary to add to the load of young minds with another new technical phrase. When the student encounters it again in pre-algebra, then the wording will be more appropriate. Right now your goal is to teach “common sense.”) You may not be familiar with the name “distributive property,” but you have used it every time you multiply by more than one digit. For example, 3^15 is the same as 3^(10+5) which is the same as 3^10 + 3^5. The 3 was distributed to each number in the ( ). Using the distributive property I can sometimes multiply up to 2 digits ^ 2 digits in my head.

12^34 = (10+2)(30+4) 10^30 + 10^4 + 2^30 + 2^4 300 + 40 + 60 + 8 = 408 29^75 = (20+9)(70+5) 20^70 + 20^5 + 9^70 + 9^5 1400 + 100 + 630 + 45 = 2175 Distribution is used a lot in algebra, so it is good to introduce the concept with plain numbers to make it easier to grasp. The basic idea is that distribution lets you split the main problem into smaller problems, solve them, then combine their subtotals into a final answer. For example, 8^12 = 8^(10+2) = 8^10 + 8^2 = 80 + 16 = 96 The key point not to miss here is that distribution works only when the group of numbers in the ( ) are being combined. If the problem looks like this: 3^(10^5) then distribution does not apply because it is not needed. I really don’t think of the distributive property as another rule to memorize. It is really

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just a common sense interpretation of what is written. It is always wise to break down big problems into smaller ones and the distributive property gives us a label to help correctly describe what we are doing. I use the word “fill” when I teach equations and sometimes the visual of a sprinkling can helps students “see” what is happening no matter what they call it. So here is the complete idea. Every number in one group must multiply every number in the other group. Look at the examples. Of course, if one is careful, the part with the parentheses can be skipped, and the four subproblems can be written out in any form, as long as the student remembers to combine their subanswers. Powers of 10 When multiplying big numbers much time can be saved if the student knows how handle powers of ten. 0’s at the end of the numbers (not in the middle) can be written down as part of the answer right away. Then normal multiplication can be performed on the remaining digits and their answer written on the left side of the answer 0’s. This shortcut works because of the distributive property. For example, 5^100 is 5^1 and two 0’s for an answer of 500. 20^30 is 2^3 and two 0’s for an answer of 600. Vertical Multiplication Whether you have your students break up the problem as in the examples or use the standard vertical method, the underlying principle is the distributive property--everything ^ everything. Now let’s look at vertical multiplication and a couple different ways to do it. The normal carrying can be used, or else the full subanswers can be written out in the middle area. I recommend the latter method because it is cleaner and easier to read. Young children often get confused about where and why to put their carries and which carry is the current one to use.

15 ^12 10 2^5 20 2^10 50 10^5 100 10^10 180

68 ^37 56 7^8 1 420 7^60 240 30^8 1800 30^60 1516

200 500

68 ^37 1 476 2040 1516

Arithmetic: Multiply

22) Multiplying Bigger-Smaller

55



The purpose of this lesson is to develop “math sense” in the student. They do not need to do the actual multiplying. They just need to look at the problem and tell if the first number is being made bigger or smaller. For example, if I multiply 8 by 2 the 8 is getting bigger, but if I multiply 8 by 1/2 the 8 is being made smaller. This skill helps in estimating and in double checking word problems to see if the answer makes sense. The key is to look at the second number to see if it is bigger or smaller than 1. If the first number is multiplied by a number bigger than 1, then the answer is bigger than the first number. If the first number is multiplied by a number smaller than 1, then the answer is smaller. A fraction is smaller than 1 if the top number is smaller than the bottom number. A decimal is smaller than 1 if there are all 0’s on the left of the decimal point. A percent is smaller than 1 if it is less than 100%.

23) Multiplying Decimals



Multiplying decimals is just like multiplying integers with one extra step. At the end we must figure out where the decimal point belongs. The rule to use is M2 on the rule sheet: Combine add-ons. In this case, it is the decimal places added on to integers. This means that we should count the decimal places in the first number, count the decimal places in the second number, then add them. The total number of decimal places in the problem is the same amount we put in the answer. Let’s progress from integers to see how this works. 5^3=15 --there are no decimal places in the problem or answer 5^30=150 --an extra 0 in the problem makes an extra zero in the answer 5^.3=1.5 --1 place in the problem makes 1 place in the answer 50^30=1500 --2 0’s in the problem make 2 0’s in the answer .5^.3=.15 --2 places in the problem make 2 places in the answer Do you see the pattern? Just like multiplying by an extra zero is like multiplying by 10, so multiplying by a decimal place is like dividing by 10. Each decimal place has its effect and so must be accounted for in the answer. For a really nitty gritty explanation we can expand the numbers.

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Action Algebra

.5^.3 = 5^.1 * 3^.1 = 5^3 * .1^.1 = 15^.01 = .15 So we see that multiplying decimals is the same as multiplying them like integers, then putting the same number of decimal places in the answer as were in the problem.

No need to line up decimal places, count them

35 ^.2 7.0

.74 ^1.1 .814

1.2 ^.12 .144

.049 ^ .02 .00098



24) Multiplying Fractions

Now let’s extend the idea of multiplying decimals. A decimal is really a shorthand fraction. For example, .4 is 4/10 and .19 is 19/100 So the basic idea is the same: the division built into the problem must be accounted for in the answer. In practice, all that means is we should multiply straight across. If I divide by the bottom number of the first fraction, then divide the result again by the bottom number of the second fraction, then those divisions accumulate. For example if I cut a number in thirds, then cut those thirds into eighths, that is just the same as if I cut the original number into 24ths. That’s why we just multiply straight across. Frontnum Tag

2 /3 ^5 /8 10 /24

2 7

4

8

* 3 = 21

4 7 28 5 * 9 = 45

No need to line up fractions, multiply straight across

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57

Note: I have referred to division, but we have not yet covered division. Therefore, we have not yet covered reducing. All the answers in this chapter are not reduced on purpose.

25) Multiplying Tags by Merging



There are two basic steps to multiplying any compound number, including fractions. 1) Multiply the frontnums in the normal manner. Use NOPE to figure the sign. 2) Merge the tags. If there is any division, put those factors after a fraction bar. For example: 2ab ^ 3b = 6abb 5/x ^ 8/y = 40/xy -7m/xy ^ 3az/x = -21azm/ xyx (Note: In the last example you could Sort the azm into amz and the xyx into xxy, but remember not to sort across the division sign. Also, we will not cover exponents until prealgebra, so do not use them yet. However, you could point out the need for a shorter way of writing xx.)

-4&^-3& = 12&& 5a^-+2b = -10ab 7/a*6 = 42/a 2a/b*8/c = 16a/bc

This looks like complicated algebra, but only in looks. The merge concept is a simple, one-step preparation that will be repeated and reviewed many times before the student actually begins algebra. At that time, this skill will be automatic--even tedious--and will lead naturally and correctly into exponents and other higher order operations. You might think of merging tags as forming compound words--just stick them together and you’re done. Now let’s apply our merging skill to a variety of problems. If something is “missing” a tag, all that means is that there is an invisible ^1 there. You can use the Show Action to write it in, if you want. However, the thinking required at this level does not need to be that much. Just merge what is there, if anything.

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Action Algebra Frontnum Tag

Frontnum Tag

5 % 6 x ^20 ^1 ^6 x 100 % 36 xx

Frontnum Tag

-6 & ^6 & -36 &&

Next, let’s look at a possible question that may arise with percents. I can’t recall any problem of this type in real life or in the textbooks, but some sharp student of yours will surely think of it, so here it is. What if we have 5% ^ 20% ? You might be tempted to think that the answer is 100% but follow the rule of multiplying nums and merging tags. 5% ^ 20% = 100%% Yes, that is %%. It is not a typo and it is actually a correct answer for students at this point. To see that it is correct try it on your calculator. You may have to convert it to decimals first, which would be .05^.20 for an answer of .01 Now let’s compare that to our answer of 100%%. Remember that % means /100. Therefore, 100% = 100/100 = 1. Therefore, 100%% = 1%. Next, 1% = 1/100 = .01 So you see, merging tags is correct. It is unfinished, but it is correct.

26) Finding Common Multiples



We have arrived at our last lesson of multiplying and it is a preparatory lesson for the full handling of fractions in the next chapter and algebra topics to come later. Common multiples is simply finding all the matches in two lists of multiples. For example, if I write a list of the first ten multiples for both 3 and 4 they look like this:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, .... 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ....

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Now taking those two lists I look for numbers that show up on both.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, .... 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, .... Because both lists can be extended to infinity, the number of common multiples can be infinity, but we are not going to go that high! Almost always we are interested only in the first or lowest common multiple (LCM). However, just for practice, textbooks often ask students to find the first two or three common multiples. Common multiples can also be found visually using grids. The examples show how the common multiples of 3 and 5 can be found. Again, we find the first ten multiples.

10

50

5

10

30

45

27

40

24

35

21

30

18

25

15

20

12

15

9

10

6

5

3 3

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Arithmetic: Divide This chapter is definitely something new. Even many adults are confused by division because of the mental reversals involved. And if there is a keyword to this chapter, it would be “reverse.” Reverse operations with adding and subtracting seem natural and almost invisible to many, but not with multiplying and dividing. Make sure the student is ready for this chapter! Growing out of reverse multiplication, and helping with it, is a major Action. Shift is introduced in this chapter as a very powerful concept and the key to understanding many moves in the game of math. The Shift Action is the key to fractions and many other mysteries.

27) Dividing on a Grid



arrow

Let’s review the concept of multiplication all the way to its logical end--division. We introduced multiplication with the illustration of a copier, which was a must faster way of combining numbers. Our copier needed an original number and a copy number. We also saw that those two roles are easily interchangeable. In fact, they are so easily exchanged that their order does not matter and so we often lose sight of their roles. However, multiplying two numbers in real life causes an interesting side effect--tag merging. This causes a new kind of thing to be made. For example, feet ^ feet produces square feet. Speed ^ time produces distance. Items ^ individual price produces total cost. Tag merging is the reason why Sort does not work on division. Division is multiplication in reverse, but its order cannot be reversed! Here is why. A number is an arrow. Two numbers multiplied together are two arrows at right angles to each other, but look at what they form! Something entirely different from

rectangle arrow

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arrows is produced--a rectangle! Now think carefully, if arrow ^ arrow = rectangle, then it only makes sense that rectangle _ arrow = arrow. Furthermore, any other combination does not make sense. What is arrow _ arrow? Nonsense. What is arrow _ rectangle? Nonsense. Changing the word “arrow” to “length” makes no difference. Length _ length and length _ rectangle are still nonsense. So what is the point? The first number in a division problem ALWAYS represents the rectangle. The numbers that follow ALWAYS represent the lengths (arrows). Order matters with division. It can never be reversed.

8 3

24

24 8

3

Therefore, if you are given a problem like 24_8, then 24 is the area of the rectangle and 8 is one of the sides. (It does not matter which side, because sides have no order as we learned in the multiplication chapter.) The answer will be 3, which must be the other side. This visual demonstration should be proof why the Sort Action says, You may rearrange the objects in a level at any time, but never change a division part. In other words, you can Sort a problem from this 4+9_3 to this 9_3+4, but you cannot Sort the division part from 9_3 to 3_9. Not only does that give you two entirely different answers, but in real life you get two entirely different meanings. So when drawing a division problem on a grid there are two possible correct answers as shown above, and putting the 24 anywhere else is wrong.

28) Learning How to Shift



No, this is not a lesson on how to drive a car that does not have an automatic transmission! This lesson is about a very important and widely used Action called, Shift.

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It is based on the principle of equality which says that every value change must have a counterchange. As I said in the chapter on basic principles, before must equal after. We can neither create nor destroy. We cannot inject new values into the problem, we can only work with what is there according to the basic rules of math which I have summed up into the ten Actions. You see, the process of math is so important that we must understand and do it correctly, because we have no way other way of checking ourselves along the way. Sometimes we don’t have a really good way of checking ourselves at the end, either. For example, let’s say you are plowing a long field. You want your rows to be straight, so you keep your eye on a fence post at the other end. All along the way you keep making minor corrections and countercorrections to keep the tractor straight. When you get to the other side, you can stop and turn around, then check yourself. Math is the same way. We start on the problem side of the field and we try to plow a straight line to the answer side of the field. All along the way we change and counterchange. We have no easy and direct feedback in the middle of all the solution steps, so we must make sure we always make a right move. Those right moves are Actions. If we always make only right moves then we might not go at the quickest pace across the field to the answer, but we are guaranteed to sooner or later find the right answer, instead of a wrong answer. Now let’s look at Shift. The Shift Action says, You may change the value of an object at any time if you counter it with an equal, opposite change within that object. 2 opposite effects, 1 side. In arithmetic, we are always working with expressions in preparation for working with equations. So we are always working on 1 side. Now let’s start with a simple, but rarely used, example: 5+2. We know that it equals 7, so we know that whatever change we make, we must make a counterchange to bring the total value back to 7. That is the original problem, so that must be our final answer. Shift says I can change the value, so I decide to add 1 to 5 to make it 6. However, Shift also says I must make an equal, opposite change. Therefore, I must subtract 1. I could subtract 1 from 5, but that would bring me back to where I started so that would be useless. I could subtract 1 from the 2 which would make it 1. I think I will do that! I now have Shifted 5+2 into 6+1. Before the Shift I had 7, and after the Shift I have 7. Everything is good! Why did I that? Just to demonstrate to you what a Shift is. Perhaps in real life someone finds it easier to work with 1’s instead of 2’s or 6’s instead of 5’s, then they might want to make this same Shift. Actually, we rarely Shift combining problems. It is when we multiply or divide that Shift is the most helpful.

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6^4 becomes 12^2 6^2=12 and 4_2=2 ^2 and _2 are equal, opposite changes This type of Shift is more for educational purposes then for arithmetic usefulness. However, in algebra we will use it a lot with factors. What this illustrates is the equal but opposite effects. I made something bigger then compensated for it by making something else smaller. I also call the Shift Action the “bigger smaller principle” or the “greater lesser principle.” This held true in the first example because the 5 got bigger while the 2 got smaller. A question arises at this point, Why can’t I add some number, say 3, to the 6 and subtract 3 from the 4? Isn’t that right, because of bigger smaller? Well, two answers. First, try it and see if you get 24. Remember, equality says before must equal after. 6+3 is 9 and 4-3 is 1, but 9^1 = 9, so the answer is, Don’t do that! Second, you will always Shift by using what is there. In the case of 5+2, that is a combining problem, so I use combining to Shift. With 6^4 that is a multiplication problem so I use multiplication or reverse multiplication (division). Try to Shift another way and you will almost always be wrong. (Coincidence will once in a while lead you to falsely believe you are right.) With that in mind let me show you some more examples for you to figure out.

11-8 becomes 13-10

3^9 becomes 1^27

-10^4 becomes -5^8

-12^5 becomes 6^-10

Did that last one stump you? I divided the -12 by -2, so I multiplied the 5 by -2. I have not given you any examples of division, because that is the idea behind reducing fractions which I cover in lesson 36. There are also many more ways of Shifting which will unfold as we progress through math. This is enough to get us started.

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29) Dividing and Bigger-Smaller



This lesson is the counterpart to lesson 22, Multiplying and Bigger-Smaller. The purpose of this lesson is also to develop “math sense” in the students. They do not need to do the actual division. They just need to look at the problem and tell if the first number is being made bigger or smaller. They should keep in mind, however, that the effect will be the reverse of multiplying. That fact can be used to figure out the answer. For example, if I multiply 8 by 2 the 8 is getting bigger, but if I divide 8 by 2 the number is made smaller. If a student forgets what division will do, just figure out the question using multiplication, then reverse the answer. This skill helps with Shifting. Just like with multiplication, the key is to look at the second number to see if it is bigger or smaller than 1. If the first number is divided by a number bigger than 1, then the answer is smaller than the first number. If the first number is divided by a number smaller than 1, then the answer is bigger. If this seem counter intuitive, it is! It is COUNTER intuitive. It is reversed, always! Try examples on your calculator until you are convinced!

15_50% is bigger than 15 23_.01 is bigger than 23 A fraction is smaller than 1 if the top number is smaller than the bottom number. Therefore, dividing by it will make a number bigger. A decimal is smaller than 1 if there are all 0’s on the left of the decimal point. Therefore, dividing by it will make a number bigger. A percent is smaller than 1 if it is less than 100%. Therefore, dividing by it will make a number bigger.

30) Speed Division



We need to change focus here for a few lessons to actually teach how to divide numbers. Then we can put everything together in the last half of the chapter to finish arithmetic. Long division still has a use because a calculator is not always available, algebra has problems that use it, and the concept itself is used often in pre-algebra and upward. However, I do not see much use in doing lots of digits that take all day to figure. So I will show you a way that quickly handles single digits that some of you might not have seen.

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65

Then we will do the regular long division with two digits and decimals.

0 028959.25 4 115837.00 3

3 2

3

1

2

1) Write answer above 2) Put remainder on left of next number

I call the method for dividing by 1 digit “speed division” because I have not seen a name for it anywhere and it is faster than long division. In speed division by 1 digit you follow the same basic steps as in long division. You start at the left and collect a second digit if you need to. If you need another digit at any time, put a zero over the one you skipped. As you look at the digit(s) you collected, ask yourself how many times the outside will go into the subnumber. Write the answer directly above and put any remainder as a small digit to the left of the next digit. Now look at that remainder and digit as a regular number. In the example above you can see 35, 38, 23, 37, 10, then 20. Repeat this same process at each digit until you get a remainder, or it comes out even, or until you can round according to your instructions.

31) Long Division



I wish there were an easier, less tedious process to show you, but this is the best one I know for both remainder and decimal answers. The basic idea is that you are trying to find out how many times the number outside multiplies into the number inside. The way I remember it is that the outside number has a knife and is trying to get through the door to cut the inside number into pieces. How many pieces is the answer you want. If the outside does not go into the inside evenly then you also want to know how many crumbs are left over, in fraction or decimal form. On the left, under the 37, notice that I doubled three times. That is a handy list to help me see what multiple of 37 goes into my number inside. If I need a multiple not listed, I just add from the next lowest. That is what I did in the lower left with the 74+37=111. Some people just make a list for all multiples up to 9. Also, notice that I bring down all the numbers inside on every step. That helps those students who lose their place. This process is time-consuming so we want it to be accurate! Here are the steps: 1) Start at the left of the inside number. Collect digits one at a time until you make a

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subnumber bigger than the outside. 2) Pick the biggest number from your list that will fit in your subnumber. Subtract it from your subnumber. 3) Write the multiple you used directly above the 1’s digit of your subnumber. Fill in any blanks with 0 if you are not at the start of your answer. 4) Bring down all the other unused digits from the inside and put them next to your answer. 5) Start over on step 1 with your new inside number. Wasn’t that fun?! About as fun as getting a tooth pulled? I understand. =(

32) Long Division with Decimals



What do we do if we are dividing by a number with a decimal in it? We Shift it. Move the outside decimal all the way to the right end of the number. Then move the inside decimal the same number of places to the right. If you move the decimal past the end of the number, just fill the blanks with 0’s. Now put the answer decimal directly above your new inside decimal and you are ready to divide. This trick works because it is a Shift. For every place we move the decimal to the right that is like multiplying by 10. Since we move the decimal the same number of places inside and out, that means we are multiplying by the same number inside and out. If you move both decimals one place, you are multiplying by 10. If you move both decimals two places you are multiplying by 100. If they both move three places, you are multiplying by 1000. Making both numbers bigger by the same multiplication is inflating the factors. More on that later in this chapter. You might have noticed that I wrote out the example in lesson 31 a little differently than in this lesson. Here I filled in all the blank spots with 0’s (or whatever digit is to be brought down), while in the previous lesson I left them blank. This is a matter of preference and organization. Some children and even older youth have problems keeping columns lined up. Filling in with 0’s may help them.

Arithmetic: Divide

34.40000 2.5 860.00000 .57 75000.000 11004.000 10000.000 1004.000 1000.000 00.000

33) Factoring

67

1630.000 9300.0000 57000.000 36004.000 34200.000 1804.000 1710.000 90.000



Factoring is a kind of division we use far more often than long division. Factoring breaks up any kind of number into its multiplication parts. In a compound number both the front number and the tag are factors, because frontnum ^ tag = compound number. All the things in a tag are also factors, because a factor is any thing or group of things that multiplies or divides with other factors to make compound numbers. Nothing tells us what to divide by. We start thinking of multiplication pairs that equal the number, with no remainder, and put them in a list. For example, 4 can be broken into 4^1. Also, 4 can be divided into 2^2. You have just factored 4. To factor 8 we can break it into 4^2 or 2^4, order does not matter when you multiply, so that counts as one pair in the list. To keep track of things and make sure I have found all possible pairs, I start with 2 and work up. So my pairs always have the smaller number listed first. Also, as the first number is working up, the last number is working down. When I meet in the middle then I know I have found them all.

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Number Factors

6

2^3

12

2^6, 3^4

9

18 24

3^3

Factoring a number breaks it into a list of smaller numbers that multiply each other

2^9, 3^6

2^12, 3^8, 4^6

For example, in the 18 list below, I found that 2, then 3, worked, but 4 and 5 did not. Since I already found the 6 with the 3, I have met in the middle and I am done. One pair you will quickly find to be useless. Don’t waste your time on a number ^ 1. Every number can times by 1, but it really doesn’t break the number down into smaller numbers, so we usually ignore it. I also put the Shift icon here because this is another subtle example of Shifting. For example, 12, which is 12^1, Shifts by a ^2 and_2 to 6 and 2.

34) Prime Factoring



Factoring looks for pairs of numbers that multiply into the original number. Sometimes, however, you must factor a number all the way down to its prime factors, which means there might be more than two factors. 8=2^4 but that breaks down further to 2^2^2. A prime factor is a prime number, which means it cannot be broken down any further. 2, 3, 5, and 7 are the first four prime numbers. They cannot be broken down any further. Remember, 1 ^ the number never counts. When prime factoring a number I check 2, 3, 5, and 7 in that order, unless something obvious and easy appears. Sometimes I also need to check 11 and 13. Let’s prime factor 36. Because it is even it divides by 2 to 18, so I put 2 on my list and start working on 18. It is even, so a put another 2 on my list and work with 9. I see that 2 is now done, so I move to 3. 9 divides by 3, so I put 3 on my list. I am left with 3, and since 3 is a prime number, I am done. My list has 2^2^3^3 which are all prime numbers that multiply to 36. Done! Quick tests: 2 is a factor if the number is even. 3 is a factor if the digit sum of the

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number divides evenly by 3. 5 is a factor if the number ends in 5 or 0. I don’t know of easy shortcuts for 7, 11, and 13 that are any faster than doing the actual division.

48=2^24=2^12=2^6=2^3 so 48=2^2^2^2^3 70=7^10=2^5 so 70=2^5^7 98=2^49=7^7 so 98=2^7^7

I sorted all the lists from small to big to compare lists easier.

35) Finding Common Factors



The main reason we factor is so that we can factor two numbers and compare their lists. Of course, we do this a lot with fractions because they have two numbers. One comparison that fractions often need is finding common factors.

