MathMammoth_Grade4A

Copyright 2008-2010 Taina Maria Miller. EDITION 1.25 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, or by any information storage and retrieval system, without permission in writing from the author. Copying permission: Permission IS granted for the teacher to reproduce this material to be used with students, not commercial resale, by virtue of the purchase of this book. In other words, the teacher MAY make copies of the pages to be used with students. Permission is given to make electronic copies of the material for back-up purposes only.

Please visit www.MathMammoth.com for more information about Maria Miller's math books. Create free math worksheets at www.HomeschoolMath.net/worksheets/

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Contents Foreword ........................................................................... Concerning Challenging Word Problems ........................

6 7

Chapter 1: Addition, Subtraction, Graphs and Money Introduction ...................................................................... Addition Review ............................................................... Adding in Columns .......................................................... Subtraction Review ......................................................... Subtract in Columns ....................................................... Mental Math Workout and Pascal's Triangle .............. Subtraction Terms .......................................................... Word Problems and Models ...................................... Missing Addend Solved with Subtraction .................... Order of Operations ....................................................... Bar Graphs ..................................................................... Line Graphs .................................................................... Rounding ......................................................................... Estimating ....................................................................... Reviewing Money ........................................................... Review .............................................................................

9 12 15 16 19 22 25 27 30 33 35 38 41 45 47 50

Chapter 2: Place Value Introduction .................................................................... Thousands ....................................................................... At the Edge of Whole Thousands .................................. More Thousands ............................................................. Practicing with Thousands ............................................ Place Value with Thousands .......................................... Comparing with Thousands ..........................................

3

51 53 56 58 60 62 64

Adding & Subtracting Big Numbers ............................ A Little Bit of Millions ................................................... Multiples of 10, 100 and 1000 ........................................ Review ..............................................................................

67 72 75 77

Chapter 3: Multiplication Introduction .................................................................... Multiplication Concept .................................................. Multiplication Tables Review ....................................... Scales Problems ............................................................. Multiplying by Whole Tens and Hundreds ................. Multiply in Parts ............................................................ Multiply in Parts with Money ....................................... Estimating Products ....................................................... Multiply in Columns - the Easy Way ........................... Multiply in Columns - Standard Way .......................... Multiply in Columns, Practice ...................................... Error of Estimation ....................................................... Order of Operations Again ........................................... Money and Change ........................................................ So Many of the Same Thing .......................................... Multiply by Whole Tens and Hundreds ...................... Multiplying in Parts with a 2-Digit Multiplier ........... The Standard Multiplication Algorithm with a 2-Digit Multiplier .............................................. Multiplying a Three-Digit Number by a Two-Digit Number ............................................... Review ............................................................................

4

79 81 83 86 90 95 99 100 102 105 110 112 114 117 119 122 124 128 131 133

Chapter 4: Time and Measuring Introduction .................................................................... Time Units ...................................................................... The 24-Hour Clock ........................................................ Elapsed Time or How Much Time Passes ................... Temperature 1 ............................................................... Temperature 2 ............................................................... Remember Fractions? ................................................... Measuring Length ......................................................... More of Measuring Length ........................................... Inches, Feet, Yards and Miles ...................................... Metric Units For Measuring Length ........................... Measuring Weight ......................................................... Measuring Weight in the Metric System ..................... Customary Units of Volume ......................................... Metric Units of Volume ................................................. Review ............................................................................

5

136 138 143 145 150 153 155 156 159 161 163 165 167 169 171 173

Foreword Math Mammoth Grade 4-A and Grade 4-B worktexts comprise a complete math curriculum for the fourth grade mathematics studies. In the fourth grade, students focus on multidigit multiplication and division, learning to use bigger numbers, solving multi-step word problems that involve several operations, and get started in studying fractions and decimals. This is of course accompanied by studies in geometry and measuring. The year starts out with review of addition and subtraction, graphs, and money. We illustrate word problems with bar diagrams and study finding missing addends, which teaches algebraic thinking. Children also learn addition and subtraction terminology, the order of operations, and statistical graphs. Next come large numbers -- up to millions, and the place value concept. At first the student reviews thousands and some mental math with them. Next are presented numbers till one million, calculations with them, place value concept and comparing. In the end of the chapter we find more about millions and an introduction to multiples of 10, 100, and 1000. The third chapter is all about multiplication. After briefly reviewing the concept and the times tables, the focus is on learning multidigit multiplication (multiplication algorithm). The children also learn why it works when they work on multiplying in parts. We also study the order of operations again, touch on proportional reasoning, and do more money and change related word problems. The last chapter in part A is about time, temperature, length, weight, and volume. Students will learn to solve more complex problems using various measuring units and to convert between measuring units. In part B, we first study division. The focus is on learning long division and using division in word problems. The geometry chapter introduces students to measuring angles, and we do lots of drawing of different shapes and circles. Area and perimeter are other important topics in geometry. Fractions and decimals are presented last in the school year. These two chapters practice only some of the basic operations with fractions and decimals. The focus is still on the conceptual understanding, building a good foundation towards 5th grade math, where fractions and decimals will be in focus. When you use these books as your only or main mathematics curriculum, they can be like a “framework”, but you do have some liberty in organizing the study schedule. Chapters 1, 2, and 3 should be studied in this order, but you can be flexible with chapters 4 (Time and Measuring) and 6 (Geometry) and schedule them somewhat earlier or later if you so wish. Chapter 3 (Multiplication) needs to be studied before long division in Chapter 5. Many topics from chapters 7 and 8 (Fractions and Decimals) can also be studied earlier in the school year; however finding parts with division should naturally be studied only after mastering division. This product also includes an HTML page that you can use to make extra practice worksheets for computation. I wish you success in your math teaching! Maria Miller, the author

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Concerning Challenging Word Problems I would heartily recommend supplementing this program with regular practice of challenging word problems and puzzles. You could do that once a week to once every two weeks. The goal of challenging story problems and puzzles is to simply develop children's logical and abstract thinking and mental discipline. Fourth grade is a good place to start such a practice because students are able to read the problems on their own and have developed mathematical knowledge in many different areas. Of course I am not discouraging people from doing such in earlier grades, either. I have made lots of word problems for the Math Mammoth curriculum. Those are for the most part multistep word problems. I have included several lessons that utilize the bar model for solving problems and tried to vary the problems. Even so, the problems I've created are usually tied to a specific concept or concepts. I feel children can also benefit from problem solving practice where the problems require “out of the box” thinking, or are puzzle-type in nature, or are just different from the ones I have made. Additionally, I feel others are more capable of making very different, very challenging problems. So I'd like for you to use one or several of the resources below for some different problems and puzzles. Choose something that fits your budget (most of these are free) and that you will like using. Math Kangaroo Problem Database Easily made worksheets of challenging math problems based on actual past Math Kangaroo competition problems. http://www.kangurusa.com/clark/pdb/ Primary Grade Challenge Math by Edward Zaccaro The book is organized into chapters, with each chapter presenting a type of problem and the ways to think about that problem. And then there is a series of related story problems to solve, divided into 4 levels. $25, ISBN 978-0967991535 You can find this at Amazon.com or various other bookstores. http://www.amazon.com/Primary-Grade-Challenge-Edward-Zaccaro/dp/0967991536/ Problem Solving Decks from North Carolina public schools Includes a deck of problem cards for grades 1-8, student sheets, and solutions. Many of these problems are best solved with calculators. All of these problems lend themselves to students telling and writing about their thinking. http://community.learnnc.org/dpi/math/archives/2005/06/problem_solving.php Math Stars Problem Solving Newsletter (grades 1-8) These newsletters are a fantastic, printable resource for problems to solve and their solutions. http://community.learnnc.org/dpi/math/archives/2005/06/math_stars_news.php

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Mathematics Enrichment - nrich.maths.org Open-ended, investigative math challenges for all levels from the UK. Find the past issues box down in the left sidebar. Use Stage 2, 1-star or 2-star problems for 4th grade. http://nrich.maths.org/public/ http://nrich.maths.org/public/themes.php lets you find problems organized by mathematical themes. Figure This! Math Challenges for Families Word problems related to real life. They don't always have all the information but you have to estimate and think. For each problem, there is a hint, other related problems, and interesting trivia. Website supported by National Council of Teachers of Mathematics. http://www.figurethis.org/ MathStories.com Over 12,000 interactive and non-interactive NCTM compliant math word problems, available in both English and Spanish. Helps elementary and middle school children boost their math problem solving and critical-thinking skills. A membership site. http://www.mathstories.com/ “Problem of the Week” (POWs) Problem of the week contests are excellent for finding challenging problems and for motivation. There exist several: z

z

z

z

z

z

Math Forum: Problem of the Week Five weekly problem projects for various levels of math. Mentoring available. http://mathforum.org/pow/ Math Contest at Columbus State University Elementary, middle, algebra, and “general” levels. http://www.colstate.edu/mathcontest/ Aunty Math Math challenges in a form of short stories for K-5 learners posted bi-weekly. Parent/Teacher Tips for the current challenge explains what kind of reasoning the problem requires and how to possibly help children solve it. http://www.auntymath.com/ Grace Church School's ABACUS International Math Challenge This is open to any child in three different age groups. http://www.gcschool.org/pages/program/Abacus.html MathCounts Problem of the Week Archive Browse the archives to find problems to solve. You can find the link to the current problem on the home page. http://mathcounts.org/Page.aspx?pid=355 Math League's Homeschool Contests Challenge your children with the same interesting math contests used in schools. Contests for grades 4, 5, 6, 7, 8, Algebra Course 1, and High School are available in a non-competitive format for the homeschoolers. The goal is to encourage student interest and confidence in mathematics through solving worthwhile problems and build important critical thinking skills. By subscription only. http://www.mathleague.com/homeschool.htm

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Chapter 1: Addition, Subtraction, Graphs and Money Introduction The first chapter of Math Mammoth Grade 4-A Complete Worktext covers addition and subtraction topics, word problems, graphs, and money problems. At first, we review the “technical aspects” of adding and subtracting: mental math techniques plus adding and subtracting in columns. If these are fairly easy for your student(s), you can choose to skip some problems. Going beyond those, the chapter includes lessons in addition and subtraction terminology. These lessons are already preparing your child for algebraic thinking. In the next lessons, the student reviews the addition/subtraction connection, and solves word problems with the help of bar models. Next, we solve simple missing addend equations using subtraction, such as x + 20 = 60. We use bar models to illustrate these and connect them with fact families. The lesson on the order of operations contains some review but it goes beyond that. In many of the problems, the student builds the mathematical expression (calculation) needed for a certain real-life situation. Going towards applications of math, the chapter contains lessons on bar graphs, line graphs, rounding, estimating, and money problems.

The Lessons in Chapter 1 page

span

Addition Review ................................................ 12

3 pages

Adding in Columns ..................................................

15

1 pages

Subtraction Review ..................................................

16

3 pages

Subtract in Columns .................................................

19

3 pages

Mental Math Workout and Pascal's Triangle ......................................................

22

3 pages

Subtraction Terms ....................................................

25

2 pages

Word Problems and Bar Models ..........................

27

3 pages

Missing Addend Solved With Subtraction ...............

30

4 pages

Order of Operations .................................................

33

2 pages

Bar Graphs ...............................................................

35

3 pages

Line Graphs .......................................................

38

3 pages

Rounding .................................................................

41

4 pages

Estimating ................................................................

45

2 pages

Reviewing Money .............................................. 47

3 pages

Review ...............................................................

1 page

50

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Helpful Resources on the Internet Calculator Chaos Most of the keys have fallen off the calculator but you have to make certain numbers using the keys that are left. http://www.mathplayground.com/calculator_chaos.html ArithmeTiles Use the four operations and numbers on neighboring tiles to make target numbers. http://www.primarygames.com/math/arithmetiles/index.htm Choose Math Operation Choose the mathematical operation(s) so that the number sentence is true. Practice the role of zero and one in basic operations or operations with negative numbers. Helps develop number sense and logical thinking. http://www.homeschoolmath.net/operation-game.php MathCar Racing Keep ahead of the computer car by thinking logically, and practice any of the four operations at the same time. http://www.funbrain.com/osa/index.html Fill and Pour Fill and pour liquid with two containers until you get the target amount. A logical thinking puzzle. http://nlvm.usu.edu/en/nav/frames_asid_273_g_2_t_4.html Estimate Addition Quiz Scroll down the page to find this quiz plus some others. Fast loading. http://www.quiz-tree.com/Math_Practice_main.html Mental Addition and Subtraction A factsheet, quiz, game, and worksheet about basic mental addition and subtraction. http://www.bbc.co.uk/skillswise/numbers/wholenumbers/addsubtract/mental/ Shop 'Til You Drop Get as many items as you can and be left with the least amount of change, and practices your addition skills. The prices are in English pounds and pennies. http://www.channel4.com/learning/microsites/P/puzzlemaths/shop.shtml Change Maker Determine how many of each denomination you need to make the exact change. Good and clear pictures! Playable in US, Canadian, Mexican, UK, or Australian money. http://www.funbrain.com/cashreg/index.html Cash Out Give correct change by clicking on the bills and coins. http://www.mrnussbaum.com/cashd.htm

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Piggy bank When coins fall from the top of the screen, choose those that add up to the given amount, and the piggy bank fills. http://fen.com/studentactivities/Piggybank/piggybank.html Bar Chart Virtual Manipulative Build your bar chart online using this interactive tool. http://nlvm.usu.edu/en/nav/frames_asid_190_g_1_t_1.html?from=category_g_1_t_1.html An Interactive Bar Grapher Graph data sets in bar graphs. The color, thickness and scale of the graph are adjustable. You can put in your own data, or you can use or alter pre-made data sets. http://illuminations.nctm.org/ActivityDetail.aspx?ID=63

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Addition Review Remember addition?

You can write any number as a SUM of the different units such as whole thousands, whole hundreds, whole tens, and ones.

5,248 = 5,000 + 200 + 40 + 8 thousands

You can add in parts:

hundreds

tens

Add in any order:

56 + 124

7 + 90 + 91 + 3

= 100 + 50 + 20 + 6 + 4

= 7 + 3 + 90 + 91

= 100 + 70 + 10 = 180

= 10 + 90 + 91 = 191

ones

Trick: add first a bigger number, then subtract to correct the error:

76 + 89 = 76 + 90 − 1 = 166 − 1 = 165

1. Add mentally. You can add in parts (tens and ones separately). a. 70 + 80 = ___

b. 140 + 50 = ___

c. 50 + 60 = ___

d. 80 + 90 = ___

77 + 80 = ___

141 + 50 = ___

54 + 65 = ___

82 + 93 = ___

77 + 82 = ___

144 + 55 = ___

58 + 62 = ___

88 + 91 = ___

2. Write the numbers as a sum of whole thousands, whole hundreds, whole tens, and ones. a. 487 =

b. 2,103 =

c. 8,045 =

d. 650 =

3. Solve the problems. a. Two of the addends are 56 and 90. The sum is 190. What is the third addend? b. Four of the addends equal 70 and five other addends equal 80. What is the sum?

12

4. Add and compare the results. The addition problems are “related”! a. 7 + 8 = ___

b. 4 + 9 = ___

c. 6 + 8 = ___

57 + 8 = ___

34 + 9 = ___

16 + 8 = ___

70 + 80 = ____

40 + 90 = ____

600 + 800 = ____

700 + 800 = ____

240 + 90 = ____

560 + 80 = ____

5. Write here four different addition problems that are “related” to the problem 5 + 8 = 13. See examples above!

6. Add in parts. a. 80 + 5 + 2 + 30 + 4 + 44

b. 127 + 500 + 4 + 3 + 9 + 90

7. Explain an easy to way to add 99 to any number. For example, explain how to do easily 56 + 99 and 487 + 99. 8. Add in parts, or use other “tricks”. a. 71 + 82 = ____

b. 42 + 47 = ____

c. 89 + 92 = ____

37 + 42 = ____

64 + 64 = ____

82 + 19 = ____

57 + 64 = ____

12 + 99 = ____

51 + 98 = ____

9. Continue the patterns. a. 600

b. 900

c. 100

d. 500

+ 600 =____

+ 900 =____

+ 75 =____

+ 45 =____

+ 600 =____

+ 900 =____

+ 75 =____

+ 45 =____

+ 600 =____

+ 900 =____

+ 75 =____

+ 45 =____

+ 600 =____

+ 900 =____

+ 75 =____

+ 45 =____

+ 600 =____

+ 900 =____

+ 75 =____

+ 45 =____

+ 600 =____

+ 900 =____

+ 75 =____

+ 45 =____

13

10. Double and halve the numbers. Half the number

10 20

Number

90

110

120

480

500

900

1,600

4,010

788

950

999

40

Its double

11. a. There are five people in the Brill family and they went to a concert. Children's tickets were $20 each and the two parents' tickets were $28 a piece. What was the total cost of the tickets for the family?

b. In another concert, adult ticket cost $30 and children's tickets were half that price. What was the total cost for the Brill family?

12. Fill in the table - add 999 each time.

n

56

69

125

156

287

569

n + 999

John is writing very simple “missing addend” problems for first graders. For example, he wrote the problem 2 + ___ = 8. The first addend is given, and the second addend is missing. John uses whole numbers from 0 on up to the number that is the sum. a. How many such problems can he write when the sum is 8? b. How many such problems can he write when the sum is 10? c. How many such problems can he write when the sum is 20? d. You should see a pattern in the above answers. Now use the pattern to solve this: How many such problems could he write when the sum is 100 (for second-graders)?

14

Adding in Columns 1. Add in columns. Check by adding the numbers in each column in different order (for example from down up). a.

b.

384 2912 2008 209 + 26

c.

$1.8 2 4 0.5 9 9.9 7 1 0.2 9 1.0 9 + 0.4 3

245 139 30 2931 594 9593 + 526

2. Write the numbers under each other carefully, and add in columns. a. 5,609 + 1,388 + 89 + 402 b. $8.05 + $0.29 + $38.40 + $293 + $203.20 + $46.49 + $94

3. The map shows some Kentucky cities and distances between them. For example, from Louisville to Frankfort is 54 miles. The one distance not marked is written below the map: from Frankfort to Lexington is 28 miles. Calculate the total driving distance, if a family goes on a field trip like this: a. Covington - Lexington - Paducah - Lexington - Covington b. A round trip from Lexington via Covington, Louisville, and Frankfort, and back to Lexington.

15

d.

1738 2390 1078 364 2803 211 + 99

Subtraction Review Marie: “I subtract in parts: first to the previous whole ten, then the rest.” Compare the methods.

35 − 7 = (35 − 5) − 2 =

30 − 2 = 28

John: “I use a helping problem.” 15 − 7 = 8 is the helping problem for 35 − 7. The answer to 35 − 7 also ends in “8” and is in the previous ten (the twenties). So, 35 − 7 is 28.

1. Subtract from whole hundreds. You can subtract in parts. a.

b.

c.

d.

100 – 2 = ____

200 – 4 = ____

500 – 5 = ____

400 – 7 = ____

100 – 20 = ____

200 – 40 = ____

500 – 50 = ____

400 – 70 = ____

100 – 22 = ____

200 – 45 = ____

500 – 56 = ____

400 – 71 = ____

2. Subtract. Use the helping problem. a.

b.

c.

d.

13 – 7 = ____

15 – 9 = ____

12 – 6 = ____

16 – 8 = ____

63 – 7 = ____

150 – 90 = ____

82 – 6 = ____

1,600 – 800 = ____

3. Subtract and compare the results. The problems are “related” – can you see how? a. 12 – 8 = ____

b. 15 – 9 = ____

c. 13 – 7 = ____

42 – 8 = ____

75 – 9 = ____

73 – 7 = ____

120 – 80 = ______

150 – 90 = ______

1300 – 700 = ______

520 – 80 = ______

650 – 90 = ______

430 – 70 = ______

4. Write here four different subtraction problems that are “related” to the problem 14 – 8 = 6. See the examples above!

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705 − 99 Trick: subtract first a bigger number, then add back some to correct the error:

140 − 88

= 705 − 100 + 1

= 140 − 90 + 2

= 605 + 1 = 606

= 50 + 2 = 52

5. Fill in the table - subtract 99 each time.

n

125

293

346

404

487

510

640

849

n – 99

Strategy: Add up to find the difference of two numbers. To solve 93 – 28, start at 28 and add until you reach 93. However much you added is the difference.

+ 2 28

+ 60 30

+ 3 90

+ 40 93

93 – 28 = (2 + 60 + 3) = 65

16 0

+ 200 2 00

+ 20 4 00

4 20

420 – 160 = (40 + 200 + 20) = 260

6. Subtract in parts, use a helping problem, add up to find the difference, or use other “tricks”. a. 91 – 82 = ______

b. 100 – 82 = ______

c. 56 – 29 = ______

42 – 37 = ______

100 – 56 = ______

61 – 39 = ______

77 – 64 = ______

96 – 48 = ______

84 – 38 = ______

d. 250 – 180 = ______

e. 1,000 – 555 = ______

f. 500 – 82 = ______

440 – 390 = ______

1,000 – 56 = ______

612 – 70 = ______

730 – 290 = ______

1,000 – 208 = ______

540 – 48 = ______

7. Fill in the table - subtract 27 each time.

n

120

140

160

180

200

n – 27

17

8. Subtract the same number repeatedly. Multiplication tables can help! a. 240

b. 1600

c. 540

– 40 =

200

– 200 = ______

– 60 = ______

d. 490 – 70 = ______

– 40 =

160

– 200 = ______

– 60 = ______

– 70 = ______

– 40 = ______

– 200 = ______

– 60 = ______

– 70 = ______

– 40 = ______

– 200 = ______

– 60 = ______

– 70 = ______

– 40 = ______

– 200 = ______

– 60 = ______

– 70 = ______

– 40 = ______

– 200 = ______

– 60 = ______

– 70 = ______

The table of 4 has a similar pattern.

The table of ____ has a similar pattern.

The table of ____ has a similar pattern.

The table of ____ has a similar pattern.

Repeated Subtraction Game! Jane and Jim are playing a repeated subtraction game. Each player has various number cards. A player pairs his cards together, two by two. With each two cards, the player subtracts the smaller number as many times as possible from the bigger number. For example, Jane pairs together cards 20 and 4. Jane subtracts 20 – 4 – 4 – 4 – 4 – 4 = 0. Jim pairs the cards 45 and 11, and subtracts 45 – 11 – 11 – 11 – 11 = 1. He can't subtract any more. Each player gets as many “points” as is the “remainder” number (the final difference). Above, Jane got 0 points and Jim got 1. The player who first accumulates 25 points loses the game. Write the subtractions that Jane does with these cards:

a.

b.

With four cards, you need to choose which two will make a pair. Pair the cards for subtractions so that you will get the least possible points. Then write the subtractions.

c.

d.

e. Play the game yourself! Try number cards from 2-30 for an easier game. Try numbers from 2 to 60 for a challenge. Give each player 4-8 cards, depending on the difficulty level you wish to have.

18

Subtract in Columns 1. This is review. Subtract in columns. Check by adding! a.

Add to check:

−

519 346

b.

Add to check:

728 − 519

+ 346

c.

Add to check:

1350 − 782

+ 519

+ 782

It is time to review borrowing over zeros! You can't subtract 3 from 0. You can't borrow a ten - there are none!

First borrow one hundred. You get 10 tens in the tens column.

Then borrow 1 ten into the ones column. Now you can subtract. 9 7 10 10

7 10

8 0 0 – 2 5 3

8 0 0 – 2 5 3

8 0 0 – 2 5 3 5 4 7

You can't borrow from the tens nor from the hundreds. So borrow 1 thousand.

Next, borrow one hundred into the tens column.

Then borrow one ten into the ones column. You're ready to subtract!

9 6 10 10

6 10

7 0 0 2 – 4 9 3 3

9 9 6 10 10 12

7 0 0 2 – 4 9 3 3

7 0 0 2 – 4 9 3 3 2 0 6 9

2. Subtract in columns. Check by adding! a.

Add to check:

−

700 356

+ 356

b.

Add to check:

5000 − 1236

+ 1236

19

c.

Add to check:

6004 − 678

+ 678

3. Subtract in columns. Check by adding! a.

Add to check:

−

506 289

+ 289

d.

5070 − 2356

Add to check:

4090 − 3785

+

9000 − 3420

Add to check:

+ 3420

f.

$80.00 − 56.70 +

h.

+

c.

+ 3785

e.

g.

4005 − 2391

b.

$600.00 − 230.50 +

i.

$400.00 −198.99 +

4. Look again at the Kentucky map. How many miles longer is a. a round trip from Lexington to Ashland and back than a round trip from Lexington to Covington and back? b. a trip from Lexington to Paducah and back than a triangular trip from Lexington via Covington, Louisville, Frankfort, and back to Lexington?

20

$109.40 − 78.65 +

7 10 11 10

8120 – 2653 – 754 = ?

8 1 2 0 –2 6 5 3

When subtracting two numbers, you can continue the subtraction under your first answer. Check by adding the answer and all the numbers you subtracted.

Check:

5 4 6 7 – 7 5 4

4 7 1 3 7 5 4 +2 6 5 3

4 7 1 3

8 1 2 0

5. Write the numbers under each other carefully, and subtract in columns. a. 4,400 − 2,745 − 493 b. 5,604 − 592 − 87 c. $45.60 − $12.36 − $1.69

6. You can solve the problem 5,200 − 592 − 87 − 345 − 99 by subtracting the numbers one at a time. That means four separate subtractions. Can you find a quicker way?

Little Hannah has almost learned to read the (analog) clock, but she can't remember which hand is the hour hand and which is the minute hand. So when the time is 1:15, she might say, “It is 3:05”, mixing the hours and the minutes. One day mom was lying in bed, sick, and she asked Hannah what time it was. Hannah said, “It is 2:20.” Just a few minutes later mom asked again for the time. Hannah claimed it was now 4:25. Remembering that each time Hannah either tells the time right, or mixes the hour and minute hands, mom was able to figure out what time it was in reality. Can you?

21

Mental Math Workout and Pascal's Triangle 1. Fill in the table - add 29 each time.

n

9

18

27

36

45

54

480

420

n + 29 2. Fill in the table - subtract 39 each time.

n

660

600

540

n – 39 3. Subtract - and be careful! a.

b.

c.

d.

500 – 3 =

600 – 2 =

300 – 3 =

1,000 – 7 =

500 – 30 =

600 – 20 =

400 – 40 =

1,000 – 70 =

500 – 300 =

600 – 200 =

500 – 5 =

1,000 – 700 =

500 – 33 =

600 – 22 =

600 – 60 =

1,000 – 77 =

500 – 303 =

600 – 202 =

700 – 7 =

1,000 – 707 =

4. Figure out the patterns and continue them.

+

5 –

1000

+

28

+

51

+

74

+

+

____

____

+

__

+

____

–

–

–

–

–

–

–

900

810

730

660

____

____

____

22

____

____

5. Continue the patterns. + 300

3,000

+ 300

___

– 400

10,000

+ 300

___ – 400

___

+

+

+

+

+

____

____

____

____

____

– 400

___

–

–

–

–

–

____

____

____

____

____

___

___

6. This will be a Pascal's triangle but you need to fill it in. On the left and right sides are ones. Any other number is gotten by adding the two numbers right above it (slightly to the right and to the left). For example, the colored number 3 comes from adding the 1 and 2 above it.

23

7. a. After filling the triangle, add the numbers in each row and make a list. For example, the first row just has 1. In the second row, add 1 + 1 = 2. In the third row, add 1 + 2 + 1 = 4. The row sums are: 1, 2, 4, ____, ____, ____, ____, ____, ____, ____, ____, ____. What do you notice about these numbers? b. Can you find a diagonal with the numbers 1, 2, 3, 4, 5, 6, 7? c. Can you find a diagonal with triangular numbers? (Triangular numbers start like this:)

Read more about Pascal's triangle and its patterns at http://ptri1.tripod.com/

Below you will find an empty Pascal's triangle to explore with. You can fill it with some other number on all the sides, such as 2, 3, or 20.

24

Subtraction Terms Remember subtraction terms? Just like “m” comes before “s” in the alphabet, the minuend comes before the subtrahend.

1. The minuend is missing! Find a general idea that always works to solve these kind of problems. a. ____ − 8 = 7

____ − 4 = 20

b. ____ − 15 = 17

____ − 24 = 48

c. ____ − 22 − 7 = 70

____ − 300 − 50 = 125

2. The subtrahend is missing! Find a general idea that always works to solve these kind of problems. a. 20 − ____ = 12

6 − ____ = 5

b. 55 − ____ = 34

100 − ____ = 72

3. a. Write three subtraction problems where the difference is 10. b. The subtrahend is 12 and the difference is 58. What is the minuend? c. The minuend is 55 and the difference is 17. What is the subtrahend? 4. Explain an easy to way to subtract 999 from any number mentally. For example, explain how to do easily 1,446 – 999.

5. The difference of two numbers is 20, and one of the numbers is 25. What can the other number be?

25

c. 234 − ____ = 100

899 − ____ = 342

Subtraction is used: z

To find the difference

z

In “less than” or “more than” situations

z

In “take away” situations

z

To find one part when you have a “whole” and several “parts”.

6. Solve the problems. You will need addition AND subtraction. a. A package of cheese costs $6 and a package of ham costs $2 less. How much do the two cost together? b. One alarm clock costs $11 and another costs $8 more. How much would the two cost together? c. Of the 45 students, 18 are girls. How many are boys? How many more boys are there than girls? d. Jack gave the clerk $50 for his purchases, and got $13 as his change. How much did his purchases cost? e. It rained five days in June and six days in July. How many non-rainy days did those two months have? f. Amy is 134 cm tall and her mom is 162 cm tall. What is the difference in their heights? g. Jack bicycled his favorite 28 km route on Tuesday and on Wednesday. On Thursday and Saturday he bicycled along a route that was 6 km shorter. How many kilometers did he bicycle all totalled?

Find the missing numbers. a. 200 − 45 − ____ − 70 = 25 b. _____ − 5 − 55 − 120 = 40

26

Word Problems and Bar Models Bar models help you see how the numbers in a problem relate to each other. Whenever you get stumped by a word problem, try drawing a bar model. On Monday, Dad drove 277 miles, and on Tuesday he drove 25 miles more than he did on Monday. How many miles did he drive in the two days?

Monday Tuesday

On Tuesday he drove 277 + 25 = 302 miles. Altogether he drove 277 + 302 = 579 miles.

The bracket “}” means addition or the total of the two bars. We do not know the total or the sum of the two days' journey, so it is marked with a question mark.

After driving 20 miles, Dad says, “I still have 15 more miles to go to the half-way point.” How long is the trip? 20 mi + 15 mi = 35 miles, and that is the first half of the trip. So, the total trip is 2 × 35 = 70 miles.

We do not know the total length, so it is marked with “?”.

Mark the numbers given in the problem in the diagram. Mark what is asked with “?”. Then solve the problem. 1. Jake worked for 56 days on a farm, and Ed worked for 14 days less. How many days did Ed work?

2. Of his paycheck, Dad paid $250 on taxes, and spent $660 on other bills and purchases. Then, half of his paycheck was gone. How much was his paycheck?

3. Dad bought two hammers. One cost $18 and the other cost $28 more. What was his total bill?

27

Angi and Rebecca split a $100 paycheck so that Angi got $10 more than Rebecca. How much did each one get? The bar diagram shows the situation. Angi got $10 more than Rebecca, and together they earned $100. To solve it, you can think this way. If you took away (subtracted) the “additional” $10, then the total would be $90, and we would only have the two equal parts (the two green parts). So, $90 ÷ 2 = $45 gives us the amount Rebecca got, and then Angi got $45 + $10 = $55. Here's another way of looking at the same situation. We draw just one bar for the paycheck, and divide it into two halves in the middle (the dashed line). Then we draw half of the $10, or $5, on either side of that middle line. We can then see Angi got $50 + $5 = $55 and Rebecca got $50 − $5 = $45.

Mark the numbers given in the problem in the diagram. Mark what is asked with “?”. Then solve the problem. 4. Mary and Luisa bought a $46 gift together. Mary spent $6 more on it than Luisa. How many dollars did each spend?

5. Henry bought two circular saws. One saw was $100 cheaper than the other. His total bill was $590. What did each saw cost?

6. Eric and Angela did yard work together. They earned $80 and split it so that Eric got $12 more than Angela. How much did each one get? Draw a bar diagram.

28

You can solve the rest of the problems any way you like best. 7. Mark bought four towels for $7 each, and a blanket for $17. He paid, and the clerk handed him back $5. What denomination was the bill Mark used to pay?

8. One plain yogurt costs $2.40, strawberry yogurt costs $0.15 less than plain yogurt, and plum yogurt costs $0.30 more than plain yogurt. What is your total bill if you buy all three?

9. Erica was 132 cm tall when she was 9 years old. In the next year, she grew 6 cm, and the next year 2 cm less than the previous year. How tall was she at the age of 11?

10. John's monthly phone service bill is $48. John said that with the money he earned on his summer job, he could pay his phone service for two months, spend $120 for a bike, and still have half his money left. How much did he earn?

11. Melissa found a nice shirt for $11.50, another for $2.55 less, and yet another for $2 less. If she buys all three, what will her total bill be?

29

Missing Addend Solved with Subtraction From this simple diagram, we can write two addition and two subtraction sentences. Those four form a fact family. x stands for a number, too. We just don't know it yet. Which fact in the family makes it easy to find the value of x?

Here is missing addend problem:

z

x + 15 = 56

z

56 – x = 15

z

15 + x = 56

z

56 – 15 = x

You can solve it by subtracting the one part (769) from the total (1,510):

769 + x = 1,510.

x = 1,510 – 769 = 741

1. The missing addend is solved with subtraction. Solve.

a. 78 + x = 145

x = 145 – 78 = _____

b. 128 + x = 400

x = ____ – ____ = _____

c. x + 385 = 999

x = ____ – ____ = _____

2. Write a missing addend sentence using x, and a subtraction sentence to solve it. a. A car costs $1,200 and dad has $890. How much more does he need?

b. The school has 547 students, of which 265 are girls. How many are boys?

30

3. a. Write a fact family using these three numbers: x, 59, 124. (Remember, x stands for a number too.) b. Solve for x.

4. Write a missing addend sentence with x. Solve. a. A school's teachers and students filled a 450-seat auditorium. If the school had 43 teachers, how many students did it have?

students + teachers = total ______ + ______ = ______ x=

b. Mom went shopping with $250 and came back home with $78. How much did she spend?

spent + left = had originally ______ + ______ = ______ x=

c. Janet had $200. She bought an item for $54 and another for $78. How much is left?

item 1 + item 2 + left = total ______ + ______ + ______ = ______ x=

d. Jean bought one item for $23 and another for $29, and she had $125 left. How much did she have initially?

______ + ______ + ______ = ______ x=

5. Which number sentence fits the problem? Find x. a. Jane had $15. Dad gave Jane her allowance (x) and afterwards Jane had $22. $15 + x = $22

OR

b. Mike had many drawings. He put 24 of them in the trash. Then he had 125 left.

$15 + $22 = x

125 – 24 = x

c. Jill had 120 marbles, but some of them got lost. Now she has 89 left. 120 – x = 89

OR

OR

x – 24 = 125

d. Dave gave 67 of his stickers to a friend and now he has 150 left.

120 + 89 = x

150 – 67 = x

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OR

x – 67 = 150

6. Pick a number sentence that you can use to find x. Then solve for x. a. Problem: 253 + x = 2056 2056 – 253 = x

OR

b. Problem: x + 148 = 397 x – 253 = 2056

148 – 397 = x

c. Problem: x – 23 = 45 45 – 23 = x

OR

OR

397 – 148 = x

d. Problem: 120 – x = 55 45 + 23 = x

120 – 55 = x

OR

120 + 55 = x

7. Solve for x. b. 23 + 56 + x = 110

x 1,750 |⎯⎯⎯⎯⎯⎯ 4,900 ⎯⎯⎯⎯⎯| a.

8. Write the numbers and x to the picture. Write a missing addend sentence. Solve. a. The Jones' family had traveled 420 miles of their 1,200-mile journey. How many miles were left to travel?

b. The store is expecting a shipment of 4,000 blank CDs. Two boxes of 500 arrived. How many are still to come?

|⎯⎯⎯⎯⎯⎯|⎯⎯⎯⎯⎯⎯⎯⎯|

|⎯⎯⎯|⎯⎯⎯|⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯|

c. A 250 cm board is divided into three parts: two 20 cm parts at the ends and a part in the middle. How long is the middle part?

d. After traveling 56 miles, Dad said, “We have 118 miles left.” How long is the journey?

|⎯⎯|⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯|⎯⎯|

|⎯⎯⎯⎯⎯⎯|⎯⎯⎯⎯⎯⎯⎯⎯|

32

Order of Operations 1. Do operations within ( ) first. 2. Then multiply & divide, from left to right.

30 − 6 − 11 + 5 = 24 − 11 + 5 = 13 + 5 = 18

4 + 3 × (6 − 2) = 4 + 3× 4 = 4 + 12 = 16

7+3×5 = 7 + 15 = 22

70 + (80 − 5) = 70 + 75 = 145

3. Then add & subtract, from left to right. Make sure you understand the examples on the right.

1. Do the calculations in the right order. a. 500 – 30 – 30 =

b. 250 + (100 – 50) + (100 – 50) =

500 – (30 – 30) =

250 + 100 – 50 + 100 – 50 =

500 – 30 + 30 =

(250 + 100) – (50 + 100) – 50 =

500 – (30 + 30) =

250 + 100 – (50 + 100 – 50) =

2. Calculate in the right order. a. 2 × (5 + 3) =

b. 2 × 5 + 3 × 1 =

c. 2 × 5 + 3 × 0 =

20 – 3 × 3 =

(10 – 3) × 3 + 1 =

(20 – 16) × 3 + 2 =

50 – 1 – 2 × 10 =

50 – 1 × 7 + 2 × 3 =

2 × (2 + 2) – 3 =

3. Match the description with the right number sentence. Then calculate. First multiply 5 times 10 and subtract from the result 7.

5 × (10 – 7)

Add to 10 the difference of 100 and 20.

5 × 10 – 7

First subtract 7 from 10, and then multiply the result by 5.

100 – (20 + 10)

From 100 subtract the sum of 20 and 10.

(100 – 20) + 10 90 – 20 + 20

4. You cut off two 20-cm pieces of a 90-cm piece of wood. Which calculation tells you the piece that is left?

90 – 2 × 20 (90 – 20) × 2

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5. A clerk in the store rings up all the items the customer buys, gets the customer's money, and figures out the change. a. Which of the calculations on the right best matches figuring out the change?

i. $50 – $1.26 – $6.55 – $0.22 – $5 ii. $50 + $1.26 + $6.55 + $0.22 + $5

b. Which calculation of the three would give you the wrong answer for the change?

iii. $50 – ($1.26 + $6.55 + $0.22 + $5)

6. Describe a shopping situation where you need to do these calculations:

a. $10 + $2.10 + $45

b. 4 × $1.20

c. $10 – 4 × $1.20

7. Put operation symbols +, – , or × into the number sentences so that they become true. b.

a.

4

1

8 = 12

2

50

5

10 = 0

100

10 (15

1

c.

2 = 14 17)

1 = 68

3

3

(2

5)

3=6 2 = 14

8. Every day, James feeds the kennel dogs 5 kg of dog food. He bought a 100-kg bag of dog food. How many kilograms are left after four days? Write a single number sentence to solve that.

9. Parking costs $2 per hour during the day and $3 per hour during the night. Write a single number sentence that tells you the cost of parking a car for 5 daytime hours and 2 nighttime hours. Solve it.

10. Write a single number sentence that tells you the change if you buy a book for $7, a ball for $5, and pay with a $20 bill.

See also the Choose Two Operations game at http://www.homeschoolmath.net/operation-game.php

34

Bar Graphs 1. Beverly asked her classmates how many hours they watch the TV each day. The results are below; she already organized them in order. 001111111111122223333444556 Each number above is someone's answer to Beverly's question. So two people answered that they watched TV for 0 hours. Quite a few answered that they watch TV about 1 hour per day. With such a bunch of numbers, we need to make first a frequency table. In a frequency table, we count how frequently or how often a certain number was in our list of data. After counting all that, we can make a bar graph. In Beverly's data above, the number zero (0 hours TV) appeared two times. The number two (2 hours TV) appeared four times. Finish the frequency table and the bar graph.

Hours of TV Frequency

0h

2

1h 2h

4

b. How many classmates did Beverly question? c. What was the most common response to Beverly's question? d. How many of these kids watch TV 1 hour or less? e. How many kids watch TV 3 hours or more? f. Are there more kids who watch TV 3 hours a day than kids who watch TV 2 hours a day? g. Are there more kids watching TV 2 hours or more, than kids watching TV less than 2 hours?

35

2. a. Beverly also asked some people about their favorite color. Make a bar graph. Color Frequency red

2

orange

1

yellow

4

green

5

blue

7

purple

4

black

2

white

2

b. How many people did Beverly question? c. Were the “warm” colors or the “cold” colors more popular? (Warm colors are red, orange, and yellow. Cold colors are green, blue, and purple.) 3. The numbers are students' quiz scores. 1 3 5 3 6 4 9 8 6 4 8 7 5 3 9 8 6 2 1 8 9 10 2 9 7 6 a. Make a frequency table and a bar graph. Test score Frequency

b. What was the most common quiz score?

How many students got that score?

c. What was the least common quiz score?

How many students got that score?

d. How many students got a score from 5 to 8? e. How many students did excellent (got a score of 9 or 10)? f. The teacher said after the test, “Anyone with a score of 4 or less will need to retake the test, and anyone with a score of 5 or 6 will get extra homework.” How many students need to do the test again? How many will get extra homework?

36

4. a. Make a bar graph out of the data in the frequency table on the right.

Height in cm Number of people

Height in cm Number of people

120...129

4

160...169

95

130...139

10

170...179

61

140...149

41

180...189

39

150...159

82

190...199

6

b. How many people were short (less than 140 cm tall)? c. How many were very tall (180 cm or taller)? d. Most adults are 160 cm tall or taller. Use this fact to guess (estimate) how many children and how many adults were in this group. e. Could this data come from z

a group of elementary school children?

z

a group of people who were at the swimming pool at 5 pm on a certain Tuesday?

z

a group of elderly women in an old people's home? Explain your reasoning.

37

Line Graphs A line graph shows how something changes over time, such as over several hours, days, weeks, months, or years. The data values are often drawn as dots. Then the dots are connected with lines. The x-axis and the y-axis are the two lines that frame the picture. The time units are written under the x-axis. To read a line graph, look “up” from the time unit until you find the dot. Then draw an imaginary line from that dot to the y-axis For example, in July Amy had saved $90. 1. Look at the line graph about Amy's savings. a. How many dollars had Amy saved in May? b. How many dollars had Amy saved in August? c. How many dollars had Amy saved in September? d. In which month had she saved up $75? e. In September Amy used up her savings to buy a used bike. How much did the bike cost? 2. The graph shows a puppy's weight for 10 days after birth. Notice how the two axes are named as “day” and “grams”. a. About how many grams did the puppy weigh on day 1? ________ Day 2? ________ Day 3? ________ Day 4? ________

b. What is the first day that the puppy weighed 600 g or more? c. What is the first day that the puppy weighed 700 g or more?

38

3. Look at the graph about the monthly retail prices of strawberries in 2004, given in dollars per pound. The retail price is the price you see in a grocery store or the price the customers pay.

a. Describe the price changes as the year progresses. Do you know why the price is lower in the summer? b. Find the highest price per pound and the lowest price per pound. What is the difference of these two? c. How much did it cost to buy 2 lb of strawberries in August? In November?

4. Becca's mom wrote down an “x” mark for every bad behavior she did during the day. The table shows the list of her x-marks. a. Make a line graph. Remember to name one axis as “days” and the other as “x-marks”. b. Did Becca's behavior improve?

Day

x-marks

Mon

10

Tue

8

Wed

9

Thu

6

Fri

3

Sat

4

Sun

2

39

Month

Price ($ per lb)

Jan

2.48

Feb

2.33

Mar

2.12

Apr

1.66

May

1.67

Jun

1.85

Jul

1.63

Aug

1.82

Sep

1.84

Oct

2.60

Nov

3.19

Dec

3.60

5. The table gives the average maximum temperatures for each month in New York.

Month

Max. Temp.

Month

Max. Temp.

Month

Max. Temp.

Jan

3°C

May

20°C

Sep

26°C

Feb

3°C

Jun

25°C

Oct

21°C

Mar

7°C

Jul

28°C

Nov

11°C

Apr

14°C

Aug

27°C

Dec

5°C

a. Make a line graph. Three values are already done for you. b. What are the coldest months? c. What are the warmest months? d. What is the difference in maximum temperature between the coldest and the warmest month?

6. Do a line graph from some data that you gather yourself! Just remember, it has to be something that changes over time. You can also “make up” data from your own head. Here are some ideas: z

outside temperature from the morning till the evening

z

your savings in the past 6 months, or an imaginary child's savings in 6 or 8 or 12 months

z

how many hours of schoolwork (or housework or playing etc.) you do each day of the week

z

how many pages of a book you read each day of the week

z

your height from year 0 to year 9 of your life

You can also use this neat online tool for creating your graph: http://nces.ed.gov/nceskids/createagraph/ To use it, you need to have your data ready. It will not give you any data. It just draws the graph.

40

Rounding When you are rounding to the nearest ten, look at the ONES DIGIT. z z z

If the ones digit is 0, 1, 2, 3, or 4, then round down. If the ones digit is 5, 6, 7, 8, or 9, then round up. If you round up, the tens digit increases by one.

When the number is exactly in the middle, round up. 85 ≈ 90.

The sign “ ≈ ” is read “is about”, or “is approximately”.

(This is just a convention.)

You can draw a line after the digit whose place you are rounding to. The digit or digits after the line will become zeros. 25 6 ≈ 26 0 (up)

8 4 ≈ 8 0 (down)

3,28 7 ≈ 3,29 0 (up)

9,85 4 ≈ 9,85 0 (down)

Notice carefully: If you are rounding up, and the tens digit is already 9, look at the two digits just before your line, and increase that “number” by one: 3,29 7 ≈ 3,30 0 (up) It is as if the “29” formed by the hundreds and tens changes into “30” - exactly one more. (In reality it is “29” tens changing to “30” tens.)

79 5 ≈ 80 0 (up)

3,09 8 ≈ 3,10 0 (up)

The “79” changes to “80”.

The “09” changes to “10”.