15=3^5 18=2^3^3

CF=3

18=2^3^3 CF=2^3=6 24=2^2^2^3 14=2^7 28=2^2^7

CF=2^7=14 CF is short for Common Factor

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A factor is common to both lists if it shows up in both lists. We also want to know how many times it shows up in both lists. For example, let’s find common factors of 8 and 10. 8=2^2^2 and 10=2^5. The only common factor is 2. It shows up once in both lists. It shows up 3 times in the 8 list, but only once in BOTH lists. Now let’s compare 8 and 12. 12=2^2^3. Now we have a common factor of 2 twice. 2 shows up twice in both lists. The third 2 in the 8 list must be ignored because the 12 list does not supply enough 2’s to partner with it. A factor in one list can only be partnered with a matching factor in another list once. No double partnering. No two timing! Once a factor is partnered it is then unavailable. After you have matched ALL the partners, multiply them to get the greatest common factor. Since this is a mouthful for many students I often refer to it as just, common factor. Since we work with just one pair of factors or all of them, students don’t seem to get confused.



36) Reducing Fractions

The first use of factoring is in reducing fractions. A fraction is reduced when all the common factors divide to 1. Why do we do this? Because we prefer smaller, simpler numbers that are easier to work with. A factor over itself, like a number over itself, divides to 1. When this happens we say the factors cancelled themselves. We don’t need to show them anymore, because a fraction multiplied or divided by 1 is still the same fraction. So canceled factors can safely disappear. (Actually, they are still there, but hidden. See the Show Action.)

9 12

= 3^4 = 4 3^3

3

18 12

= 2^2^3 = 2 2^3^3

3

15 27

= 3^3^3 = 9 3^5

5

Canceling is a Shift. You are dividing the top and bottom by the same number(s). You are making an equal change and counterchange. Whenever you look at a fraction you should think of two things automatically. First, there are two numbers, so you will have a change/counterchange pair. Second, the two numbers are linked by reverse multiplication, so you will use multiplication or division to change the fraction. Factoring never works with combining, because they are on different zoom levels. A question often arises, How come we Shift by ^ or + twice, but at other times, such

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as with fractions, we ^ once and _ once? The answer is bigger-smaller. The goal is not multiply or divide once or twice, the goal is to make one part of the fraction (or other term) bigger while making the other part smaller. It is the bigger-smaller that balances things out and keeps the number equal before and after. Multiplying and dividing are just the tools you use if and when you need them. Let’s compare problems. The overriding concern is to start and end with the same final value, 40 on the left and 3 on the right. Shift helps me change my numbers to a form I prefer while keeping that final value the same. Whether or not I multiply or divide is not the main concern. Try all the combinations of ^ and _ in both problems and you will see there is only one way that works. More on Reducing The reason why reducing fractions looks backwards from what you expect is because there is division happening already inside it. As you know, whenever you get anything with division in it, all your normal expectations will get turned backward. When you think things should get bigger they will get smaller. When you think they will get smaller they will get bigger. But here is the good news! Division will always reverse what’s normal. So whenever you see division, reverse your normal expectations. Remember, division is reverse multiplication. A fraction is top _ bottom. When you make the top bigger you multiply by a bigger number, but when you make the bottom bigger you divide by a bigger number. So the two effects cancel each other out. They change the numbers without changing the fraction. This big-small effect is exactly what Shift does.

8^5=40 becomes 4^10=40 _2 and ^2

18_6=3 becomes 9_3=3 _2 and _2

Even though you divide top and bottom by a common factor, their effects cancel each other. You are multiplying by a smaller number on top, but you are also multiplying a smaller number on the bottom. It is like pizza. If you get a whole pizza on your plate, or several small pieces, you end up with the same amount. What is the difference between 1 whole pizza and 4 quarter slices?

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9_3 3 = 12_3 4 18_6 3 = 12_6 2

If you eat the whole pizza does it matter how many pieces you cut it into? A big number on top that gets smaller means you have less pieces, but a big number on the bottom that gets smaller means the pieces are bigger! That’s fair!

15_3 5 27_3 = 9

=

=

37) Dividing Fractions Using Reciprocals



A reciprocal is the upside-down version of a fraction. For example, the reciprocal of 3/5 is 5/3 The reciprocal of an integer is 1 over the number. 6, which is 6/1, has a reciprocal of 1/6. Reciprocals come in handy in several places in algebra, but they also help us divide fractions. All you need to do is flip the second fraction and turn the _ into ^. So we really don’t divide fractions, we turn them into fraction ^ reciprocal, then we calculate. Why does this work? Because of Shift.

1

4_4=1 and 4^ 4 =1 1

4x4=16 and 4_ 4 =16

3 2 15 8 _ 5 = 16

3 5 15 8 ^ 2 = 16

Prove both of these problems to yourself on your calculator! Dividing by a number is the same as multiplying by its reciprocal.

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Look at the example above. You can rewrite the division problem to look like a fraction problem. Next we multiply top and bottom by the same number, similar to reducing. If you have doubts if this is a correct Shift, what is any number (including fractions) divided by itself? 1. So multiplying the “super” fraction (called a complex fraction) by 1 is perfectly legal. But why did we multiply by 4/3 on the top and bottom? Because we are clever! On the next step the two fractions on the bottom cancel each other out, leaving us whatever is on top. And lo and behold! what is on top but multiplying by the reciprocal! So _ by a number = ^ by its reciprocal.

5 3 2_4=

5 2 3 4

4

^3 ^4 3

5 4 =2^3

The 3/4 ^ 4/3 on the bottom entirely cancels out to 1. Since dividing by 1 changes nothing, you can hide it, this leaves the top fractions.

Whether or not you followed that technical explanation, here is the rule to follow. It is D3 on the rule sheet: Reverse last, then multiply. To reverse the last fraction you flip it.

38) Making Like Fractions



Now that we know about factors, multiples, and reducing, we can fill in a gap from the chapter on combining. We can now combine fractions with different bottom numbers, because now we can change their tags to match each other and make COLT happy. (By the way, my guess is that like is short for “alike.” It is the math word for same, identical, etc...) How do we do that? By using multiples and the same process as reducing, except we will make the numbers bigger instead of smaller. We have no technical name for it. I call it “inflating.” I think of reducing fractions to smaller numbers as deflating, so inflating works for me! Let’s say you want to combine 3/8 + 1/2 We will need to inflate 1/2 until the 2 matches the 8. Because I am in a fraction made by reverse multiplication, I Shift the fraction by multiplying by 4 on top and bottom. 1/2 now become 4/8. Now I have same tags, so now I can combine like things to get a like thing. 3/8 + 4/8 = 7/8 (Do not add the 8’s to get 16, nor multiply them to get 64. Remember, nickels + nickels = nickels, not dimes nor quarters.)

74

2 1 2*2 1*3 4+3 7 3 + 2 = 3*2 + 2*3 = 6 = 6

Action Algebra

1) Find common multiple of bottom numbers 2) Times top and bottom by missing factors 3) Calculate answer over common multiple

1 5 3 1*2 5*4 3*3 2+20+9 31 6 + 3 + 4 = 6*2 + 3*4 + 4*3 = 12 = 12

+1 /6 +5 /3 +3 /4 stop0

+2 /12 +20 /12 +9 /12 +31 /12

39) Combining Fractions



The heavy lifting to combine fractions has been done, so now those that have negatives or that subtract each other is no different than adding positive fractions. Follow the same steps and do what you already know about combining positive and negative numbers and counting a series of signs. The only new advice you need is that if you see a negative on the bottom, simply move it to the top and then proceed as normal. Because a fraction is a compound number, it does not matter where the negative(s) is, but it is so much easier to work with the negatives on the top that I won’t even bother explaining how to work with

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negatives on the bottom. Note: The process of using common multiples of the bottom numbers to find a number that all the bottoms will multiply into is technically known as finding the lowest common denominator (LCD). I thought you would like to know that.

-2 -1 -2*2 -1*3 -4-3 -7 3 + 2 = 3*2 + 2*3 = 6 = 6 -2 -1 -2*2 -1*3 -4+3 -1 3 - 2 = 3*2 - 2*3 = 6 = 6 3 7 -3*2 7 -6-7 -13 -5 - 10 = 5*2 - 10 = 10 = 10

40) Canceling Tags



The idea of canceling is so easy (anything over itself cancels to an invisible 1) that we can teach it to young children to complete their knowledge of division. Variables, constants, or whatever is on both the top and bottom cancels. Even though we don’t know what actual number each variable holds, we do know one thing. All x’s hold the same number. That means x over x is a number over itself. That means it will cancel to 1. Therefore, it can disappear. One caution: Don’t double cancel. When a factor cancels, it is gone. You cannot use it to cancel with another matching factor. If you partially cancel a number you can cancel with the smaller factor, but you cannot use the original number again.

76

6xyx 3xyy

15abc 10ab 8xyx 4xxy

5ab 10ab

6xy& 3xyy 8x&x 4xx&

Action Algebra

2x =y 3c =2

= =

2 1 1 2

2& =y

=

2

3 6

12x 3 = 8x 2

4 2

Cancel a number or variable only once and when it cancels it cancels to 1, not 0

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Pre-Algebra: Exponents With pre-algebra we definitely ascend another level to multi-step problems and new notations, such as exponents and roots. Any weaknesses in arithmetic will show up here, so be prepared to pause for extra review once in a while. From now on, I will use the standard word “term” instead of “compound number.” You will also see the dot notation much more than in arithmetic, * instead of ^. This is because we will be using x much more often and we don’t want to confuse x with ^.



41) Basics

Exponents and powers are the same thing. You recognize them because they are the little numbers that sit on the upper right of other numbers or groups, called the “base.” The exponent tells us how many times to multiply the base by itself. 43 = 4*4*4 = 64 (not 4^3!!)

xxyxyyx=x4y3 aabbcc=a2b2c2 aaaaxxxxx=a4x5 (-x)(-x)(-x)(-x)=(-x)5 (-7)(-7)(-7)(-7)=(-7)5 Exponents were invented as a type of shorthand to give us a way to collapse those long strings of variables we get when we multiply. Instead of xxxxx, we can write x5. Instead of aaa, we can write a3. Count the bases, then write the base once and write the count as an exponent. If the

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79

negative gets repeated with each base, be sure to use ( ) so that the power affects the sign. -4*-4*-4*-4 = (-4)4 -2*-2*-2=(-2)3 -5*5*5*5=-54 When we collapse we are not making a calculation. We are just rewriting a multiplication problem as an exponent problem. We are Morphing (converting) from times to power. We have not done anything except to make the same problem shorter. We are really just changing from one function to another. The exponent affects just what it touches. If it touches a group, the whole group gets affected. This means the negative sign may or may not get affected by the exponent. This is an exception to treating the sign as part of the number. -42 = -(4)(4) = -16 (-4)2 = (-4)(-4) = 16 Another correct way of looking at the negative sign is that it is part of a separate, invisible -1 that is multiplying the base. -42 = -1*42 = -1*4*4 = -16

30 = 1

x1 = x

x2 = xx

31 = 3

-x1 = -x

-x2 = -xx

32 = 9

(-x)1 = -x

(-x)2 = (-x)(-x)

33 = 27

-(x)2 = -(x)(x)

If a base has an exponent of 1, or the exponent is invisible, that means the base occurs just once. It doesn’t multiply anything. Nothing happens. 51 = 5 -171 = -17 (-17)1 = -17 If a base has an exponent of 0, then the base automatically becomes 1, even if the base is negative. 60 = 1 2980 = 1 -90 = -1 (-9)0 = 1 x0 = 1 -x0 = -1 (-x)0 = 1 You may be asking why the 0 exponent makes everything into 1. I think the next lesson on negative exponents will help us answer that question. Note: I use the funnel icon with exponents and roots because they act like functions. They are like little machines into which you feed the base and out comes a different number.

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42) Negative Exponents



Negative exponents are the reciprocals of their positive counterparts. For example, 23 = 8 and 2-3 = 1/8 So the basic idea is that when you see a negative power, flip it from the top to the bottom, or the bottom to the top, and change its sign. 104 = 10000 103 = 1000 102 = 100 101 = 10 100 = 1 10-1=.1 10-2=.01 10-3=.001 10-4=.0001

Where do negative exponents come from? Look at the sequence on the left to figure out the pattern the numbers make. Negative exponents were not invented, they were discovered when someone kept dividing past the 0 exponent. Notice that as you go down the list, the exponents get lesser by 1 as the numbers keep dividing by 10. When the power becomes 0, the answer becomes 1. All numbers, not just 10, follow this same pattern. That is why any number to the 0 power = 1. Now keep dividing by 10, which lowers the exponent by 1. Each step lower adds another decimal place, and decimal places are just fractions. 10-2 = .01 = 1/100 10-3 = .001 = 1/1000

What happens when we have a negative base with a negative exponent? The same pattern applies, but watch how the negative sign changes as the exponent changes from odd to even. The negative sign of the exponent has nothing directly to do with the sign of the base. The negative sign of the exponent makes a fraction. The number part of the exponent then tells how many times to multiply the base. Multiplying the base determines the final sign of the base. As you can see, there is a little extra work involved to use exponents, but their ability to collapse long strings of numbers and letters make it worth it.

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1 1 9 -2 2 =1_3 =1_ =1^ =9=3 9 1 -2 3 22=4 21=2 20=1 2-1=1/2 2-2=1/4

Negative exponents make fractions, not negative numbers

(-2)3 = (-2)(-2)(-2) = -8 (-2)2 = (-2)(-2) = +4 (-2)1 = (-2) = -2 (-2)0 = +1

(-2)-1 = 1/(-2) =

-1/

(-2)-2 = 1/(-2)(-2) =

2 +1/

(-2)-3 = 1/(-2)(-2)(-2) =

4 -1/

8

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43) Multiplying Bases



Because exponents are just another way of writing multiplication, at any time you can rewrite an exponent problem as a multiplication problem. It will take longer, but you will be sure you have the right answer. I mention this because it is not uncommon for students to forget the exponent rules. To multiply matching bases, combine the exponents. (Rule M2) 52*56=58 73*7-5=7-2 x3*x7=x10 -84*89=-813 (The - is not part of the base) Nothing magical is happening here. If we write a problem out the long way, then collapse it, we get exactly the same answer. So why not figure out the pattern of what is happening and do it the short way?! 52*56 = 5*5^5*5*5*5*5*5 = 58 The exponent pattern is part of a bigger pattern. When we multiply any number we count negative signs. When we multiply decimals we count decimal places. Counting is combining one at a time. So when you multiply numbers you combine the signs, places, or powers that go along with them. This is why the rule says: When multiplying, combine add-ons.

102*105=107 23*22*22=27 x3*x3=x6

x3*x-3=x0=1

(-4)2*(-4)-5=(-4)-3

(-2)3*(-2)-3=(-2)0=1 x2y4*x5y7=x7y11

Notice that I am combining the exponents just as they are when the bases multiply. In the next lesson where I divide bases, I will change the sign of the last exponent. This is why textbooks say to add exponents when ^ bases and subtract exponents when _ bases.

Pre-Algebra: Exponents

44) Dividing Bases

83



To divide matching bases, change the sign of the last power, then combine. (Rule D3) 52_56=5-4 (-7)3_(-7)-5=(-7)8 x9_x5=x4 Notice that in every example the bases matched. It must be that way because exponents are shorthand for long strings of the same base. So, if you have bases that are different you must make them the same or else stop. For example, you can pull negatives out and deal with them separately as a string of -1’s. (-8)4*89 = (-1)4*84*89 = 1*813 = 813 There is something else you need to notice--the parentheses around _ problems with more than one base. The ( ) are needed to make sure everything in the last part is being divided. For example, in ab_xy, only the x is dividing. The y is multiplying. In fraction form it looks like this: ab/xy The y is actually on top, not the bottom as you might think. To make sure both the x and the y are on the bottom use ( ) like this (ab)_(xy) It is important to be clear to write what we mean and mean what we write!

102_105 = 102-5 or 102*10-5 = 10-3 x3_x3 = x3-3 or x3*x-3 = x0 = 1

x3_x-3 = x3+3 or x3*x3 = x6

(-4)2_(-4)-5 = (-4)2+5 or (-4)2(-4)5 =(-4)7 (x2y4)_(x5y7) = x2-5y4-7 or (x2y4)(x-5y-7) = x-3y-3

Let’s look at this one again in fraction form

x2y4 xxyyyy x-3x-3 1 = = = 5 7 x y xxxxxyyyyyyy x3x3 Once again we see that division reverses things. When we multiply, we just combine

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the exponents as they are, but when we divide, we must reverse the signs of the exponents after the division sign or fraction bar. This is because the exponents are not expanding the string of bases, but because they are canceling them. Look at the rule sheet and compare M2 with D3, and look at examples M9 and D7.

Groups with one term

(5x)(-6y) (5x*-6y)

(3m2y*2m-3) -4x(-8xy)

(

x2y4 x5y7

)

Groups with multi terms

(-5x-6y-2x) (3m2-4m2) (x2+x-7) (x+5)+9

Factors ^ and _ in terms Terms + and - in expressions Groups box anything

45) Zoom Levels Now it is time for students to be introduced to zoom levels. We need to focus on groups, terms, and factors. As we have already seen, factors multiply or divide each other to make up a term. Terms combine with each other. 5x+2y are two terms with two factors each, but 5x*2y is one term with four factors.

Pre-Algebra: Exponents

(22)3=222222=22*3=26=64

85

(x2)3=x2x2x2=x2*3=x6

(x4y2)2=x4y2*x4y2=x8y4 (x4+y2)2=stop for now

If we zoom out one level from factors we see terms coupled together with + and - signs. It is important to recognize the difference between terms and factors because exponents will cause different effects. If there is only one term inside a set of parentheses, a completely different answer is made compared to two or more terms inside the parentheses. Now we want to focus on groups and terms. Groups use ( ) or [ ] or { } to pack a term, a part of a term, or multiple terms. Groups occur at any level. They are like boxes because they make something happen to everything in the box, not just one of the items. We need to recognize when a box holds a bunch of factors in one term or if it is holding a bunch of separate terms. It makes a big difference! For example, (3abc) is one term with four factors, but (3+a+b-c) are four terms. Probably the trickiest part in seeing zoom levels correctly is paying attention to the difference between a - sign being used for combining and a - sign that is part of a number being multiplied. For example, look at the first two lines on the left above. Notice that -6y is not being combined with the 5x, rather it is being multiplied. There is a ( and a * in front of them. Now compare that with the first line on the right. There is nothing in front of the -6y to tell us to multiply, therefore it is being combined. See the difference?

x3 xxx 3-2 1 1 x x = = = = = x2 xx x2-3 x-1

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46) Groups with exponents

Action Algebra



Some textbooks call this “power of powers” or “multiply powers.” A special “shortcut” is given, but it is really just basic exponent properties. Sometimes the base of an exponent is not a number or variable, but a grouping symbol like ) or ] or }. If there is only one term in the box, then you multiply all the inner exponents by the outer exponent, including any invisible 1’s. (x4y2z-3)5 = x20y10z-15 (xy3)2 = x2y6 This works because the outside power tells you how many groups you have, and inside each group is the same string of repeated variables. (x2y4)3 = (xxyyyy)(xxyyyy) (xxyyyy) = x6y12 You can think of this as our regular exponents on steroids. If the variable looks plain, like it does not have an exponent, that means it has an invisible exponent of 1. Remember the basics, anything to the 1 power is itself, so anything that is itself is to the 1 power. For example, (xy2)4 = (xyy)(xyy)(xyy)(xyy) = x4y8 Why do it the long way, if you just remember the x is really x1 and gets multiplied by the 4 like any other exponent. If there is more than one term in the box or in the fraction in the box, you should just stop for now. Multiplying the outside power by all the inside powers does not apply. In the polynomials chapter we will learn how to solve this type of problem.

47) Exponents in Fractions



Multiplying bases on the top or bottom of a fraction is no different than multiplying bases that are not in fractions. What gets interesting is dividing bases in fractions. You still follow the same rule you learned, but you have two ways of doing it. You can start the division from the top or from the bottom, but where ever you start dividing is where your answer goes.

(

Pre-Algebra: Exponents

)=

x2y4 3 x5y7

x6y12 x15y21

87

x2y-3 x2-4 x-2 x-2y-8 1 = 5+3 = = = 4 5 8 y x y y x2y8 x2yx3y-4 x5y-3 x5+3 x8 x8y-6 1 = = = = = xy5x-4y-2 x-3y3 y3+3 y6 x-8y6 Look at the examples again and you will see that the answer ends up where the division started. Where the division started is the first exponent and where the division ended is the last exponent. So if you do top _ bottom, then the bottom exponent changes sign and the answer belongs on the top. If you do bottom _ top, then the top power changes sign and the answer belongs on the bottom. What this all leads to is another shortcut. You can flip a base between the top and bottom if you change the sign of its exponent. What you are doing is finding reciprocals. Let’s investigate the above example a little more. Remembering that exponents are just shorthand for multiplication, it makes sense that dividing exponents are related to cancelling. Therefore, cancelling two pairs of x’s leaves one x on the top, which is x3-2. But what about the x2-3 on the bottom? To get an answer only on the bottom means we must cancel everything from the top, but how can we do that if we are one x short on the bottom? By going in the hole. We “borrow” an x and say that we will pay it back later. If that sounds like combining, it is! Just remember, because we are working across a fraction bar (division) we need to reverse the sign of the exponent(s) on the other side from which we start.

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48) Scientific Numbers A handy application of exponents is scientific numbers or “scientific notation.” They were invented by scientists (duh!) to handle very big or very small values. Scientifics can handle the largest numbers, like googol. Instead of writing a 1 followed by 100 zeros, you just write 1 followed by a ^10 with an exponent of 100, like this: 1x10100 A lot easier and shorter!

7^102=7^100=700

0.25^104 = .25 ^ 1000 = 2500

4.8^105 = 4.8^ 100000 = 480000

8.03^10-3 = 8.03 ^ .001 = .00803 75^10-3 = 75 ^ .001 = .075

A positive power moves decimal bigger A negative power moves decimal smaller Of course, scientific numbers are compound numbers. The frontnum is only the number in front, and the tag starts at the ^ sign. To morph a scientific to a decimal all you need to do is move the decimal the same number of places as the exponent. If the power is positive move the decimal that many places to the right. If the power is negative move the decimal that many places left.

5^102 = 5^100 = 500

Decimal moved two places right

3.9^103 = 3900 105.3^10-2 = 1.053

3.9^10-3 = .0039 .0027^103 = 2.7

One last, important point: A true scientific follows a precise pattern. It starts with one digit then has a decimal point and 1 or more digits (including 0) after it, followed by

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89

^10power. That means the frontnum must be 1.0 or greater and less than 10. So that means you sometimes end up with a scientific-looking number and not a true scientific. You need to learn to adjust a scientific-looking number to make it be a true scientific: digit decimal ^ power of 10. That is the next lesson. Notice how Shift is involved here. When the integer or decimal in front gets smaller, the ^10power gets bigger. It goes from a negative exponent up to an invisible ^100 which is 1. When the integer or decimal gets bigger, the power gets smaller by going down to ^100. Shift is involved in so many things. It pays to watch for it so that you will then be inclined to use it. It was even used in the canceling of factors and exponents in the previous lesson!