1. Round the numbers to the nearest ten. The number line can help.

a. 294 ≈ ______

b. 315 ≈ ______

c. 278 ≈ ______

d. 285 ≈ ______

e. 315 ≈ ______

f. 296 ≈ ______

g. 304 ≈ ______

h. 207 ≈ ______

2. Round these numbers to the nearest ten. a. 526 ≈ ______

d. 197 ≈ ______

g.

b. 34 ≈ ______

e. 705 ≈ ______

h. 5,971 ≈ ______

k. 2,282 ≈ ______

c. 181 ≈ ______

f. 392 ≈ ______

i. 9,568 ≈ ______

l. 4,003 ≈ ______

41

440 ≈ ______

j. 4,061 ≈ ______

Find the whole hundred that is nearest to 539. Rounded to the nearest hundred, 539 ≈ _________. When you are rounding to the nearest hundred, look at the TENS DIGIT. z z z z

If the tens digit is 0, 1, 2, 3, or 4, then round down. If the tens digit is 5, 6, 7, 8, or 9, then round up. The rounded result is a whole hundred so it ends in two zeros. The hundreds digit changes by one if you round up.

You can draw a line after the digit whose place you are rounding to. The digits after the line will become zeros. 5 62 ≈ 6 00

2 48 ≈ 2 00 (down)

1,2 90 ≈ 1,3 00 (up)

5,4 28 ≈ 5,4 00 (down)

Notice carefully: If you are rounding up, and the hundreds digit is already 9, look at the two digits just before your line, and increase that “number” by one: 5,9 92 ≈ 5,5 00 (up) It is as if the “59” formed by the thousands and hundreds changes into “60” - exactly one more.

6,9 71≈ 7,0 00 (up)

12,9 61≈ 13,0 00 (up)

The “69” changes to “70”.

The “29” changes to “30”.

3. Round the numbers to the nearest hundred.

a. 3,520 ≈ ______

b. 3,709 ≈ ______

c. 3,935 ≈ ______

d. 3,541 ≈ ______

e. 3,962 ≈ ______

f. 3,425 ≈ ______

g. 3,847 ≈ ______

h. 3,656 ≈ ______

4. Round these numbers to the nearest hundred. a. 526 ≈ ______

d. 197 ≈ ______

g. 2,907 ≈ ______

j. 3,032 ≈ ______

b. 54 ≈ ______

e. 706 ≈ ______

h. 5,971 ≈ ______

k. 2,959 ≈ ______

c. 761 ≈ ______

f. 365 ≈ ______

i. 7,543 ≈ ______

l. 4,014 ≈ ______

42

Rounded to the nearest thousand, 4,772 ≈ ________. When you are rounding to the nearest thousand, look at the HUNDREDS DIGIT. z z z z

If the hundreds digit is 0, 1, 2, 3, or 4, then round down. If the hundreds digit is 5, 6, 7, 8, or 9, then round up. The rounded result is a whole thousand so it ends in three zeros. The thousands digit changes by one if you round up.

You can draw a line after the thousands digit. The digits after the line will become zeros. 2, 723 ≈ 3,000 (up)

9, 804 ≈ 10,000 (up)

7 288 ≈ 7,000 (down)

457 ≈ 0 (down)

5. Round the numbers to the nearest thousand.

a. 3,520 ≈ ______

b. 6,709 ≈ ______

c. 5,499 ≈ ______

d. 7,230 ≈ ______

e. 2,800 ≈ ______

f. 4,087 ≈ ______

g. 3,602 ≈ ______

h. 4,555 ≈ ______

6. Round these numbers to the nearest thousand. a. 526 ≈ ______

d. 4,197 ≈ ______

g. 2,907 ≈ ______

j. 9,605 ≈ ______

b. 54 ≈ ______

e. 5,672 ≈ ______

h. 5,502 ≈ ______

k. 2,553 ≈ ______

c. 761 ≈ ______

f. 3,099 ≈ ______

i. 9,397 ≈ ______

l. 1,047 ≈ ______

7. Round these numbers to the nearest ten, nearest hundred, and nearest thousand.

n

55

2,602

9,829

3,199

rounded to nearest 10 rounded to nearest 100 rounded to nearest 1000

43

495

709

5,328

The rounding rules remain the same even with money amounts. Rounding to the nearest dollar, look at the ten-cents DIGIT (tenth of a dollar). z z z

Rounding to the nearest ten dollars, look at the dollars DIGIT (ones digit).

If it is 0, 1, 2, 3, or 4, then round down. If it is 5, 6, 7, 8, or 9, then round up. The rounded result is in whole dollars so omit the decimal point and the cents.

z z z

If it is 0, 1, 2, 3, or 4, then round down. If it is 5, 6, 7, 8, or 9, then round up. The ones digit becomes zero. Omit the decimal point and the cents.

$12. 72 ≈ $13

$59. 92 ≈ $60

$4 7.26 ≈ $50

$56 2.94 ≈ $560

$452. 34 ≈ $452

$3,480. 55 ≈ $3,481

$39 5.60 ≈ $400

$4,53 9.50 ≈ $4,540

8. Round these numbers to the nearest dollar. a. $3.17 ≈ ______

b. $97.99 ≈ ______

c. $3.29 ≈ ______

d. $1,680.25 ≈ ______

e. $47.38 ≈ _____

f. $125.59 ≈ ______

g. $13.70 ≈ ______

h. $977.50 ≈ ______

9. Round these numbers to the nearest ten dollars. a. $45.70 ≈ ______

b. $7.99 ≈ ______

c. $73.78 ≈ ______

d. $6,289.40 ≈ ______

e. $43.27 ≈ ______

f. $169.49 ≈ ______

g. $255.55 ≈ ______

h. $564.00 ≈ ______

10. Round these numbers to the nearest one, nearest ten, and nearest hundred. n

$129.78

$455.09

$69.42

$591.95

rounded to nearest dollar rounded to nearest ten dollars rounded to nearest hundred dollars

11. Round the prices, and use the rounded prices to estimate the total bill. a. pencils $2.28, paper $5.90, notebook $4.76, books $12.75. b. Chairs $126.70, table $195.99, bed $256, mattresses $346.60.

44

$1,285.38

$6,089.90

Estimating You can estimate the result of a calculation. Round the numbers, and then calculate (add or subtract) using the rounded numbers. Your result is not exact. That is why it's called an estimate. Use the symbol “≈” (is approximately) instead of equality “=” when you change from exact numbers to rounded numbers.

567 + 89 – 413

$4.12 + $27.90 + $5.99

≈ 600 + 100 – 400 = 300.

≈ $4 + $28 + $6 = $38.

1. First estimate by rounding the numbers to the nearest hundred. Then find the exact answer. a. 967 + 231 + 4,792 Estimation:

b. 320 + 405 + 587 Estimation:

c. 1,029 – 372 – 105 Estimation:

d. 3,492 – 1,540 – 211 Estimation:

2. The table lists the costs of running the student recess time snack bar. Estimate the total cost over these five weeks by rounding the numbers to the nearest ten. Week 37 Week 38 Week 39 Week 40 Week 41 $147

$164

$182

$129

$131

45

3. Mary's family is going to rent an apartment for a 3-week vacation. They have two choices: one apartment costs $289 per week, and the other costs $327 per week. a. Estimate the cost of each one. b. How much approximately would the family save by choosing the cheaper rental?

4. Solve these problems with estimation. You don't need to find the exact answer! a. Each bus can take 47 passengers. About how many passengers are in four buses?

b. A gallon of gas is $2.87. How many gallons can you get with $20?

c. A book is $4.87 and another is $6.95. What is the total approximately?

d. You have $10. How many ice creams could you buy that cost $1.97 each?

5. The chart lists the number of loans that Charleston library had on the weeks of May and June. From this chart, you cannot read the exact numbers of loans, but you can find the approximate numbers of loans. Estimate to the nearest ten, the total number of loans for a) weeks 18-21 b) weeks 22-25.

46

Reviewing Money 1. Write the dollar amounts as cents or vice versa. a. $0.25 = _____ ¢

b. $0.70 = ______ ¢

c. $1.25 = _______ ¢

d. $5.60 = ______ ¢

e. $31.55 = ______ ¢

f. $_______ = 76¢

g. $_______ = 20¢

h. $______ = 154¢

i. $_______ = 859¢

j. $______ = 419¢

k. $80.34 = _______¢

l. $_______ = 104¢

2. Round to the nearest dollar. a. $1.05 ≈ ______

b. $7.72 ≈ ______

c. $35.17 ≈ ______

d. $165.83 ≈ _______

e. $94.90 ≈ ______

f. $99.09 ≈ ______

g. $99.90 ≈ ______

h. $100.56 ≈ _______

3. You bought items for $1.50, $12, and for $2.20. You paid with a 20-dollar bill. How much was your total? How much was your change? 4. Make change. Mark how many of each bill/coin you need. Item cost

Money given

a. $56

$70

b. $29

$50

c. $78

$100

d. $129

$200

Change needed

$5 bill

$20 bill

$50 bill

$1 bill

5. Make change. Mark how many of each bill or coin you need. Item cost

Money Change given needed

a. $2.56

$5

b. $8.94

$10

c. $7.08

$10

d. $3.37

$10

$5 bill

$1 bill

47

25¢

10¢

5¢

1¢

6. Solve. Write a number sentence for each problem. a. Mike had $99. He spent $34 , and he has $56 left.

a. $99 − $34 = $56.

b. Dad had ______. He spent $250, and has $170 left.

b.

c. Mom had $280. She spent $45, and now has ______ left.

c.

d. Greg bought a $45-tool and now he has $15 left. Originally he had _______.

d.

e. Alice had $12. She bought an item, and now she has $3.56. The item cost ________.

e.

7. Match the situations (a), (b), and (c) with number sentences (i), (ii), and (iii). Then solve for the unknown number x in each situation. a. Andy had $60 and he bought a tool set for $48. How much does he have left?

i. $60 − x = $48

b. Elisa bought food for $60 and now has $48 left. How much money did she have initially?

ii. $60 − $48 = x

c. Greg had $60 when he went to the store. He came back home with $48. How much did he spend?

iii. x − $60 = $48.

8. Solve the word problems. a. Mike had $38, and after Grandma's gift, he had $158. How much did Grandma give him?

b. Ashley spent half of her $88 in town. How much does she have now?

c. Greg bought two $15 books with his birthday money ($60). How much did he have left?

d. Jill bought three $4 magazines with her birthday money, and now she has $28. How much was the birthday money?

e. You bought 4,000 marker pens at $0.98 each, and 1,000 whiteboard erasers at $1.02 each, Estimate the total using rounded numbers.

f. Dad bought a $0.60 ice cream cone for each of the three kids, and an $0.80 ice cream cone for himself. How much was the total? What was his change from $10?

48

Discounts Often the store lowers the price of an item. That is called discounting the item. If a shirt first costs $10, and the store then puts a new price of $9 on it, the shirt is discounted by $1. The discount is how many dollars the price changed. This time the discount was $1. A TV costs $650. Now it is discounted by $100.

A flower vase was discounted by $2.10. The new price is $6.

The new price is $650 − $100 = $550.

Add to find the original price, which is of course higher: $6 + $2.10 = $8.10

9. How much is the discount, the new price, or the original price?

a. Old price $5.25 New price $4.50 Discount ______

b. Old price $1.56 New price $1.32 Discount ______

e. A jacket cost $54.99 at first; the new price is $47.95. How much is the discount?

c. Before $500 / month Now _____ / month Discount $23

d. Before $______ Now $29.50 Discount $5.50

f. A $1,199 TV-set has a $200 discount. What is the new price?

10. The chart lists some Disney World ticket prices. For each ticket there is an adult and child price, normal (gate) price and discount price. Ticket type Normal price Discount price

1 Day Adult costs $______ more than 1 Day Child.

4 day Adult

$235

$225.31

2 Day Adult costs $______ more than 2 Day Child.

4 day Child

$200

$193.38

3 day Adult

$221

$218.73

3 day Child

$189

$186.81

2 day Adult

$165

$162.20

2 day Child

$143

$141.70

1 Day Adult

$103

$103

1 Day Child

$92

$92

For 4 Day Adult ticket, the discount is __________. For 4 Day Child ticket, the discount is __________. For _________________ and _________________ tickets, there is no discount.

11. You're a family of 2 adults and 2 children. a. How much would it cost for your family to spend 2 days in Disney World using the discount tickets? b. Can you spend three days there if you can afford to spend $800 at the most?

49

Review 1. a. Write a subtraction problem where the difference is 15 and the minuend is 100. b. Write an addition problem where one addend is 339 and the sum is 2,193 2. Solve x + 283 = 1,394.

3. Amanda and Abigail weeded a garden together, and shared the pay so that Amanda got $50 more than Abigail, because she spent more time in weeding it. If their total pay was $300, how much did Amanda get and how much did Abigail get?

4. Calculate in the right order. a. 5 × (2 + 4)

(50 – 20) × 2 + 10

b. 120 – 20 – 2 × 0

5×3+2×7

c. (80 – 44) + (80 – 34)

10 × (4 + 4) – 4

3 × $13 – $2

5. Which expression matches the problem? Find the cost of three $13-hammers when they are discounted by $2.

$13 – 3 × $2 ($13 – $2) × 3

6. How many feet do ten dogs and 20 chickens have in total? Write a single number sentence to solve.

7. After spending $15.20 on food and $34.60 on gasoline, Mom had $70.20 left in her purse. How much did she have originally?

8. Alberto bought two pairs of skis; one cost $48.90 and the other cost $25 more. What was his total cost?

50

Chapter 2: Place Value Introduction The second chapter of Math Mammoth Grade 4-A Complete Worktext covers large numbers (up to 9 digits) and place value concepts with those. The first lessons only deal with thousands or numbers with a maximum of four digits. These are for review and for deepening the student's understanding of place value. It is crucial that the student understands place value with these numbers before moving on to larger numbers. Yet again, these larger numbers can be very easy as long as the student understands the basics of how our place value system works. Besides the concept of place value, the chapter contains lessons on comparing numbers, adding and subtracting in columns, mental math problems, and the idea of multiples.

The Lessons in Chapter 2 page

span

Thousands .......................................................

53

3 pages

At the Edge of Whole Thousands ...................

56

2 pages

More Thousands .............................................

58

2 pages

Practicing with Thousands ..............................

60

2 pages

Place Value with Thousands ..........................

62

2 pages

Comparing with Thousands ............................

64

3 pages

Adding & Subtracting Big Numbers ............... 67

5 pages

A Little Bit of Millions ...................................

72

3 pages

Multiples of 10, 100 and 1000 ........................

75

2 pages

Review ...........................................................

77

2 pages

51

Helpful Resources on the Internet Place Value Payoff Match numbers written in standard form with numbers written in expanded form in this game. http://www.quia.com/mc/279741.html Megapenny Project Visualizes big numbers with pictures of pennies. http://www.kokogiak.com/megapenny/default.asp Keep My Place Fill in the big numbers to this cross-number puzzle. http://www.mathsyear2000.org/magnet/kaleidoscope2/Crossnumber/index.html Place value puzzler Place value or rounding game. Click on the asked place value in a number, or type in the rounded version of the number. http://www.funbrain.com/tens/index.html Estimation at AAA Math Exercises about rounding whole numbers and decimals, front-end estimation, estimating sums and differences. Each page has an explanation, interactive practice, and games. http://www.aaamath.com/B/est.htm Can you say really big numbers? Enter a really big number, try say it out loud, and see it written. http://www.mathcats.com/explore/reallybignumbers.htm

52

Thousands one (o)

Look at the pictures. How many... z z

ten (t)

z

ones go to a ten? ____ tens go to a hundred? ____ hundreds go to a thousand? ____

That is why our way of writing numbers is called the base ten system. hundred (h)

thousand (th)

th h t o 7,284 has

7 thousands, 2 hundreds, 7 2 8 4 8 tens, and 4 ones.

Writing the number 5,608 in expanded form means we write out the number as a sum of whole thousands, whole hundreds, whole tens, and ones. You see all of it right from the number: z

It has 5 thousands = 5,000.

z

It has 6 hundreds = 600

z

It has 0 tens = 0.

z

It has 8 ones = 8.

Now write it as a sum: 5,608 = 5,000 + 600 + 0 + 8

1. Write the numbers in expanded form. a. 8,325 = 8000 + 300 + 20 + 5

b. 4,935 =

c. 4,039 =

d. 3,002

e. 2,090 =

f. 9,405

2. Write in normal form. a. 4000 + 500 + 90 + 3

b. 2000 + 90

c. 3000 + 200

d. 8000 + 5

e. 1000 + 80 + 7

f. 5000 + 600 + 9

g. 6 hundred 4 thousand

h. 8 tens 4 thousand

i. 3 ones 7 thousand 2 hundred

j. 4 hundred 5 ones 1 thousand

k. fifty, 7 thousand

l. 4 thousand, 5

m. 9, sixty, 4 thousand

n. 8 hundred, 3 thousand, 9

53

The 7, 2, 8, and 4 are called digits of the number 7,284. But 7 in the number 7,284 actually means seven thousand. The value of the digit 7 is 7,000. The 2 in the number 7,284 actually means two hundred. The value of the digit 2 is 200. The value of the digit 8 is eighty or 80. The value of the digit 4 is four. The value of the digit depends on WHERE it is in the number. Look where NINE is in these numbers: 690

“9” in 690 means ninety

The value of the digit “9” is 90.

“9” is in tens place.

9,055 “9” in 9,055 means nine thousand The value of the digit “9” is 9,000. “9” is in thousands place. 419

“9” in 419 means just nine.

The value of the digit “9” is 9.

1,970 “9” in 1,970 means nine hundred. The value of the digit “9” is 900.

“9” is in ones place. “9” is in hundreds place.

In other words, the value of the digit 9 depends on where it's at, or where its place is. That is why this system of writing numbers is called the place value system. If nine is in the hundreds' place, then its value is 900 (for example in number 5,900). If nine is in the tens place, then its value is 90 (for example in number 498). 3. What is the value of the digit 5 in the following numbers? a. 3,859 fifty

b. 65

c. 549

d. 2,506

e. 5,012

f. 3,050

4. Write the value of the underlined digit. a. 509 five hundred

b. 9,843

c. 940

d. 2,088

e. 1,200

f. 4,002

g. 7,008

h. 405

i. 4,400

j. 90

5. a. What is the largest possible number you can build by using the digits 2, 5, 8, and 4? b. What is the least possible number you can build by using them? 6. What is the difference between the largest and the least possible number you can build using the digits 6, 9, and 1?

54

What is 4,769 + 10?

4,769 has 6 tens. One ten more means there will be 7 tens: 4,779.

What is 2,958 + 100? 2,958 has nine hundreds. One hundred more means there will be 10 hundreds, but that makes a thousand. Our answer number will have 3 thousands, with no hundreds: 3,058.

7. Fill in the table - add 10, 100, or 1000. If in doubt, you can add in columns.

n

1,056

2,508

342

4,009

59

6,980

723

n + 10 n + 100 n + 1000 8. What is missing?

a. 4,036 = 4000 + ____ + 30

b. 483 = 80 + 3 + ______

c. 9,328 = 300 + 9,000 + ____ + 20

d. 8,005 = 5 + ______

e. 5,320 = 20 + _____ + 300

f. 7,609 = 9 + ______ + 7,000

9. If you add 1 thousand, 1 hundred, 1 ten, and 1 to this number, it becomes 9,000. What is the number?

For the digits given, build the largest and the least possible number you can. Then find their difference. In which multiplication table can you find each of the differences? a. 7 and 5

b. 2 and 9

c. 4 and 5

d. 8 and 3

75 and 57 difference: 18 Do the same as above, but now with three digits. For each difference you find, add its digits. If you then get a two-digit number, add its digits as well. What do you notice? e. 7,1, 5

f. 9, 4, 7

g. 8, 9, 7

751 and 157 difference 594 5 + 9 + 4 = 18 1+8=9 You can also try the same with four digits!

55

h. 4, 1, 8

8,299

At the Edge of Whole Thousands Just one is missing from thousand:

Ten is missing from thousand:

999 + 1 = 1,000

990 + 10 = 1,000

1. How much is missing from thousand? Write an addition sentence.

a. ______ + ____ = 1,000

b. ______ + ____ = 1,000

c. ______ + ____ = 1,000

a. ______ + ____ = 1,000

b. ______ + ____ = 1,000

c. ______ + ____ = 1,000

2. We have 900-something. Complete a thousand. a. 999 + 1 = 1,000

b. 980 + ____ = ______

c. 930 + ____ = ______

992 + ___ = ______

985 + ____ = ______

937 + ____ = ______

a. 1,920 + ____ = ______

b. 1,990 + ____ = ______

c. 6,950 + ____ = ______

1,999 + ____ = ______

7,940 + ____ = ______

4,900 + ____ = ______

2,998 + ____ = ______

5,970 + ____ = ______

3,995 + ____ = ______

3. Complete the next whole thousand.

56

4. Subtract from whole thousands. a. 2,000 − 1 = 1,999

b. 5,000 − 3 = ______

c. 6,000 − 50 = ______

2,000 − 4 = ______

4,000 − 10 = ______

9,000 − 30 = ______

2,000 − 7 = ______

7,000 − 20 = ______

10,000 − 100 = ______

Mental math trick: Add up to find the difference to the next whole thousand. First fill the next whole ten, the next whole hundred, and then the next whole thousand. + 8

6,7 82

+ 10

6,7 90

+ 200

6,8 00

+ 50

7,0 00

5,7 50

6,782 + 218 = 7,000

+ 200

5, 800

6,0 00

5,750 + 250 = 6,000

5. Round the numbers to the nearest thousand, and write down the rounding error. That is the difference between the number and the rounded number. Number Rounded number Rounding error

Number Rounded number Rounding error

4,993

8,029

7,890

5,113

9,880

2,810

6. Solve. Use the top problem to help you in the bottom ones. a. 2,000 − 100 = ______

b. 5,000 − 200 = ______

c. 9,000 − 500 = ______

2,000 − 150 = ______

5,000 − 230 = ______

9,000 − 580 = ______

2,000 − 250 = ______

5,000 − 280 = ______

9,000 − 680 = ______

7. Mark bought a computer for $1997 and a monitor for $995. a. Estimate his total bill in whole thousand dollars. b. How many dollars short of that estimate is the exact bill? 8. What is the rounding error, if the sum 1,982 + 3,950 is rounded to 6,000?

57

More Thousands

On this number line you see whole thousands from one thousand till fifteen thousand. 7 8,0 0 0

Read: 78 thousand

The colored digits are the “thousands period” and count as the whole thousands. Read the numbers is as if you say the word “thousand” for the comma.

1 5 3,0 0 0

Read: 153 thousand

8 0 2,0 0 0

Read: 802 thousand

We continue with whole thousands until reaching a thousand thousands.

9 9 0,0 0 0

Read: 990 thousand

9 9 9,0 0 0

Read: 999 thousand

That number has a new name: one million.

1,0 0 0,0 0 0 Thousand thousand = 1 million

1 7,5 4 4 6 0 9,2 3 0 7 0,0 8 0 9 0 2,0 0 5

The rest of the digits tell us our hundreds, tens, and ones just like you have learned.