49) Adjust Scientifics



Scientifics are compound numbers with two basic parts, a decimal and a power of 10. These are the two parts we will often Shift so we can combine scientifics and make them fit the official format. The big-small principle applies here. If the power goes up, the decimal goes down. If the power goes down, the decimal goes up.

9150^101 = 9.15^104 9150^10-10 = 9.15^10-7 .0108^104 = 1.08^102 .0108^100 = 1.08^10-2

Notice that the power becomes greater so that the decimal can become smaller, or vice versa. It is not decimal times exponent. It is decimal times 10 with an exponent.

7.25^104 = 72.5^103 = 725^102 Power stepped down 1, so decimal stepped up 1 place.

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6.4^105 = .64^106 = .064^107 Power stepped up 1, so decimal stepped down 1 place. Of course, you are not limited to adjusting the power one step at a time. You can change it by any amount you like, as long you Shift the decimal an equal, but opposite, amount.

8.4^109 = 84000^105 = .000084^1014 Why does this work? Every decimal place smaller or bigger is a change by 10 and every step up or down in the exponent is a change by 10. Since the power and the decimal are in the same compound number, the scientific does not actually change value. Notice that the decimal and the power are factors in the same term. Factors ^ or _ and we are using ^ and _ to make adjustments.

50) Multiply Scientifics



Because multiplying can handle different tags, we can easily multiply any scientific number times any other scientific. Multiply the front nums as usual, and combine the exponents. Even if one or both of the exponents is negative, combine them. This is no different than multiplying variables with exponents. It might help to Sort the scientifics differently to make this plain.

4^105*9^107 4*9*105*107 36^1012 3.6^1013

3.6^102 * 2.8^104 10.08^106 1.008^107

2^105 * 7^10-8 2*7*105*10-8 14^10-3 1.4^10-2

8^10-3 * 6^10-7 48^10-10 4.8^10-9

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91

3^105*7^102 = 3*7*105*102 = 21^107 3^x5*7^x2 = 3*7*x5*x2 = 21^x7 Notice that we are not combining or changing in any way the 10’s. They are just the base for the exponents. Only the exponents get combined. The 10 remains a 10, just like the x stays x. When you get an answer, you will probably need to adjust it. Often it only looks like a scientific, but is not a true scientific.

(5^103)2 25^106 2.5^107

(2^10-2)-4 1/ ^108 16 .0625^108 6.25^106

51) Powers of Scientifics

(3^10-6)3 27^10-18 2.7^10-17



A scientific raised to a power is no different than a group of variables to a power. Use the same logic to solve this problem. Distribute the outside power to the inside power. (8^104)2 = 82^108 = 64^108 = 6.4^109 or (8^104)2 = (8^104)(8^104) = 8^8^104^104 = 64^108 = 6.4^109 You can write problems out the long way to prove it to yourself. (4^103)4 = 4*4*4*4*1000*1000*1000*1000 = 256^1012 = 2.56^1014

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52) Dividing Scientifics



Just like dividing variables with exponents, change the sign of the last exponent then combine.

7^104 _ 2^108 = 3.5^10-4 3^10-2 _ 8^10-6 = .375^104 = 3.75^103 There are two right ways and one wrong way to show division with scientifics. The two right ways are on the left. Use a fraction or use ( ), if you don’t use ( ) you are not actually dividing the whole scientific, just the front number. 8^102 _ 4^107 is not saying what you mean. Only the 4 is doing any dividing. Only the 4 is on the bottom of the fraction. The 107 is still on top. You can still solve this problem, but it won’t be what you might expect: 2^109 not 2^10-5

(8^102)_(4^107) 8^102 4^107 2^10-5

20^103_5^106=4^109 No ( ) so only the 5 was divided

The cause of this “problem” is the Sort Action. We can sort anything at anytime, except division. Since the 107 at the end was not in ( ) we can Sort it to the beginning of the line where it multiplies with the 8. Only the 102_4 are permanently stuck in division.

53) Combining Scientifics



Just like fractions, it is the second part of the compound number, the tag, that tells us whether or not they can be combined. The bottom number of the fraction and the power of 10 are the parts to be matched. Similar to fractions, we need to “make common denominators” so we can combine them. Of course, we won’t literally make common denominators. Instead, we will adjust the powers so that they match, then we can combine

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scientifics. A good habit is to adjust the lesser power up to the greater. Often, but not always, it will save you a step. Compare the first two examples below on the left and you will see what I mean. The top problem I adjusted to 102 then had to adjust again to 105. Whichever method you use, always remember to adjust your answer so it becomes a true scientific.

3.1^102 + 2^105 3.1^102 + 2000^102 2003.1^102 2.0031^105 3.1^102 + 2^105 .0031^105 + 2^105 2.0031^105 8^103 - 8^107 .0008^107 - 8^107 -7.9992^107

Frontnum Tag

+4 ^103 +5 ^103 +9 ^103

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Pre-Algebra: Morphs Morph is my cool, hip, and up-to-date word for convert. I also needed a one-syllable word because all the other Actions ended up with one syllable! Morphing (converting) is not so much a problem in itself as it is a tool to help solve other problems. Sometimes a problem is given to me in mixed numbers, but fractions are much easier to work with, so I convert the mixed numbers to fractions, work the problem, then morph the answer back to mixed. With that in mind, this chapter will be a collection of techniques. A related idea that fits in with morphing is units. We often have to convert 24 inches to feet and 78.5 meters to centimeters and so on. That will be covered in this chapter as well.

54) Fractions and Mixed Numbers



Fraction to mixed number. Divide the top by the bottom and the answer becomes the whole number and the remainder becomes the new number on top. The bottom never changes.

14 4 =2 5 5 2 5 14 10 4

-23 2 =-7 3 3 7 3 23 21 2

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Mixed number to fraction. Multiply the bottom ^ the whole number, then add the top. Mixed numbers are a rare exception to the notation of multiplying by touching. Of course, two digits touching each other is a number, but a whole number next to a fraction is a mixed number where the whole number adds to the fraction rather than multiply.

4 37

55) Rounding

7*3+4 = 7

25 = 7



This is the one time where we will change the value of a number without making a counterchange. This is technically illegal, but we consider the change so small compared to the bother of writing a long number, that we allow this loophole. In the real world, because of imprecision in measuring we don’t even consider this a loss of value. Many decimals are the result of dividing two numbers that never reach an even answer. The division keeps going and going with remainders that never reach 0. If there is a repeating pattern we can at least turn it into a fraction, but some have no pattern and no end. Those are irrational numbers. They are insane! They wander endlessly and aimlessly. Who wants to write all that?! This is why we have rounding. You know, who cares if you lose a few billionths! Textbooks and teachers are different, but in my classroom, the rule was to round to the 4th place during a problem, then round the answer to the 2nd place, the pennies place. We kept the 4th place during a problem because of all the steps and operations that could lose accuracy. To round to the 4th place I must first look at the 5th place. If the 5th place is 4 or lower, I will just chop the number past the 4th place. If the 5th place is 5 or higher I will make the 4th place 1 higher, then chop the number. If the 4th place has a 9, then rounding up makes it a 10, so carry the 1 to the 3rd place.

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Round to 2nd place (nearest hundredth)

13.7063 13.71 4.89517 4.90 .27483 .27

Round to 4th place (nearest ten-thousandth)

Look to the right of your place.

5 rounds it up,

and 4 does nothing.

Chop off the number at your place.

13.706319 13.7063 4.8999501 4.9000 .27483522 .2748



56) Fractions and Decimals

Fraction to decimal. This is one of the times where you look at a fraction as a division problem, top _ bottom. The number on top goes outside. Round your answer according to the instructions of your textbook. My classroom rule is to round decimals to the 4th place during work and then to the 2nd in the final answer. That way enough accuracy is kept along the way for the final answer to be accurate.

Fraction to decimal: Top divided by bottom

3.4 17 = 5 17.20 5

Decimal to fraction. Because it does not change the number, every number is invisibly divided by 1. So Show the decimal over 1. Now you have a decimal in a fraction, so Shift it. Inflate the fraction by moving the decimal all the way to the right, then add as many zeroes to the 1 as places you moved. If you moved the decimal once, add one 0. If you moved the decimal twice, add two 0’s, and so forth. Now you have a normal looking fraction! Reduce it, if necessary.

Pre-Algebra: Morphs

Decimal to fraction: Decimal over 1, Shift, reduce

97

3.4= 3.4 = 34 =17 1 10 5

57) Fractions and Percents



Percents are just fractions over 100 and they are decimals to the hundredths place. They don’t appear by themselves. They take a Percent to fraction: part of something else. For example, 10% of $5, Number over 100, reduce or 25% off of the price, or 50% of the profits. Like mixed numbers, we don’t work with them in problems. At the beginning, you will convert the percent to a fraction or decimal, which ever seems easiest to fit in with the rest of the problem. You will then work the problem as normal, then convert back to a percent at the end, if needed. Since percents always appear as a “percent of” something else, multiply the percent times the other thing. For example, translate 8% of 22 into 8/100 ^ 22 or into .08 ^ 22. Translate 20% of $32.78 into .20 ^ 32.78. In the last example, I did not bother with making the percent into a fraction, because the money was in decimal form. I could turn the percent into a fraction, but that would make the problem harder for me. Percent to fraction. % means /100 so put the number over 100, then reduce, if needed. Fraction to percent. Think of this as part of a combining fractions problem where the other fraction is over 100. Your “common denominator” is 100, so multiply the bottom of your fraction to make it 100. Then, because of Shift, multiply the top by the same amount. Now you have a fraction over 100, which is a % by definition.

25% = 25/100 3% = 3/100

3^25 = 75 =75% 4^25 100

Fraction to percent: Multiply bottom to 100, multiply top by the same you now have percent

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58) Decimals and Percents



If you can remember the direction the decimal should move, you will find it easy to Morph decimals and percents. First, realize that the decimal always move two places. Remember what the two 0’s in the % sign mean? They are dividing by 100, which is two places. Now the only question is, Should the decimal move left or right? Percent to decimal. Shift answers the question. A % sign divides a number 100 smaller, so what happens if you take it away? The number will get 100 times bigger by returning to normal. That is a change. Now we need a counterchange, so make the number 100 times smaller. That is, move the decimal left two places. So when you take the % away from the end of a number, move the decimal away from the end two places.

Percent to decimal: Percent goes away, decimal goes away two places

72% = .72

4.9%=.049

This Morph makes me think of magnets attracting or repelling each other. Remember playing with those as a kid?! If you take the % magnet away from the right, there is nothing holding the decimal, so it goes away to the left by two places. When you bring the % magnet in on the right, it pulls the decimal towards it two places. Decimal to percent. The % comes in at the right end, so the decimal moves to the right two places. This is a Shift because the % is making the number 100 times smaller. Moving the decimal right two places makes the number 100 times bigger. Everything that applies to decimals and percents also applies to integers and percents, because integers have an invisible decimal at the right end.

Decimal to percent: Percent comes in, decimal comes in two places

.45 = 45%

3.8 = 380%

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

59) Units

In arithmetic I used the phrase “compound number” to help expand the students’ view of numbers and the things they count. Now we must complete that view. A unit (inches, meters, gallons, pounds, etc...) is the third part of a compound number, or term. It tells us how the things are being counted. For example, we can measure distance in inches, feet, miles, kilometers, etc... Units really don’t have an effect until we have two compound numbers with units. Then we must figure the ratio between the units, then multiply or divide by that ratio. The units can then be dropped and regular math proceed. For example, if we have 5 yards that is nice to know, but if we want to know what 5 yards + 8 feet equals, then we have some calculating to do. First, the ratio of feet to yards is 3 because 3 feet = 1 yard. (3 of the smaller unit fits into 1 of the bigger unit.) Second, I need to decide if I want to change yards to feet or feet to yards. Sometimes the textbook tells me what to do, sometimes it is left up to me. In this case, it is left up to me and I usually find it easiest to change all units to the smallest. So 5 yards will become feet. Third, how do I know whether to multiply or divide? Look what happened when I changed yards to feet. The unit got smaller. Therefore, the Shift Action tells me that the number must get bigger so I don’t lose any of the original length. Obviously, I multiply 5^3 to get 15 feet. Now I can combine 15 feet with 8 feet for an answer of 23 feet. Shift comes in all over the place! Are you catching on to its importance and power? Changing from one unit to another is a Shift operation. If the unit gets changes to a bigger unit, then the number must get smaller. If the unit gets smaller then the number must get bigger. For example, to change 48 inches to feet, inches gets 12 times bigger, so the number must get 12 times smaller. The answer is 4 feet.

Convert 73 inches to feet Inches to feet means unit gets bigger, so the number gets smaller, so I divide: 73_12 = 6.08 feet What if I don’t know how many inches are in a foot? Look it up in the Unit Conversion Chart. The chart tells you how many little units fit into the big unit. That is the number you will use to multiply or divide. If you can’t find a pair of units that match what you need,

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make the match in steps. As you work your way from small to big, multiply the numbers along the way, until you have the multiplier/divider you need. For example, to convert inches to miles, find inches to feet. That number is 12. Then find feet to miles. That number is 5280. Multiply both numbers 12^5280=63360 inches in a mile. The 63360 is what you will use in your problem.

How many days in 4 centuries? Centuries to days means unit gets smaller, so number gets bigger, so I multiply: 4^365^100 = 1,460,000 or 1.46^106 Convert 24 hours to seconds Hours to seconds means unit gets smaller, so number gets bigger, so I multiply: 24^60^60 = 86,400 or 8.64^105

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Unit Conversion Chart Small unit inch foot foot yard mile ounce cup pint quart gallon ounce pound ton second minute degree degree degree right angle semi-circle second minute hour day month week year year year year century

Abbreviation How many in Big unit LENGTH in. " 12 foot ft. ' 3 yard ft. ' 5280 mile yd. 1760 mile mi. LIQUID oz. 8 cup 2 pint pt. 2 quart qt. 4 gallon gal. WEIGHT oz. 16 pound lb. 2000 ton tn. ANGLES, CIRCLES sec. " 60 minute min. ' 60 degree deg. 360 circle deg. 90 right angle deg. 180 semi-circle 4 circle 2 circle TIME sec. " 60 minute min. ' 60 hour hr. 24 day 365 year mo. 12 year 52 year yr. 10 decade yr. 20 score yr. 100 century yr. 1000 millenium cen. 10 millenium

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

60) Metric Units

I’ve taught many classes, but one class I have never taught is chemistry. However, I know chemistry uses scientifics a lot. I have seen chemistry students using worksheets to figure their metric numbers. It seemed like a long process to me when I would need to help them with the math. After a while we came up with a shorter process that works for any metric problem in any class. You just need the Metric Conversion Chart and the following three steps. 1) Write the power number above each prefix and a - above “to” 2) Copy the number, attach a ^10, and the power will be the answer to step 1 3) Adjust the decimal and power if needed. always put a - in between

0

-

-2

numbers come from chart

6.8m to cm = 6.8^102cm -3

-

3

3.1mg to kg = 3.1^10-6kg 6

-

-9

87Ml to nl = 87^1015nl

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Metric Conversion Chart abbr. yotta zetta exa peta tera giga mega kilo hecto deka __m __l __g deci centi milli micro nano pico femto atto zepto yocto

prefix Y Z E P T G M k h da meter liter gram d c m µ n p f a z y

number 1,000,000,000,000,000,000,000,000 1,000,000,000,000,000,000,000 1,000,000,000,000,000,000 1,000,000,000,000,000 1,000,000,000,000 1,000,000,000 1,000,000 1,000 100 10 0

power 24 21 18 15 12 9 6 3 2 1 0

rough size beyond universe galaxy diameter 100 light years solar system past Saturn 3 trips to moon Texas 6 blocks soccer field big room long yardstick

.1 .01 .001 .000 001 .000 000 001 .000 000 000 001 .000 000 000 000 001 .000 000 000 000 000 001 .000 000 000 000 000 000 001 .000 000 000 000 000 000 000 001

-1 -2 -3 -6 -9 -12 -15 -18 -21 -24

handwidth fingerwidth thick fingernail bacteria molecule atom proton

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Pre-Algebra: Calculate We have looked at the pieces and parts, but now it is time to put them together to calculate real world problems, use formulas, and solve common word problems from standardized tests. After introducing the basic principle of multi-step problems, I will then show two ways to approach them (I let my students choose), then we will start the lessons. IN FUN MUD COLT We have done all the basic operations with all the kinds of numbers, but what do we do if we have more than one kind of operation in the same problem? What happens when functions and parentheses are also present? That is when order of operations helps us. It tells us the order of importance of all the functions, operations, and grouping symbols. The basic principle is to calculate the complicated first. Inside parentheses is before functions which is before multiply and divide which is before combine. In the example, I worked inside the parentheses first even though combining is the least complicated. The ( ) made it the most important. Then I went outside and looked for the most important thing to IN FUNny MUD is a COLT do. The 32 was the most important to do next because it was the most complicated. I learned about it 2 after combining and after multiplying. That brought me to the third line where I had to choose between adding 2 the 1 or multiplying the last 3 numbers. Multiplying won because it was more important and complicated. Lastly, I added the 1 to get the right answer of 19. If I had just worked left to right and forgotten about order of importance, I would have calculated an answer of 82. That is way off from 19! So now that I have demonstrated the importance of the order of operations, how do I do it and teach it

1+2*3 (1+0) 1+2*3 (1) 1+2*9(1) 1+18 19

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systematically? There are two ways that I call Copy or Calc, and Multi-Pass. Both methods have their advantages and can be used together in the same problem. The worksheet solutions use the Multi-Pass method. Copy or Calc The Copy or Calc method is what most students start doing naturally, but do not always use it successfully. Part of the reason is that they use it out of laziness and/or not thinking. Starting at the left you work left to right, but you look one operator or function ahead to decide if you copy or calc the number you are at. For example, I start at the 4 in 4+2*57*2. My current spot is on the 4 and it appears that the first thing I should do is combine the 2, but now I look ahead to see what is happening on the other side of the 2. It is multiplying the 5. So now I decide, copy or calc the 4? The correct thing to do is copy it to the next line, because the 2 has something more important to do on its right with the 5 than it does with the 4 on its left. So copy 4 to the next line and move to the 2, then start the process over again. Your new spot is on the 2. Look ahead to the 5. Should it combine with the 7 on its right or multiply with the 2? MUD comes before COLT, so go ahead and calc +2*5. You now have 4+10-7*2 Now your next spot is -7 and you can calc it with the *2 because you are at the end. Now you have 4+10-14. Since all you have left is combining you can finish the problem and get your answer of 0. Notice that this example had no parentheses. That can throw students off if they are not paying attention. All calculating should stop, everything before the ( ) should be copied to the next line. Then start fresh inside the parentheses. Multi-Pass The multi-pass method basically scans the line from left to right first looking for any parentheses, then starting again looking for functions, then for multiplying/dividing, then for combining. This method appears to be slower (but is not), but it is safer because the student tends to be more focused and aware of what s/he is looking for. Let’s look at the example from Copy or Calc. 4+2*5-7*2 I first scan the problem looking for ( ), but I see none. Then I look again for functions, such as exponents, trigonometry, and logarithms, but I see none. Then I look for multiplying/ dividing. I first see the 2*5, so I write down the next line 4+10-7*2. Then I scan again for multiplying and see the 7*2, so I write down the next line 4+10-14. Scanning again for multiplying gives me nothing, so all I have left is combining, so the answer is 0. Earlier, I said the two methods could be used together. In the example above, at the first multiplying pass, I could have done both multiplications at the same time because they don’t interfere with each other. 4+2*5-7*2 becomes 4+10-14 on the second line, and the answer of 0 is reached on the third line. This combined method is the one I use.

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Equal importance What do you do when you look ahead and find something of equal importance? In other words, the spot you are on is ready to combine, then you look ahead and the next operation is combining also. What do you do? Work left to right for habit’s sake. (It doesn’t matter what order in which you combine, just keep the signs with their numbers.) When multiplying or dividing, definitely work only left to right. It makes a difference! Look at these examples. 8_4^2 = 2^2 = 4 but 8_4^2 = 8_8 = 1 The first problem I worked left to right and got 4. The second one I worked right to left and got 1. Since there are two different answers, mathematicians have voted that we work left to right. If you want someone to get 1 for the answer use parentheses, like this, 8_(4^2). Now they have to do the 4^2 first. Exception alert! Treat fractions like single compound numbers, not number _ number. Try these on your calculator. 8_4/2 = 4 is correct, and 8_4_2 = 1 is correct. So if you see a fraction, use your fraction button, not your division button.

2^12_4^3 24_4^3 6^3 18 YES!

2^12_4^3 24_12 2 NO

2^12_4^3 2^12_12 2^1 NO 2

Thankfully, most of the confusion in these types of problems is avoided in the textbooks, because they use parentheses. But I said “most” not “always.” If you stay in the habit of always working left to right, you won’t get caught ignorantly making mistakes by working out of order.

61) MUD before COLT



Combining is the last thing you do because it was the first thing you learned. Another way of saying that is multiplying and dividing is more complicated than combining, and calculate the complicated first. Think of it visually. Multiplying and dividing use a grid with two number lines. Combining uses only one number line.

Pre-Algebra: Calculate

3+4*5 5-3*7+2-1*4 -2*4+8*3-5 3+20 5-21+2-4 -8+24-5 23 -18 11

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In the following problems notice I handle equal importance of multiplying and dividing by working left to right.

7-10*4_2-5 7-40_2-5 7-20-5 -18

3+8_2*6-1 3+4*6-1 3+24-1 26

62) FUN before MUD

12_3*4+2*9 4*4+18 16+18 34



Even more complicated than multiplying and dividing are functions like powers, roots, trig, and logs. Therefore, functions are more important than multiplying and dividing. After all, how do you know what to multiply until you put the numbers into their functions and get an answer?

3*log100+2*92 3+4*52 1-9*23*cos(0)+5 3*2+2*81 3+4*25 1-9*8*1+5 6+162 3+100 1-72+5 168 103 -66 If there are two functions next to each other by one of the four operations, + - ^ _, then you can calculate them as independent units. They are not interfering with each

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other, so order does not matter. However, it is always good to stay in the habit of left to right. I will not cover any details of log or trig functions (that’s the next book!) except to say the worksheets will only include problems where only button pushing on the calculator is required. Roots, however, will be explained in the next chapter. For now, just button pushing!

2*@ 36 2*6 12

63) IN before FUN

5+3* @49-28_7 5+3*7-4 5+21-4 22



Parentheses override everything else and makes you start over on the inside. Parentheses can come in three flavors ( ) or [ ] or { } or they can be fraction bars. These are all grouping symbols. Each pair of symbols puts numbers and functions in a box, and you cannot touch what is in the box until you calculate it. It is best to think of these boxes as being made out of frosty glass. You can see the numbers inside, but you can’t clearly see the answer. And we must know the value of the box before we let something on the outside do it’s thing to the box. For example, say you have 3 boxes filled with candy bars. Someone asks you how many candy bars you have, but all you can honestly reply is, 3 boxes worth. Until you open the boxes and count the bars, you have no real clue how many total candy bars you have. So boxes are the most important and complicated of all. They wrap their answers in mystery until you go inside and figure them out. All other calculations must pause and wait for the answer from inside the parentheses.

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102*(1+1) 102*2 100*2 200

9+5(3-8)*42 2*6(log10+32) 9+5(-5)*16 2*6(1+9) 9-25*16 2*6(10) 9-400 120 -391 6+3*4*22 6+3*4*4 6+48 54 = = = -52*3-7 -25*3-7 -75-7 -82

64) Order with fractions



With fractions, the process is no different than with integers and decimals, but they make everything look so much more complicated it is worth focusing on them. You follow the same order: IN FUN MUD COLT. What is important to remember is that you treat fractions as a single compound number, rather than number _ number. Use the fraction button on your calculator instead of the division button.

1/

4 2 1 2 3+ /5*10-( /3+ /6) 1/ +4/ *10-(5/ )2 3 5 6 1/ +4/ *10-25/ 3 5 36 1/ +8-25/ 3 36 275/ 36

17 3 1 + * 5 5 4 17 3 + 5 20 71 20

110

65) Nesting

Action Algebra



It is possible that you will see parentheses inside of parentheses. This is called nesting. That is why there are different flavors. It is easier to see what matches. Go to the innermost box and work your way out until you get to the outside. Along the way, always use FUN MUD COLT. Some books will change brackets [ ] and curly braces { } to parentheses ( ) as the inner boxes disappear. Whether or not your book changes them, you know it all means the same thing, just different flavors.

8*2-[4{2cos(3*30)+6}+32] 8*2-[4{2cos(90)+6}+9] 8*2-[4{2*0+6}+9] If you want, 8*2-[4{0+6}+9] you can safely calculate the 8*2-[4{6}+9] 8*2 at the start 8*2-[24+9] because the - to 8*2-33 the right is lower importance 16-33 -17

66) Absolute Value



Absolute value signs | | are special parentheses that do more than just group. They turn any negative answer inside them positive. If the inside answer is positive, it leaves it positive. So this means that you will ALWAYS get a POSITIVE answer from an absolute value group. Going all the way back to what we learned about the size and direction of numbers in

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the first chapter, absolute value is only concerned about size, and size is always plain or positive. |-2|=2 |+2|=2 In both of these examples, the final answer could also be +2. Absolute value makes only the ANSWER, not all the signs in the problem, into a positive. |-5+2|=|-3|=3 Do NOT do this |+5+2| Do ALL the inside calculations first, then change the sign of the answer. Never change the signs in the problem. |7-4+3|=|0|=0 |-6-2+3|=|-5|=5 |4*-3+7|=|-12+7|=|-5|=5 Absolute value signs behave just like parentheses. Which means you follow IN FUN MUD COLT. Find the answer inside first, then make it positive, then calculate outside. 3|-9+7|=3|-2|=3(2)=6 2-4|1-6|=2-4|-5|=2-4(5)=2-20=-18

3|4-8| 3(4-8) 3|-4| 3(-4) 3(4) -12 12

67) Formulas

2-|3-9| 15_|-3+8| 2-|-6| 15_|5| 2-(6) 15_5 -4 3



Formulas are equations or relationships between numbers that usually come from real life. For example, someone discovered that it is always true that the total distance travelled on a journey equals the average speed times the number of hours. So now we have the formula: d=rt. A formula is useful when you have all the parts except one. If you are missing two pieces then you will need two equations, three missing parts, then three formulas, and so on. Right now, we will concentrate on just one unknown. To use a formula, just substitute the values you know for their corresponding variables in the equation, then calculate. For example, you can calculate the distance a car goes if it drives at 50 mph for 4 hours. Substitute the 50 for r (rate). Substitute 4 for t (time). You now have the equation: d=50*4=200 The formulas are sometimes made up to give practice before entering chemistry, physics, and other classes.

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For example, the instructions may say that x=2 and y=-5. Then it asks the student to find the answer to 3(x-y)+8. So Sub to get: 3(2--5)+8. Following the order of operations (IN FUN MUD COLT) you get 3(7)-8=21-8=13

A rectangle has width=15 and length=20. What is the area? A=lw, so A=15*20=300 You run 8mph for 3 hours. How far do you run? D=rt, so D=8*3=24 In the simplest equations it looks like overkill, but a good idea is to put ( ) around each variable, then Sub into those parentheses. This helps students see the all-in-or-all-out principle of Substituting, especially when the new value is negative

Let x=-4 and y=10. Find 5xy-8x. 5(x)(y)-8(x)=5(-4)(10)-8(-4)=-200+32=-168 The brackets in this next problem serve the same purpose as the parentheses in the previous problem, but I want to distinguish them. I could use ( ) in ( ), but that might be too confusing.

Let a=-3, b=-1, c=6. Find b(c-a).

[b]([c]-[a]) = [-1]([6]-[-3]) = -1(6+3) = -1(9) = -9

68) Units in Formulas



In the previous lesson the units were selected so that they all matched. That is, time was in hours and speed was miles per hour. We did not have time in minutes in one place and in hours in another place. This must always be the case before you start solving any

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problem. All your lengths must be in the same unit. All your times must be in the same unit. All your weights and temperatures must also be in the same unit. Never try to work a problem with both meters and feet, or pounds and tons, or days and hours.

Use only one unit for time, length, weight, and temperature. Pick a unit for each type of measurement (time, length, weight, etc...) that makes it easy for you. Work the problem with those units. Then, after finding your answer, convert your units to the units the problem requires. Now let’s look at an example. How far does a car travel if several drivers drive 50mph for 3 whole days? First, we notice that the speed uses hours, but the driving time is in days. Hours must be changed to days or days to hours before we calculate. You know that 3^50 can’t be right!

Unlike variables, units only merge if they match. Let’s change 3 days to 3*24 hours for 72 total hours. Now we can substitute in our formula d=rt. D=50*72=3600 miles. Since our problem was based on miles then our answer is in miles. If we change hours to days we will still get the same answer. 50mph ^ 24 hours = 1200 miles per day times 3 days gives us 3600 miles. So you see, it does not matter what unit you use, but it must be only one unit before you calculate.

What is the area of a rectangle with a width of 6 inches and height of 2 feet? A = 6in*24in = 144in2 A = .5ft*2ft = 1ft2 1 square foot = 144 square inches Both units give correct answers!

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69) 2D Shapes For the next several lessons we will not introduce new math concepts, instead we will apply what we already know to common visual and word problems. Two dimensional shapes are surfaces with length and width, or height and width, both are correct. What they lack is depth, also called height. (Now we wonder why kids get confused!) The area of any surface is measured in square units, such as square inches, square centimeters, square miles, etc... Notice that “square” is represented with an exponent of 2. This corresponds with 2D, so it is easy to remember. The previous examples could be abbreviated: in2, cm2, and mi2. This works for all surfaces like paper, floors, land, etc... This even works for curved surfaces like balloons and cylinders. And don’t be fooled by 3D surfaces like cereal boxes. Just “unfold” them so they lay flat and total up all the parts.

area = &r2 circumference (perimeter) 2&r

width

radius

height

length

base area of triangle = 1/2bh area of rectangle = lw

There are three basic shapes to learn, because they can be cut and/or combined to make up more complex shapes. (Of course, we are not including reeeeally complex curves and angles which require calculus and trig.) The shapes are circles, rectangles, and triangles. Notice that the area of a triangle is derived from a rectangle. The base of a triangle is the same as the length of its containing rectangle and the height of a triangle is the same as the width of its containing rectangle. Now look at the left and right halves of the triangle and then of the rectangle. See how the triangle is half? It always works that way. You may have memorized formulas for the areas of squares and parallelograms when you were younger, but these are just special rectangles with the same formula. Trapezoids

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are just combinations of a rectangle and 1 or 2 triangles, but if you want a formula, it is: vertical height^.5^(top+bottom) One more thing just to be sure I cover my bases. You can have kids memorize formulas for the perimeters of different shapes, but except for the circle, it is superfluous. Just add up all the sides and you are done.

70) 3D Shapes

height

3D shapes are able to enclose volume. They can hold air or water or solids. They have length, width, and height. Looking at them from a different point of view you can also say they have length, width, and base depth. Either way, there are 3 dimensions to be multiplied which will give you a 3 exponent on your unit: mm3, km3, ft3. base There are two categories of shapes that concern us. Those shapes that have the same outline on top directly height over the bottom (vertical, with no skewing or slanting, as in the top diagram), and those that come to a point on top (cones or pyramids as in the bottom diagram). The first category is called “right solids.” They are made by drawing a base, then lifting it exactly vertical (right angle) to a height. Their volume is found by multiplying the area of the base ^ height. This works for rectangles, triangles, other polygons, and circles. Pyramids have a polygon for a base and cones have a circle for a base. Their formulas are the same. 1/3 ^ base area ^ height. So if you have an ice cream cone that exactly fits in a tin can, find the volume of the tin can and divide by 3. A sphere (perfectly round ball) is our last object and it is 2/3 the volume of the cylinder into which it exactly fits. So its formula is 4/3&r3. This comes from 2/3^2r^&r2. That is translated from 2/3 ^ height(which is the diameter) ^ area of circular base.

base

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71) Averages Average, also called “mean,” is the idea of “leveling” a group of numbers to find their value if all of them were the same. It is like having four buckets with different amounts of sand in each one, then pouring them into a big barrel, then pouring the sand back into the buckets in equal amounts. That new amount in each bucket is the average. The average evens out the fast and slow parts of a trip. For example, you might drive 45mph to get out of town, then stop 10 minutes at the gas station, then drive 70mph to the campground. Overall, it takes you 2 hours to travel 100 miles, so your average speed is 50mph. The average of a group of numbers is found by combining the numbers, then dividing by the number of numbers. To find the average of 10, 12, and 14 we write it like this: (10+12+14)/3 Notice the use of parentheses to make sure I am dividing the total by 3, rather than just dividing the last number by 3. The correct answer is 12, not 26.67.

N1+N2+N3+...+Nx average = x N1 is the first number and N2 is the second number and so forth until you reach the last number. X is the count of the numbers. So if you have 5 numbers, divide by 5. If you have 8 numbers, divide by 8.

72) Rates Rate, or speed, problems ask questions that link how fast for how long with how far. The formula, or equation, is d=rt. An example is, How far do you go (d) if you travel at 30mph (r) for 3 hours (t)? Believe it or not, rate (speed) problems are very much like the box problems we did back in our arithmetic days. How many jars do I have if I have 3 boxes that each have 30 jars? Each box, and therefore each group of 30 jars, is repeated 3 times. The same repetition happened with the rate problem. Each hour, and therefore the 30 miles travelled each hour, was repeated 3 times. The answer is the same for both problems, 90: 90 jars or 90 miles.

Pre-Algebra: Calculate

1 hour

30 miles

117

1 hour

30 miles 90 miles in 3 hours

1 hour

30 miles

What might be hard for some students to comprehend is the fact that time can be a container. Instead of a visible box holding visible jars, the container is a unit of time that holds the number of miles travelled or the number of widgets produced. This shows a small part of the power of math. By looking at the same idea in a slightly different way, we can solve new problems. There is yet more. We can prepare the student for the need of modifying equations by introducing problems where a variable is missing from the side with more than one variable. That is, rather than looking for d in d=rt, what if we ask the student to find r or t? They do not yet know how to solve equations for a variable, but these are simple enough that they can do one step backwards to find the answer. If I travel 200 miles in 4 hours, how fast did I drive?

1 hour

50 miles

1 hour

1 hour

50 miles 50 miles 200 miles in 4 hours

1 hour

50 miles

Because the total time is divided into four parts, then the total distance must also be divided into four parts. This just makes sense, and that math-sense is what we want the student to develop right now, before they learn to just mechanically plug numbers into an equation. Developing math-sense will make the eyes light up, but mechanical math leads to sloppiness and loss of motivation, which results in lowered performance. Another thing: Notice the arrows lining up. What does that remind you of? Adding numbers on the number line? Yes! A student can “do the math” or they can “draw the math.” The point I make here is that if students develop their own practical sense of doing and seeing math, they are not bound by the one method of memorize-the-formula-anddo-it. Just like us, they often forget, especially under test pressure. If they understand how all math grows from the basics, they can return to the basics and solve a problem their own way. Sure, it’s slower, but it develops creativity, inventiveness, and American ingenuity. I did it myself on some tests, just to be sure or to double check my work!

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73) Ratios Rates and ratios are related. In face, some teachers say they are the same thing. I won’t argue, because I look at them like multiplication and division, one is the reverse of the other. If a problem does not appear to me to be a repeating problem, like rates, then I look to see if is a ratio, or division, problem. My habit is to look for the easiest possibility first! A ratio problem often uses “per” or “for each” as in a simple division problem. However, such wording is not always used, so my failsafe backup is to find two things that equal each other, then I will know they form a fraction and I will do other things with the other numbers present. Let’s look at this problem. 10 bananas cost 5 dollars. How much will 30 bananas cost? A) Right away I see that if I triple the bananas, then I must triple the dollars, so my answer is 15. B) Of course, it won’t always be that easy or obvious, so I look for a rate. I don’t see anything repeating, but it seems I must figure out the rate, or cost, of a single banana. To find that I need a ratio or fraction. Which 2 of the 3 numbers should I make a fraction from? The first sentence tells me. 10 bananas = 5 dollars because when I go to the checkout counter I will give the clerk 5 dollars and then she will let me walk out of the store with 10 bananas. I can’t make a fraction with 30 bananas, because I don’t know anything else that equals them. So now which fraction do I make? 5/10 or 10/5? It doesn’t really matter. Just be consistent all the way through. To help students I first have them make a word fraction on the left that will be the model they will follow for all the other fractions they might make.

bananas 10 30 = = dollars 5 x

dollars 5 x = = bananas 10 30

So both ways of setting up the problem will work. I like it when that happens. That means I don’t have to remember which way is the right way and that means there is less chance I will make a mistake. Why doesn’t it matter? Because I am keeping my units the same all the way across, and each fraction is saying the same thing, either bananas per dollar or dollars per banana. This is the equality principle! So now my problem is a simple reducing or inflating fractions problem. To inflate the 10 to a 30, I need to multiply by 3. Shift tells me to counterchange and multiply the x by

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3. Therefore, my answer is 15 dollars. Actions have more value than just guiding the math part of problems. Actions come from real life, so Actions can help us think about real life, which helps us set up our problems. Changing Rates Ratios can help us with problems in which the rates change. Of course, we will not be able to make equal fractions, but the idea of keeping the units organized will still help us. Miranda can ride 80 miles in 2 days. If she doubles her speed, how long will it take her to ride 400 miles? Because the speed changes, I cannot say 80/2 = 400/x, but I have other options. A) Find her rate for 1 day, then divide. 80 miles in 2 days means she rides 40 miles each day. Doubled means she now rides 80 miles per day. 400 miles divided by 80 miles per day is 5 days. B) Double the distance. Doubling the days means she would take longer to cover the same miles, therefore she is slower. So I want to double the distance in the same time to show a doubling of her speed. Now I can set up equal fractions.

miles 160 400 = = days 2 x

days 2 x = = miles 160 400

160 is inflated by 2.5 to equal 400 (400_160). 2 days times 2.5 is 5 days. C) Other ingenious approaches might divide the 400 by 2 because that is the equivalent of doubling the speed. Or, you might write the fraction 80/2 next to your model, but just not put an equal sign next to it. By looking at 80/2 it might be more obvious how to show doubling of speed instead of halving of speed by mistake.

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Pre-Algebra: Roots This chapter completes what the Exponents chapter started. Roots are just the reverse of exponents and fractional exponents put powers inside of roots. The same rules used by exponents are used by roots, because every root can be turned into a fractional exponent.



74) What is a Root?

Since roots are just the inverse (the “undo” function) of exponents, there are more roots than the common square root. The default root is the square root. This means it has an invisible power number of 2, like this: Ć If you want to show it, you can. It won’t hurt. Look at the examples below and you will see how roots “undo” exponents. 22 = 4 so @4 = 2 23 = 8 so ć8 = 2 24 = 16 so Č16 = 2 25 = 32 so č32 = 2 26 = 64 so đ64 = 2

32 = 9 so @9 = 3 33 = 27 so ć27 = 3 34 = 81 so Č81 = 3 35 = 243 so č243 = 3 36 = 729 so đ729 = 3

52=5*5=25 23=2*2*2=8 @25=@5*5=5 ć8=ć2*2*2=2 Odd roots can have negative answers

(-2)3=(-2)(-2)(-2)=8 ć-2=ć-2*-2*-2=-2

Why is there no root 1? Let’s look at the pattern of what is happening inside the roots. Č16=Č2*2*2*2=2 ć8=ć2*2*2=2 @4=@2*2=2 1@2=2 If this last part looks

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weird, that’s because it is. A root is a division problem with identical factors, but a 1st root just has the number itself without any division! There is no root 1 because it is pointless.



75) Reducing Roots

Reducing roots resembles reducing fractions. Break up the number inside the root into factors. If a factor repeats itself, pull the pair outside and multiply one of those partners by what is already there. The key is to remember to multiply by only one partner, not both. Why? Inside the root sign, the numbers are “bloated.” They are squares. Squares = root ^ root. When you take a square from the inside to the outside, it becomes its root, not its root ^ root. For example, 9=3^3, so @9 cannot equal 3^3 also. @9=3, just 3. This is another Shift Action. 3 gets inflated to 9, but we can’t just change without a counterchange. That is what the @ sign is for. It is the counterchange. The 9 is just a bloated puffer fish borrowing air from the @. It looks like a regular 9, but not really. It is an inside 9, which means it is smaller than an outside 9. When the @ goes, the 9 goes also and returns to being just a regular 3.

@ 12 = @2*2*3

= 2@3

4 inside = 2 outside

Roots turn 2 duplicate inside factors into 1 outside factor

2 @ 45 = 2 @3*3*5 = 6 @ 5

9 inside = 3 outside ^ the 2 already there

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76) Combining Roots

Roots are just another factor that can appear in the tag of terms (compound numbers). The number in front, which might be an invisible 1, is the frontnum. Of course, Combine Only Like Tags applies here as everywhere else. If the tags don’t match, you cannot combine them. However, you may be able to reduce some or all of the roots which will then make some roots match. Then you can combine them. Once again, Combine Only Like Tags to get a like tag. 6@5+2@3 = stop 6@5+8@5 = 14@5 9@3-7@3 = 2@3 5@4+2@9-7@25 = 10+6-35 = -17 @12+@75 = 2@3+5@3 = 7@3 8@9-2@5+3@16 = 8*3-2@5+3*4 = 24-2@5+12 = 36-2@5 4@8+5@18 = 4*2@2+5*3@2 = 8@2+15@2 = 23@2

@ 12 + @ 27 @ 2*2*3 + @ 3*3*3 2 @ 3 +3 @ 3 5@ 3

Frontnum Tag

+2 @5 +7 @5 +9 @5

8@5-2@3+7@5+3@3 15@5+@3

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77) Multiplying Roots



You can merge roots by multiplying together the numbers inside. This may enable you to do some reducing you were not able to do in the beginning. So @3^@3 = @9 = 3 If there are numbers in front, you will multiply the nums together, then merge the tags. For example, 4@2*5@2 = 20@4 = 20*2 = 40 Multiplying two identical roots is squaring a square root, so they cancel each other leaving a single partner outside. For example, @4 ^ @4 = 4, because @4=2 and 2^2=4

3@ 5 *4 @ 7 =12 @ 35

@ 6 * @10 * @15 = @ 900 = @ 9*100 = 3*10 = 30 8 @ 48 _2@ 2 = 4@ 24

=4@ 2*2*2*3 =4*2@ 6 =8@6 Dividing works like multiplying. @22_@2 = @11 @28_@7 = @4 = 2 @32_@2 = @16 = 4 20@8_4@2 = 5@4 = 5*2 = 10 24@18_6@6 = 4@3

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78) Fractional Exponents

We have seen how a root is really a division problem. Now what if we have a multiplying problem within the division? In other words, what if we have an exponent inside a root? That situation leads us to fractional exponents. The top number is the inside power and the bottom number is the outside power. The numbers in a fractional exponent are behaving just as they do in a regular fraction. The top number is multiplying and the bottom number is dividing. Since exponents are short for multiplication and roots are short for dividing, the top number is the exponent and the bottom number is the root.

ć52

=5

2 3

č37

=3

7 5

Top is inside power, bottom is outside root

ć8 = 8

@

34

ć82

1 3

č3 = 3

= @81 = 9 = 3 = ć64 = 4 = 8

2 3

1 5

4 2

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79) Rationalize Roots



It is considered bad manners to have a root in the bottom of a fraction. So we rationalize the fraction to take care of the problem. This means we will square a square root to have only an outside number. Of course, if we change the bottom, we must counterchange the top. Therefore, like reducing or inflating regular fractions, we will Shift a fraction with a root on the bottom. 1) Multiply top and bottom by the square root on the bottom. 2) Multiply on top as far as possible. 3) The root cancels itself by squaring, leaving you a plain bottom number. Notice that this is somewhat similar to canceling factors, except that you are canceling functions. Roots undo squares, and squares undo roots. An exponent 2 cancels a root 2. Note: This lesson only shows how to rationalize a root multiplying another factor. The next chapter will show how to rationalize a root combining with another term.

3 @5 3 @5 3 @5 = = 5 @5 @5 @25 7 @6 7 @6 7 @6 = = 6 @6 @6 @36

5 @2 5 @2 5 @2 = = 6 3@2 @2 3@4

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80) Roots with Same Base



There is nothing really new in this lesson. It is the cumulation of the other lessons in this chapter. We will do some reducing, a little multiplying, and then some combining. Of course, the regular order of operations needs to be followed. First do functions (roots and exponents). You might need to multiply unreducible roots to make a root that will reduce. After reducing, you may see like roots (tags) that you can combine.

5@2*@6-8@3 5@12-8@3 5*2@3-8@3 10@3-8@3 2@3

3@5*@10+7@2 3@50+7@2 3*5@2+7@2 15@2+7@2 22@2

3@12+5@27 3*2@3+5*3@3 6@3+15@3 21@3

4@8-2@18 4*2@2-2*3@2 8@2-6@2 2@2

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Algebra: Polynomials Polynomial means “many terms.” In other words, many compound numbers combining with each other. These terms have certain limitations and must fit this pattern, axn. Any number can be “a,” but the exponent, n, of the variable, x, must be non-negative (0 or greater) and must be a whole number. Also, the terms can be combined or multiplied, but not divided. Therefore, a/x is not permitted. Don’t be frightened. These limitations actually make our job easier because we will have less to deal with. Polynomial expressions will prepare us for equations, although most students find polynomials more difficult than equations. So, in a way, this chapter represents the final ascent up Math Mountain. After this, it gets easier, not downhill, but more like a gentle slope up a long ridge. Because we will mostly be starting and stopping with variables, we will not be able to get a single number for our answer. Rather, our answers will look like 2x or x+4/6 We will take problems as far as they can go and then just stop. Don’t stretch this next statement too far, but our main goal in this chapter is not so much to get a single answer as it will be to learn the processes and use the Actions correctly. These problems are really part of larger problems for those who go further in math. Therefore, a student must correctly learn the steps along the way if s/he wants to arrive at the right answer. And speaking of “right answer.” That phrase can take on a whole new meaning when we leave the world of arithmetic and enter the world of algebra.