Read: 17 thousand 544 Read: 609 thousand two hundred thirty Read: seventy thousand eighty Read: 902 thousand five

1. Place a comma into the number. Fill in missing parts. a. 1 6 4 0 0 0

b. 9 2 0 0 0

c. 3 0 9 0 0 0

d. 3 4 0 0 0

e. 7 8 0 0 0 0

____ thousand

____ thousand

____ thousand

____ thousand

____ thousand

2. Place a comma into the number. Fill in missing parts. Read numbers aloud. a. 1 6 4,4 5 3

b. 9 2 9 0 8

c. 3 2 9 0 3 3

d. 1 4 0 0 4

164 thousand 453

____ thousand ____

____ thousand ____

____ thousand ____

e. 5 5 0 0 5 3

f. 7 2 0 0 1

g. 8 0 0 0 0 4

h. 3 0 0 3 6

____ thousand ____

____ thousand ____

____ thousand ____

____ thousand ____

58

3. Read these numbers aloud. a. 456,098

b. 950,050

c. 23,090

d. 560,008

e. 78,304

f. 266,894

g. 219,513

h. 306,700

4. Think in whole thousands and add! a. 30,000 + 5,000 =

b. 200,000 + 1,000 =

think: 30 thousand + 5 thousand

c. 400,000 + 30,000 =

d. 710,000 + 40,000 =

e. 300,000 + 600,000 =

f. 700,000 + 70,000 =

5. Add and subtract, thinking in whole thousands. a. 35,000 + 5,000 =

b. 210,000 + 10,000 =

c. 420,000 + 30,000 =

d. 711,000 + 10,000 =

e. 300,000 – 60,000 =

f. 700,000 – 70,000 =

g. 30,000 – 5,000 =

h. 200,000 – 6,000 =

i. 723,000 – 400,000 =

j. 500,000 – 1,000 =

6. On the number line below, 510,000 and 520,000 are marked (at the “posts”). Write the numbers that correspond to the dots.

7. Make a number line from 320,000 to 340,000 with tick-marks at every whole thousand, similar to the one above. Then mark the following numbers on the number line: 323,000 328,000 335,000 329,000 330,000

59

Practicing with Thousands 35 thousand 4 35,004 thousands H T O

There are no hundreds nor tens.

There are no hundreds nor ones.

203 thousand sixty 203,060

So we put 0 in the hundreds and tens place.

So we put 0 in the hundreds and ones place.

thousands H T O

1. Break these numbers down to whole thousands, hundreds, tens, and ones. a. 49,015

b. 206,090

49 thousands ____hundreds ____ tens ____ ones

____ thousands ____hundreds ____ tens ____ ones

c. 107,802

d. 88,030

____ thousands ____hundreds ____ tens ____ ones

____ thousands ____hundreds ____ tens ____ ones

e. 790,302

f. 903,000

____ thousands ____hundreds ____ tens ____ ones

____ thousands ____hundreds ____ tens ____ ones

g. 250,067

h. 300,070

____ thousands ____hundreds ____ tens ____ ones

____ thousands ____hundreds ____ tens ____ ones

2. Write the numbers. a. 20 thousand 4 ones

7 hundreds

b. 204 thousand 8 tens

c. 101 thousand 6 hundred

d. 540 thousand 4 ones

e. 230 thousand 7 tens

f. 9 thousand 6 hundred 7 ones

g. 873 thousand 5 tens

h. 40 thousand 4 hundred

60

3 hundred

i. 59 thousand 5 ones

6 tens

3. Write the numbers. It's all mixed up, be careful! Remember, after thousands are three more digits! a. 4 tens 25 thousand 7 ones 3 hundred

b. 2 tens 700 thousand 6 hundred 4 ones

c. 8 hundred 1 thousand 60 thousand 8 ones

d. 50 thousand 6 tens 3 thousand

e. 42 thousand 7 ones 8 tens

f. 9 thousand 600 thousand 50 thousand 4 tens

g. 90 thousand 4 tens 200 thousand

h. 20 thousand 9 hundred 7 thousand 5 ones

i. 500 thousand 4 thousand 8 ones

4. Continue the patterns. a.

b.

45,000

134,000

c. 800,000

d. 400,000

45,500

134,200

750,000

390,000

46,000

134,400

700,000

380,000

5. Add. a. 30,000 + 50

b. 254,000 + 300 + 5

c. 133,000 + 200 + 50

d. 77,000 + 4

e. 2 + 60,000

f. 120,000 + 3 + 60

g. 5,000 + 10,000 + 20

h. 4,000 + 6 + 20,000

i. 300 + 30,000 + 90

j. 400 + 86,000 + 70 + 1

61

Place Value with Thousands 728 thousand 401 hth tth th h

t

Each of the six digits has its own place (“box” in the picture).

o

7 2 8, 4 5 1 z z z z z z

In the charts: “hth” means hundred thousands “tth” means ten thousands “th” means thousands

Each of the “places” has its own value.

7 is in the “hundred thousands” place. The value of “7” is seven hundred thousand. 2 is in the “ten thousands” place. The value of “2” is twenty thousand. 8 is in the “thousands” place. The value of “8” is eight thousand. 4 is in the “hundreds” place. The value of “4” is four hundred. 5 is in the “tens” place. The value of “5” is fifty. 1 is in the “ones” place. The value of “1” is one.

728 thousand 401 hth tth th h

t

o

7 2 8, 4 5 1 7 0 0, 0 2 0, 0 8, 0 4

0 0 0 0 5

0 0 0 0 0 1

501 thousand 029

728,401 in expanded form is

hth tth th h

700,000 + 20,000 + 8,000 + 400 + 50 + 1.

t

o

5 0 1, 0 2 9 5 0 0, 0 0 0 1, 0 0 0 2 0 9

501,029 in expanded form is 500,000 + 1,000 + 20 + 9

1. Fill in the place value charts. a.

hth tth th h

t

o

8 7, 0 1 5

b.

hth tth th h

t

c.

o

4 0 3, 2 8 0

hth tth th h

t

o

6 9 2, 0 0 4

2. Write the numbers from exercise (1) in expanded form.

62

d.

hth tth th h

t

o

7 0 0, 2 0 4

3. Write the numbers in expanded form. a. 32,493 b. 172,392 c. 25,600 d. 109,020 e. 220,000 f. 900,701 4. Find the missing number. It's all mixed! a. 26,290 = 90 + _______ + 200

b. 205,500 = 200,000 + 500 + ________

c. 80,020 = 80,000 + ______

d. 707,070 = 70 + 700,000 + _________

e. 778,090 = 90 + 8,000 + _______ + 700,000 f. 917,500 = 900,000 + 500 + 10,000 + __________ g. 30,239 = 9 + 200 + 30 + ___________ 5. What is the value of the digit 5 in the following numbers?

a. 513,829 five hundred thousand

b. 400,065

c. 700,549

d. 59,906

6. Write the value of the underlined digit.

a. 1,209

b. 19,843

c. 89,605

d. 208,000

e. 302,600

f. 300,027

g. 210,408

h. 5,425

i. 921,993

j. 300,094

7. What are these numbers? a. 8 is in tens place, 5 is in hundred thousands place, and 7 is in ones place. b. 4 is in hundreds place, 8 is in tens place, and 2 is in ten thousands place.

63

Comparing with Thousands Which is more, 399,393 or 393,939? You are used to comparing small numbers. When comparing big numbers, use the same principles: z

Check if one number contains bigger place value units (or is “longer”). For example, 675,000 > 95,239 because 95,239 does not have any hundred thousands, but 675,000 does.

z

If the numbers have the same amount of digits (are equally “long”), then you need to compare the digits in the different “places”. Compare the digits starting from the BIGGEST place value. Though you don't have to use them, place value charts can help. t.th th

h.th t.th th

3 3

h

t

o

place value (ten thousands)

place value (hundred thousands)

{

o

↑ start comparing from the BIGGEST

↑ start comparing from the BIGGEST

{

t

2 7 0 4 5 2 7 0 5 4

9 9 3 9 3 9 3 9 3 9

{

h

{

Start at the hundred thousands place. The digits are the same (3). Both numbers have 300,000. At the ten thousands place, the digits are the same. Both numbers have 90,000 At the thousands place, one number has 9, the other has 3. The upper number has 9,000 while the other has only 3,000!

{

{

{

Therefore 399,393 > 393,939

Start at the ten thousands place. The digits are the same (2). Both numbers have 20,000. At the thousands place, the digits are the same. Both numbers have 7,000. At the hundreds place, the digits are the same. At the tens place, one number has 4, the other has 5.

Therefore 27,045 < 27,054

1. Write < or > between the numbers. These are fairly easy!

a. 45,200

54,000

b. 18,700

d. 78,111

77,001

e. 5,605

g. 1,788

17,880

h. 392,000

191,000 605,000 365,000

2. Write the numbers in order from smallest to greatest. a. 18,309; 81,390; 8,039; 818,039 b. 52,000; 5,020; 250,000; 520,000

64

c. 22,029

202,000

f. 34,092

43,200

i. 493,239

521,000

3. Find the largest number.

45,500

a. 54,000

52,400

134,000

d. 144,000

143,400

7,887

b. 8,708

7,708

c. 10,101 11,001

5,606

e. 5,556

5,599

f. 8,099 8,909

11,101 8,009

4. Write < or > between the numbers. Use the place value chart now if you need to.

a. 78,187

77,817

b. 21,089

21,098

c. 23,392

23,293

d. 349,309

343,909

e. 493,605

465,093

f. 199,909

20,900

g. 545,055

545,405

h. 909,808

908,809

i. 200,189

200,210

5. Look at the number lines and mark the following numbers (approximately!) there with a little circle.

15,090

15,131

15,678

15,430

15,878

15,923

16,050

34,896

34,950

35,254

35,599

35,020

34,631

35,117

6. a. Make a number line from 67,000 till 68,000 with tick marks at every whole hundred.

b. Mark these numbers (approximately!) on your number line. 67,250 67,030 67,510 67,780 67,940 67,370 67,049

67,703

c. Write the above numbers in order on the lines below.

_______ < _______ < _______ < _______ < _______ < _______ < _______ < _______

65

7. Find the largest number. It helps to place the comma that separates the thousands in the numbers.

383800

49830

a. 39903 d. 93024

398039

110293

290290

3420

b. 92022 e. 301481

99029

c. 600606 606660

606066

30420

f. 379444 390200

390002

8. Write the numbers in order. a. 500 5,600 5,406

5,505

1,500 1,459

b. 87,600

8,708

78,777

78,707 77,988 7,800

_______ < _______ < _______ < _______ < _______ < _______

_______ < _______ < _______ < _______ < _______ < _______

9. Find the number that fits in place of x.

a. 400,000 + x = 500,000

b. x + 30,000 = 100,000

c. x + x = 10,000

d. 500,000 – x = 300,000

10. Continue the patterns for six more numbers.

a. 81,400

b. 162,400

c. 1,000,000

d. 600

81,950

168,600

880,000

1,200

82,500

174,800

760,000

2,400 4,800

66

Adding and Subtracting Big Numbers 1. Adding in columns happens exactly the same way as with smaller numbers. See how well you can do!

905,091 + 40,510

b.

d.

608,781 + 230,911

e.

g.

289,300 120,000 + 409,436

h.

a.

c.

78,402 + 13,770

321,866 + 34,770

f.

60,066 + 477,770

89,502 45,987 13,770

i.

560,421 340,060 + 4,987

+

+

29,313 407,616

2. Continue the patterns. Use mental math.

480,000

29,100

906,500

162,700

485,000

29,300

916,600

172,700

490,000

29,500

926,700

182,700

67

9 7 10

9 9 7 10 10 10

7 10 10

800,000 – 510,065 Borrow over zeros...

800,000 – 513,065

800,000 – 510,065

Subtraction happens the same way as with smaller numbers. Just be careful with lots of borrowing!

Keep borrowing... (Complete the problem.)

3. Subtract. a.

120,091 – 34,510

b.

199,136 – 79,160

c.

670,000 – 1,300

d.

234,688 – 167,991

e.

65,570 – 23,677

f.

90,080 – 5,025

g.

554,600 – 128,000

e.

600,000 – 223,065

i.

400,000 – 18,344

4. Match the expressions (calculations) that have the same value. a.

b.

419,000 + 1,000

150,000 + 40,000

500,000 – 3,000

140,000 + 70,000

500 + 36,000

20,000 + 400,000

189,000 – 80,000

97,000 + 400,000

189,000 + 1,000

36,100 + 400

40,600 – 500

20,000 + 20,100

40,500 + 500

180,000 − 2,000

250,000 – 40,000

100,000 + 9,000

177,300 + 700

36,000 + 5,000

77,700 – 7,000

100,000 – 29,300

68

Line up the ones, tens, hundreds, etc. - even the commas.

134,607 + 3,065

134,607 + 3,065

457,934 37 ,921 + 24

457,934 37,921 + 24

THIS IS OFF! (sloppy writing, numbers not lined up)

(Complete the problem)

NOT THIS WAY! This is good!

5. Calculate. Line up all the place values carefully. a. 300,145 + 2,399 + 345

b. 560,073 + 81,400 + 98

c. 23,000 + 456 + 3,256

d. 345 + 870,077 + 32 + 5,801

6. Add a thousand, a ten thousand, or a hundred thousand.

n

13,000

78,000

154,000

n + 1,000 n + 10,000 n + 100,000

69

275,000

500,000

640,500

Line up the ones, tens, hundreds, etc. - even the commas.

509,032 –219

509,032 – 219

245,032 – 37,921

NOT THIS WAY!

245,032 – 37,921

118 111 THIS IS OFF! This is good!

(Complete the problem)

(errors in borrowing)

7. Calculate. a. 509,788 – 82,345

b. 30,760 – 2,906

c. 26,509 – 1,208

d. 984,044 – 329

8. If the two expressions (calculations) are equal, put an equal sign “=” into the box between them. If they are NOT equal, put a not equal sign "≠" between them. a. 660,000 + 30,000

620,000 + 40,000

d.

b. 499,000 + 2,000

501,000 – 1,000

e.

c. 125,000 – 4,000

119,000 + 2,000

f. 10,000 – 1,200

70

1,990 + 11 5,000 – 300

1,999 + 2 6000 – 1,300 6,000 + 2,500

9. Both anatomy and astronomy often contain big numbers! If you travel around the earth one time on the equator, your trip is 24,900 miles long! If you laid out an adult human's blood vessels, they'd go for 93,200 miles! The Moon lies at an average distance of 238,857 miles from the earth. a. How many whole loops around the earth would those blood vessels go? (You can use estimation.)

b. How many adults' blood vessels would you need to lay out to reach from the Earth to the moon?

c. About 10,000 average-sized human cells can fit on the head of a pin. About how many would be on ten pinheads? On twenty? On thirty?

d. One cubic milliliter of blood (TINY TINY amount) normally has between 4,000 and 10,000 white blood cells, and between 150,000 and 400,000 platelets. Suppose someone has 50,000 white blood cells and 10,000 platelets in that amount of blood. Their white blood cell count is (high/low) and their platelet count is (high/low). Would that be normal or is the person sick with something?

e. There are 100,000 hairs on the top of your head. You lose 100 of them when you brush your hair. How many do you have now? Should you worry about getting to be bald if this continues for a while?

f. Check your science book for other big numbers.

71

A Little Bit of Millions If you count by whole thousands... (read aloud)

994,000 995,000 996,000 997,000 998,000 999,000

1,0 0 0,0 0 0 ...what comes after 999 thousand?

A thousand thousands! It is called ONE MILLION.

The little comma separates the millions places (digits) from the rest. 5,0 0 0,0 0 0

6 9,0 0 0,0 0 0

9 6 7,0 0 0,0 0 0

Read: 5 million

Read: 69 million

Read: 967 million

Every number listed above has 6 zeros - they are whole millions! After the millions, the rest of the number is read just like you have learned before. 3 4 7,5 0 0,0 0 0

1 9,0 2 0,0 0 0

3 4 7,0 4 0,3 2 6

347 million 500 thousand

19 million 20 thousand

347 million 40 thousand 326

Simply read the word “million” at the first comma, and “thousand” at the second comma.

1. Place two commas into the number: one to separate the thousands' places, and another to separate the millions. a. 7 2 4 0 0 0 0 0 0

b. 5 1 2 0 0 0 0 0 0

c. 4 0 4 0 0 0 0 0 0

______ million

______ million

______ million

d. 4 0 0 0 0 0 0

e. 8 6 0 0 0 0 0 0

c. 8 3 4 5 0 0 0

______ million

______ million

______ million _____ thousand

g. 2 2 9 0 6 0 0 0

h. 5 1 4 3 1 0 0 0 0

i. 4 0 3 0 0 0 0 0

______ million _____ thousand

______ million _____ thousand

______ million _____ thousand

2. Write the numbers. a. 18 million

,

b. 906 million

,

c. 2 million 400 thousand

,

,

,

d. 70 million 90 thousand

,

,

72

,

3. Place commas into the number. Fill in missing parts. Read the numbers aloud. a. 7 7 9,4 5 3,2 3 0

b. 9 2 9 0 8 0 7

779 million 453 thousand 230

_____ million _____ thousand _____

c. 5 2 9 0 7 0 3 3

d. 5 5 0 0 1 4 5 3

_____ million _____ thousand _____

_____ million _____ thousand _____

e. 7 2 0 2 5 0 9 0

f. 2 2 8 0 1 0 2 0 0

_____ million _____ thousand _____

_____ million _____ thousand _____

In the following, there are NO thousands - so we don't even say the word “thousand”. g. 1 0 7 0 0 0 4 5 3

h. 7 2 0 0 0 0 9 0

_____million __thousand ______

_____million __ thousand ______

i. 2 8 0 0 0 0 0 6

j. 37 0 0 0 0 018

_____million __ thousand ______

_____million __ thousand ______

4. Write the numbers. a. 41 million 456 thousand 200

,

b. 80 million 80 thousand 80

,

,

c. 5 million 6 thousand 170

,

,

d. 299 million 3 thousand 9

,

,

,

5. Count - until you reach a million. a.

b.

c.

d.

900,000 910,000

300,000 400,000

999,990 999,991

999,200 999,300

73

6. Compare and write < or > between the numbers. a.

6,111,050

5,990,099

b.

d. 18,000,0000

181,000

g.

6,090,045

6,009,056

2,223,020

2,222,322

c. 192,130,659

192,130,961

e. 13,395,090

13,539,099

f.

2,367,496

988,482

h.

1,001,000

i.

17,199,066

1,000,999

71,857,102

7. Find five large numbers in a newspaper with the help of an adult. Write the numbers here.

8. A project with large numbers. Choose one of the options below, or one of your own. Use an encyclopedia, internet, or some other similar source, and make a list in descending order - that is, the one with largest number first, and then towards the smallest ones in order. a. of United States Western states and their populations; b. of Asian countries and their populations; c. of the amount of distinct animal species in the seven continents. d. of United States Midwest states and their land areas.

Puzzle Corner warm-up questions: z

z

How many times does 100 fit into 1,000? How many times does 1000 fit into 10,000?

→ How many times does 100 fit into 10,000?

z

z

How many times does 10,000 fit into 100,000? How many times does 100,000 fit into 1,000,000?

→ How many times does 10,000 fit into 1,000,000?

Jack and Mary were investigating big numbers. Suddenly Jack blurted out, “Hey, I just noticed something! The number (10 × 10) goes into (100 × 100) the same number of times as the number (100 × 100) goes into (1,000 × 1,000)!” Was Jack right? Would this same idea work also with these numbers? z z

(1 × 1) and (10 × 10) ? (2 × 2) and (20 × 20) ?

z z

74

(3 × 3) and (30 × 30) ? (4 × 4) and (40 × 40) ?

Multiples of 10, 100 and 1000 70 80 90 100 110 120 130 140 150

= 7 × 10 = 8 × 10 = 9 × 10 = 10 × 10 = 11 × 10 = 12 × 10 = 13 × 10 = 14 × 10 = 15 × 10

500 600 700 800 900 1000 1100 1200 1300

= 5 × 100 = 6 × 100 = 7 × 100 = 8 × 100 = 9 × 100 = 10 × 100 = 11 × 100 = 12 × 100 = 13 × 100

8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 16,000

= 8 × 1000 = 9 × 1000 = 10 × 1000 = 11 × 1000 = 12 × 1000 = 13 × 1000 = 14 × 1000 = 15 × 1000 = 16 × 1000

...because you These are get them if you multiples multiply some of 10... number by 10.

...because you These are get them if you multiples multiply some of 100... number by 100.

...because you These are get them if you multiples multiply some of 1,0000... number by 1,000.

850 is also a multiple of ten, because 850 = 85 × 10.

4,000 is also a multiple of 100, because 4,000 = 40 × 100.

56,000 is also a multiple of 1,000, because 56,000 = 56 × 1,000.

3,480 is also a multiple of ten, because 3,480 = 348 × 10.

7,600 is also a multiple of 100, because 7,600 = 76 × 100.

Do you notice a similarity? ALL multiples of 10 end in a zero!

ALL multiples of 100 end in two zeros!

392,000 is also a multiple of 1,000, because 392,000 = 392 × 1,000. ALL multiples of 1000 end in three zeros!

1. a. Write four distinct multiples of ten, different from those above. Write also what number times 10 they are. b. Write four distinct multiples of 100, different from those above. Write also what number times 100 they are. c. Write four distinct multiples of 1,000, different from those above. Write also what number times 1,000 they are. 2. Multiply. a. 11 × 100 = ________

b. 19 × 10 = ________

29 × 100 = ________

70 × 10 = ________

73 × 1,000 = ________

50 × 100 = ________

99 × 10 = ________

493 × 1,000 = ________

124 × 100 = ________

100 × 10 = ________

50 × 1,000 = ________

75

c. 6 × 1,000 = ________

3. Write these using numbers. a. 49 thousands __________

b. 20 tens __________

c. 37 tens __________

49 hundreds __________

20 hundreds __________

37 hundreds __________

49 tens __________

20 thousands __________

37 thousands __________

4. Write number expressions with numbers inside parenthesis. a. “Columbus landed in America in fourteen hundred ninety two. (__________)” b. “Andrew's car cost twenty-five hundred dollars (__________) when he got it but he is going to sell it for twelve hundred (__________).” c. “My great-great-grandfather was born in the year nineteen hundred (__________), and died in the year nineteen hundred sixty (__________).” 5. Write with numbers. What do we usually call... a. 10 tens __________

b. 10 hundreds __________

100 tens __________

100 hundreds __________

c. 100 thousands __________ 1,000 thousands __________

6. a. How many dollars do you have in a stack of forty 100-dollar bills? b. How many dollars do you have in a stack of fifty 10-dollar bills?

In division problems, ask “How many times does the divisor go into the dividend?” 8,000 ÷ 1000 = ?

720 ÷ 10 = ?

4,500 ÷ 100 = 45

2,000 ÷ 100 = 20

How many times does 1,000 go into 8,000? 8 times.

How many times does 10 go into 720? 72 times.

How many times does 100 fit into 4,500?

How many times does 100 fit into 2,000?

7. Divide. We will practice this more later. a.

b.

c.