Thinking in Algebra To understand algebra you must think like an algebratician. Algebratician?! Is that even a word? No, but if it were, it would mean a mathematician generalized. Imagine Einstein and Sherlock learning 2+2, then they learn another problem, 2+3. Then they learn another problem, 3+3, and so on. What would happen when they encounter 200 + 200? If they did not know how to think in algebra, then they would have to learn about adding these two numbers as if they had nothing to do with all the adding problems they had already solved. So, you have halfway done algebra already by generalizing about adding and all the other operations, but now you need to understand adding x + x.

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We have also been introduced to algebra in two other ways. When we worked with tags and exponents we used variables in the examples. We did algebra, but without understanding it from an algebra point of view. Also, when we worked with formulas, we were using a prime example of algebra. Rather than re-inventing a new formula for every word problem we meet, we use one that someone else has already figured out. The variables in a formula are the placeholders that anyone can use at anytime to drop in their own custom numbers for their own unique problem. (Motivation alert: Too often students are solving somebody else’s problems instead of encountering and experiencing their own. A formula takes on more meaning when it helps you solve something important, rather than simply being a means to someone else’s end.) So algebra helps us expand our thinking beyond this problem with this number to working with patterns and any number at all. Of course, this is a reflection of life. What if my mother tells me to wash my hands before supper tonight? What if she told me to wash my hands before supper last night and the night before and the night before that? What if it never occurs to me to generalize her command from this night to all nights? That would be all right for a night or two, but wouldn’t you consider me a slow learner if I never caught on after 5 nights or 50 nights? Just as my own thinking abilities should extend my mother’s command from a few nights to any and all nights, so algebra extends arithmetic from common numbers to any number, even unknown numbers. Of course, the answer to such a question usually looks like “any number” instead of a specific number, but we try to get as close as we can. Let’s look at a problem like 3+3. Since both numbers are the same, we can rename 3 to x and say x+x. Of course, we can do this with any other two numbers that match. But now what? What do we do with x+x? Since x could be any number, I can’t arbitrarily say x+x=6 or 10 or -25! What if x is 7? Then x+x=14. My three earlier guesses would be wrong. In fact all my guesses (even 14) could be wrong if I don’t know what x equals. So what good is algebra if my odds of guessing the right number is 1 in infinity? Algebra is not about guessing the right number. It is all about figuring out the right pattern which sometimes happens to be a number. Let’s get back to our example, 3+3 and x+x. What do we know about our example and what can we generalize about it? Isn’t 3+3 the same as 2^3? Yes, and isn’t 4+4 the same as 2^4, and 5+5 the same as 2^5, and so on? Because x+x fits that pattern, wouldn’t it fit the 2^x pattern as well? Wouldn’t ANY number fit those patterns? Wouldn’t ANY number added to itself equal two times itself? Therefore, we have discovered something. x+x=2x. We can now say the answer to the problem, x+x, is 2x. If “answer” seems like a strong word to use, we could also say 2x is another way of looking at x+x, or we could also say, x+x can be rewritten as 2x. In and of itself this discovery seems small, but it could be the key to helping us crack a bigger

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problem. More importantly, this simple problem is helping us right now think in algebra. Let’s pause for a moment and think about what an “answer” is. We have been conditioned to think only one way. That is, we think an answer can only be a single number. Usually, that is the case, but why? More often than not, but not always, a single number is desirable because we want to know how many cups of flour to put in the cake batter, or how many tickets were sold, or what the net worth of a company is at the present moment. One number is easy, concrete, and we can nail it down. However, is it always best to have just one number? A high school graduate takes the SAT and earns a score of 1200. Did he do reasonably well on both math and English, or did he do superb on one and bomb the other portion of the test? 1200, a single number, does not, and cannot, tell us. Something like 800+400 or 600+600 would be more helpful. Another example, what if I want to write out a gazillion math problems for my eager math students to do in one night? I would write a computer program that would work something like the following. 1) Generate a list of random numbers. 2) Pick two numbers at a time (a and b), print them with a + sign in between. 3) On the answer key, reprint the problem and the calculated answer. This brings us back to the use of variables as placeholders, but notice that the answer to my problem are the two numbers that are a problem for the student. Just printing an answer would be printing a list of random numbers. Instead, I need a list of random problems in the form a+b. The “answer” is a matter of perspective. The “answer” is what is most helpful, useful, and informative. So now let’s go way back to the x+x example. Let’s say you have collected a lot of data from doctors around the world about eye exams. You then need to make a report about the number of eyes examined. Simple, you think, just add the number of exams to itself to figure the number of eyes. However, in this realistic, but greatly simplified scenario, you discover that your computer will take 5 days to generate the report if you add all your numbers. Your CPU is not optimized for adding, but it excels at multiplication. Thinking algebraically, you reason that 2x is the same as x+x. The computer can generate the report in 5 hours, in time to meet the deadline. What did not seem like an “answer” before is now your lifesaver. There is a cost to everything we do. It pays to discover new ways of thinking and doing because one day the reward will be greater than the cost. We have no idea now how our ingenious insights might build a bridge, launch a rocket, or save a life, but all the technology we have came from looking at problems and answers differently. Think outside the box of arithmetic. That is thinking in algebra.

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As Few Variables as Possible It takes little comprehension to see what is going on with 3+3. Count out three dots * * *, then extend it with three more dots * * * for a total of 6 dots * * * * * *. There it is, right there before my very eyes. But what if the numbers are different? What if we don’t know either of the numbers? We can’t say x+x, because we can’t be sure they are the same. We need to say x+y, because it might be 5+8 or 2+7 or -1+4. Therefore, I can’t say the answer is 2x or 2y. I am forced by simple logic to say x+y=z. (z could equal x or y if either of them is 0, but the odds of that happening are low.) So what do I do with x+y=z? Nothing. Too many unknowns. Too little specific information. I can’t nail anything down. Let’s say a student of yours thinks they can finally stump you. He asks you, “What is x+y?” You ask, “What is x and y?” He says, “I don’t know.” Then you say, “Oh, I know the answer then! It is z.” He asks, “Really? What is z?” You say, “I don’t know numerically, but I know it is the right answer to your question!” The point is that there are limits to algebra and we are the ones that need to put limits on it. If our problem does not have enough concrete and unchangeable numbers in it, then we can’t deal with it. The basic rule of thumb is to avoid making a new variable whenever possible, because for each additional variable you will need another equation, if you want to find its value. This dilemma is the topic of the last chapter on systems of equations. Until then, in our math problems and word problems, we work with just what we are given and relate the information we have to itself. For example, a common word problem goes something like this. The second box has twice as many widgets as the first box. If their sum is 24, how many widgets are in each box? The first inclination is to write, x+y=24. However, if we are ever able to write the problem with fewer variables, we should. We can relate the second box to the first box by describing it as 2x. Why not?! After all, the problem said the second box has twice the amount of the first! So now our problem can be written as x+2x=24. This problem we can solve when we get to the next chapter on linear equations. For now, the answers are 8 and 16. Avoiding making new variables applies to the type of problem which we will delve into next, multiplying with variables. We have already multiplied variables by merging them. 5x*2y=10xy, but what about variables in groups? (x+5)(x-2) Without inventing any new variables, we can do this operation using the distribution.

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

81) Distribution

To distribute means to hand out or spread around. Let’s say you are helping with an earthquake relief effort. You distribute aid to all who need it. In the same way, the distributive property tells us that whatever is in front of the parentheses gets distributed to all that need it inside. Every term, every compound number, gets multiplied once by whatever is in front. This is rule M11 on the rule sheet: all terms ^ all terms. This really isn’t a technical rule. It is just common sense. If you have 4 boxes then you have 4 of everything that is in the boxes. 4(2) means you have 4 boxes that each have a 2 in them so you have 8 items total. If you have 4(x+2), that means you have 4 boxes that each have an x+2 in them. I cannot combine x+2, because I don’t know what x is. Neither do I want to call it y. Therefore, I just bring the x+2 outside of the box and lay out all 4 sets. 4(x+2) gives me 4x and 4*2, which is 4x+8. I still do not know what x+2 equals, but at least now I have a formula for my total that I can use once someone tells me x. You should always get an answer term for every possible pairing of terms in the problem. If there is one term outside (or in the parentheses in front) and two inside, then there are 1*2 pairings, which means you should get 2 terms in your answer. If there is 1 outside and 3 inside you will get 1*3 terms in the answer. You can also have more than 1 term in front if there are ( ). (x+1)(x+2) means you have 2 terms times 2 times for a total of 4 terms in the answer. (3x-5)(x2-6x+8) is 2 terms ^ 3 terms for a total of 6 terms in the answer.

3(x+5)

x(4-x)

2(x-9)

3x+15

4x-x2

2x-18

All terms ^ all terms = answer terms

(x+5)(x-8) = x2-8x+5x-40 (x-3)(x2+4x-7) = x3+4x2-7x-3x2-12x+21

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

82) FOIL

There is a special case of distributing that is worth special attention because it is used so often. When there are 2 terms times 2 terms there is a visual pattern formed that has been nicknamed FOIL for First Outer Inner Last. This is just a special mnemonic (memory aid) to make sure you don’t overlook something when multiplying all terms by all terms. In many cases, but not all, the outer and inner pairs give you like terms which can then be combined. You actually get 4 answer terms (from 2^2 terms), but they collapse to 3 in the end. All the rules still apply, but when two of the terms get combined, some students think something else happened and get confused. So, even if your textbook does not show the step with 4 terms, it still got 4 terms then combined to 3.

first

last

(x+5)(x-4) inner

outer

2 x -4x+5x-20 2 x +x-20

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83) Rationalize with conjugates



Thanks to our distributive property (all terms times all terms), we can now take care of the complication that is introduced when the root on the bottom of a fraction combines with another term, rather than multiplies with another factor. We multiply by the conjugate, then distribute. Whoa! The conjugate?! What is that? It is the almost identical twin to the two terms on the bottom. Only the middle sign is reversed, everything else is exactly the same. Why do we multiply by such a strangely named beast? Because it makes our life turn out easy. The important point is not to remember the name, but to think one step ahead toward your goal. Your goal is to have no roots on the bottom. So we must Shift the fraction in such a way that all roots cancel themselves, and that is precisely what the conjugate does. Let’s call this “clever Shifting!” Let’s say you have 5-@2 on the bottom. If you multiply by 5+@2 then you will get these answers: 25+5@2-5@2-@4. Notice that 5@2-5@2 = 0. Those roots are gone! Now notice that @4 reduces to an integer, 2. All the roots are gone! This same pattern will always happen if you multiply by the conjugate. If you don’t change the sign in the middle, then the roots in the middle will add instead of subtract to 0.

x 2+@3

2-@3 2-@3

2x-x@3 4-3

=

=

2x-x@3 4+2@3-2@3-@9

2x-x@3 1

=2x-x@3

Reverse the sign in the middle, then distribute, then combine

Algebra: Polynomials

84) Common Factoring

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

Common factoring with variables works the same as common factoring without variables. You are looking for factors that are contained by all terms. When you figure out what the common factor is, go ahead and divide each term and write down the answers. The answers go in parentheses with the common factor out front. Make sure that your common factor will multiply evenly into EVERY term, not just some of them. If you cannot find a common factor that works with all terms, then just stop. Why? Because the answer must equal the problem. You should be able to distribute your factor to all the terms inside and return to your problem. Factoring is always a Shift Action and Shift always works in both directions.

6x3+9x2-24x 3x 3x 3x

3x(2x2+3x-8)

8x3+4x2-10x 2x 2x 2x

2x(4x2+2x-5)

Just like with plain numbers, a common factor can be divided out of ALL terms

85) Bifactoring



Bifactoring is FOIL in reverse. Of course, it is a little trickier because you don’t know what factors to use to multiply. You are not told. Instead, you must figure out two factors that multiply into the last number, and also combine to give you the middle number, including signs. When we say, “first, middle, and last,” that always assumes the expression has been Sorted into descending order of x. The x2 term should always come first, then the x term, then the plain number term. Usually it is pre-sorted, but sometimes you need to Sort it. This type of problem can lead to a lot of guessing, but there is a way of using a short list. Start with the last number, the plain number. Make a list of all possible factor pairs,

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starting with 1 and the number, then moving up until you meet in the middle. For example, a list for 12 starts with 1 and 12, then 2 and 6, then 3 and 4. Because 3 and 4 touch each other, we know we are done. Your answers will be found in that list. An answer list for 24 is 1 and 24, then 2 and 12, then 3 and 8, the 4 and 6. 5 doesn’t work, but it is in the middle so we are done. The next number is 6, but we have it already. An answer list for 28 is 1 and 28, 2 and 14, skip 3, 4 and 7, skip 5, skip 6, done. 7 is next and we have it already.

x2+7x+12 first

middle

x2-9x+18 (x-3)(x-6)

last

(x+4)(x+3) first

last

outer inner

first

last

inner outer

first ^ first = first last ^ last = last outer + inner = middle

x2-5x+24 (x+3)(x-8)

Multiply your answer. It must equal problem.

x2+12x-13 (x+13)(x-1) 13x -1x

Let’s look at the last example more closely. Students usually see the x*x=x2 right away. Most have little difficulty finding two numbers to multiply to equal the last number. However, getting all the signs right and having the same two numbers also combine to equal the middle is difficult for most. This is another reason why at least the small number arithmetic should be learned without a calculator. If it is not automatic, algebra is almost impossible. Another way of looking at these bifactoring problems is as a word problem where you are asked to find two numbers that multiply to -13, but also combine to +12. It may help to approach the problem this way for those who are not visual learners.

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86) Bifactor when a>1



Remember the pattern that a polynomial must fit before it can be bifactored? There must be an x2, then an x, then a plain number. That pattern is usually described this way: ax2+bx+c, where a, b, and c are whole numbers (constants), including invisible 1’s or 0’s. In the previous lesson, the leading number, a, was always 1. Now we want to bifactor when a is greater than 1. The procedure is the same, but involves a lot more trial and error until you find the numbers that make FOIL work. Most of the time, a will be small and prime. Therefore, put the number and x in one set of ( ) and a plain x in the other ( ). That helps you nail something down. Now work with your list of factors of the last number, c. However, don’t forget that one of them must get multiplied by a. This is what doubles your number of combinations and can double your time in finding the right one that works.

3x2+2x-8 first

middle

last

(3x-4)(x+2) first

last

outer

inner

first

last

inner outer

2x2-19x+35 (2x-5)(x-7)

first ^ first = first last ^ last = last outer + inner = middle Multiply your answer. It must = problem.

2x2+10x+12 (2x+4)(x+3)

In this case, first common factor by 2. It’s easier. 2(x2+5x+6) 2(x+3)(x+2)

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87) Bifactor- other



Bifactoring can happen with more than the x2+x+number pattern. After sorting the expression to be in alphabetical, descending order of powers, here are some patterns to look for. If the first exponent is twice the second exponent you can bifactor, as in these examples:

x4-5x2+6

x6+4x3-12

(x2-3)(x2-2)

(x3+6)(x3-2)

x4-2x2-3x2+6

x6-2x3+6x3-12

Bifactoring also works if the middle uses the same variables as the first and last, as in these examples.

x2+5xy+6y2

a2+5ab+6b2

(x+3y)(x+2y)

(a+3b)(a+2b)

x2+2xy+3xy+6y2

a2+2ab+3ab+6b2

The gray lines are the answers multiplied so I can check if I am right. It also helps me to understand the method. Study the examples until it becomes clear that they are nothing but FOIL problems with a twist. The twist is the higher power exponents or the extra variables. Also remember that all this FOIL business is really just a tool for the underlying principle of all terms ^ all terms, which is the distributive property, which is multiplying variables without making new ones.

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88) Squares



Two more special bifactoring patterns are called perfect squares and difference of squares. A perfect square is when the terms in the parentheses match so that they can be collapsed to a single group with a 2 exponent, like this: (x+5)(x+5)=(x+5)2

x2+12x+36

x2-4x+16

(x+6)(x+6)

(x-4)(x-4)

(x+6)2

(x-4)2

Difference of two squares is when the problem has a 0x in the middle, which is usually left invisible, and the two terms you see are square - square. This tells you that the bifactors will have the square roots in them with opposite signs, like this: x2-25 = (x-5)(x+5)

x2-36

x2-81

(x-6)(x+6)

(x-9)(x+9)

x2+6x-6x-36

x2+9x-9x-81

Again, to figure these out and to check them, multiply your answer factors and you should get back to the problem.

89) Double factoring



Double factoring lets us handle an x3 pattern we are unable to handle with the factoring techniques we have learned so far. Double factoring uses common factoring in parts,

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rather than common factoring the whole problem. Of course, make sure the expression is Sorted in descending order, then try to common factor the first two terms, then common factor the last two terms. This should leave you with the same terms in parentheses.

x3-5x2+2x-10 = x2(x-5)+2(x-5) Now look at each ( ) with a term in front of them as a compound number or a sort of “super-term.” The frontnum part of each compound number forms a bifactor while the tag of each compound number collapses to form the other bifactor.

(x2+2)(x-5) What you have really done is factored the tag from each compound number. Putting it first in your answer may make it look more familiar. (x-5)(x2+2)

x3+7x2+2x+14

x3+4x2-3x-12

x2(x+7)+2(x+7)

x2(x+4)-3(x+4)

(x2+2)(x+7)

(x2-3)(x+4)

90) Quadratic Formula



When all the other types of factoring fail (and even when they succeed) the quadratic formula will give you both factors. It always works for any quadratic equation or expression. As you may have noticed, not all polynomials can be factored by one of our methods. As one advances deeper and deeper into math, all-in-one methods are harder to find, not impossible, but much more abstract thinking is required. One example is the quadratic formula. A quadratic equation looks like ax2+bx+c=0. A quadratic expression looks like ax2+bx+c. The a, b, and c represent the numbers (possibly an invisible 1) in front of the variables. You will substitute only the numbers, but not the variables, in the formula. After you Sub the numbers work the formula as if it is two formulas, one with the + and the other with the -. The two formulas will give you two answers. You may be asked

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to leave them in root form or you may put them in decimal form. Note that the quadratic formula is meant to solve quadratic equations, not quadratic expressions. There is a difference, but it is easy to deal with. Let’s say you get 1.97 and .38 for your answers. Those answers can be turned into factors by reversing their signs and putting an x in front of them. So the answer 1.97 becomes the factor (x-1.97), and the answer .38 becomes (x-.38). If your answers are -.25 and 9, then your factors are (x+.25)(x-9) [Be aware that there are possible inaccuracies due to rounding. Therefore your answer may not quite check.) Factors are the answers to an expression problem, while numbers are the answers to an equation problem. If the “discriminant,” the b2-4ac expression inside the root, calculates to a negative number, then your answer is a complex number. This is another number type in addition to those introduced so far in this book and is a whole topic in itself. For now, with expressions, stop. The expression cannot be factored.

-b±@b2-4ac 2a

ax2+bx+c=0

3x2+6x-8=0 a=3 b=6 c=-8

-3±@62-4*3*-8 2*3

ax2+bx+c

5x2-7x+9 a=5 b=-7 c=9

7±@(-7)2-4*5*9 2*5

2 answers in one formula First from the +, second from the Factors have opposite signs from the answers

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

91) Reduce fractions

Now we start putting your factoring skills to greater use. By factoring the polynomials in a fraction, we can turn the tops and bottoms into factors that multiply each other. Which then means we can cancel matching factors that divide each other, because any factor over itself is 1. Remember that you can only cancel factors. You cannot pick and choose among terms. Any factoring method that works can be used.

x2+7x+12 2x+8

=

(x+3)(x+4) (2)(x+4)

=

x+3 2

x2-6x-16 x2-4x-32

=

(x+2)(x-8) (x+4)(x-8)

=

x+2 x+4

x2-36 x2+12x+36

=

(x-6)(x+6) (x+6)(x+6)

=

x-6 x+6

In every case, I factored the top and bottom as far as possible and put the factors in parentheses. This is a good safeguard to help me to cancel factors and not the individual subterms inside the factors. Why can’t you cancel terms? Remember how reducing plain fractions works. We do not add or subtract numbers on the top and bottom, we multiply or divide. Likewise, I must multiply or divide an algebra fraction by the same factor on the top and bottom. You can split the fraction into two smaller fractions to see that you are really multiplying a fraction by 1.

(x-6)(x+6) (x+6)(x+6)

=

(x-6)*(x+6) (x+6)*(x+6)

=

(x-6) (x+6) *1

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

92) Multiply fractions

Just like in arithmetic, multiply fractions straight across. What you especially need to be aware of is that a fraction bar is a grouping symbol. It puts the entire top into one set of invisible ( ) and the bottom into another set of invisible ( ). So the whole top is one factor divided by the whole bottom, which is another factor. You must deal with the tops and bottoms as factors. You can’t just cancel a term or two here and there. If you have a hard time seeing this, draw ( ) around the top and bottom until you “see” the invisible ( ) all the time. Dividing polynomial fractions follows the same rule of “reverse last and multiply.”

(x+5) *(x-2) (x-2) (x+7)

=

(x+5)(x-2) (x-2)(x+7)

=

x+5 x+7

x-4 * x-7 x+3 x-4

=

(x-4)(x-7) (x+3)(x-4)

=

x-7 x+3

x-3 _ x-3 x+1 x+6

=

(x-3)(x+6) (x+1)(x-3)

=

x+6 x+1

If this looks like reducing fractions, it is! These are all just factors multiplying and dividing each other.

93) Combine fractions



Just like in arithmetic, Shift the fractions so their tags (bottom) match, then combine. Again, the entire top and the entire bottom are factors.

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term

term

x+1 + x-4 = (x+1)(x+2)+(x-4)(x-3) x-3 x+2 (x-3)(x+2) term

Because of space, the examples don’t show the middle step where each fraction is multiplied top and bottom by what its bottom is missing, but the result shows the two fractions in one and each has the missing factor. The left fraction shows up in the left “circle” having (x-4) on top and bottom. The right fraction shows up in the right “circle” having (x-5) on top and bottom. This is just like combining number fractions!

term

term

x-1 + x+1 = (x-1)(x-4)+(x+1)(x-5) x-5 x-4 (x-5)(x-4) term

1 5

+

1 4

=

1*4+1*5 5*4

When combining you will always get more than one term on the top. To cancel a factor, you must cancel it out of all terms on top and bottom, or none.

94) Long Division



A polynomial can be divided by another polynomial by a process very similar to long division. (And just as painfully!) 1) Divide the first term inside by the first term outside. Write the answer above the term

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145

that matches. 2) Multiply the rest of the terms outside by the answer you just wrote. Write those answers below the matching terms inside. 3) Subtract the line you just wrote from the one above it and repeat the process. Any number at the end is the remainder. So the first example can be written this way:



x+5 x+2 x2+7x-8 2 x2+2x 3 5x-8 (x2+7x-8)_(x+2)=x+5 r-18 5x+10 -18

If your problem is “missing” a term when you put it in descending order, remember that the frontnum is really just 0. So write it in.



1

x2+1x-4 x-1 x3+0x2-5x-9 x3-1x2 +1x2-5x-9 +1x2-1x -4x-9 -4x+4 -13

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Algebra: Linear Equations

95) Recognize equation types There are four basic equation types that we want to focus on now. You can tell them apart by how many kinds of variables they have and by the highest power of a variable. Kinds of variables: If an equation only has x for its variables, then the equation is a single variable equation, V1. If the equation has two different variables, then it is a 2 variable equation, V2. If it has 3 kinds of variables, then it is V3, etc.... Degree: Ignore any numbers that have exponents. All we care about are variables with exponents, and we want to know the greatest power any variable has. For example, if an equation has an x and an x2 then the degree is 2, D2. If an equation has an x, x2, and x3 then it has a degree of 3, D3. Also, the variables can have no fractional exponents or roots.

V1D1 V1D2 V1D3 V2D1 V2D2

4x-9=-11+7x 8-2x2+3x=-x+1-5x2 7x3+6x2-5x+34=0 y=8x-3 3x2+4x-y=6

The third equation has an exponent of 4, but its base is a number, so x3 sets degree

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FA Method for linear equations Here is one of the more difficult equations to show you something for each step. This method has solved every linear (V1D1) equation I have encountered in the textbooks. As you can see, this method is based on Actions, but also tells you when to use which Action. As long as you go in order, you can’t go wrong. You may need to repeat a step or you may need to skip a step, but never go backwards. Here is the meaning of the letters and the next lessons will look at each step one at a time.

F) Fill parentheses or Flip complex fractions or Figure functions E) Eliminate fractions or decimals D) Decouple like terms (letters left, use Switch Sides Switch Signs) C) Combine like terms (COLT) B) Break variable term (divide both sides by coefficient using OOOS) A) Answer! check it, reduce it, round it

F E D C B A

-.2(1.5x+5)=-2(.35x+.3) 10(-.3x-1)=(-.7x-.6)10 -3x-10=-7x-6 7x-3x=-6+10 4x=4 x=1

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

96) Answer!

A) Answer! check it, reduce it, round it Before students hand in their homework or test for grading, they can know for sure if they got their equations right. The Sync Action, based on balance and equality, is the key. The whole premise that makes equations work is “equal.” An equation must start and finish equal on both sides. Therefore, if both sides equal the answer is correct. Take the answer and substitute it back into the original equation. Then calculate the left and right sides separately. If their answers equal other, then the answer being checked is right. If the sides do not equal, then it is time to find and fix the mistake. Fraction answers should be reduced. Decimal answers should be rounded according to the instructions. In my classes, I always had my students round their answers to the nearest hundredth because that is like the nearest penny.

Here is the problem from the previous page

-.2(1.5x+5)=-2(.35x+.3) We found this answer: x=1

We check it by subbing it back in -.2(1.5[1]+5)=-2(.35[1]+.3) -.2(1.5+5)=-2(.35+.3) -.2(6.5)=-2(.65) -1.3=-1.3

The sides match, so we are right! Note: If your answer ends up being a fraction with 0 on top then the answer is 0. However, if the 0 is on the bottom, the answer is “null” (Ø) or “no answer exists.” This is the classic “divide by zero” error. Anything divided by 0 is infinity, not a number. Try dividing any number by 0 on your calculator!

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

97) Break variable term

B) Break variable term (divide both sides by the number in front of x using OOOS) Break means that we are breaking the bond between the number in front of the variable (the coefficient). We want to know what x is, not what 3x or -5x or 2/3x is. Since the coefficient is connected to the variable by multiplication, then that is what we will use to break the bond. Divide by that number or multiply by the reciprocal, on both sides, to get your answer. Opposite Operation Opposite Side is the shortcut. Because the left side will equal x, divide the other side by what is multiplying x. This gives x = answer.

2x=10 -4m=12 7x=14 x=5 x=-3 x=2 -8x=9 -8x/-8=9/-8 x=-9/8

x=4 3 3 1 * x=4* 1 x=12 1 3 1 3

There is a strong similarity and a key difference between Sync and Shift. As the Action chart showed, Sync uses the same effect on opposite sides, while Shift uses opposite effects on the same side. Their purpose is identical, however, in that they seek to maintain balance by making a counterchange for every change. Multiplying or dividing a side by any number (except 1) obviously changes the value of the entire side. To make a counterchange on the side you just changed would be counterproductive. Therefore the counterchange is made on the other side. This is different from our previous problems because for the first time we are working with both sides of an equation, rather than just an expression on one side. Equations are the “rest of the story” for expressions. All the thinking you and your students have developed in Shifting can now be extended

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to Syncing. The point and purpose of both is the same, only the mechanics are different. The big/small move now needs to become same/opposite. The best picture is the “scales of justice” type of scale with two hanging plates. If one side sinks lower under a larger weight, then the other side needs to sink lower under an equally larger weight to bring it back in balance. That is Syncing. If you are limited to working only on one plate, like we must when working only with expressions, then you must make a big/small transaction on one side. Larger weight on one part of the plate is countered by smaller weight on another part. The end result is that the plate does not move and the scale stays in balance. That is Shifting. So the purpose of both Syncing and Shifting is the same--to maintain balance. They differ only in how they achieve that goal. Sync uses same/opposite. Shift uses big/small.



98) Combine like terms

C) COmbine Like Terms (COLT) Nothing new here. The reason you do step D before step C is to get terms separated according to kind. Letters go left, and plain numbers go right. That sets you up for step C, because now each side is made up of only like terms. Combine the variable terms and combine the numbers and you are done with this step. You will always have whole numbers at this step because step E previously eliminated any fractions and decimals.

7x-12x=-1+5

4x-9x+2x=6+11-8

-5x=4

-3x=9

-x-6x+2x=-5-1+7

3x+8x-2x=-4+17

-5x=1

9x=13

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99) Decouple like terms



D) Decouple like terms (letters left, use Switch Sides Switch Signs) Our goal is to get x on the left equal to a number on the right. Right now, we have x’s and numbers all over the place. So use the Sync shortcut, Switch Sides Switch Signs, to get letters left and numbers right. Switch the signs of ONLY those terms that jump the = sign. Just moving a term around on the same side does not switch its sign. Same Side Same Sign goes along with Switch Sides Switch Signs.

-5+7x=12x-1 7x-12x=-1+5 -11-9x-6=-4x-2x+11 4x-9x+2x=6+11-8 1-x=-5+6x+7-2x -x-6x+2x=-5-1+7 The SSSS shortcuts are based on Sync which is based on equality is just a special case of OOOS (Opposite Operation Opposite Side). Both the D and the B steps are similar because they end up moving something from one side to the other. The B step used multiplying and dividing, while the D step uses combining. These shortcuts are well worth teaching students. Some books teach them to write out all the details of the combining on both sides, but the end result is a simple move. So I teach them to make the move based on the Sync Action. This results in simpler thinking and less writing, which results in less mistakes. Sometimes writing things down clarifies a student’s thinking and helps reduce mistakes, but not here. Writing down the simple and obvious actually increases mistakes. Of course, some students insist, so fine, I let them.

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

100) Eliminate decimals

E) Eliminate decimals by multiplying both sides by biggest number of places The rule is abbreviated for easier memorization and needs explaining. You can’t multiply by a decimal place, but you can multiply by 1 with a 0 for each place. So if your equation has a number with 1 decimal place, then multiply by 10, which is a 1 with one zero. If your equation has a number with 2 decimal places, then multiply by 100, which is a 1 with two zeroes. If you find a number with 3 places, multiply by 1000. Make sure you use enough zeroes to match the number with the most places. It is better to multiply by a number with too many zeroes than too few so that you make sure to eliminate all decimal places. Reducing or rounding later will automatically take care of too many places. Also make sure not to forget to multiply the numbers with invisible decimal places. Everything on both sides must be multiplied for a proper Sync. Also, make sure you have done step F first. Filling the parentheses will change the decimal places! So step E should never take place if ( ) are still there.

places

1

0

1

0

10(.2-3x)=(1.4+6x)10 2-30x=14+60x

Notice that I wrap each side in ( ) to remind myself that I must change the whole side when I Sync. Of course, Sync is required because I am inflating the sides by 10, 100, or whatever. Shifting, by the distributive property, then takes place when I multiply the sides. Notice also that the E step on top, when done completely, gives me a clean D step on the next line. This is the purpose of E and F--to reduce the clutter and eliminate the complications. The E and F steps fill a similar role to In and Fun with expressions. They clean up the complicated, leaving us with just the basic operations for the D-A steps.

In both equations I multiply by a 1 with as many 0’s that match the most decimal places I found

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153

2

0

1

0

100(.37+x)=(2.1+8x)100 37+100x=210+800x

101) Eliminate fractions



E) Eliminate fractions by multiplying both sides by common multiple of the bottoms The technical names of what we are looking for is least common denominator or least common multiple. Remember that the denominators are the bottom numbers and we want to eliminate them so that the fractions become whole numbers, which means the bottoms must all become 1. To achieve that we must multiply all the fractions by a number that all the bottom numbers multiply into. If that seemed like a lot of words, basically we are going to look for common denominators, but stop short of combining the fractions. If you have bottom numbers of 2 and 3, then times by 6. If you have bottoms of 3 and 4 and 12, then times by 12. Any number will work if all the bottoms go into it evenly. Just like with decimals, it is better to multiply by a number to big, than one too small. Any common multiple will work, not just the lowest. Reducing in step A will auto-adjust. Don’t forget the invisible 1’s on the bottom of whole numbers. The integers need to get multiplied by your number, just like the visible fractions. Just like with decimals, make sure you have done step F before step E, because any parentheses will change your fractions.

6(4x+ 1 )=(5- 5 x)6 3 2 24x+2=30-15x

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In both equations I multiplied by a number that all bottom numbers go into evenly. This Shifts every fraction and turns them into whole numbers.

12(- 1 + 3 x)=( 5 x+ 1 )12 2 4 6 3 -6+9x=10x+4

102) Fill Parentheses



F) Fill parentheses by multiplying outside ^ inside (distribute) There is nothing new here. This is the familiar Shift Action applied to parentheses. The outside number is the sprinkling can that waters every term inside and makes them grow. The only thing you need to carefully notice is the change to fractions and decimals during this step. Because they change is why you must wait to eliminate them until the next step.

4(9x-6)=-2(13-6x) 36x-24=-26+12x These examples show why it is so important to do step F before step E. Also, if students try to do step E before step F, they usually multiply the outside factor and the inside factors, which is often wrong, because it is a double multiplication to that superterm. Other terms outside any ( ) they will only multiply once. That puts their equation out of Sync. In the fraction example below, they will ^20 inside and ^20 outside, which is multiplying by 400. If they do that on both sides, they are fine. However, let’s say they have a problem where there is a +4 next to the (x-7). They will multiply the 4 by 20 only once, while all the ( ) terms are ^ 400. That is out of Sync. This is why F is before E, to help them avoid that minefield of complexity altogether.

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It looks like I should multiply by 20 on this line, but after the ( ) I see I really need to multiply by 40.

3 2

( 3 + 3 x)= 1 (x-7) 5 4 2 9 + 9 = 1 x- 7 10

8

2

2

Should I multiply by 2 places or 3 places?

.3(.4x-1.6)=.25(2x+.5) .12x-.48=.5x+.125 These questions, and risk of mistakes, can be avoided by doing step F before step E.

103) Flip complex fractions



F) Flip complex fractions, this turns them into regular fractions Once in a great while I have seen complex fractions in equations. These are very easily turned into regular fractions and then you can continue with the regular FA steps. A complex fraction is fractions within a fraction. Instead of number over number, it is fraction over fraction. Just as the top number is divided by the bottom number, so the top fraction is divided by the lower fraction. So really, this is just a fancy way of writing fraction _ fraction. Therefore, flip the bottom fraction and turn it into two regular fractions multiplying each other.

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7 2

x= 3 x+ 5

3 4

7 2

1 4 1 7

* 4 x= 3 x+ 1 * 7 3

5

4

1

Fraction over fraction turns into fraction ^ flipped fraction multiply right away

104) Figure functions



F) Figure functions The reason for step F is to take care of the complicated first. Step E cleans up what step F can’t handle. That is the basic principle that applies even to things not covered by step F. Simplify whatever looks complicated so you can turn the equation into something familiar and easy. Use the calculator! The one point not to miss is that this applies only to numbers. If you have a power or root of x, or a trig or log function of x, you can do nothing. The equation is not even a V1D1 equation. The degree is more or less than 1, but it is not 1. These FA steps apply only to degree 1.

3x*cos45=92x+log28 3x*.7071=81x+3

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You may not at first see how to solve an equation, but do what you can to get it to a point where it looks familiar and solvable

(sin90)x=43+log100*x 1x=64+2x



105) Proportions

Proportions are a special type of equation that is simply fraction = fraction. These occur often enough in real life and on tests that it is worth learning a shortcut for them. Of course, you can use the FA method starting on step F, but the shortcut is faster and is the natural result of step F. The basic idea is shown here.

a b

=

c d

becomes ad=bc

If you were to follow the normal F step and multiply both sides by the common denominator, bd, then cancel, you end up with the same result. This works with polynomials of any degree in the fractions. The only limitation is that it must be fraction = fraction. There can be no other terms combined with either fraction. The shortcut is commonly called cross-multiplying, similar to cross-canceling with fractions. Once the cross-multiplying is done you end up with a regular equation that is usually on step D or B. You might even have a quadratic (x2) equation on your hands. We will cover those in the next chapter.

x+1 x = 4 7

Ž 7(x+1)=4x

x+1 8 2 =x-4

Ž (x+1)(x-4)=16

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Algebra: Quadratic Equations Quadratic equations are V1D2 equations. They have one variable with degree two. By Syncing and Shifting you can make it look like ax2+bx+c=0. Many of these equations will factor and all of them can be solved with the quadratic formula.

106) Fill, Flip, or Figure



This step is identical to step F for linear equations. You can ignore the x2 for now because you are dealing only with the numbers. Just as with method 1, fill parentheses, flip complex fractions, or figure functions. Everything should look and feel the same, except there will be an x2 along with the x.

4x2*tan45=9x+log216 4x2*1=9x+4

.3x(.4x-1.6)=.25(2x2+.5) .12x2-.48x=.5x2+.125 4(9x-6)=-2x(13-6x) 36x-24=-26x+12x2

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107) Eliminate fractions or decimals



Again, this step is identical to step E for linear equations. Continue to treat the x2 as a regular x, because you are dealing only with the fractions and decimals.

12(- 1 + 3 x2)=( 5 x+ 1 )12 2 4 6 3 -6+9x2=10x+4 places

2

0

1

0

100(.37+x)=(2.1+8x2)100 37+100x=210+800x2

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108) Descending order = 0



Here is where the FA method for quadratic equations takes off in a different direction from linear equations. Because you have an x2 in the equation, instead of decoupling the letters to the left and the plain numbers to the right, bring everything to the left. Combine like terms if you need to. Then put the terms in descending order of exponents in the familiar pattern of ax2+bx+c=0. This is just like what you did in the polynomial chapter. You are now ready for factoring or for the quadratic formula.

5x-8x2=7+2x -8x2+5x-2x-7=0 -8x2+3x-7=0 1-2x=9+2x2-4x -2x2-2x+4x+1-9=0 -2x2+2x-8=0

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

109) Common factor

This is nothing but the old familiar common factoring from the polynomials chapter. You may not be able to find a common factor, but if you can find one, you should use it. If the number in front of the x2 is negative, then you should factor by a negative, because that will make the next steps easier for you.

4x2+6x+18=0

-2x2+2x-8=0

2(2x2+3x+9)=0

-2(x2-x+4)=0

2x2+3x+9=0

x2-x+4=0

Now notice something. You can divide both sides by the common factor. Since the right side is 0, it entirely disappears. Look at your equation as if it is at the B step in FA method 1. The common factor in front of the ( ) is the coefficient, and the ( ) is the tag. Dividing both sides by the coefficient cancels it out. You are now prepared to bifactor.

Divide both sides by 5 and it disappears because of the 0 on the right (More clutter gone!)

15x2-5x-20=0 5(3x2-x-4)=0 3x2-x-4=0

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

110) Bifactor

Again, this is a repeat of the bifactoring from the polynomials chapter. The new part is that if you can bifactor, then go ahead and set each factor equal to 0, then solve those miniequations. Notice that the answers will always be the opposite sign of the factors from which they come.

x2-7x+10=0 (x-5)(x-2)=0 x-5=0 x-2=0 x=5 x=2

x2+8x+12=0 (x+6)(x+2)=0 x+6=0 x+2=0 x=-6 x=-2

Factor ^ factor = 0, therefore when either factor is 0 the equation is true. So set each factor = 0. 2x2-5x-33=0 (2x-11)(x+3)=0 2x-11=0 x+3=0 x=11/2 x=-3

3x2-11x-20=0 (3x+4)(x-5)=0 3x+4=0 x-5=0 x=-4/3 x=5

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

111) Answer formula

Of course, this is the quadratic formula. If none of the factoring got you to the answers then the quadratic formula will. It always will give you an answer, even if it has to use imaginary numbers. Just make sure your equation looks like ax2+bx+c=0

ax2+bx+c=0 ax2+bx+c=0 3x2+6x-8=0 5x2-7x+2=0 a=3 b=6 c=-8

-3±@62-4*3*-8 2*3 x=1.41

x=2.41

b2-4ac

-b±@ 2a

a=5 b=-7 c=2

7±@(-7)2-4*5*2 2*5 x=1

x=.4

2 answers in one formula First from the + Second from the -

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Algebra: Other Equations

112) Linear Technically, any equation of degree 1 is a linear equation because its graph is a straight line. However, this lesson is focusing on the equations that are based on y=mx+b. Most algebra textbooks will have at least one whole chapter dedicated to these types of equations. They come in different forms and you usually have to change them from one form to another. This is where the FA method comes in handy. If you can think of the y as being the variable you want and the x as a constant, like &, then you can use the FA method for any degree 1 equation. The D step will look a little different, but the concept is the same. The variable you want, y, goes on the left. The unwanted variable, x, and all the numbers, go on the right. The C step is often skipped, but if there are like terms, go ahead and combine them. You just won’t be able to combine everything on the right side to a single term. Combine them as far as you can and put the x term first followed by the plain number term.

F D B A

3x=1/4(2y-5) 3x=1/2y-5/4 -1/2y=-3x-5/4 y=6x+5/2

fill parentheses

y goes left, all else right

^ both sides by -2 put in desired format

D 4=4y+8x D x=2y+5 B -4y=8x-4 B -2y=-x+5 A y=-2x+1 A y=1/2x-5/2

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113) Rational These equations look almost like proportions. They look enough like proportions that many students get fooled and start doing very wild forms of cross multiplying. What throws them off is the fact that there are variables on the bottom, but a proportion is only fraction = fraction and nothing else. These other equations are really just step E equations that could be linear or quadratic. They reveal themselves after step E is done.

Step E: Eliminate fractions Common denominator is 4x(x+2). The x in the second fraction is factor of 4x already. x2 is extra. All tops must be ^ by factors they are missing on the bottom, just like regular and polynomial fractions. Follow steps E through A as normal.

3 1 x+2 x

2

= 4x

4x(x+2)( x+2 3

1 x

= 4x ) 2

3(4x)-4(x+2)=2(x+2) 12x-4x-8=2x+4 12x-4x-2x=8+4 6x=12 x=2

The above equation proved to be a linear equation, but not this next one. At first, it is not obvious that this equation contains an x2 but after the E step it is clear. Therefore, switch to the FA method for quadratic equations.

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Step E: Eliminate fractions Common bottom is 2x(x-3). 2x-6 factors to 2(x-3). All tops must be ^ by factors they are missing on the bottom, just like regular and polynomial fractions. Follow steps E through A as normal.

x 4 2x-6 2x

4

= x-3

2x(x-3)( 2x-6 - 2x = x-3 ) x

4

4

x(x)+4(x-3)=4(2x) x2+4x-12=8x x2+4x-8x-12=0 x2-4x-12=0 (x-6)(x+2)=0 x=6 and x=-2

Algebra: Other Equations

167

114) Multi-variable A linear equation is a multi-variable equation with two variables, but you will also encounter equations with three or more variables. The directions will tell you to solve for x or solve for n or some other variable. If the degree of the variable you need to find is 1, then you can use the same basic steps of the FA method for linear equations. Treat all the unwanted variables as constants.

Solve for x 5b-3x+6=8x-b+7 -3x-8x=-5b-6-b+7 -11x=-6b+1 x= 6b-1

all but x goes right COLT _ both sides by -11 fraction is reduced

Solve for n D 4n-6m+s=7s-2+2n C 4n-2n=6m-s+7s-2 B 2n=6m+6s-2 A n=3m+3s-1

all but n goes right COLT _ both sides by 2 no more to do

D C B A

11

168

Action Algebra

115) Exponential A special set of equations that appear in many textbooks are exponential equations. These equations have variables in their exponents. This makes it impossible to use a calculator because how do you enter something like 4x ? Neither you nor the calculator knows what x is. But the equations are setup to have similar bases, which means their exponents must equal. For example, 6x=62 We now use the fact that this equation is already in Sync. Therefore, x=2 because the exponents must be equal to keep the sides equal and in Sync.

8x+1=84 Because the equation is in Sync and the bases are the same, the exponents must equal. Therefore, x+1=4, so x=3. Check it on your calculator! If the bases are not equal, then Shift and use the exponent rules to make them equal. Then you can set the left exponent = to the right exponent.

2x-4=8 becomes 2x-4=23 Now you can set the exponents equal to each other. x-4=3 therefore x=7.

if nexpression1=nexpression2 then expression1=expression2 Now solve the new equation using the FA method

4x+2=45

6x-.3=67

D

x+2=5

x-.3=7

E

C

x=5-2

10x-3=70

D

A

x=3

10x=70+3

C

10x=73

B

x=7.3

A

Algebra: Other Equations

169

116) Inequalities Inequalities are just regular equations with one small twist. Whenever you Sync both sides by multiplying or dividing by a negative number, you need to reverse the direction of the arrow in the inequality. That’s it! Remember this one thing and solve inequalities just like equations.

-4x Þ 36 x Ý -9

-5x Û -20 x Ü4

Whenever you sync by ^ or _ with a negative number reverse the direction of the inequality arrow.

x+6 Ü 9+4x x-4x Ü -6+9 -3x Ü 3 x Û -1

2x+8 Ý 9x-6 2x-9x Ý -6-8 -7x Ý -14 x Þ2

170

Action Algebra

117) Radical Algebra is just the beginning of a whole new world of math. There are many new types of problems beyond what you have seen that will require new knowledge and new strategies. However, their foundation will be the ten Algebra Actions. One example is a type of equation known as radical equations. This is when a variable, not a number, is inside the root symbol. In this type of equation, you will “solve it twice.” In the first stage, pretend that the root is a plain x and follow the FA steps down to the B step. Now Sync by squaring both sides. Do not individually square the terms on each side, but put ( ) around each entire side and square the whole side at once. In really complicated cases you will need to use distribution. After squaring, you can then start the second stage at step F. One note of caution. If you ever get a square root equal to a negative number, then you need to stop or use imaginary numbers if you know them. In the world of real numbers there is no such thing as a square root of a negative, because - ^ - = + and + ^ + = +. Not even your calculator can give you the square root of a negative number!