500 ÷ 100 = ______

90 ÷ 10 = ______

2,000 ÷ 1,000 = ______

1,000 ÷ 100 = ______

100 ÷ 10 = ______

30,000 ÷ 1,000 = ______

2,100 ÷ 100 = ______

700 ÷ 10 = ______

342,000 ÷ 1,000 = ______

900 ÷ 100 = ______

340 ÷ 10 = ______

1,000 ÷ 1,000 = ______

76

Review 1. Write the numbers. a. 13 thousand 4 ones 9 tens

b. 300 thousand 5 tens 6 thousand

c. 1 million

6 thousand

2. Write the numbers. a. 78 million 50 thousand 3 hundred

b. 206 million 7 thousand eighty

3. What is the value of the digit 3 in the following numbers?

a. 213,047

b. 94,032

c. 5,300,049

d. 93,229,255

4. Round these numbers to the nearest hundred and nearest thousand.

n

78

5,367

558

4,409

2,603

rounded to nearest 100 rounded to nearest 1000

5. Round these money amounts... a. to the nearest ten dollars:

$34.69 ≈ ______

b. to the nearest dollar:

$4.92 ≈ ______

c. to the nearest hundred dollars:

$3,156.50 ≈ ______

6. First estimate the result of 5,076 – 2,845 – 675 by rounding the numbers to the nearest hundred. Then find the exact answer. Estimation: Exact answer:

77

3,359

7. Find the missing number. a. 40,505 = 5 + _______ + 40,000

b. 796,000 = 96,000 + ________

c.. 4,605,506 = 500 + 5,000 + 4,000,000 + 6 + _________________ 8. Write < or > between the numbers. a. 5,406

5,604

b. 49530

49553

c. 605748

9. Write the numbers in order from the smallest to the greatest.

5,905,544

95,695

495,644

496,455 145,900

590,554

10. Calculate. Line up all of the place values carefully. b. 490,213 − 45,344

a. 355,399 + 2,455 + 34,200

11. At a certain point while counting the votes that people had cast in an election, Candidate A had received 638,344 votes and Candidate B had received 584,042 votes. The last 48,388 votes had not yet been counted. If all of those last votes were for Candidate B, would he win?

12. A grocery store pays monthly $145,600 in salaries and $12,390 in other expenses. Will its total costs for June, July, and August exceed half a million dollars?

78

60584

Chapter 3: Multiplication Introduction The third chapter of Math Mammoth Grade 4-A Complete Worktext covers multi-digit multiplication and some related topics. While the first lessons briefly review the multiplication concept and the times tables, the focus in fourth grade is on multi-digit multiplication (also called algorithm of multiplication, or multiplying in columns). We start out by multiplying by whole tens and hundreds. After this is mastered, comes the very important concept of multiplying in parts. This essentially means that 4 × 63 is done in two parts: 4 × 60 and 4 × 3, and the results are added. The whole algorithm of multiplication is based on this principle, so it is important to master it. I don't want kids to multiply in columns “blindly”, without understanding what is going on with that algorithm. Before showing the traditional form of multiplying in columns, the lesson Multiply in Columns - the Easy Way shows a simplified form of the same, which is essentially just multiplying in parts. You may skip that lesson at your discretion or skim through it quickly if your child is ready to understand the standard form of the algorithm, which comes next. Other lessons in this chapter practice estimation and the order of operations, and multiplying with money. Many kinds of word problems abound. The lesson “So Many of the Same Thing” could be entitled “Proportional Reasoning” but I wanted to avoid scaring parents and children with such a high-sounding phrase. The idea in that lesson is really simple, but it does prepare for proportions as they are taught in 7th grade and in algebra. After that, we multiply by whole hundreds in order to prepare for double-digit multiplier problems, and to understand the algorithm of multiplication with more digits.

The Lessons in Chapter 3 page

span

Multiplication Concept ..................................... 81

2 pages

Multiplication Tables Review ..........................

83

3 pages

Scales Problems ...............................................

86

4 pages

Multiplying by Whole Tens and Hundreds ......

90

5 pages

Multiply in Parts ..............................................

95

4 pages

Multiply in Parts with Money ..........................

99

1 pages

Estimating Products ......................................... 100

2 pages

Multiply in Columns - the Easy Way ............... 102

3 pages

Multiplying in Columns, Standard Way .......... 105

5 pages

Multiplying in Columns, Practice .................... 110

2 pages

79

Error of Estimation ........................................... 112

2 pages

Order of Operations Again ............................... 114

3 pages

Money and Change .......................................... 117

2 pages

So Many of the Same Thing ............................. 119

3 pages

Multiply by Whole Tens and Hundreds ........... 122

2 pages

Multiplying in Parts with a 2-Digit Multiplier .......................................... 124

4 pages

The Standard Multiplication Algorithm with a 2-Digit Number Multiplier .................... 128

3 pages

Multiplying a Three-Digit Number by a Two-Digit Number ........................................... 131

2 pages

Review .............................................................. 133

3 pages

Helpful Resources on the Internet Math Playground Learn how to think algebraically with these clever weighing scales. http://www.mathplayground.com/algebraic_reasoning.html Thinking Blocks Thinking Blocks is an engaging, interactive math tool that helps students learn how to solve multistep word problems. Scroll down to Multiplication and Division. http://www.mathplayground.com/thinkingblocks.html Rectangle Multiplication An interactive tool that illustrates multiplying in parts using the area model. Choose the “common” option for multiplying in parts. http://nlvm.usu.edu/en/nav/frames_asid_192_g_2_t_1.html Interactive Pan Balance Each of the four shapes is assigned a certain weight. Place shapes on either side of the pan balance and figure out their relationships. http://illuminations.nctm.org/ActivityDetail.aspx?ID=131 Scales Problems from Math Kangaroo Problem Database http://www.kangurusa.com/clark/pdb/quiz.pl? dir=./kangur/output&y1=2002&l1=0304&i1=10&y2=2004&l2=0304&i2=10&y3=2005&l3=02&i3=19&n Multiplication Games A list of times tables games and activities to practice multiplication facts. http://www.homeschoolmath.net/math_resources_2.php#multiplication

80

Multiplication Concept z

Multiplication has to do with many groups of the same size: 3 × 5 means three groups of 5. You can find the total by adding: 3 × 5 = 5 + 5 + 5 = 15.

z

Multiplying by 1 means you have just one group: 1 × 17 = 17.

z

Multiplying by 0 means “no groups”: 0 × 82 = 0

z

The order in which you multiply does not matter: 3 × 6 and 6 × 3 are both 18.

3 groups of 6 or 6 groups of 3.

Multiplication terms The numbers being multiplied are factors. The result is called a product. There may be more than 2 factors. For example, in 4 × 5 × 2 = 40, the numbers 4, 5, and 2 are all factors.

1. Write the addition sentences as multiplication, or vice versa. Solve. a. 2 + 2 + 2 + 2 = ___×___ = ____

b. 80 + 80 + 80 = ___×___ = ____

20 + 20 + 20 + 20 = ___×___ = ____

8 + 8 + 8 = ___×___ = ____

c. ___________________ = 4 × 5 = ___

d. _________________ = 2 × 12 = ___

___________________ = 4 × 50 = ___

_________________ = 2 × 120 = ___

2. Write two multiplications! b. ___ rows, ___ columns: ___ × ___ = ___

a. ___ rows, ___ columns: ___ × ___ = ___ ___ rows, ___ columns: ___ × ___ = ___

___ rows, ___ columns: ___ × ___ = ___

3. Solve. a. 8 × 2

8×0×7

b. 3 × 5

c. 2 × 8

1×2×5

2×2×2 81

d. 3 × 10

3×3×3

4. Find the products. You can often use addition. a. 2 × 24

d. 2 × 150

g. 4 × 1,000

j. 2 × 34

b. 14 × 0

e. 3 × 2,000

h. 5 × 200

k. 3 × 21

c. 16 × 1

f. 4 × 3,000

i. 3 × 211

l. 4 × 50

5. Solve the problems. Write a multiplication fact for each. a. Seven children have _____ toes.

b. Four cows have _____ feet.

c. Eight bicycles have _____ wheels.

d. Three dozen eggs are _____ eggs.

e. Five people have _____ hands.

f. Nine cars have _____wheels.

6. a. Write the terms.

2 × 23 = 46 ↑ ↑ ↑ ___________ ___________

b. Write a multiplication problem with factors 4 and 8.

c. What happens if one of the factors is zero? The ______________ is _____. d. In one multiplication problem, two factors are 2 and 6. The product is 60. What is the third factor? 7. Write a single calculation to solve these problems. Your calculation will use several operations, not just one. a. Mom had three dozen eggs in cartons and five in a bowl. How many eggs did she have in all? b. Jack bought six packages of magazines. Each had 10 magazines. He opened one package and gave three magazines to his friend. How many magazines does Jack have left? c. Anna put crayons into boxes. Into four of the boxes, she put 10 crayons each, and into three boxes she put only six. How many crayons were there in all? d. Ernest bought three books for $11 each, and paid with $50. What was his change?

82

Multiplication Tables Review So WHY is it important to learn your multiplication tables? Why couldn't you just use addition or other ways to find what is 6 × 9 or 7 × 8 or 4 × 7 ? The reason is, the knowledge of multiplication tables is also needed in the “opposite” sense. You need to know them so you can divide quickly problems such as 54 ÷ 6 or 56 ÷ 7 or 48 ÷ 8. You need to know those in order to do long division. Also, a little later when you study fractions, you need to be able to immediately notice that in the fraction

56 , both numbers you see are in the table of 8. When you see 64

28 , you need 49

to immediately “see” that both numbers are in the table of 7. Without that, fraction operations such as addition and fraction simplification will be a “pain” to do.

1. Fill in the multiplication tables below and answer the questions.

1×5= 2×5= 3×5= 4×5= 5×5= 6×5=

7×5= 8×5= 9×5= 10 × 5 = 11 × 5 = 12 × 5 =

1 × 10 = 2 × 10 = 3 × 10 = 4 × 10 = 5 × 10 = 6 × 10 =

7 × 10 = 8 × 10 = 9 × 10 = 10 × 10 = 11 × 10 = 12 × 10 =

1 × 11 = 2 × 11 = 3 × 11 = 4 × 11 = 5 × 11 = 6 × 11 =

To find a number times 5, first multiply that number by 10, and take half of that. So for 7 × 5, first go 7 × 10 = 70 and take half of that.

7 × 11 = 8 × 11 = 9 × 11 = 10 × 11 = 11 × 11 = 12 × 11 =

Elevens are as easy as a pie!

What same answers do you find in tables of 5 and 10? Why?

1×2= 2×2= 3×2= 4×2= 5×2= 6×2=

7×2= 8×2= 9×2= 10 × 2 = 11 × 2 = 12 × 2 =

What same answers (products) do you find in the tables of 2, 4, and 8?

1×4= 2×4= 3×4= 4×4= 5×4= 6×4=

7×4= 8×4= 9×4= 10 × 4 = 11 × 4 = 12 × 4 =

To find a number times 4, you can double twice: 7 × 4 = ?? Double 7 is 14, then just double that to get 28.

83

1×8= 2×8= 3×8= 4×8= 5×8= 6×8=

7×8= 8×8= 9×8= 10 × 8 = 11 × 8 = 12 × 8 =

Here you can double thrice: 6 × 8 = ?? Take double 6, and double that, and double that. 5, 6, 7, 8 - fifty-six is 7 times 8. Color ones digits one color and tens digits another. You will see a pattern.

2. Fill in the multiplication tables below and answer the questions.

1×3= 2×3= 3×3= 4×3= 5×3= 6×3=

7×3= 8×3= 9×3= 10 × 3 = 11 × 3 = 12 × 3 =

1×6= 2×6= 3×6= 4×6= 5×6= 6×6=

7×6= 8×6= 9×6= 10 × 6 = 11 × 6 = 12 × 6 =

1×9= 2×9= 3×9= 4×9= 5×9= 6×9=

7×9= 8×9= 9×9= 10 × 9 = 11 × 9 = 12 × 9 =

What same products do you find in tables of 3 and 6? Why is that?

Table of 9 has special things!

To find a number times 6, you can double the corresponding one from table of 3:

Color all the ones digits yellow (of the answers). Color all the tens digits red (of the answers).

6 × 7 = ?? Go 3 × 7 and double that.

Add the digits of each answer. What do you notice?

These are harder ones... but remember you can change the order of multiplication. 8 × 7 is the same as 7 × 8, which is 56.

1×7= 2×7= 3×7= 4×7= 5×7= 6×7=

7×7= 8×7= 9×7= 10 × 7 = 11 × 7 = 12 × 7 =

1 × 12 = 2 × 12 = 3 × 12 = 4 × 12 = 5 × 12 = 6 × 12 =

7 × 12 = 8 × 12 = 9 × 12 = 10 × 12 = 11 × 12 = 12 × 12 =

3. Time to test your knowledge with missing factor problems! a. ____ × 7 = 49

b. ____ × 6 = 48

c. ____ × 8 = 64

d. ____ × 9 = 72

____ × 7 = 28

____ × 6 = 30

____ × 8 = 48

____ × 9 = 54

____ × 7 = 56

____ × 6 = 54

____ × 8 = 56

____ × 9 = 63

e. ____ × 5 = 45

f. ____ × 3 = 27

g. ____ × 4 = 28

h. ____ × 2 = 18

____ × 5 = 35

____ × 3 = 18

____ × 4 = 36

____ × 2 = 16

____ × 4 = 32

____ × 2 = 12

____ × 5 = 40 i. ____ × 7 = 35

____ × 3 = 21 j. ____ × 5 = 60

k. ____× 6 = 36

l. ____ × 8 = 72

____ × 7 = 63

____ × 5 = 25

____ × 6 = 72

____ × 8 = 16

____ × 7 = 21

____ × 5 = 30

____ × 6 = 42

____ × 8 = 32

84

4. Fill in the table.

×

0

1

2

3

4

5

6

7

8

9

10

11

12

0 1 2 3 4 5 6 7 8 9 10 11 12

5. A school hired 8 minivans that take 7 passengers each and one bus to take all 90 students to a swimming pool. If all minivans were full, how many students went in the bus? 6. Let's practice the order of operations again. a. 4 × 7 + 5 = ____

c. 4 × (7 − 5) = ____

e. (4 + 5) × (5 + 2) = ____

b. 2 × (5 + 6) + 4 = ____

d. 100 − 5 × 6 = ____

f. 70 − (5 + 6) × 4 = ____

7. Fill in the missing numbers so that both sides of the equal sign “=” have the same value. Example: 2 × 12 = 8 × 3 because

24 = 24

c. 3 × 10 = 6 × ____

a. 2 × 6 = 4 × ____

b. 6 × 6 = 4 × ____

12 = 12 d. 2 × 20 = 10 × ____

85

e. 5 × 12 = 6 × ____

Scales Problems This is a pan balance or scales. Things go into the two “pans”, and the heavier pan will go down, like in a seesaw. If the two things weigh the same, the balance stays balanced.

1. Solve how much each geometric shape “weighs”. You can use either pounds or kilograms.

a. The square weighs _____

b. The square weighs _____

c. One ball weighs _____

d. One rectangle weighs _____

e. One pentagon weighs _____

f. One oval weighs _____

g. One square weighs _____

h. One square weighs _____

86

If there are “unknown shapes” on both sides, use this “trick”: Take away the same amount of unknown shapes from both sides. The scale WILL continue to stay balanced!

Take away two diamonds from both sides. Then we see that three diamonds weigh 15.

2. Solve.

a. One pentagon weighs _____

b. One oval weighs _____

c. One triangle weighs _____

d. One triangle weighs _____

3. Solve. These are trickier. Use both balances to figure out the two unknown shapes.

a. One rectangle weighs _____

b. One circle weighs _____

One circle weighs _____

One diamond weighs _____

87

4. A few more with double scales...

a. One circle weighs _____

b. One square weighs _____

One square weighs _____

One triangle weighs _____

c. One square weighs _____

d. One circle weighs _____

One circle weighs _____

One triangle weighs _____

In mathematics, the equal sign “=” is like a scales that is balanced. Something is on the right side, and something is on the left side, and they are equal or “balanced” 5+7=2×6

5. Find the unknown number that goes on the empty line. a. 78 + ____ = 148

b. 7 + 6 + 6 = ____ – 10

c. 2 × 50 = 40 + ____

160 = ____ + 90

5 + 5 + 5 + ____ = 2 × 12

7 × 6 = 2 × ____

50 – ____ = 32

16 + 19 = 2 × ____ + 1

4 × 6 – 7 = 2 × ____ + 1

On the next page you will find empty scales pictures. You can print out the page and devise your own problems. But be careful! If you just make random problems, the solutions are likely to be fractions. See also: http://www.mathplayground.com/algebraic_reasoning.html - weighing scales game that practices algebraic reasoning http://illuminations.nctm.org/ActivityDetail.aspx?ID=33 - an interactive pan balance with shapes.

88

89

Multiplying by Whole Tens and Hundreds 1. a. Ten tens make a hundred. How about 20 tens or more?

b. Ten hundreds make a thousand. How about 20 hundreds or more?

10 tens = 10 × 10 = ____

10 hundreds = 10 × 100 = ____

13 tens = 13 × 10 = ____

12 hundreds = 12 × 100 = ____

20 tens = 20 × 10 = ____

15 hundreds = 15 × 100 = ____

21 tens = 21 × 10 = ____

18 hundreds = 18 × 100 = ____

37 tens = 37 × 10 = ____

20 hundreds = 20 × 100 = ____

56 × 10 is the same as 10 × 56. Both are 560. 92 × 100 is the same as 100 × 92. Both are 9,200. To multiply a number by 10, just tag a zero in the end. To multiply a number by 100, just tag two zeros in the end.

10 × 56 = 560

100 × 47 = 4700

10 × 481 = 4,810

100 × 2,043 = 204,300

Note especially what happens when the number you multiply already ends in a zero. The rule works the same; you still have to tag a zero or two zeros.

10 × 60 = 600

100 × 20 = 2,000

10 × 500 = 5,000

100 × 3,400 = 340,000

2. Multiply. a. 10 × 315 = ____

b. 100 × 62 = ____

c. 10 × 25,000 = ____

3,560 × 10 = ____

10 × 1,200 = ____

100 × 25,000 = ____

35 × 100 = ____

100 × 130 = ____

10 × 5,060 = ____

90

What is 20 × 14?

What is 200 × 31?

Imagine the problem without the zero. Then it becomes 2 × 14 = 28. Then, just tag a zero to the end result: 20 × 14 = 280.

Imagine the problem without the zeros. Then it becomes 2 × 31 = 62. Then, just tag two zeros to the result: 200 × 31 = 6,200.

Why does that work? It is based on the fact that 20 = 10 × 2. For example,

Why does that work? It is based on the fact that 200 = 100 × 2. For example,

20 × 14 = 10 × 2 × 14

200 × 31 = 100 × 2 × 31

In that problem, first multiply 2 × 14 = 28. Then multiply by ten:

In that problem, you can multiply first 2 × 31 = 62. Then multiply by a hundred:

10 × (2 × 14) = 10 × 28 = 280.

100 × (2 × 31) = 100 × 62 = 6,200.

3. Multiply by 20 and 200. a. 20 × 8 = ____

b. 200 × 7 = ____

c. 20 × 12 = ____

d. 20 × 16 = ____

4 × 20 = ____

5 × 200 = ____

35 × 20 = ____

42 × 200 = ____

20 × 5 = ____

11 × 200 = ____

200 × 9 = ____

54 × 20 = ____

The same principle works if you multiply by 30, 40, 50, 60, 70, 80, or 90. You can imagine multiplying by 3, 4, 5, 6, 7, 8, or 9, and then tag a zero into the end result. Similarly, if you multiply by some whole hundred, imagine multiplying without those two zeros, and tag the two zeros to the end result. 50 × 8 = 400

90 × 11 = 990

300 × 8 = 2,400

12 × 800 = 9,600

4. Multiply. a. 40 × 3 = ____

8 × 20 = ____ e. 200 × 9 = ____

7 × 400 = ____

b. 70 × 6 = ____

c. 80 × 9 = ____

50 × 11 = ____ f. 700 × 6 = ____

600 × 11 = ____

30 × 15 = ____

12 × 40 = ____

g. 200 × 12 = ____

h. 3 × 1100 = ____

15 × 300 = ____

91

d. 60 × 11 = ____

8 × 900 = ____

It even works this way:

5. Multiply.

In a problem 40 × 70 you can just multiply

a. 20 × 90 =

4 × 7, and tag two zeros to the result:

b. 60 × 80 =

70 × 300 =

30 × 900 =

c. 400 × 50 =

d. 80 × 800 =

40 × 70 = 2,800 In a problem 600 × 40 you can multiply 6 × 4, and tag three zeros to the result:

200 × 200 =

200 × 500 =

e. 100 × 100 =

f. 800 × 300 =

600 × 40 = 24,000 In a problem 700 × 800 you can multiply 7 × 8, and tag four zeros to the result.

40 × 30 =

700 × 800 = 560,000

90 × 1100 =

6. Write different factors for these products, using whole tens and whole hundreds. Have you noticed?

a. 6 × ___ = 420 and

b. ___ × ___ = 350 and

7 × 80 = 560 and 70 × 8 = 560 !! c. ___ × ___ = 280 and

___ × ___ = 280 Have you noticed?

6 × 400 = 2,400 and 60 × 40 = 2,400 and 600 × 4 = 2,400 !!

60 × ___ = 420 d. ___ × ___ = 400 and

___ × ___ = 400 f. 2 × ___ = 1,800 and

___ × ___ = 350 e. ___ × ___ = 990 and

___ × ___ = 990 g. ___ × ____ = 5,400 and

20 × ___ = 1,800 and

___ × ____ = 5,400 and

200 × ___ = 1,800

____ × ___ = 5,400

h. ___ × ____ = 3,000 and

i. ___ × ____ = 3,600 and

j. ___ × ____ = 3,600 and

___ × ____ = 3,000 and

___ × ____ = 3,600 and

___ × ____ = 3,600 and

____ × ___ = 3,000

____ × ___ = 3,600

____ × ___ = 3,600

92

7. Find the missing factor. Think backwards of how many zeros you need. a. ____ × 3 = 360

b. 40 × ____ = 320

____ × 50 = 450

5 × ____= 600

d. ____ × 30 = 4,800

e. 40 × ____ = 2,000

____ × 200 = 1,800

c. ____ × 40 = 400

____ × 2 = 180 f. ____ × 800 = 56,000

6 × ____= 4,200

____ × 20 = 12,000

8. Here is another method for finding ten times a number. We will find 10 × 88 in steps, and start out by finding 2 × 88.

Find 9 × 88 by adding 88 to your previous result.

Find 2 × 88 by adding: 88 + 88 = _____ Find 4 × 88 by doubling the previous result.

9 × 88 = _____

4 × 88 = _____ Find 8 × 88 by doubling the previous result.

Find 10 × 88 by adding 88 to your previous result.

8 × 88 = _____

10 × 88 = _____

Did you get 880? Do you prefer using the shortcut?

9. These questions help you find how to multiply money amounts by 10. a. What is 10 × 40 ¢ in dollars?

d. What is 10 × 80 ¢ in dollars?

b. What is 10 × $2?

e. What is 10 × $11?

c. What is, therefore, 10 × $2.40?

f. What is, therefore, 10 × $11.80?

g. What is 10 × 6 ¢ in cents?

k. What is 10 × 5 ¢ in cents?

h. What is 10 × 20 ¢ in dollars?

l. What is 10 × 10 ¢ in dollars?

i. What is 10 × $8?

m. What is 10 × $13?

j. What is, therefore, 10 × $8.26?

n. What is, therefore, 10 × $13.15?

Based on the questions above, can you discover a shortcut for multiplying money amounts by 10? It is found on the next page.

93

To multiply a money amount by 10, move the decimal point by one digit. Tag one zero so you have two digits to show the cents.

10 × $2.40 is $24.00

10 × $45.30 is $453.00

10 × $1.56 is $15.60

10 × $17.82 is $178.20

Make sure your whole dollar amounts were multiplied by 10. For example, if you have $3 first, after multiplying by 10 you need to have $30:

10 × $3.42 is $34.20.