B A

5@x=15 (@x)2=(3)2 x=9

@x+4=7 (@x+4)2=(7)2 B x+4=49 A x=45

2@x-5=8 @x-5=4 (@x-5)2=(4)2 x-5=16 x=16+5 x=21

B

D C

A

Algebra: Other Equations

171

System of Equations Visually, a system of equations is two or three lines on a grid that intersect at a point. We solve the equations to find the coordinates of that point. We are not going to graph in this course. Graphing and other topics, like trigonometry, will be covered in another book. What we want to look at in this chapter are systems of two or three linear equations with or without quadratic equations. Some textbooks include other combinations, but these are the major systems all books cover as well as both methods of solving. The overall strategy to solve a system is not always one of the Actions, but along the way the equations are Subbed then solved using the FA method.

118) Systems by Substitution This method will solve all the standard problems in textbooks and on tests, but can take longer than the elimination method that sometimes works. It’s a matter of choice. We are looking for the unique combination of x and y that will make two equations work. In other words, we want that special (x,y) coordinate that is on both lines at the same time. It stands to reason that if we find the x that works, then we can substitute that value back into the equations to find the y that works. By solving one of the equations for x I will find all the y’s that go with it. Then if I Sub that value of x into the other equation I will find the y that works in that equation as well. I now have half my answer. By going back to the first equation with my y answer and Subbing it, I will find x. Now I am done! Here is an example and the steps to take.

The system of equations to solve

x+3y=11 -2x-2y=-10

Solve an equation for a variable

x+3y=11 x=11-3y

172

Action Algebra

The solved equation always gets Subbed into the other equation

-2(11-3y)-2y=-10

Because x=11-3y, then 11-3y can go where x was

4y=12

-22+6y-2y=-10 y=3

Now that I know y, I can Sub it back into the other equation to find x

x+3(3)=11 x+9=11 x=2

I now have both answers, x=2 and y=3 Notice how I kept alternating my use of equations. I solved the top equation, then Subbed into the bottom equation to solve it. Then I took that answer and Subbed it into the first equation. If I Sub into the same equation I solve, I end up with correct, but useless, information. For example, if I Sub back into the first equation I solved I would get (11-3y)+3y=11. Solving that gives me 0y=0. Really?! 0=0 How amazing! The substitution method also solves a linear-quadratic system such as the one below, which is a line crossing a parabola and hitting it in two points.

The system of equations to solve

4x-y=-6 y-x2=-6

Solve an equation for a variable (solving for the x2 will take longer)

y-x2=-6 y=x2-6

Algebra: Other Equations

173

The solved equation always gets Subbed into the other equation

4x-(x2-6)=-6

Because I have a degree 2 equation, I have two answers. Sub them both into the other equation.

x2-4x-12=0

Finding 2 x's means I find 2 y's My answers are (6,30) and (-2,-2) in coordinates

4x-x2+6=-6 (x-6)(x+2)=0 x=6 and x=-2 y-(6)2=-6 y-36=-6 y=30 y-(-2)2=-6 y-4=-6 y=-2

Where the substitution method gets more complicated is when the first equation you choose to solve uses fractions. Yet, it can be done! Just keep alternating your solving and Subbing as well as your equations.

174

Action Algebra

5x+2y=9

3) solve 1) solve

3x-2y=-1

2) sub

y=3/2x+1/2

5x+2(3/2x+1/2)=9 5x+3x+1=9 8x=8 4) sub

x=1

5) solve 3(1)-2y=-1

3-2y=-1 -2y=-4 y=2

119) Systems by Elimination The elimination method is also called the addition method. It will not work with linearquadratic systems, but works nicely with linear systems. Let's look at the previous example again, but solve it using elimination.

Algebra: Other Equations

The idea is to combine one of the variables so it = 0. That lets you find the other variable, which you then Sub back into an equation.

175

5x+2y=9 3x-2y=-1 8x

x=1

=8

3(1)-2y=-1 3-2y=-1 -2y=-4 y=2 In this example the y's eliminated themselves without us needing to do anything. Usually you will have to multiply one or both equations to make a variable become a common multiple in both equations. If you keep the variables the same sign, then subtract the equations, else add them. Whatever you do, the goal is to combine them to 0. Study the example on the next page, then read my confession. I cannot tell you why this method works. It has never been explained to me and I cannot find one on the web. I suspect the proof for it lies in some book of advanced algebra not accessible by the common person.

176

Action Algebra

Equations to solve

I chose to eliminate the x so I made its nums = a common multiple of 6. Of course I Synced the whole equation. Because the 6x are both +, I subtract them and all the other terms to get 0x Now I can solve for y First answer! Sub y into either of the original equations and solve

3x-4y=1 2x+5y=16 2(3x-4y)=(1)2 3(2x+5y)=(16)3 6x-8y=2 6x+15y=48 -23y=-46 y=2 3x-4(2)=1 3x-8=1 3x=9

Second answer!

x=3

Algebra: Other Equations

177

120) Systems of Three While quadratics could be included, I have only seen linear equations used to make a system of three equations. Both substitution and elimination can be used and will be demonstrated. Of course, there are more steps. The basic elimination strategy is to pick two different pairs of equations and eliminate the same variable from each. This will leave you a system of two equations. After solving that new system for those two variables, you can then substitute the answers back into one of the original equations to find the third variable. Easy?!

The three equations to be solved I chose to eliminate y in the first two. First equation ^ 2. Added to second.

First equation for stage two with x and z. Eliminate y in second and third equations.

Multiply each, then add. Second equation for stage two with x and z.

2x+y-3z=-9 -x-2y+2z=3

2x-3y+2z=-2 2(2x+y-3z)=(-9)2 4x+2y-6z=-18 -x-2y+2z=3 3x-4z=-15

-3(-x-2y+2z)=(3)*-3 2(2x-3y+2z)=(-2)2 3x+6y-6z=-9

4x-6y+4z=-4 7x-2z=-13

Now we go to stage two with the two equations that have x and z. Notice that I keep using the word "choose." It really is up to me to choose a pair of equations to work with and a variable to eliminate according to what I think might be easiest. There is no one exact way.

178

Action Algebra

Multiply by two,

so I can subtract to get. Equation with only x.

Sub x into either equation with x and z. Solve for z. Sub x and z into any original equation with 3 variables to find y.

2(7x-2z)=(-13)2 14x-4z=-26 3x-4z=-15 11x=-11 x=-1

3(-1)-4z=-15 -3-4z=-15 -4z=-12 z=3

2(-1)+y-3(3)=-9 -2+y-9=-9 y=2

That was fun, wasn't it?! Yes, it is easy for students to get lost, but this is good exercise for the brain and organizing and tracking abilities! Now we will solve the same system using substitution. Here my strategy is to pick a variable for which to solve. I then Sub that into the other two equations to give me a new system of two equations with the same two variables. I can then solve that smaller system using Substitution or elimination. Finding those two variables, I now Sub them back into an original equation to find the third.

The three equations to be solved. I solved the first one for y. It was easiest.

2x+y-3z=-9 -x-2y+2z=3

2x-3y+2z=-2 y=-2x+3z-9

Algebra: Other Equations

179

Sub for y in the second equation.

First equation with x and z for stage two.

-x-2(-2x+3z-9)+2z=3 -x+4x-6z+18+2z=3 3x-4z=-15

Sub for y in third 2x-3(-2x+3z-9)+2z=-2 equation. 2x+6x-9z+27+2z=-2 Second equation with x 8x-7z=-29 and z for stage two. Stage two avoids fractions by eliminating.

8(3x-4z)=(-15)8

Subtract to eliminate x.

24x-21z=-87

Solve for z.

Substitute z into any equation with x and z to find x.

Sub x and z into any of the original equations to find y.

3(8x-7z)=(-29)3 24x-32z=-120 -11z=-33 z=3

3x-4(3)=-15 3x-12=-15 3x=-3 x=-1

2(-1)+y-3(3)=-9 -2+y-9=-9 y=2

Actions Explained This section focuses on Actions rather than on problems. I will bring together samples of different problems to help emphasize the Action that connects them and provide an overview for those who skipped the earlier chapters wanting to get to the heart of things. However, if you skipped the Basic Principles chapter, I encourage you to read that before this chapter as the Actions are based on those practical principles. You might also want to look at the Action chart for a glimpse of the big picture first. If you have read through the chapters, you have already seen the Actions applied, and so this section may seem redundant, but think of it as a reference.



Action 1: Sync

2 Opposite Sides - 1 Effect: You may do whatever you want to an equation as long as you do the same thing to both entire sides. If you change one side, then you must immediately sync the other side.

4+5=6+3 2*(4+5)=(6+3)*2 x+13=8-2x 13+(x-13)=(8-2x)+13 Visualize it this way. You have one of those balance scales hanging from your hand just like the blindfolded Lady Justice. You don’t know how much is on each side, but if you increase both sides by the same amount of weight, then you know it will remain balanced. In the same way, if you remove the same amount of weight from each side then it will remain balanced. You can also double or triple each side. You can cut each side in half or reduce them 10% each. You do not need to know how much is on each side, you just need to apply the same force to both sides. The icon tries to remind you of this idea because it looks similar to a balance scale or

a teeter totter. If the forces on both sides are equal, then it is in balance. A good thing to do when learning to use this Action is to put giant parentheses around both sides and then outside the parentheses perform whatever operation you desire. This helps you make sure the whole side gets affected, not just part of the side. Shortcuts Shortcuts are good, if you understand the principle behind them and don’t make so many that they become hard to remember. They can also shorten how much you need to write at each step and that can help reduce mistakes. So here are two shortcuts based on the Sync Action. Opposite Operation Opposite Side (OOOS). This is just the net result of syncing. If you have a small equation like 5x=15, then you divide both sides by 5, then you can cancel the 5’s on the left, then you divide 15 by 5 on the right, you end up with x=3. You can skip all the error-prone writing in between and simply jump to the next line by writing x= then doing 15/5 in your head and writing down 3. You know the 5’s will cancel on the left, that is why you chose to divide by 5. So just cancel what is in the way on one side and immediately write down the answer of the operation on the opposite side. Switch Sides Switch Sides (SSSS). This is just a special application of OOOS that applies only to terms. Move the term to the opposite side and change its sign at the same time, instead of writing down all the intermediate adding or subtracting. The mirror image to Switch Sides is Same Side Same Sign. This is a good mnemonic to remember when students Sort on the same side of the equal sign but think they still should change signs.

x+7=12

4x=32

x=5

x=8

same result above as long way below

same result above as long way below

x+7-7=12-7

4x/4=32/4

x=5

x=8

x+7=12

4x=32



Action 2: Shift

1 Side - 2 Opposite Effects: You may change an object at any time if you counter it with an equal and opposite change to that same object. An object, such as a term or factor or set of parentheses, is always on one side only. Often you have problems where you have no equal sign, and therefore only one side. You can’t use Sync, because there is not another side to sync with. That is when Shift comes to the rescue. Shift comes from the idea of shifting your weight from one foot to the other. Your weight did not change, you just transferred it within yourself. If you happen to be standing on a scale, it doesn’t change or become unbalanced, because weight did not change. The Shift icon is like squeezing a water balloon. All the water is still there, one part just gets smaller while the other gets bigger. The water shifts around without increasing or decreasing.

(4)(5)=20 (4/2)(5*2)=20 (2)(10)=20 Another good name for this Action is Big-Small. You do something to one part of the object to make it bigger, then you do an equal and opposite operation to make the other part of the object smaller. You can shift 8^3 into 4^6 because the 8 got halved while the 3 got doubled. You can also shift 8^3 into 16^1.5 by doubling the 8 and halving the 3. In both of these examples, one number got bigger and the other got smaller by an equal but opposite operation.

7+8=15 7-3+8+3=15 4+11=15 Force, not amount Let’s be absolutely clear what is going on with the Shift Action. It is what you DO that must be equal and opposite, not the new numbers that you get. It is the FORCES you apply to the side of the scale with the old numbers that must be equal and opposite, not

the differences between the new numbers. Also, how do you know when to add and subtract versus times and divide? By using what is already there. If the two numbers use + or - then you will also. If the two numbers use ^ or _, then you will also. Focus on what you DO, not the amounts the numbers go up or down.

100*6=600 (100*3)(6_3)=600 300*2=600

Don’t be fooled by the 100 soaring way up to 300 while the 6 goes down only a little to 2. What I DID was opposite and equal, it does not matter what the numbers do. In the end, the total of 600 remains unchanged. That proves I did a good thing! Let’s look at two similar problems using ( ) to make it obvious: (10)+(4) and (10)(4) The first expression joins the 10 and the 4 by combining, so I will use combining to Shift them by any amount I want. (10+1)+(4-1) which gives me 11+3, which gives me 14, which is the same total I started with. The key to using Shift is to apply opposite but equal forces on the scale, +1 and -1, to keep the original total the same. The second problem uses ^, so I will ^ and/or _. (10^4)(4_4) = (40)(1) = 40 After ^4 and _4 I end up with the same total with which I started. Even though 10 went way up to 40 and 4 only moved down to 1, that does not matter. Opposite and equal forces were applied to the scale and the original answer was kept unchanged. I Shifted the numbers without changing the total weight.

4x2+6x = 1(4x2+6x) = 2x(2x+3)

All factoring is Shifting (outside bigger, inside smaller)

Action 3: In (Inbox, Inside first)



Parentheses ( ) and brackets [ ] and curly braces { } all serve the same purpose. They put things in boxes. They group things. They look different only so we can more easily tell where the boxes start and stop, especially if we have boxes within boxes. One

might just as easily use different colored parentheses to make things clearer. Just like wrapped presents for your birthday, parentheses have a way of making everything harder to get at. You can’t just play with your toys, you have to unwrap them first. You don’t even know what gift is inside until you unwrap it. And like money stored in hidden accounts, you have to get at each account first before you can figure how much money you have in total. So if you see parentheses in a problem, go there first.

3^5+2

3^(5+2)

15+2

3(2)

17

6

Math gives us a kind of x-ray vision, because we can see the numbers inside the boxes, but what we don’t know is their answer. So as you work left to right and you run into the start of a box, ( or { or [, then stop, figure out the answer inside, then return to the rest of the problem.

8+3^5+2 (8+3)^(5+2) 8+15+2

11^7

25

77

Once inside, you follow the regular order of fun, mud, colt. Only after you are done working inside can you go out to play! The examples above show that putting boxes in problems can change their answers by a large amount. Pay attention to them!

Action 4: Fun (Funny Functions)



A function is like a mini factory. You feed raw materials in one end and different items come out of the other. The icon uses a FUNnel that takes in a circle and spits out a triangle. Functions are little math factories that intake a single number, or sometimes several numbers, or a group of numbers in a box. After the factory does its unique process, it spits out another number in the form of an answer. You then take that answer and use it in further calculations on the way to your final answer. You never work with the input

number(s). You always feed it into the function, then work only with the answer it gives you. There are gazillions of functions in the world of math, but what you will see in most algebra courses are exponents and roots, trigonometry functions, and logarithms. You will learn the basics of how they work, but pretty much they are telling you to get your calculator out. Most calculators let you enter the function and the number in the same order you see them, then you hit the equal button and you’re done!

5+3*23

5+(3*2)2

5+3*8

5+(6)2

5+24

5+36

29

41

9+2*cos(90)

(9+2)*sin90

9+2*0

(11)*sin90

9+0

11*1

9

11

The ( ) with functions serve as "mouths" into which to feed the input. Sometimes ( ) are not used, in that case, only the first number after the function gets fed into it.

Intro to Combining vs. Multiplying



Combining is adding and subtracting, while multiplying includes dividing, because dividing is multiplication in reverse just as subtracting is adding in reverse. Multiplication is a fast shortcut for addition, but it is also more than that. Multiplication can do something that combining cannot. Multiplying can bond together more than one type of thing. Combining can only work with one kind of thing at a time. COLT is Combine

ONLY Like Things, but you can MUltiply and Divide any kind of thing. Combining lets you go up and down, or left and right, on only a single number line. This is why you can only deal with the same kind of things. You combine apples only with apples and oranges only with oranges. Multiplying, however, comes from a grid, which is the result of two number lines, called “axes.” One axis can represent years, while the other can represent inches of snow. Putting them together makes sense. We want to know how many inches of snow fell each year. Adding them would not make sense. Years + inches is nonsense. So when you see different things together, they were put together only by multiplication, never by combining.

5ft*5ft=25ft2 3*3=32=9 x*x=x2=x2 4a*7b=28ab

You can not only multiply different things, but different things are made when you multiply

28yz/4y=7z 12ft2/4ft=3ft Action 5: MuD



Mud gets everywhere and gets on everything! That means you can MUltiply and Divide everything. (In the right order, of course!) Only like terms can be combined, but anything can be multiplied simply by merging them together. For example, you cannot combine 6x and 2y, but you can multiply them by merging them together like this, 12xy. (12 times x times y.) Division is just reverse multiplication so you can write 6x_2y as 3x/y. Multiplication and division are both very tolerant and flexible in this regard, while combining is very picky.

3 nickels + 2 nickels = 5 nickels = 1 quarter

but: 3 nickels ^ 2 nickels = 6 quarters

The reason for the differences was explained on the previous page. In practical living it gets applied this way. When you measure a room with a tape measure, you measure one wall at a time. You discover their lengths are 15 feet and 20 feet. Notice that you used a tape measure which looks like a number line, but the walls are at right angles to each other. Two number lines at right angles make a grid. So now you can also figure out how much carpet the room needs.

15ft^20ft=300ft2 Did you notice that feet got turned into square feet! A length on a tape measure that looks like a number line, gets turned into an area that looks like carpet. MUD everywhere means MUltiply and Divide everything because it even makes new things! NOPE! Do I like mud everywhere? NOPE! That is how I -=remember what to do with the signs. NOPE stands for Negative Odd Positive Even. Translated into English that --=+ means Negative answers come from an Odd amount of ---=negatives in the problem, and Positive answers come ----=+ from an even amount of negatives in the problem. I only count negatives. When multiplying or dividing you can -----=ignore positive signs. They have no effect. For example, ------=+ -3*-5=15 In the numbers I multiplied, I counted two negatives, and since two is even, my answer is positive. -------=The answer is still positive even if the problem is written this way -+3*-5 or this way -++-3*+--5 Both problems turn out positive because I ignored + signs and counted an even amount of negatives. Study the examples until you catch on to the pattern.

-3*+2=-6

-3*-2=+6

+3*-2=-6 +3*+2=+6

If you are a visual person, you will be reminded of what to do with the signs every time you see the icon. Remember that the numbers you multiply are arrows. So put them on the grid with their tails at the origin (the intersection in the middle) and going at right angles to each other. If the rectangle they make goes up to the right or down to the left, the answer is +, else it is -. Notice that ^ looks like a + sign and that _ looks like a - sign.



Action 6: COLT

2 +1 3

COLT stands for Combine Only Like Things (or Terms or Tags).

Add nickels get nickels Combine same things get same thing

Combining is adding or subtracting, but it is very picky. It won’t add just anything. It will only add things that are of the same kind. You can add any two numbers but you must line up their (invisible) decimals so that 1’s add with 1’s and 10’s with 10’s and so forth. If you have 2 nickels and 3 pennies, you need to change the 2 nickels to 10 cents then add it with the other 3 cents to get an answer of 13 cents. Nickels and pennies are not like things. They are not same kinds, so they must be changed into similar kinds. If you can’t make two different things into the same kind, then you cannot combine them. For example, 2x+3y is stuck. We don’t know what x is or what y is, so we don’t know how to change them. Therefore, we just have to stop. Don’t make the mistake of accidently multiplying them. If you answer 6xy, that is correct for multiplication, but not for combining! When you combine like things you get a like thing, ALWAYS! Nickels add up to nickels, and quarters add up to quarters, always. After you get a nickel answer you can change it to dimes, but a nickel problem always give you a nickel answer. Combining x’s gives you an x.

4x+2y+5x+3y = 9x+5y 7x-3x=4x not 4x2

2y-8y=-6y not -16y2

SSADDL SSADDL your COLT. Same Signs Add Differents Destroy, Largest: Combining looks for the LARGEST number, then takes that sign for the answer. Combining looks at the signs of BOTH numbers, asks if they are the same or different, then decides. If the signs are both + or both - you will add the numbers to get a larger number (farther from 0). If

the signs are different, + and -, then subtract to get a smaller number (closer to 0).

+6 +6 -6 -6 -2 +2 -2 +2 +4 +8 -8 -4

1) Always write largest number on top 2) Same Signs Add, Differents Destroy 3) Answer sign is Largest sign (top)

Let’s be practical. If you have $2 in your pocket, but owe somebody $3, you are really $1 in the hole at -1. Notice that 3 was the largest and negative, so it put you in the hole. Where do you end up if you start on the ground floor of a building, then walk up 3 floors, then walk down 5 floors? 2 floors in the basement. You walked down more than you walked up, so you finished in the basement. Think of positive numbers as cherries and negative numbers as Pacmans that eat the cherries. If you have 4 cherries, but 6 Pacmans want to eat, you are going to have two hungry and angry Pacmans leftover, -2. If you combine cherries with cherries or Pacmans with Pacmans, you have more of the same. Same Signs Add. If you put cherries with Pacmans, your cherries get destroyed. Different signs Destroy. But whatever you do, the answer sign always matches the sign of the Largest number.

Action 7: Show



This means that you may show or hide invisible objects at any time. For example at the right end of every whole number is an invisible decimal. Whether you hide it or show it does not change the value of the number. 1 = 1. = 1.0 = 1.00000 = 0000001.000000 That last 1 looks bigger than the rest, but it is just like a bloated puffer fish, as hollow as all the useless 0’s. Here is a list of many, but not all, invisible things.

Positive signs: 4 = +4 Multiply by 1: 5 = 5*1

Divide by 1: 3 = 3/1 Add 0: 8 = 8 + 0 Subtract 0: 8 = 8 - 0 Square root: @4 = Ć4 Exponent of 1: 7 = 71 Fraction bar parentheses:

3+x x-1

(3+x)

= (x-1)

All of these changes can happen anytime for any reason. Why? They never change the value of what is already there. Also, since a value change never took place, a counterchange is not needed. We only changed the looks, not the values. So if it makes your life easier, show it!

Action 8: Sort



You may re-arrange objects at any time, except division. The order of combining and multiplying do not matter, but division order matters. 2/10 is a lot different than 10/2, but 2-10 is the same as -10+2, and 2*10 = 10*2. In algebra, it is good practice to keep things sorted in alphabetical order. Sort 5y+2x into 2x+5y. Sort 3b-5c+2a into 2a+3b-5c.

1+2-3+4=-3+2+4+1

8b+2c-4a=-4a+8b+2c 9x+7+5x2=5x2+9x+7 -3-14_7=-14_7-3

but never sort the division part! You cannot sort -14_7 to 7_-14

Some formulas, like the quadratic formula, require you to have things sorted so that you put the right numbers in the right places in the formula. Sort 7-2x+3x2 into 3x2-2x+7. Make a habit of sorting things alphabetically, then by descending order of exponents. Once again, notice that we changed no values, so no counterchange was made. We only changed looks.