10. Multiply money amounts by 10.

a. 10 × $2.20

b. 10 × $35.10

c. 10 × $1.87

d. 10 × $22.45

e. 10 × $45

f. 10 × $167.50

g. 10 × $9.16

h. 10 × $299.99

11. a. Mark bought 10 pairs of socks for $3.70 each, and 10 pairs of mittens for $5.50 each. What was his total bill? b. Mike bought ten pencils for 89 cents each, and paid his purchases with a $10 bill. What was his change? c. Which is cheaper, to buy a 10-pack of cans of juice for $9.99, or to buy ten individual cans of juice for $0.99 each? What is the price difference?

John wanted to prove that 40 × 70 is indeed 2,800 by breaking the multiplication into smaller parts. He wrote 40 as 4 × 10 and 70 as 7 × 10, and then multiplied in a different order: 40 × 70 = 4 × 10 × 7 × 10 = 10 × 10 × (4 × 7) = 100 × 28 = 2,800. You do the same, and prove that 60 × 50 is indeed 3,000.

94

Multiply in Parts Multiply 3 × 46 Break 46 into two parts: 40 and 6. Then multiply those two parts separately by 3: 3 × 40 is 120, and 3 × 6 is 18. Then add these two partial results: 120 + 18 = 138. Here is another way of showing the same thing, using ten-bundles.

3 × 40 = 120

3 × 6 = 18

46

46

46

3 × 46

120 + 18 138

Study these examples. Multiply tens and ones separately:

8 × 13 (10 + 3)

5 × 24

7 × 68

(20 + 4)

(60 + 8)

8 × 10 and 8 × 3

5 × 20 and 5 × 4

7 × 60 and 7 × 8

80 and 24 = 104

100 and 20 = 120

420 and 56 = 476

1. Multiply tens and ones separately. Then add to get the final answer.

a. 6 × 27 (20 + 7)

b. 5 × 83 (

c. 9 × 34 )

(

)

6 × ____ and 6 × ___

5 × ____ and 5 × ___

9 × ____ and 9 × ___

____ and ____

____ and ____

____ and ____

= _____

= _____

= _____

95

2. Break the second factor into tens and ones. Multiply separately, and add. a. 6 × 19

6 × 10 = 6×9 =

60 + 54

b. 3 × 73

3 × ___ 3 × ___

c. 4 × 67

+

114 d. 5 × 92

e. 9 × 33

f. 7 × 47

3. Multiply in parts. You can write the partial products under the problems, if you wish. a. 5 × 13 = ____

b. 9 × 15 = ____

c. 5 × 33 = ____

d. 8 × 21 = ____

e. 4 × 22 = ____

f. 4 × 36 = ____

g. 6 × 42 = ____

h. 7 × 51 = ____

i. 5 × 25 = ____

4. Solve. Write a number sentence for each problem. a. How many seconds are there in one hour? b. Jack bought 8 shirts for $14 each. What was his total bill? c. Mary and Harry set up nine rows of seats in the school auditorium, with 14 seats in each row. After that, they had 56 seats unused. How many seats were there in all? d. A package of small spoons costs $13. A whole silverware set is four times as expensive. How much do both items cost together?

96

Break 329 into three parts: 300 and 20 and 9.

It works with larger numbers, too:

Then multiply those parts separately by 7: 7 × 300 is 2,100, and 7 × 20 is 140, and 7 × 9 = 63. Lastly add the partial results:

2,100 140 + 63 2,303

5. Multiply hundreds, tens, and ones separately. Then add to get the final answer. a. 3 × 127 (100 + 20 + 7)

b. 5 × 243 (

)

3 × ____ and 3 × ____ and 3 × ___

5 × ____ and 5 × ____ and 5 × ___

____ and ____ and ____

____ and ____ and ____

= _____

= _____

c. 7 × 314 (

d. 4 × 607 (

)

)

7 × ____ and 7 × ____ and 7 × ___

4 × ____ and 4 × ____ and 4 × ___

____ and ____ and ____

____ and ____ and ____

= _____

= _____

6. Break the second number (factor) into hundreds, tens and ones. Multiply separately, and add. a. 4 × 128

4 × 100 = 4 × 20 = 4×3=

b. 8 × 151

400 80 + 12

d. 6 × 317

c. 3 × 452

+

e. 8 × 212

+

f. 6 × 198

+

7. Solve the word problems. 97

+

+

a. Katie prepares crafts for a craft club that has 23 kids. For the upcoming club meeting she needs to get at least 10 cm of string, 3 sheets of paper, and two toilet paper rolls for each kid. Write down her list of needed supplies.

b. A guitar class costs $18. Ernest paid for eight classes from the $200 that he has saved. How much does he have left?

c. Susie orders roses for her flower shop in bunches of six dozen (72 flowers) at a time. She needs a new batch once a week. How many roses will Susie order in 5 weeks?

d. One batch of six dozen roses costs her $70. How much will the roses she orders in five weeks cost her?

8. Compare. Write < , > , or = in the boxes between the number expressions. a.

10 × 10

d. 100 × 26

9 × 11

b.

6 × 12

5 × 14

c. 8 × 22

5 × 27

40 × 70

e. 5 + 195

40 × 5

f. 4 × 72

300

9. The expressions are supposed to be equal, but something is missing. Fill in the missing numbers. a. 6 × 6 = 9 × ___

b. ___ × 10 = 5 × 24

c. 20 + ___ = 4 × 10

d. 6,000 = 30 × _____

e. 120 − 75 = 5 × ____

f. ____+ 750 = 5 × 300

The equal sign “=” means that whatever is on the left side and on the right side of the sign are supposed to be equal:

10 + 10 = 5 + 15 2×6=3×4

g. 2,000 − 200 = 30 × _____

18 − 3 = 5 × 3

98

Multiply in Parts with Money Break money amounts in parts, and multiply the parts separately. When multiplying cent-amounts, remember to change them to dollar-amounts.

3 × $1.70

8 × $4.28

3 × $1 is $3, and 3 × 70¢ is 210¢ or $2.10.

8 × $4 is $32. 8 × 8¢ is $0.64 and 8 × 20¢ is $1.60. Lastly add:

Lastly add: $3 + $2.10 = $5.10.

$32.00 $0.64 + $1.60

4 × $15.22 $60

+

(4 × $15)

$0.88 = $60.88

$34.24

(4 × $0.22)

1. Multiply cent-amounts. Write the answers as dollar-amounts. a. 6 × 30¢ = 180¢ = $1.80

b. 5 × 50¢ = _____¢ = $______

c. 8 × 70¢ = _____¢ = $______

d. 3 × 90¢ = _____¢ = $______

e. 5 × 18¢ = _____¢ = $______

f. 6 × 41¢ = _____¢ = $______

2. Break the money-amount into dollars and cents. Multiply separately, and add. a. 6 × $2.80

b. 5 × $4.70

______ + ______ =

______ + ______ =

(6 × $2)

(5 × $4)

(6 × $0.80)

c. 4 × $12.50

(5 × $0.70)

d. 7 × $5.61

e. 6 × $6.75

f. 7 × $14.09

g. 6 × $11.85

h. 5 × $2.93

i. 11 × $9.45

6 × $11 6 × $0.80 6 × $0.05

+

99

Estimating Products If you don't need an exact result, you can estimate. To estimate a product (= an answer to a multiplication problem), round the factors so that they become easy to multiply mentally. There are no hard and fast rules as to how exactly you should round. Just so that your new rounded numbers are easy to multiply in your head. Estimate 8 × 189.

Estimate 42 × 78.

Estimate 7 × $4.56.

189 can be rounded to 200. The estimated product is 8 × 200 = 1,600.

42 ≈ 40 and 78 ≈ 80. The estimated product is 40 × 80 = 3,200.

$4.56 ≈ $4.50. Multiply in parts: 7 × $4 = $28 and 7 × 50¢ = $3.50. 7 × $4.50 = $31.50.

1. Estimate the products by rounding a factor or both factors to the nearest ten. Don't round both if you can calculate in your head just by rounding one factor! a. 5 × 69

b. 11 × 58

c. 119 × 8

d. 27 × 52

e. 7 × $4.15

f. 8 × $11.79

g. 25 × $42.50

h. 9 × 17

i. 63 × 897

b. 512 Popsicles at 19¢ each

c. 210 yards of wire at $1.29 per yard

2. Estimate the cost. a. 24 chairs at $44.95 per chair

3. Solve. a. Estimate the cost of six tennis balls that cost $3.37 each and two rackets that cost $11.90 each. b. A can of beans costs $0.29. A bag of lentils costs $0.42. Estimate which is cheaper: to buy 8 cans of beans or to buy 5 bags of lentils. c. Jackie needs to buy 8 ft of string for each of the 28 students in the craft class. The string costs $0.22 per foot. Estimate her total cost.

100

How many times does it “fit”? Use estimation. Many word problems can be solved using multiplication and estimation. Study the examples. If each bus can seat 57 passengers, how many buses do you need to seat 450 people? One bus seats 57 passengers. Two buses seat 114 passengers.

Ten buses seat 570 passengers. Eight buses seat 8 × 57 passengers.

With how many buses will your answer be 450 or a little more? This problem could be solved by division (450 ÷ 57) but instead, you can estimate using multiplication. Round the number 57 to 60, and quickly calculate: 7 × 60 = 420 and 8 × 60 = 480. It looks like 8 buses are needed. However we need to check it using the exact number 57: 8 × 57 = 400 + 56 = 456, so yes, eight buses is the answer. A sticker collection costs $2.39. How many collections can Jill buy if she can afford to spend $70? First round the price to $2.40. Two collections cost $4.80. TEN collections cost $24. Twenty collections cost $48, and thirty cost $72. Twenty-nine collections would cost a little less than $70 ($72 − $2.40) so that is how many she can afford. However, we used the rounded price $2.40 to find that. The difference between $2.40 and the real price $2.39 is 1 penny. For 30 collections we make an error of 30 cents. This is not big enough to “throw off” the estimation (in reality 30 collections cost only 30 cents less than what we estimated). 4. Solve the word problems using estimation. a. How many $0.79 pens can you buy with $5?

b. An advertisement in a newspaper costs $349. How many ads can Bill buy with $2000?

c. Renting skates at a skating rink costs $2.85 per hour. How many whole hours can Sandra skate for $25?

d. You earned $6.50 for weeding the vegetable garden. How many times do you need to weed before you can afford a paint set that costs $38.90?

101

Multiply in Columns - the Easy Way 38 × 6

Let's multiply 6 × 38 in parts, writing the numbers under each other.

38 × 6

38 × 6

38 × 6

48

48 180

48 +180

First multiply 6 × 8.

Multiply 9 × 2.

Multiply 9 × 80.

82 × 9

82 × 9

18

18 720

228

Then multiply 6 × 30 and write the result under the 48. Remember, the "3" is in the tens' place in the number 38 so it actually means 30. Add.

Lastly, add.

Multiply 3 × 7.

Multiply 3 × 40.

82 × 9

47 × 3

47 × 3

47 × 3

18 +720

21

21 120

21 +120

738

Add.

141

1. Multiply. a.

b.

c.

d.

e.

f.

g.

h.

102

Multiplying a 3-digit number happens in a similar way. You multiply in parts: first the ones, then the tens, then the hundreds. Lastly, add. Ones: 7×6

tens: 7 × 20

hundreds: 7 × 500

Add.

Ones: 9×8

tens: 9×0

hundreds: 9 × 200

526 × 7

526 × 7

526 × 7

526 × 7

208 × 9

208 × 9

208 × 9

42

42 140

42 140 3500

42 140 +3500

72

72 0

72 0 1800

3682

Add.

×

208 9

72 0 +1800 1872

2. Multiply. a.

b.

c.

d.

e.

f.

g.

h.

3. Solve. Remember multiplication is done before addition and subtraction. a. 58 × 5 + 291

b. 1,000 − 3 × 145

103

You can also multiply money amounts in parts.

$23.57 × 3

$5.18 × 4 4 × 0.08 (cents) 4 × 0.10 (ten-cents) 4 × $5 (dollars)

Multiply first the individual cents, then whole ten-cents, then whole dollars, and so on.

3 × 0.07 3 × 0.50 3 × $3 3 × $20

0.32 0.40 + 20.00 $20.72

0.21 1.50 9.00 + 60.00 $70.71

4. Multiply. a.

b.

d.

c.

5. a. You bought two notebooks for $1.45 each and two packs of crayons for $2.85 each. What was the total cost?

b. Jennie has 55 marbles and Sue has four times as many. How many marbles do the girls have together?

6. Figure out what was multiplied. a.

b.

×

7

1 4 4 9 0 + 7 0 0

c.

×

4

×

9 8 1

2 0 0 +1 2 0 0

+6 3 0 0

1 4 3 2

6 3 8 1

104

Multiplying in Columns, Standard Way Here we learn the standard algorithm of multiplication. It is based on the same principle of multiplying in parts: you simply multiply ones and tens separately, and add. In the standard algorithm the adding is done at the same time as multiplying. The calculation looks more compact and takes less space.

×

63 × 4

1

1

63 4

63 × 4

2

252

Multiply the ones first. 4 × 3 = 12 Place 2 under the line at the ones place, but the tens digit (1) is written above the tens column as a little memory note. This is called carrying to tens.

12 + 240

Then multiply the tens, and add the 1 ten that was carried over. 4 × 6 + 1 = 25 There is a total of 25 tens, which actually signifies 250. Write the 25 in front of the ones digit (2).

252 Compare to the method of “multiplying in parts” that you learned previously, where the adding is done separately.

(In the calculation 4 × 6 + 1 = 25, the 6 and the 1 are actually tens. So in reality we calculate 4 × 60 + 10 = 250.)

Look at other examples. In each case, some tens are carried as a result of multiplying the ones. 2

2

6

6

2

2

27 × 4

27 × 4

69 × 7

69 × 7

54 × 6

54 × 6

8

108

3

483

4

324

4 × 7 = 28

83 × 9

9×3=

4 × 2 + 2 = 10

83 × 9

9×8+2=

7 × 9 = 63

7 × 6 + 6 = 48

4

4

77 × 7

77 × 7

7×7=

105

6 × 4 = 24

38 × 5

5 × 6 + 2 = 32

38 × 5

Compare the earlier method with the one in this lesson:

×

75 8

40 +560

4

4

75 × 8

75 × 8

0

600

5 × 8 = 40, 4 is carried.

7×8+4= 56 + 4 = 60

OR

600

You can choose which one you use. Discuss it with your teacher. 1. Multiply. Be careful with the carrying. a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

m.

n.

o.

p.

q.

r.

s.

t.

106

With a 3-digit number you might have to carry twice, to tens and to hundreds. 1 3

1 3

238 × 4

238 × 4

238 × 4

2

52

952

3

Multiply the ones first. 4 × 8 = 32 Place 2 under the line and carry the tens digit (3) to the tens' column.

Then multiply the tens' digit, and add the 3 tens that were carried over. 4 × 3 + 3 = 15 Place the 5 in the tens' place and carry the 1 into the hundreds' column.

Then multiply the hundreds digit, and add the 1 hundred that was carried over. 4×2 + 1=9 Place the 9 in the hundreds' place.

Look at other examples. Compare to the earlier method of multiplying in parts.

1 2

1 2

127 × 4

127 × 4

127 × 4

8

08

508

2

4 × 7 = 28

4 × 2 + 2 = 10

127 × 4 28 80 + 400

4×1+1=5

508

4 3

4 3

496 × 5

496 × 5

496 × 5

496 × 5

0

80

2480

3

5 × 6 = 30

5 × 9 + 3 = 48

5 × 4 + 4 = 24

_

_

729 × 4 _ multiply ones

729 × 4

729 × 4

_ multiply tens and add what was carried

multiply hundreds and add what was carried

107

30 450 +2000 2480 729 × 4 36 80 +2800

2. Multiply. a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

m.

n.

o.

p.

108

3. Solve the word problems. Write a number sentence for each one. a. The school has 304 students. To go to the museum, they hired buses which can each seat 43 passengers. How many buses did they need?

b. The school also has 24 teachers. How many seats were left empty when all of the students and all of the teachers joined the trip?

c. Each package of paper contains 250 sheets. Marie needed 1300 sheets. How many packages did she need to buy?

d. Mick earned $345 from strawberry picking, and Jeanine earned three times as much. How much did they earn in all?

e. Emily solved 17 crossword puzzles, and Elaine solved three times as many. How many more did Elaine solve than Emily?

Find the missing numbers in these multiplications:

1

1 ×

4 4 6 8

4

3 ×

×

7 9

8 7 0

109

3 6

9 3

× 5

Multiplying in Columns, Practice Estimate the answer before you actually calculate. If the estimated answer is very different from the calculated answer, you can suspect an error.

Estimation:

Estimation:

5 × 45

45 × 5

≈ 5 × 50 = 250

2025

≈ 7 × 400 = 2,800

The estimated answer 250 is VERY different from the calculated answer 2025. There must be an error!

7 × 418

418 × 7 2926

The estimated answer 2,800 and the calculated answer 2,926 are fairly close. This does not prove the answer is correct though, but if there is an error, it is a “smaller” one.

1. Estimate the products, and then multiply to find the exact result. a. Estimation:

b. Estimation:

c. Estimation:

d. Estimation:

43 × 9

72 × 8

68 × 3

89 × 5

e. Estimation:

f. Estimation:

g. Estimation:

h. Estimation:

72 × 5

126 × 6

771 × 3

808 × 4

≈ 9 × 40 = 360

2. a. A first grade math book has 187 pages, and an 8th grade math book is three times as long. How many pages does it have? b. If it takes three hours to drive 180 km, how many hours would it take to drive 600 km?

110

3. Solve where the kids go wrong. Find out the right answers, too. 4

a. Look at Minnie's calculations. Figure out what error she is doing every time she multiplies.

b. This is Andy's math work. Where did he go wrong?

2

4 2

78 × 3

38 × 9

133 × 4

252

297

811

28 × 3

45 × 5

25 × 3

624

2025

615

5

c. And Ann does things wrong, too. How?

6

2

48 × 7

48 × 8

239 × 3

286

484

697

4. The expressions are supposed to be equal, but something is missing. Fill in the missing numbers. a. 5,400 = 90 × _____

b. ___ × 20 = 8 × 40

c. 7 × 49 + ____ = 8 × 49

d. 24,000 = 300 × _____

e. 7 × 13 = 5 × 13 + ____

f. ____ − 500 = 5 × 200

Fill in the missing numbers in these multiplications:

6

0 ×

3 3 1 5

7

3 ×

×

9

3 7 0 2

111

8 4

×

6

4

4 5 7

Error of Estimation Let's estimate 8 × 78. 78 ≈ 80 and 8 × 80 = 640.

Let's estimate 6 × $4.35. $4.35 ≈ 4.50 and 6 × 4.50 = $27.

The exact calculation gives us 8 × 78 = 624.

The exact calculation gives us 6 × $4.35 = $26.10.

The difference between these two results is 640 − 624 = 16. That is the error of estimation.

The difference between these two results is $27 − $26.10 = $0.90. That is the error of estimation.

The error of estimation is the difference between the estimated result and the exact result. The error tells you how much “off” you were.

1. First estimate the products, then calculate the exact result, and then find the error of estimation. a. Estimation:

4 × 91 ≈ 4 × 90 = 360 Error of estimation 4 c. Estimation:

6 × 34 ≈

b. Estimation:

Exact:

5 × 67 ≈

91 × 4 364

9 × 68 ≈

Exact:

d. Estimation:

7 × 59 ≈

34 × 6

8 × 242 ≈

Exact:

59 × 7

Error of estimation _____ f. Estimation:

Exact:

9 × 113 ≈

68 × 9

Error of estimation _____ g. Estimation:

67 × 5

Error of estimation _____

Error of estimation _____ e. Estimation:

Exact:

Exact:

113 × 9

Error of estimation _____ Exact:

h. Estimation:

5 × 693 ≈

242 × 8

Error of estimation _____

Error of estimation _____

112

Exact:

693 × 5

2. First estimate the total cost. Then find the total cost exactly. Then, find the error of estimation. a. Jack bought two train sets for $56.55 each.

b. Elisa bought for her mom three books for $11.58 each.

c. Angi bought 9 candles for $0.58 each.

You can estimate even if there are many operations. The goal is to round the numbers in such a way that you can then calculate mentally. You can also round numbers up to the “middle 5” if you can calculate mentally with it.

1,124 − 2 × 243

7 × $14.85 + $41.95 ≈ 7 × $15 + $42 = $105 + $42 = $147.

≈ 1,100 − 2 × 250 = 600

3. Estimate first, then find the exact result. Remember multiplications are done before additions and subtractions. a. 6 × 78 + 129

b. 1,754 − 5 × 139

c. 2 × $4.85 + 3 × 0.73

4. a. Calculate.

b. Find the missing factors.

c. Find the missing factors.

6 × 800 = _____

____ × 4 = 360

____ × 60 = 480

50 × 90 = _____

70 × ____ = 2,800

5 × ____ = 450

300 × 8 = _____

7 × ____ = 490

90 × ____ = 18,000

400 × 20 = _____

____ × 50 = 4,000

8 × ____ = 5,600

113

Order of Operations Again 1. Calculate anything within parentheses ( ).

20 − 2 × 5 + 9

(20 − 2) × 5 + 9

2. Do multiplications and divisions from left to right.

= 20 − 10 + 9

= 18 × 5 + 9

3. Do additions and subtractions from left to right.

= 10 + 9 = 19.

= 90 + 9 = 99

1. The following calculation has four operations. In which order are they done?

1. First calculate the sum ____ + ____. 2. Next multiply that sum by ____.

650 − 9 × (23 + 31) + 211

3. Subtract that result from ______. 4. Then, add ____ to the subtraction result.

2. Multiply in any order. Try to find the easiest, of course.

a. 2 × 3 × 300 =

b. 6 × 5 × 7 =

10 × 5 × 3 =

c. 40 × 10 × 8 =

d. 20 × 70 × 2 =

10 × 0 × 40 =

70 × 4 × 20 =

5 × 8 × 50 =

3. Solve mentally. a. 500 – 2 × 200 =

b. 70 × 30 + 2,000 =

c. 60 × 10 + 20 × 20 =

500 + 2 × 200 =

70 × 30 – 2,000 =

30 × 40 – 40 × 20 =

e. 90 + 15 + 2 × 7 =

f. 500 – 7 × 70 – 10 =

d. 8 × 200 – 200 – 500 =

800 – 2 × 20 + 100 =

90 × 10 + 120 – 40 =

10 × 7 × 5 + 100 + 250 =

4. Some more with parentheses. a. (500 – 200) × 2 =

b. 70 × (30 + 20) =

(500 + 200) × 2 =

70 × (30 – 20) =

c. 60 × (10 + 20) × 2 =

d. 8 × (200 – 100) – 500 =

30 × (40 – 40) × 2 =

(800 – 200) × 20 + 100 =

e. 90 + (15 + 5) × 7 =

f. (500 – 50 – 50) × 7 – 100 =

90 × (10 + 20) – 40 =

10 × 7 × (50 + 30) + 200 =

114

5. Solve for N. a. 5 × N = 150

N = ____

b. 3 × N × 3 = 27

c. 20 × 3 × N = 180

N = ____

d. N × 2 × 100 = 2,400

N = ____

N = _____

6. Write with numbers and solve. a. the sum of ten, a hundred, and a hundred and forty

b. the difference of 1,400 and 200

c. the product of seventy and forty

d. ten times the difference of 75 and 50

e. the sum of 5 and 8, multiplied by 2

f. 20 times the sum of 45 and 15

7. Calculate in the right order. a. 5 × 98 − 2 × 87

b. 2819 − 4 × (28 + 138)

c. 8 × (281 − 133) − 4 × 15

8. Find a matching number sentence (expression) for each problem and solve.

4 × ($2 + $3) a. Lisa bought four cards for $2 each and three shirts for $3 each. What was her total bill?