Action 9: Morph

The common word for this Action is convert, but morph sounds cooler! Besides it is just one syllable like all the other actions. The Morph Action lets you convert an object from one format to another at any time. This is like one person who wears many masks. Only the looks change, but not the value.

1/2 = 3/6 = 5/10 = 50/100 = .5 = 50% 3 1/4 = 3.25 = 3 2/8 = 13/4 = 325% 5 = 500% = 5/1 = 50/10 = 5 0/1 = 4 1/1 I think you get the idea. A number can morph from a whole number or decimal to a fraction to a mixed number to a percent to a scientific number to an algebraic expression. The number never really changes. It only changes how it looks. Sometimes Morph is the result of another Action. For example, to Morph 1/2 into 3/6 requires the Shift Action.

800% 16 2

8^100

8

8+0i

8/1

7

3 3



Action 10: Sub

This is just like calling in a substitute in a football or basketball or hockey game. Someone off the bench with the same kind of jersey can sub for someone on the field with a matching jersey. They are on the same team, so in that sense they are equal and can be traded for each other. The parallel does not quite hold up all the time because usually the player off the bench is not as skilled as the starting player on the field, but no parable is perfect! But in one sense this works, because player A comes off the field because he is probably tired or can’t do the thing the coach wants done on the next play. So we can say player B is equal, but better. Likewise, we make a substitution in some problems because the new object we bring in is equal, but better. It must be equal, but it also makes things easier.

-3x+6y=11 and x=7 therefore, -3*7+6y=11 So, you may replace object A with object B at any time, if both have equal value. When object A goes in, all of object B must go out. You never leave part of a player on the field! For example, 3x+5y=-7 and x=4, therefore we can take x out and put 4 in its place. 3*4+5y=-7 The wrong way to do the substitution is to leave in the x, like this 3*4x+5y=-7 One way to look at this is that the big equation is the field and the sub statement is the bench. Coach is telling us to sub x for 4. That means x must come out.

5x2+4x-13 you are also told that x=y+1. So wrap player with ( ) then sub 5(x)2+4(x)-13 5(y+1)2+4(y+1)-13

Rule Sheet Understanding and deadlines to get things done and arbitrary requirements of a system out of our control do not always line up. We also sometimes need a quick reminder of what to do in a pinch. The Rule Sheet on the next two pages condenses the Actions and their applications to numbered rules and examples for easy reference. Reinforcing the rules as well as the Actions with each new problem will help students. Sometimes understanding leads to concrete achievement and sometimes the performance must precede understanding. A printable pdf file on regular letter sized paper is available on the ActionAlgebra.com website. Print one for each student!

     

 



 



 



     

 

 

  

 

     



 





     

 

      



 

 

 

 

       

  

  

 

   

    

 

 

 

 

 

 

             

  

  



   

              



 

 

      

   

               

   

              

  



  



    

   

  

  

 

        

 

 

 

                                                                                                                                                                                         

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Goals & Methods I have presented to you the mental framework of math. That alone, combined with almost any curriculum will provide solid understanding for almost all students and help them meet graduation requirements. But I have more in mind than that. Let me explain the thinking that led me to my goals and the methods to reach them. I think it would be good if every student in the world could understand and pass Algebra II. The level of education and achievement that would entail would be astounding. The level of personal development and growth in thinking skills would be phenomenal. I would love to see such a thing happen, but I am also realistic. I am also a human realizing that everyone else is human with their own unique strengths and weaknesses, gifts and interests. Is Algebra II really needed by everyone? My real dream is that our system of education become more like a network of gardens rather than assembly lines in manufacturing plants. Yet, I am realistic about that, too. So my goal for this math presentation has been to make arithmetic and algebra as connected and understandable as possible, so that every teacher and student in their personal situations can gain as much benefit as possible. Last I checked, understanding never hurt anyone! It helps relieve the suffering of the non-math types and it helps accelerate the progress of those with a knack for math.

Encrypted Education Computer experts tell us that the best way to keep a password secret is to make the encrypted code look as random as possible. The purpose is to hide the pattern you used to encode the password. Randomness is the key to encryption, which is the science of confusion. Education is the opposite of encryption. Education tries to make things plain. It does not try to bury or hide information, but tries to make it known to everyone. Therefore, randomness is the enemy of education. Randomness encrypts, while education makes things clear and brings them out into the open. In my experience as a student and then as a teacher for over 20 years, math was mostly a random presentation. Topics were organized into chapters, but the topics themselves were mostly disconnected collections of rules and steps. There was no unifying model, no basic set of principles applied in a systematic way from start to finish. Many students

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and teachers alike see math as a random collection of problems and rules and techniques. Therefore, to them, math is encrypted. It is locked in a secret code without a key. What we need is a model for math based on the patterns that repeat themselves and simplify and connect all the steps and rules. If that model includes all the basic patterns then it will be the key to a more efficient, understandable, and enjoyable math experience. Patterns slay the randomness dragons of arbitrary drills and warm fuzzy guesswork. A model based on patterns is a great need and the true solution for math education.

What Is Understanding? Understanding is the result of education, which is the opposite of encryption. So understanding is a connected framework of concepts and practices. Understanding is “getting the big picture.” It is organized and makes sense. It is a living tree with branches and leaves, not a pile of dead leaves blowing away in the wind. Therefore, understanding is not the “do it and forget it” method of training. While a student needs to learn how to “do it” now, they should also be learning the underlying concepts and similarities so they can transfer their understanding to new problems and situations. This book takes both approaches. It teaches the concepts that explain why, but it also teaches the rules that tell how. And both concepts and rules are held together in the Action framework. Purposely, the Action framework is not “pure math” communicated to the logical level of the university professor. Young students don’t think in that manner. There is a logical development, but not in a strictly Which student has the greater chance formal way like one might find in of success: the one who tries to geometry proofs. More often, kids (and memorize a collection of 1000 rules adults) think in analogies. “This is like or the one who seeks to understand a that.” This can be good enough for a while so that a habit can be established, system of 10 Actions? which lowers stress levels, which then unblocks the understanding and connections are made. In short, this book is written for teachers and parents of elementary and high school students who are not ready at the same time for the same concept, not for college professors. Now for a quick note for any professors who may happen to pick up this book. The vast majority of American students are not graduating from high school with a solid understanding of the math they “passed.” Before lamenting the lack of “pure math”

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students entering college, we first have to increase the quantity and quality of students graduating from high school. This will increase the pool of college freshmen who can be taught “pure math,” and if those students have a framework of understanding from which to work, then the job of the math professor will be much easier and more fruitful. The high school graduate’s math understanding might not be “perfect,” but it will be correct and complete. One other note, most students do not go on to math-intensive professions, so a practical approach is more helpful more often than a theoretical one.

Readiness Just as important as teaching in a conceptual way is the student’s ability to perceive concepts. Just because we set a standard for a certain age group does not mean everyone at that age is ready to comprehend. Students vary widely and even a single student goes through spurts of slow and rapid growth. Kids are kids, not cars on an assembly line. Math is mostly an abstract mental framework of dealing with the universe. Most of math is beyond the young, immature mind. We will actually get farther You know what happened when I took the with most students by waiting a few math books from grade 2 through Algebra years beyond the standard age of 6 1 and tore out all the duplicate lessons? I or 7 to start formal, abstract math education. Before that time, most was left with just two years of content! If children will gain more by putting the we teach for mastery, then keep students paper and pencil away and doing thoroughly reviewed, there is plenty of math with real objects, money, and large visuals on large whiteboards. time for them to learn and understand. However, I know that proposal is considered too radical and politically incorrect in this time of pre-school mania. Therefore, I have tried to structure the math sequence and strategies to accommodate this force-feeding of young minds capable of cheerfully memorizing what it does not understand. (How we enjoy the early, easy years when children live to please and impress adults!) At least I hope the teacher/parent will understand the concept involved and periodically come back to first principles so that comprehension will catch up with rote learning. We as parents have too much anxiety about our young children. What’s the rush? They are kids only once and when your child reaches the middle grades most parents will be unable and/or unwilling to be fully involved with him/her. I have seen this over and over again in families and classrooms and it is really sad. This is one reason I have sought

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ways to shorten the time to learn math. Even if we don’t lessen the time requirements, I want to help relieve the pressure on the student. We as teachers have too much pressure to hand out grades, churn the kids through the system, and make them look good, thereby making us look good. But who really takes the time to care about the youth, help them to figure out their purpose, plan, and priorities in life? We enable our youth in their immaturity by constantly telling them what to do, rather than teaching them why. We as administrators and policy makers have too much ignorance about what constitutes true education. We are too focused on keep the money flowing and the gears of the system of turning. After all, what has “worked” for generations should work for us if we only do the same thing faster and more intensely! Sigh. While the wheel of life keeps grinding, the students in the garden grow up deformed and malnourished. Then we wonder what is wrong with their generation and why the garden is not green. What does this have to do with math? Everything! Why and how do we teach what math to which students?! Motivation is the ignition switch for the engine of the mind. Arbitrary requirements--no matter how good--only work for little minds until they begin to understand or sense that they arbitrary. Is it really human and educationally sensible that everyone do everything at the same time? or is it just arbitrarily convenient? Yes, tracking helps, but that deals with years, what about from week to week and day to day?

Resources Action Algebra is more than just this book. To individualize learning as far as practical and provide a manageable system for the teacher is the other goal. A framework of understanding must be communicated, practiced, and assessed. Therefore, this book is just the core of a complete curriculum. Arithmetic and Algebra are the foundation for word problems, geometry, statistics, trigonometry, science, and advanced math. Over time, the plan is to extend the thinking of Action Algebra into all these areas. Resources such as virtually unlimited worksheets to broaden or advance or review students at different learning paces are one key tool. One textbook does not fit all. The worksheets provide more than fill in the blank answers. Multiple choice, true-false, and answer columns provide test preparation and are easy to grade. Step by step solutions keyed to Actions are also provided where appropriate. A short video to explain and demonstrate each worksheet is another needed resource. Sometimes teachers need to be in five places at once. I wish I had them years ago.

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An interactive computer program that does more than drill and reward with game time is also needed. I envision a program where the student “builds” solutions with “tools” in a simulation of real life. All of these resources are, or will be, available on the website. One of the major observations that led me to the development of this approach was upper grade school students fighting the idea of negative numbers as if they were an evil impossibility. This is what happens when teachers have never been taught a framework from which to teach. To “get by” and to get kids through this year’s test, shortcuts are shortcutted even more and conceptual comprehension is the first item to be jettisoned. This has a huge effect in later years. At that time, to combat those symptoms, middle grade teachers hack their own solutions to their own problems and so math education becomes an incomprehensible mess of rote memorization and a bag of random tricks. Knowing this cycle will not be broken overnight, if ever, I am creating the resources I wish I had all the years of my classroom teaching. Sometimes a student was ready to What I memorize I soon forget, forge ahead, while others needed more time but what I understand and review with a topic. One book with only one or two pages of practice and only one or two becomes a part of me forever. approaches was not enough. Some of these resources I was able to create while still in the classroom and whenever I had them, they helped. Having different forms of a test not only stopped cheating, it encouraged many students to try again and better their score and increase their ability. It helped them overcome their fear of tests, because they ended up taking so many the tests became very familiar and lost their intimidation. Now that I have multiple choice tests in my arsenal, my effectiveness is increased to help them conquer standardized tests. The general strategy is to start with something real and relevant to the student (or at least visual in some cases) to root the concept in reality. Then the theory is explained and demonstrated. Plenty of practice and review is the next step until the student feels confident and accurate. Next, applications such as word problems are a good way to broaden the understanding and add interesting review. My goal is always to get the student through the homework as quickly as possible to the chapter and cumulative reviews (which I have in endless supply). At that point students often ask for more homework covering particular weaknesses they have discovered. So I guide them through the chapter, but then they guide themselves (to a large degree) through test preparation and test re/taking. This is far more esteem and knowledge building then standing over them with the big stick.

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My Student Is Stuck! It happens to all of us sooner or later when we learn something new. We hit a plateau. We get stuck on a level and we can’t seem to advance any higher. What do I do as a teacher when my student(s) just can’t seem to master a test after re-taking it ten times? First, think about what you would do if it happened to you as an adult. You would take a break. You would do something different for a while to get your mind off of it before attacking it again later. So why not allow your students do the same thing? But, you say, it is time for math class and everyone needs to do math. I can’t have everybody doing their own thing at their own time, that would be chaos. Yes, I agree, especially if the kids have low self-discipline. So what I do is let the student review some easier math or have them “go sideways” by doing word problems and other applications. You can do this if you have sufficient resources, like the worksheets. Print them on demand or have some printed ahead of time. But just as often, I will record the grades (usually the best 2 or 3 in a series of retakes; I never let them take just one) and let them move on. I have discovered that the next step “seals in” the previous step. It gives the student perspective so they start seeing the relevance of the previous step. Also, when I later let them return to the original test, it seems easier to them. You might be thinking, he allows his students too many re-takes! Because I have the resources to do it, this is a good thing. Whether they ace the first test or fail five in a row, they get earnest practice. Homework is often yawned at, but trying to pass a test draws forth effort and concentration. In the end, I really don’t care how many homework pages or tests the student did, once they know it, they know it. If their knowledge seems shaky, that is why I have lots of later review tests they also have to pass. Let’s face it, at can’t do any worse than the teach, forget, re-teach, forget cycle that is so common. A true story. In grade school my teachers sent me to the upper grades when it came time for math. I enjoyed it and was good at it. But a funny thing happened when I started algebra. All the x’s and y’s knocked the times table clear out of my head. I could not remember my multiplication facts. I had to take out paper and pencil every time, instead of just recalling. After several weeks of that, my times table came back and coexisted peacefully with algebra, but it was very frustrating until that time. So that experience taught me the value of repetition, perseverance, and the next step sealing in the previous step. I never forgot my times table again!

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Pre-Formal Math The best way to prepare concrete thinking children is to use concrete things. (“Concrete” as in real objects, not “cement!”) Very few children reach abstract conceptualization by the age of 7, which is the standard age of entry to first grade. Once there, they are made to sit all day and work with pencil and paper and fine motor movements. Everyone does the same worksheet at the same time, ready or not. Sure, they are “capable” of doing this, but in just a few years, or less, they start to lose motivation, hate school, and burn out. It is because they are doing without really learning. Children are more like flowers in a garden and less like cars in a factory than we like to admit. I realize the system will probably never change, but there are things we can do within the system to make it better for our children under the ages of 8 to 10. One big contribution we can make is to replace a portion of the My real dream is that our system formal teaching we do with pre-formal of education become more like a teaching. I purposely use the [invented?] word network of gardens rather than “pre-formal” instead of “informal.” Informal assembly lines in factories. gives me the impression of unplanned, unguided, uncontrolled. That is certainly not what we want. On the other hand, we want to reach the child where s/he is at, which is at the concrete, hands-on level. Pre-formal seems to get at this goal better than the other word. Therefore, in this section I will give several of many possibilities to engage children through their natural mode of learning and motivation. It is crucial to keep their motivation and interest in math as high as possible in the early years, because I have seen hundreds of times where the middle school years wipe out their early gains. The more we push kids in the early years the more we lose in the later years. We want to find the right pace for each child, but in general, too slow is better than too fast and too late is better than too early. Manipulatives, games, and large visuals tend to help us avoid those errors.

Activities The first test of readiness is when the child understands the concept of numbers. It is one thing to recognize two hands and two feet, but they don’t comprehend “two-ness” until they can set aside two of any object you call for. The objects should be of any size, shape, color (maybe flavor and smell, also!). You want to verify that they understand twoness, three-ness, etc... apart from any other properties.

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Different areas of the brain develop in different ways at different times. Quantity recognition may come after comparison for some children. By showing them how to arrange a random pile of objects into a single, straight line, they can then compare the lengths of the lines they have made to determine which has more and which has less. They do not need to know how to count or how to name numbers to do this. They are simply comparing bigger and smaller. When you think they have mastered both counting and comparing, test them by having them tell you which line has more objects: a line of 10 golf balls, or a line of 3 inflated balloons. After basic counting and imprecise comparisons, they should be ready to be introduced to money. Not only is this an excellent preparation for formal math, but it is a life skill of great value! The number system with place value can be taught using dollars, dimes, and pennies. Have your student give you 10 pennies and give them 1 dime in return. Help them to understand the idea of “packed value” (which leads to place value) in the dime. Even though it is just one coin, that coin has as much value packed into it as a stack of 10 pennies. Take them to the store and let them buy X pieces of gum using pennies, then the same number of pieces of gum using dimes. If the clerk can be brought into the scenario ahead of time, s/he can help add validity and confirmation to your teaching. Keep emphasizing the importance of pennies, dimes, and dollars as you prepare them to understand the decimal system. You are teaching them in parables until the necessary neurons connect to enable them to generalize and transfer the idea of place value to many other situations. You can use nickels and quarters, also, but explain that they are only helpers for convenience. Like five fingers on one hand, nickels can be a bridge to adding up to ten for those children who get confused by too many items. Two groups of five looks more manageable than a big group of ten. This actually has merit because many studies have shown our short term memories are usually in the range of 5 to 7 items. Remember we are focusing on children of age 7 or so, definitely under 10. So many of them will be so concrete in their thinking that they will need to count on their fingers or use objects. This is all right at that stage. I have seen a few algebra students still use their fingers, but it fades away. So I see little need for setting an arbitrary date when kids must stop using their fingers. However, I agree in the long run, that dependency on fingers is not best. To get them through that stage as quickly as possible, you might try having them count with pennies and dimes from the very beginning. This involves a little extra work than just tapping fingers, but the extra work should encourage them to learn to use their heads as soon as possible. Money can not only help you teach counting and place value (once the child gets

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above 10), but it can easily and naturally show the four basic operations. Adding is counting one pile of coins, then continuing without starting over at 0 with another pile of coins for the total or sum of the two piles. Subtracting can be approached in two ways. After counting the total of a pile, the child can count backwards with each coin removed. Where the downward count stops is the number of coins in the pile. (The discard pile can be added back in for verification.) After taking an amount away from the main pile counting could begin at 0 and go up to the total of the pile. Continued counting of the discard pile brings them back to the original total which verifies the subtraction of the discard from the main pile. Lining up the coins in the pile in number line like fashion prepares students for their first formal math lessons. Without moving any coins, your fingers could simply touch the total and the “discarded” coins so they can see the two numbers adding to make the total. This is also a step up in organized thinking leading to full abstract capabilities. Multiplying and dividing can be shown by arranging coins in a rectangular grid. A grid of 3 by 4 pennies shows that 3^4=12 and 4^3=12 and 12_4=3 and 12_3=4. This method also prepares them for the 2 dimensional graphs with x and y axes so commonly used in graphing. The rectangular grid of coins is also a great way to introduce skip counting which is the prelude to multiples and common multiples so critical to fractions and algebra factoring. So you see, all the basics of arithmetic can be taught without the rigid scheduling and confinement of small motor movement worksheets for the entire class. If you periodically have your students individually explain what they are doing and why it works, you can achieve almost the same formal results, while maintaining interest and impressing practical, relevant understanding. Playing store or bank or Monopoly (my favorite!) can be just playing a game or it can be quite educational if you participate, guide, challenge, and ask for feedback. Another tip on teaching through games is a fabulous website called Let’s Play Math! Type this address into your browser-- http://letsplaymath.net

Whiteboards and Vinyl One technique that highly recommend to teachers and parents at all levels is the personal use of large whiteboards. If you only have one, let the students do their work on it and then write down or tell you their answer. There is something about the large motor movements that do not take energy away from many students like fine motor movements do. Believe it or not, I discovered this trick during one of my pre-calculus classes with high school seniors.

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There was particularly nervous girl who just had to get everything right or her life would fall apart, so it seemed. Sitting down next to her and explaining and demonstrating and then having her show it back to me just was not working. Finally, in desperation (didn’t someone say desperation is the mother of invention, or something like that?!) I told her to take my marker and do it on the whiteboard while I remained seated. Lo and behold, she breezed through the problem with no mistakes! I had her try a new problem and she succeeded there, also. I tried it with other students and difficulties and found the time to understanding and mastery was greatly reduced. It was not a panacea for every case, but average progress was always increased, doubled or tripled even! Years later I got the idea of cutting a smooth, white showerboard from the hardware store into 8.5x11 sections and drilling holes for a 3-ring binder. This idea helped a few students, but not like a large whiteboard. So now, to increase access, I use whiteboards about 2’x2’ or larger. These are not really portable by the student, but they can be stored easily in a classroom. My latest improvement on this idea is thick, clear PVC vinyl from a fabric store. It is commonly used to cover dining tables. I can roll it up and take it with me. I can also put large graph paper behind it and write on the plastic and keep my paper re-usable at the same time.

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Grade Sheets A regular grade book with one sheet per class did not work for me when I wanted to individualize my math class. It could not handle variations in the versions or numbers of assignments the students did. Incompletes or blanks just scattered information over several sheets in the book and it was hard to see what a student was doing. Software was no better with its limited screens designed like a gradebook. What I really needed was a sheet for each student. The next three pages show what I came up with for each class. The overall progress on the sheet determines the grade. Each class worked out nicely with four chapters each followed by a fifth column for the final test itself. Abbr. is the abbreviation you create to label an assignment. Acronyms based on first letters in the title usually work well. Vers. is the three letter version code in the upper right of each sheet. Homework can be tracked as well, but I just gave a standard percentage of the grade when the student showed me completed homework for the chapter. Most of the grading was done by the students individually or in a group led by me. It was the tests that I was mainly after. I wanted to know what test a student took and when. I did not have all the unlimited worksheets like I do now and so I had to be more careful how soon a student repeated a test. I found about ten versions was usually enough. Faster students don't need that many before moving on, and slower students can't remember all the problems and answers even if they do repeat one they already had. I usually required three 80's or two 90's before letting a student move on to the next chapter. Although, since I was always constrained by a school imposed curriculum, I could not always enforce that policy. My usual compromise was to move the group as fast as possible through the chapter to leave as much time as possible for testing at the end. The importance and power of tests cannot be over emphasized. These really woke up the student and caused him/her to ask questions. Especially when they saw they were almost to the next level, they wanted to try harder. That is the point at which I could get overwhelmed as a teacher. There were not enough of me to go around! That is why I make all the videos. Progress once in a while cannot proceed in a straight line from chapter to chapter. Let the student go ahead or sideways with applications for a while, then bring them back. Before deciding, it helps to check verbally to see if the hang-up is one of understanding or one of inaccuracy in the writing of the mechanical details.

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Polynomials

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ALGEBRA Score

About the author Ed Lyons has taught math, computers, and other subjects at the middle and high school levels for over 20 years. He is an avid backpacker, skier, and nature photographer, but his favorite experiences are when his students’ eyes light up when they learn something new. Lyons earned his Bachelors degree in Secondary Education and his Masters degree in E-learning Education. Out of the regular classroom, he is now a speaker, consultant, and programmer turning the Action Algebra curriculum into an interactive computer experience. His contact information is on his website at ActionAlgebra.com