4 × $2 + 3 × $3 4 × $3 + $2

b. Mom bought for each of the four children crayons for $2 and a book for $3. What was her total bill?

c. Mark bought five packs of paper clips for $3 each and five pens for $2 each. What was his change from $50?

115

4 × $3 × 2 $50 − 5 × $3 + $2 $50 − 5 × $3 × 2 $50 − 5 × $3 − 5 × $2

9. Write a number sentence (expression) for each problem and solve. a. Elisa bought seven packs of needles for $2.55 each. What was her change from $30?

b. What is the total weight of eight 3-kg bags of strawberries and fifteen 2-kg bags of blueberries?

c. The distance from Dad's job to his home is 24 km. He drives to work and back every day, five days a week. How many kilometers does he drive in a 5-day work week?

d. An apartment house is nine stories high; each story is 9 feet tall. Another apartment house is three times as tall as that one. How tall is the second house?

e. What is the total cost of buying six books for $4.25 each and three books for 8.50 each?

f. Which is more costly, to buy five meters of material for $3.60 per meter, or to buy four meters of material for $4.80 per meter?

Put operation symbols +, – , or × into the boxes and add parentheses ( ) so that the calculations become true. a. 7

2

8 = 70

b. 80

5

10

116

5 = 55

c. 4

8

5

20 = 40

Money and Change Multiplying with money amounts happen the same way as multiplying whole numbers.

Estimation: 8 × $3.59 ≈ 8 × $4 = $32

Just add a decimal point and a dollar sign ($) in the answer. Estimate your answer first.

4 7

×

$3 . 5 9 8

$2 8 . 7 2

1. Multiply with money amounts. Estimate first. a. Estimation: 4 × _____ ≈ _____

$4 . 5 5 × 4

b. Estimation: ___ × _____ ≈ _____

c. Estimation: ___ × _____ ≈ _____

$9 . 7 0 × 4

$3 . 9 1 × 7

d. Estimation: ___ × _____ ≈ _____

$0 . 8 2 × 6

2. Study the charts. Solve. Write a number sentence under the chart in c. a. Jill bought three baskets for $7.20 each. She paid with $50. What was her change?

3 × $7.20 + change = $50 b. James bought four wheels for $29 each, and afterwards he had $51 left. How much did he have originally?

4 × $______ + $_______ = original amount c. Jack had $30, and then he bought five screwdrivers for $3.08 apiece. How much did he have left?

117

3. Now we have a bunch of word problems to solve. Just remember, in real life you will deal with money a lot, so it is an important topic to master! You can draw charts to help. a. If you buy 20 cans of fish for $1.29 each, what is your change from $30? (Hint: multiply dollars and cents separately!)

b. Mom paid $11.50 per meal for five meals. She now has $12.50 left. How much did she have originally?

c. A teacher bought 20 pencils for $0.15 and 30 notebooks for $1.09. What was the total cost?

d. Roger has been saving $45 each week to buy himself a laptop for $399. How many weeks will it take? How much will he have left over after buying it?

e. The $135-wheelbarrows were discounted by $20. A construction company bought eight. What was their total bill?

f. How many $0.13 erasers can you buy with $10? Guess and then check, until you know for sure.

g. Paul bought four computer mice for $9.80 each and a hard drive for $65, and had $25.80 left. How much money did Paul have initially?

h. A pack of four juice bottles costs $2.76. You want 20 bottles. What is your total cost?

118

So Many of the Same Thing 1. Fill in these tables. a. A bus is traveling 45 miles per hour. Fill in the table with how many miles it can travel in the given numbers of hours. Miles

45

Hours

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

5

6

7

8

9

10

12

14

16

18

20

80

90

100

b. One meter of fabric costs $5.10. Fill the table. Dollars

$5.10

Meters

1

2

3

4

c. Two cans of beans costs $3.00. Fill the table. Dollars

$3.00

Cans

1

2

3

4

d. You can get four buckets of paint for $60. Fill the table. Dollars

$60

Buckets

2

4

6

8

10

e. An earthworm can travel at the speed of 240 feet per hour. Fill the table. Feet Minutes

10

20

g. Ernie can patch four bicycle tires in an hour. Tires

Minutes

30

40

50

60

g. Maria can knit three scarves in nine days. Days

Scarves

70

g. Jack earned $75 in five hours. Hours

1

3

1

2

6

2

3

9

3

4

12

4

5

15

5

6

18

6

119

Dollars

Two sacks of the same size contain 12 kg of potatoes. How many of that size of sack would you need to get 30 kg of potatoes? To solve the problem, make up a little table as on the right:

1 sack

____kg

2 sacks

12 kg

3 sacks

3 × 6 kg = 18 kg

____ sacks

____ × 6 kg = 30 kg

First find out how many in ONE box: The total number of chocolates in five identical boxes was 30. How many chocolates would there be in two boxes?

1 box

____ chocolates

2 boxes

____ chocolates

5 boxes

30 chocolates

2. Solve the problems using tables. First find out the numbers for “ONE” of the things. a. Six flowers cost $18. How much would five flowers cost?

1 flower 5 flowers 6 flowers

$18

4 cans

800 g

3 lures

$6

b. Four cans of peas weigh 800 g. How much would three cans weigh?

c. A package of three fishing lures costs $6. How much would seven lures cost?

d. Mark can watch three episodes of the “Animal Farm” series in 90 minutes. How long would it take for him to watch 5 episodes?

e. Mark did five situps in ten seconds. How many could he do in one minute if he maintains the same speed?

120

f. Ten notebooks cost $20. What would seven cost?

g. Ann read 120 pages in four days. At that rate, how many days would it take her to read a 300-page book?

h. Six pairs of socks cost $4.50. What do 30 pairs of socks cost?

3. These problems are more challenging. a. Five collectible cars cost $35.50. What would four cars cost?

b. Margie can weed five rows of strawberry plants in one and a half hours. How long would it take her to weed nine rows?

c. Elaina can run four times around a running track in an hour. Today though, she only ran around three times, and then walked through the track the fourth time. All in all, this took her 10 minutes longer than on her normal days. How many minutes did it take her to walk around the track?

121

Multiply by Whole Tens and Hundreds Something to ponder: 5

58 × 7 406

Something to ponder: 5

7 × 58 = 406. What would 70 × 58 be?

16 × 9 = 144.

16 × 9 What would 160 × 90 be? 144

Can you guess?

Can you guess?

Don't read more until you think about the questions above! 70 × 58

160 × 90

= 10 × (7 × 58)

= 10 × (16 × 9) × 10

so the result to 70 × 58 is ten times the result to 7 × 58.

so the result to 160 × 90 is 10 × 10 or a hundred times the result to 16 × 9.

Since 7 × 58 = 406, then 70 × 58 is 4,060. Just tag a zero!

Since 16 × 9 = 144, then 160 × 90 is 14,400. Just tag two zeros!

Similarly, 700 × 58 would be 40,600. We tagged two zeros!

Similarly, 160 × 900 = 144,000. We tag three zeros!

1. Use the above method to multiply these: a. 60 × 87 b. 20 × 820 c. 510 × 400 d. 56 × 3,000 2. Solve the problems. a. A crate of apples weighs 20 kg. How much does 65 crates weigh? b. One crate contains apples laid out in four layers. There are 25 apples in each layer. How many apples are in a crate? c. A store owner sold 60 kg of apples to one customer. How many apples did the customer get? 122

See the zero in 70? You can write a zero in the ones place in the answer before calculating. Then just multiply 7 × 58 normally.

5

58 × 70 4060

5

160 × 90 14400

See the zero in 160 and another in 90? You can write two zeros in the ones and tens places in the answer before calculating. Then, just multiply 9 × 16 normally.

3. Multiply. Place a zero in the ones place in the answer before multiplying. a.

46 × 80

b.

27 × 60

c.

805 × 30

d.

179 × 40

e.

549 × 20

0 4. Multiply. Place a zero in the ones place and in the tens place in the answer before multiplying. a.

40 × 80

b.

415 ×300

c.

120 × 70

d.

231 × 800

00 5. The bus driver Mr. Hendrickson drives about 250 km each day on his route. How many kilometers does he drive in his 5-day work week? How about in his total of 300 workdays a year? 6. One side of farmer Greg's square-shaped field measures 200 m. He jogs around it seven times. How long is his jogging track? 7. Calculate. Use a notebook if needed. a. 80 × 560 + 15,000 b. 65,000 − 50 × 430 c. 20 × (85 + 126) + 2,333

If 382 × 29 = 11,078, then what is 3,820 × 29,000?

123

e.

658 × 700

Multiplying in Parts with a 2-Digit Multiplier You've learned to do 7 × 82 in parts: first multiply 7 × 80 and then 7 × 2. This same idea works even when we have two 2-digit numbers. Let's look at 25 × 34. To find 25 times some number we can find 20 times the number and 5 times the number, and then add those two. So we break 25 × 34 into two parts: 20 × 34 and 5 × 34. To find 78 × 47, break the 78 into two parts (70 and 8) and multiply by both: it is 70 × 47 and 8 × 47. 25 × 34 in parts: 20 × 34

5 × 34

78 × 47 in parts: Then add the parts.

70 × 47

2

8 × 47

4

5

Then add the parts.

34 × 20

34 × 5

680 + 170

47 × 70

47 × 8

3290 + 376

680

170

850

3290

376

3666

Study more examples. Note you have three separate calculations to do. 34 × 16 Do 30 × 16.

Then do

Don't forget the extra zero.

4 × 16.

1

29 × 35 Then add.

Do 20 × 35. Remember the extra zero.

2

1

Then do

Then add.

9 × 35 4

16 × 30

16 × 4

64 + 480

35 × 20

35 × 9

315 + 700

480

64

544

700

315

1015

1. Break the multiplications into two parts. You don't have to find the final product (answer). a. 28 × 16 = 20 × 16 and 8 × 16

b. 48 × 73 = _____ × 73 and _____× 73

c. 19 × 42 = _____ × 42 and _____ × 42

d. 55 × 89 = _____ × 89 and _____ × 89

e. 46 × 41 = _____ × _____ and _____ × _____

f. 28 × 39 = _____ × _____ and _____ × _____

g. 15 × 27 = _____ × _____ + _____ × _____

h. 93 × 16 = _____ × _____ + _____ × _____

124

2. Find the final answers for the problems in (1). a. 28 × 16

×

b. 48 × 73

×

+

×

c. 19 × 42

×

×

+

×

×

+

×

+

×

+

f. 28 × 39

×

+

×

g. 15 × 27

×

+

d. 55 × 89

e. 46 × 41

×

×

h. 93 × 16

×

+

×

3. Solve. a. Mary's Internet bill is $35 per month. How much does Mary pay in a year? b. Find the product of 1 × 2 × 3 × 4 × 5 × 6. c. A store sells large 15-kg boxes of apples for $35 a box. If you buy twelve boxes, what is their total weight?

125

Let's do the same when one number has two digits, and the other has three. To find 57 × 314. you break 57 into two parts: 50 and 7, and multiply 314 by them both. So we break 57 × 314 into two parts: 50 × 314 and 7 × 314. 57 × 314 in parts: 50 × 314

62 × 180 in parts:

7 × 314

2

Then add the parts.

60 × 180

2

2 × 180

4

Then add the parts.

1

314 × 50

314 × 7

15700 + 2198

180 × 60

180 × 2

10800 + 360

15700

2198

17898

10800

360

11160

4. Try your skills. Break each calculation into two multiplications, and then add. a. 28 × 315 = 20 × 315 and 8 × 315

×

×

b. 65 × 135

+

×

c. 19 × 472

×

+

×

+

×

+

d. 45 × 683

×

+

×

e. 73 × 394

×

×

f. 56 × 602

×

+

×

126

5. Solve. a. 45 × 83 b. 72 × 865 c. 32 × 118 d. 7 × 8 × 9 × 10

e. There are 365 days in a year. How many hours are in a year?

f. A baby sleeps 12 hours each night, and Annie sleeps eight. How many more hours does the baby sleep in a year than Annie?

Multiplying a two-digit number by a two-digit number can be broken down into four parts: 26 × 89 is first of all 6 × 89 and 20 × 89. 6 × 89 is 6 × 80 and 6 × 9.

20 × 89 is 20 × 80 and 20 × 9.

So all total, 26 × 89 is... 20 × 80 and 20 × 9 and 6 × 80 and 6 × 9 is four parts. The area of a rectangle is side times side. Where are those four parts represented in this picture? Can you explain why?

127

The Standard Multiplication Algorithm with a 2-Digit Multiplier You have learned to calculate multiplications such as 67 × 54 in parts. You did two multiplications and then added. It took three separate calculations. In the usual, traditional way of multiplying there are also three separate calculations. But this time ALL three calculations appear together.

67 × 54 2

3 2

54 × 67

54 67

54 × 67

378 3240

378 +3240

×

378 First multiply 7 × 54. (Pretend the 6 of the 67 is not there.)

3618

Then multiply 60 × 54 - but put the result underneath the 378. Remember the zero. Pretend the 7 of the 67 is not there.

Then add.

Study these examples, too. Note the extra zero needed in the ones place on the second line! 5 × 34

20 × 34

Add.

4 × 63

2

90 × 63

1

2

34 × 25

34 × 25

34 × 25

63 × 94

63 × 94

63 × 94

170

170 680

170 + 680

252

252 5670

252 +5670

850

5922

1. Fill in the missing digits and complete the calculation. a.

Add.

b.

c.

128

d.

2. Multiply. a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

m.

n.

o.

p.

q.

r.

s.

t.

129

3. Solve the word problems. Write a number sentence for each one. a. How many eggs are in 15 dozen eggs?

b. The 455 pupils in a school are going to a zoo by bus. One bus can seat 39 passengers. Are 11 buses enough to take them all? (Use multiplication!)

c. There are 60 minutes in each hour. How many minutes are there in 15 hours?

d. Each month, Brenda earns $21 from watering the neighbor's flowers. How much does she earn in a year?

e. Andrew gets paid $25 weekly. How much does he earn in a year? (There are 52 weeks in a year.)

f. Who earns more, in one year, Brenda or Andrew? How much more?

h. If Andrew wants to buy a radio that costs $78, how many months does he have to save for that?

130

Multiplying a Three-Digit Number by a Two-Digit Number You multiply the same way if one number has three digits. It is done in parts. 1

×

735 42

1470 First multiply 2 × 735.

121

46

735 42

$6.9 1 × 57

$6.9 1 × 57

$6.9 1 × 57

1470 29400

1470 + 29400

4 8.3 7

4 8.3 7 3 4 5.5 0

4 8.3 7 + 3 4 5.5 0

Then multiply 40 × 735.

30870

First multiply 7 × $6.91.

Then multiply 50 × $6.91.

$3 9 3.8 7

×

735 42

6

×

Then add.

1. Fill in the missing digits and complete the calculation. a.

b.

c.

d.

2. Multiply. a.

e.

b.

c.

f.

g.

131

d.

h.

Then add.

3. Solve the word problems. a. There are 365 days in a year. How many hours are there in a year?

b. Two cans of butter cost $8.80. What would 32 cans cost?

4. A store did inventory. The workers filled the table with numbers. a. First fill in the “Total count” column. b. Calculate what goes to the “total value” column. Use a notebook for calculations if you need more space. c. Fill the two empty boxes in the TOTALS row.

In boxes

On shelf

Total count

unit price

Total value

Cereal A

3 × 50

13

163

$2.87

163 × $2.87 = $467.81

Cereal B

2 × 40

25

$3.00

Granola A

3 × 25

6

$4.38

Granola B

2 × 25

27

$4.90

Product

TOTALS

Find a. 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 b. 9 × 10 × 7 × 4 × 0 × 6 × 2 × 1

132

Review 1. Multiply in parts. a. 5 × 43 = ____

b. 8 × 13 = ____

c. 7 × 57 = ____

2. Find a number that makes the expressions equal. a. 4 × 30 = ____ × 3

b. ____ × 500 = 250 × 4

c. 450 + 350 = ____ × 20

3. Fill in the table. Roses

1

Estimated price

$0.90

Exact price

$0.88

2

3

4

5

6

7

8

4. Solve the word problems using estimation. a. How many $0.59 toy cars can you buy with $8? b. If you earn $515 weekly, in how many weeks will you have earned more than $4000?

5. First estimate the products, then calculate the exact result, and then find the error of estimation. a. Estimation:

7 × 48 ≈

b. Estimation:

Exact:

Exact:

6 × 83 ≈

48 × 7

Error of estimation ______

83 × 6

Error of estimation ______

6. Multiply. a. 400 × 3 = ____

9 × 20 = ____

b. 70 × 60 = ____

300 × 11 = ____

133

c. 90 × 900 = ____

100 × 400 = ____

7. Find the missing factor. Think of how many zeros you need. a. ____ × 50 = 4,000

____ × 50 = 350

b. 70 × ____ = 280

c. ____ × 40 = 12,000

7 × ____= 2,800

____ × 800 = 64,000

8. Multiply. a.

b.

c.

d.

e.

9. Solve mentally. a. (1,500 – 1,000) × 4 =

d. 200 × (500 – 400) =

b. (76 + 34) × 2 × 0 =

e. 200 × (20 + 20) + 4,000 =

c. 80 × 2 × (30 + 20) =

f. 10,000 – 300 × (40 – 10) =

10. Calculate in the right order. a. 2 × 98 − 8 × 17

b. 6 × (342 + 125) − 769

134

11. Solve. a. A store owner bought 50 boxes of shirts, with 20 shirts in each box, and each shirt costs $2. What was his total bill?

b. Dad bought 15 boxes of nails for $2.30 a box. What was his change from $50?

c. Charlene bought five ice cream cones for $1.50 each. She has $12.50 left. How much did she have originally?

d. Jan bought six coconuts for $1.28 each and seven pineapples for $2.15 each. What was her total bill?

e. Jack paid for seven hours of housework, $8.20 per hour, with $100. What was his change?

f. A gazelle can run 9 miles in 15 minutes. How far could it run in 10 minutes?

g. Which weighs more, four boxes that are 44 pounds each, or five boxes that are 32 pounds each? How much more?

h. A huge roll of wrapping paper costs $140 but it was discounted by $20. How much do five rolls cost?

i. Two sheets of wrapping paper cost $3.20. What would six sheets cost?

135

Chapter 4: Time and Measuring Introduction The fourth chapter of Math Mammoth Grade 4-A Complete Worktext includes time, temperature, length, weight, and volume related lessons. The focus on fourth grade is no longer the actual act of measuring, but calculations that involve conversions between different measuring units. In time lessons, the student gets to do fairly complex calculations concerning hours and minutes. In temperature, the student is introduced to negative numbers and gets to do a few simple calculations even. The lessons concerning measuring units usually include a table that lists the units and the conversion factors. For metric units, those tables always include all of the units, even when they are not in common usage. For example, when studying metric units of volume, the chart looks like this: 10 10 10

liter

L

for larger amounts of volume

deciliter

dl

(not used much)

centiliter

cl

(not used much)

milliliter

ml for small amounts of volume

Only milliliters and liters are dealt with in the lesson. But the chart shows the two other units as well in order to get the student used to two basic ideas of metric measuring units: 1. How the units always differ by a factor of ten, 2. How the units are named consistently, with always the same prefixes such as milli-, centi-, deci-, deca-, hecto-, and kilo-. These prefixes and their meanings are NOT yet studied in detail in fourth grade; but I wanted to include the charts to familiarize the students with the terms and the ideas. You may, of course, at your discretion, explain it all to the student.

The Lessons in Chapter 4 page

span

Time Units ........................................................ 138

5 pages

The 24-Hour Clock ........................................... 143

2 pages

Elapsed Time or How Much Time Passes ...................................................... 145

5 pages

Temperature 1 .................................................. 150

2 pages

Temperature 2 .................................................. 153

2 pages

Remember Fractions ........................................ 155

1 pages

Measuring Length ............................................ 156

3 pages

136

More Measuring Length .................................

159

2 pages

Inches, Feet, Yards and Miles ........................

161

2 pages

Metric Units for Measuring Length ................

163

2 pages

Measuring Weight ..........................................

165

2 pages

Measuring Weight in the Metric System ........

167

2 pages

Customary Units of Volume ...........................

169

2 pages

Metric Units of Volume ..................................

171

2 pages

Review ............................................................. 173

2 pages

Helpful Resources on the Internet Calculating Time from BBC SkillsWise Fact sheets, worksheets, and an online game to practice time calculations. http://www.bbc.co.uk/skillswise/numbers/measuring/time/calculatingtime/ A Dictionary of Units of Measurement Explains the common measuring systems and has lots of background info of their history. http://www.unc.edu/~rowlett/units/ Measure It! Practice measuring lines with either centimeters or inches. Multiple choice questions. http://onlineintervention.funbrain.com/measure/index.html Measures Activities, revision bites, and quizzes about measuring time, weight, and capacity (in metric units). http://www.bbc.co.uk/schools/ks2bitesize/maths/shape_space_measures.shtml Measurements Online lessons with interactive exercises on metric prefixes, symbols, number values, metric mass, length, volume, US length and volume, and temperature conversions. http://www.aaamath.com/B/mea.htm Units of Measurement Quizzes Quizzes for area, distance, volume, and mass - both metric and English systems. http://www.quiz-tree.com/Units_of_Measurement_main.html Metric Measurement Matching Game Match metric terms and prefixes with the correct match http://www.quia.com/mc/4177.html Reading a tape measure worksheets Worksheet generator - you can choose to which accuracy to measure, inches, or inches & feet. http://themathworksheetsite.com/read_tape.html

137

Time Units 60 seconds = 1 minute

Pay close attention and memorize all these time units, if you don't know them yet.

12 months = 1 year

60 minutes = 1 hour

365 days = 1 year

24 hours = 1 day

366 days = 1 leap year

7 days = 1 week

1. Fill in the tables. Use your multiplication skills you've learned! Minutes

1

2

3

4

5

6

Seconds Years

Hours 1

2

3

4

5

6

1

2

3

4

5

6

Months Years

Days

Days 2. Solve the problems. a. Brian puts $120 into his savings each month. After saving for a year, he bought a keyboard for $799. How much does he have left of his savings? b. How much money do you use if you buy a candy bar for $2 every day of the year? c. Joan finished the foot race in exactly two minutes, and Jean was 24 seconds faster. What was Jean's finishing time? d. John was given an antibiotic for three whole days following his surgery. How many hours is that? e. Write a multiplication expression to find the number of seconds in one year. Use a calculator to find the product.

138

1

2

3

4

5

6

7

3. Convert minutes to hours and minutes and vice versa. a. 70 min = ____ h _____ min

b. 60 min = ____ hour

c. 5 h = ____ min

100 min = ____ h _____ min

72 min = ____ h _____ min

4 h 6 min = ____ min

170 min = ____ h _____ min

114 min = ____ h _____ min

3 h 37 min = ____ min

220 min = ____ h _____ min

145 min = ____ h _____ min

10 h 50 min = ____ min

560 min = ____ h _____ min

189 min = ____ h _____ min

12 h 3 min = ____ min

4. Solve the problems. a. Jennie helped her aunt on her strawberry farm during one busy week. She kept track of her working hours: Monday

Tuesday

Wednesday

4 h 45 min 2 h 30 min 5 h 20 min

Thursday

Friday

5 h 15 min

3h

Saturday

Sunday

2 h 30 min 3 h 40 min

How much time did she work total? She got paid $6 an hour. How much did she earn? b. It takes about 40 minutes to drive to town from Raymond's home. The family is going to spend about 3 hours shopping, and come back. What is the total amount of time they will be gone on their shopping trip?

c. Raymond's family's shopping trip didn't go as planned, as they also stopped for a visit at a friend's house for 1 hour and 10 minutes. How much time did they take for everything?

d. The batteries on a portable CD player last for 8 hours. You plan to use it every day while walking your dog, which takes about 25 minutes each day. How many days will the batteries last?

e. As a teacher, Raymond gives five 45-minute lessons each day. How many hours/minutes does he teach in a day? In a five-day week?

139

How many days are there in a month? Either 31, 30, 29, or 28 days. February has 28 days but on leap years February has 29. The rest of them have 30 or 31.See the chart. In calculations, use 30 days for one month unless it is a specific calendar month and you can know how many days it has. Here's another little mnemonic: Look at the knuckles of your fists. Count the months using both knuckles and the “valleys” between them, starting from the little finger's knuckle. The months on the knuckles have 31 days. The months on the “valleys between” have 30 days, except February which has 28 or 29.

1 2 3 4 5 6 7 8 9 10 11 12

January February March April May June July August September October November December

Notice that months 7 and 8 (July and August) both are on a knuckle, so both have 31 days. How many days are there from March 13th till July 5th, including the starting and ending days? March 13 ... March 31 is 19 days April: 30 days May: 31 days

June: 30 days July 1st ... July 5th: 5 days

Total: 115 days

5. Solve the problems. Both the starting and ending days are included. a. How many days are from June 12th till September 6th? b. How many days are from January 5th till October 5th? c. How many days are from your birthday till December 31st? d. An advertisement starts running on October 6th and runs for 120 days. When is the last day that the advertisement runs? e. The month of June has ____days, ____ × ____hours, ____ × ____ × ____ minutes, and ____ × ____ × ____ × ____ seconds = _______ seconds. (Use a calculator.)

140

31 28 31 30 31 30 31 31 30 31 30 31

How many weeks are in a year? We commonly say there are 52 weeks in a year, but that is not exact: 52 × 7 days = 364 days The whole year is normally 365 days so there is a one-day difference. That is why if your birthday is on Monday one year, the next year it is on the next weekday (unless it was a leap year, and then it would “jump” two weekdays). 6. Solve the problems. a. Jane watches TV about 7 hours a week. She swims about 6 hours a week, and does chores about two hours a day. How many hours in a year does she spend with each activity?

b. Hewitt's family homeschools all but 12 weeks of the year, five days a week, about four hours a day. How many hours do they do school in a year?

What is a leap year?

Why do we need a leap year?

It is a year that is one day longer than normal years. A leap year is 366 days long. On a leap year, February gets an extra day (29 days long).

Because the time our earth takes to go around the sun is not exactly 365 days. It is about 365 1/4 days. That is why in four years we “get off” one day and need to add that to the calendar.

Leap years occur every fourth year, except not when the year number ends in two zeros. For example, the years 1980, 1984, 1988, 1992, and 1996 were leap years. 2000 was not. The years 2004, 2008, 2012, 2016 and so on are leap years

7. a. How many days were in the years from 1997 till 2000? b. How many days were in the years from 2001 till 2005? c. Find your age in days. Remember that some years have been leap years.

d. Find how many days there are in a century (= 100 years). The right answer is NOT just 100 × 365 = 36,500.

141

Below you see a 2007 school calendar for New York City schools. The whole year from September 4th till September 3rd is of course 365 days. How many days of that do the kids spend in school and how many days off school? Remember a school week is from Monday thru Friday and not seven days. School Calendar September 4

School begins

October 8

Columbus day (no school)

November 12

Veterans day (no school)

November 22-23

Thanksgiving Recess

December 24 - January 1 Winter Recess January 15

Dr. Martin Luther King, Jr. Day (no school)

February 18 - February 22 Midwinter Recess March 21

Good Friday (no school)

April 21 - April 25

Spring Recess

May 26

Memorial Day (no school)

June 27 - September 3rd

Summer Vacation

142

The 24-Hour Clock As you know, the hour hand goes around the entire 12-hour clock face two times in one day. A day has 2 × 12 hours = 24 hours. Instead of using a.m. and p.m. to indicate which “round” we are on, we can use the 24-hour clock. The hours are simply numbered from 0 till 23 (or sometimes from 1 till 24). The afternoon hours are those from 13 till 24. The 24-hour time is commonly called the “military time” or astronomical time in the United States. In most countries of the world it is the dominant system used for bus, school, or TV schedules. How do we change a time expressed in the 12-hour clock to the 24-hour clock? z

For a.m. times the numbers don't change.

z

For p.m. times you add 12 to the hours.

a.m. / p.m. system 24-hour clock

The other way around, to change 24-hour clock times to 12-hour clock times, you subtract 12 hours from the afternoon times.

3:50 a.m.

3:50

noon

12:00

5:54 p.m.

17:54

10 p.m.

22:00

midnight

24:00

1. Change the times to the 24-hour clock times. a. 5:40 a.m.

___ : ____ e. 12:30 p.m.

___ : ____

b. 8:00 p.m.

c. 6:15 p.m.

___ : ____

___ : ____

f. 4:35 p.m.

g. 11:55 p.m.

___ : ____

___ : ____

d. 11:04 a.m.

___ : ____ h. 7:05 p.m.

___ : ____

2. Change the 24-hour times to the a.m. / p.m. times. a. 15:00

___ : ___ p.m. e. 14:30

___ : ____

b. 17:29

c. 4:23

___ : ____

___ : ____

f. 10:45

g. 16:00

___ : ____

___ : ____ 143

d. 23:55

___ : ____ h. 21:15

___ : ____

3. Study the bus schedule below. The times are given as (hours minutes) in the 24-clock time. Each column represents a bus that leaves at York Mills at a certain time, and arrives in Newmarket. There are a total of 12 different buses. Stops: York Mills Bus Terminal Yonge Street Finch GO Bus Terminal Thornhill Richmond Hill Hillcrest Mall Richmond Hill Yonge & Bernard Oak Ridges Aurora Newmarket Bus Terminal

Bus 1

Bus 2

Bus 3

Bus 4

Bus 5

Bus 6

Bus 7

Bus 8

Bus 9

15 10

15 35

15 50

16 05

16 17

16 29

16 41

16 53

17 05

17 20

17 35

17 50

15 17

15 42

15 57

16 12

16 24

16 36

16 48

17 00

17 12

17 27

17 42

17 57

15 28

15 53

16 08

16 23

16 35

16 47

16 59

17 11

17 23

17 38

17 53

18 08

15 42

16 07

16 22

16 37

16 49

17 01

17 13

17 25

17 37

17 52

18 07

18 22

15 50

16 15

16 30

16 45

16 57

17 09

17 21

17 33

17 45

18 00

18 15

18 30

16 02

16 27

16 42

16 57

17 09

17 21

17 33

17 45

17 57

18 12

18 27

18 42

16 09 16 15

16 34 16 40

16 49 16 55

17 04 17 10

17 16 17 22

17 28 17 34

17 40 17 46

17 52 17 58

18 04 18 10

18 19 18 25

18 34 18 40

18 49 18 55

16 30

16 55

17 10

17 25

17 37

17 49

18 01

18 13

18 25

18 40

18 55

19 10

a. If you need to be at Newmarket by 5 p.m., which bus should you take from York Mills? b. If you need to be at Newmarket by 6 p.m., which bus should you take from York Mills? c. Each bus takes the exact same amount of time to travel from York Mills to Newmarket. How much time is that? d. Jack was going from Oak Ridges to Newmarket. He came to the bus stop at half past five and caught the first bus that came. When was he in Newmarket? e. How many minutes does it take to travel in a bus from Thornhill to Aurora? f. How many minutes does it take to travel from Yonge Street to Oak Ridges? g. Mark lives in Thornhill and he goes to an art class in Newmarket that starts at 6:30 p.m. He has to walk for 15 minutes from the Newmarket bus stop to the art class. Which bus should he take from Thornhill?

144

Bus 10 Bus 11 Bus 12

Elapsed Time or How Much Time Passes When finding out how much time passes between two different times, we are dealing with the difference. You can find the difference by starting from the earlier time and “adding up” the elapsing time until the latter time. Imagine turning the hand of a clock from the starting time on, and keeping track of how much time passes. How long was an airplane flight if the plane took off at 12:45 p.m. and landed at 5:10 p.m. ?

From 12:45 till 1:00 15 min From 1 till 5

4h

From 5 till 5:10

10 min

Total 4 h 25 min

Time differences can also be found using subtraction. Subtract the hours and minutes separately in their own columns. How much time passes between 2:10 a.m. and 8:43 a.m.?

8 h 43 m − 2 h 10 m 6 h 33 m

How much time passes between 4:46 p.m. and 7:13 p.m.? Notice you can't subtract 46 minutes from 13 minutes. Before you even start, you need to borrow 1 hour from the hours column. 1 hour is 60 minutes so add that to the minutes you have in the minutes column. Do NOT borrow 10 or 100 minutes. How much time passes between 9:42 p.m. and 2:45 a.m.? Here the p.m. changes to a.m. It is safer to figure this in two parts: first from 9:42 p.m. to midnight, and then from midnight to 2:45 a.m. If you subtract the two numbers, you get the time difference the other way around: From 2:45 to 9:42, which is not the right answer. (Of course, knowing that you can figure out the other by subtracting the answer from 12 hours.)

6 h 73 m

7 h 13 m − 4 h 46 m 2 h 27 m 9:42 p.m....10 p.m. = 18 min 10 p.m....midnight = 2 hours Midnight...2:45 a.m. = 2 h 45 min

18 m 2h 0m + 2 h 45 m 4 h 63 m =5h3m

1. How much time passes? Solve mentally. a. From 12:30 p.m. till 2 p.m. ____h ____ min d. From 9:30 a.m. till 2:10 p.m. ____h ____ min

b. From 4:35 p.m. till 6:15 p.m. ____h ____ min e. From 7.58 p.m. till midnight ____h ____ min

145

c. From 5:19 a.m. till noon ____h ____ min f. From 11:05 p.m. till 6:35 a.m. ____h ____ min

2. How much time passes? Use subtraction. a. From 4:53 p.m. till 8:26 p.m.

b. From 6:37 p.m. till 9:03 p.m.

c. From 2:45 a.m. till 8:14 a.m.

8 h 26 m − 4 h 53 m

3. How much time passes? Do it in two parts. a. From 8:27 p.m. till 2:12 a.m.

b. From 9 a.m. till 5:16 p.m.

c. From 10:48 a.m. till 8:26 p.m.

4. How much time passes? Use the 24-hour clock. a. From 8:27 till 13:45

b. From 6:30 till 17:10

c. From 9:45 till 23:25

5. Solve the problems. a. Workers in a factory work in three shifts. How long is each shift?

Shift 1 6:00 a.m. - 2:30 p.m. Shift 2 2:00 p.m. - 10:00 p.m. Shift 3 9:30 p.m. - 6:30 a.m.

How many minutes is the overlap between two shifts?

b. Make a schedule for a doctor. He assigns 30 minutes for each patient, and after three patients, he has a 20-minute break. Use the 24-hour clock. Time Patient 1 8:00 - 8:30 Patient 2 Patient 3 break Patient 4 Patient 5 Patient 6 break

Time Patient 7 Patient 8 Patient 9 break Patient 10 Patient 11 Patient 12

c. Make a class schedule. Each class is 50 minutes, with 5 minutes between them. 146

Lunch break is 40 minutes. Class

Time

Social Studies 8:00 -

Class

Time

Lunch

Math

History

Science

P.E.

English

When will it end? The meeting starts at 2:30 p.m. and lasts for 1 hour 15 minutes. Simply add the hours to the clock time hours, and minutes to the clock time minutes: 2 hours + 1 hour = 3 hours. 30 minutes + 15 minutes = 45 minutes. Answer: The meeting ends at 3:45 p.m. Jake started playing at 3:35 p.m. and played for 45 minutes. You can still add like you did above and get 3 hours 80 minutes, but 80 minutes is more than one hour! We need to see the 80 minutes as 60 + 20, where 60 minutes makes one hour. Therefore the final answer is 4 hours and 20 minutes, or 4:20 p.m. The other way is to add the starting time and the elapsing time. If it started raining at 10:53 and it rained for 4 hours and 40 minutes, when did the rain end?

10 h 53 m + 4 h 40 m 14 h 93 m

Add the minutes and the hours separately. Note the minutes go over 60, so we need to change the 93 minutes to 1 hour and 33 minutes. The final answer is 15:33 or 3:33 p.m.

= 15 h 33 m

6. When will it end? a. Guests came at 3:40 p.m. and stayed for 2 hours and 30 minutes. b. Making pizza takes 1 hour and 40 minutes. Mom starts at 13:45. c. The pool opens at 8 a.m. and is open for 10 1/2 hours. When does it close? d. Jen's exam lasted for 2 1/2 hours, starting at 8:45. e. The airplane takes off at 18:08 and flies for 3 hours and 55 minutes. f. The food went into the oven at 5:47 p.m. and baked for 35 minutes.

147

When did it start? One more possible problem is that you know when something ends and how long it lasted. The airplane landed at 4:30 p.m. The flight lasted for 3 hours and 40 minutes. When did the plane take off? You need to go backwards from the ending time. Start at 4:30 and let the minute hand travel in your mind backwards 3 full rounds, and then 40 minutes. Where do you end up? Alternatively, subtract in columns. You will again need to borrow an hour = 60 minutes. The answer 50 minutes would mean the clock time 12:50 p.m.

3h 90 m

4 h 30 m − 3 h 40 m 50 m

Mental math is always good! A 55-minute class ended at 21:10. When did it start? If it had lasted 1 hour, it would have started at 20:10. But it was 5 minutes shorter and therefore started 5 minutes later, or at 20:15. A TV show lasted 1 h and 35 min, ending at 11:20 p.m. When did it start? Subtract in columns or think it through mentally. Again you'd need to borrow. It started at 9:45 p.m.

10 h 80 m

11 h 20 m − 1 h 35 m 9 h 45 m

7. Find the starting time. a. From ____:_____ p.m. till 2:00 p.m. is 40 minutes.

b. From ____:_____ p.m. till 8:12 p.m. is 30 minutes.

c. From ____:_____ a.m. till 4:15 a.m. is 1 hour 30 minutes.

d. From ____:_____ p.m. till 7:34 p.m. is 4 hours10 minutes.

e. From ____:_____ a.m. till 5:00 p.m. is 6 hours 20 minutes.

f. From ____:_____ a.m. till 6:54 a.m. is 5 hours 32 minutes.

g. From ____:_____ p.m. till 15:30 p.m. is 45 minutes.

h. From ____:_____ p.m. till 16:30 p.m. is 2 hours 40 minutes.

8. Solve the problems. a. The Johnson family arrived in the city at 10:30 after a 3-hour, 15-minute car ride. When did they leave home? b. When should the family leave the city to make it home by 20:00? (assuming the driving time back home is the same)?

148

c. Shannon runs through a path in the woods, and times himself. Here is the chart he made up. Fill in the chart with how much time he spent running each day.

Start: End: Running time:

Mo Wd Th Fr Sa 17:15 17:03 17:05 17:45 17:12 18:20 18:05 18:12 18:39 18:15

d. Find Shannon's total running time during the week.

e. Gordon works from 8:30 till 17:15 each day. He has a 30-minute lunch break, and two 15-minute “coffee” breaks. How many hours/minutes does he actually work?

f. Pete went to sleep at 22:15, and woke up at 7:00. But he also woke up at 3:30 and couldn't sleep till 5:10. How many hours/minutes did he actually sleep during the night?

g. The air conditioner is kept running from 7:30 a.m. till 9 p.m. How many hours does it run in a week?

h. An airplane is scheduled to take off at 3:40 p.m. and land at 5:10 p.m. The flight is delayed so that it leaves at 3:55 p.m. instead. When will it land?

149

Temperature 1 In the Celsius scale, zero degrees is the freezing point of water. Below that temperature, water turns to ice. Rain falls as snow. When the temperature drops below 0 degrees, we use negative numbers. The temperature just 1 degree below zero is “minus one degree Celsius” or -1°C. When reading negative numbers on a thermometer, you read it sort of backwards. The line just under 0 degrees matches -1°C. The line below that is -2°C, and so on. The temperature in the picture on the right is -4°C. The example on the left shows the temperature of -16°C. The table lists some benchmark figures for the Celsius scale. Water boiling

100°C

Normal body temperature

37°C

Nice inside temperature

20-25°C

Water freezing

0°C

1. Color the thermometer to show the temperature on the thermometers.

a. -5°C

b. -8°C

c. -12°C

2. Describe a situation where you might have a temperature of ... a. 12°C b. -5°C c. 31°C d. -23°C

150

d. -19°C

e. -23°C

3. Read the thermometer and write down the the temperature the thermometer shows.

a. _____°C

b. _____°C

c. _____°C

d. _____°C

e. _____°C

4. First write down the the temperature the thermometer shows. Then the temperature changes as indicated. Color the empty thermometer to show the new temperature

→ rises 3°C

a. _____°C

→ falls 5°C

_____°C

b. _____°C

_____°C

5. The temperature rises or falls, write the new temperature. temperature After rises 1°C ______ a. -9°C

temperature After falls 1°C ______ b. -9°C

temperature After rises 3°C ______ d. -13°C

temperature After falls 5°C ______ e. -7°C

f. 2°C

temperature After rises 5°C ______ g. -5°C

temperature After falls 5°C ______ h. 2°C

temperature After falls 3°C ______ i. -13°C

Now

Now

Now

Now

Now

Now

151

temperature After rises 3°C ______ c. -1°C Now

Now

Now

temperature falls 4°C

After ______

Fahrenheit scale In the Fahrenheit scale, the freezing point of water is not zero but 32°F. Anything below 32°F means that ice forms. The Fahrenheit scale does have a zero as well, and below it we again use negative numbers. The table lists some benchmark figures: Water boiling

212°F

Normal body temperature 98.7°F Nice inside temperature

70-78°F

Water freezing

32°F

Notice: The temperature where water freezes is 0°C = 32°F. 6. Describe a situation to fit these temperatures. a. 33°F b. -12°F c. 102°F

7. Write the temperature the thermometer shows. Notice the scale carefully!

a. 12°F

b. 76°F

c. 54°F 152

d. 88°F

e. 104°F

Temperature 2 1. Read the chart and fill in the table, as best as you can. Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Max Temperature

a. What is the hottest month? b. What is the coldest month? c. Find two months that have the same temperature. d. What is the difference in the maximum temperatures between May and June? e. How about between June and July? f. What is the difference in the maximum temperature between the coldest and hottest months?

153

Dec

2. Draw a line graph with the data.

Month

Minimum Temperature

Month

Minimum Temperature

Jan

-10

Jul

7

Feb

-9

Aug

6

Mar

-8

Sep

3

Apr

-2

Oct

-4

May

-1

Nov

-5

Jun

5

Dec

-7

a. Which month is colder, January or March? b. What is the difference in the minimum temperature between May and June? c. How about between October and November? d. How many degrees does the minimum temperature change from January to June?

154

Remember Fractions? One-half of the rectangle is colored.

→ how many equal parts →

Three-fourths of the rectangle is colored.

→ how many equal parts →

1 2

how many parts colored

3 4

how many parts colored

1. Color the parts to show the fraction.

a.

1 3

1 4

b.

c.

3 5

d.

5 8

2. Write the colored part as a fraction.

a.

b.

c.

d.

3. The fractions are equal because there is the same amount! Write the fractions.

1 2 2 4

a.

b.

c.

d.

e.

f.

4. Are the fractions equal or not? The diagrams above can help. a.

1 1 and 2 3

b.

1 2 and 2 4

c.

155

2 1 and 8 4

d.

3 5 and 4 8

Measuring Length Remember? This ruler measures in inches. The three lines between each two numbers on the ruler divide each inch into four parts, which are fourth parts of an inch.

The 2/4 mark is also the 1/2 mark. We normally use 1/2 instead of 2/4. This ruler measures in centimeters. The numbers signify whole centimeters. All the little lines between those are for millimeters. There are 10 millimeters in each centimeter. 10 mm = 1 cm 1. Measure the lines to the nearest fourth of an inch. Also measure them using a centimeter-ruler. a. ________ in. or _____ cm _____ mm

b. ________ in. or _____ cm _____ mm

c. ________ in. or _____ cm _____ mm

d. ________ in. or _____ cm _____ mm

e. ________ in. or _____ cm _____ mm

156

In the ruler below, each inch is divided into eight parts, which are eighth parts of an inch. Usually, the lines that show 1/2 inches are longer than other lines.

Compare it to the ruler below it, which shows fourth parts of an inch only. Notice that 2/8 inch = 1/4 inch, 4/8 inch = 1/2 inch, and 6/8 inch = 3/4 inch. This line is 1/8 inch long

This line is 3/8 inch long

This line is 2/8 or 1/4 inch long

This line is 5/8 inch long

2. Measure these lines. b. ________ in.

a. _________ in.

c. _______ in.

d. _________ in.

e. ________ in.

f. ________ in.

157

3. Draw lines using a ruler. a. 3 1/8 inches long b. 4 1/4 inches long c. 5 7/8 inches long d. 9 5/8 inches long e. 7 1/2 inches long 4. Draw lines using a ruler. a. 5 cm 3 mm b. 12 cm 1 mm c. 4 cm 4 mm d. 25 cm 7 mm e. 19 cm 9 mm 5. Practice measuring small items (such as pencils, pens, pins, erasers, the width of books) using a ruler that measures in 1/8th parts of an inch. If you don't have a ruler, cut out the ruler from the bottom of this page. Measuring tapes used for sewing often have a 1/8-inch scale on them, also. Item

Length/width

158

More of Measuring Length 1. Spread one hand wide open and let someone measure the distance from your thumb tip to your pinky tip. This distance is the definition for the measure span. So your span is _____ inches. (The official span is 9 inches.)

Now use your span to measure the height of a table (or chair): Table height = ____ spans. Now, use that to estimate the height of the table in inches. Height estimate = _____ in. Lastly measure with a measuring tape to check. Height: ______in. You can repeat this for other objects. 2. Find five small things. BEFORE you measure, make a guess of the length or width. Then, measure them in inches and centimeters both. Item

Guess (in.)

Reality (in.)

Guess (cm)

Reality (cm)

3. a. Measure all the sides of this figure to the nearest eighth-inch. b. Measure its sides also in centimeters and millimeters. c. Figure out the perimeter (“the distance all the way around”), in centimeters and millimeters.

159

4. Use your measuring tools, maybe even draw lines, to find out which is a longer distance: a. 3 cm or 1 inch

b. 2 inches or 7 cm

c. 15 cm or 6 inches

5. Change between centimeters and millimeters. a.

b.

c.

4 cm = ____ mm

6 cm 1 mm = _____ mm

4 cm 5 mm = _____ mm

17 cm = ____ mm

9 cm 9 mm = _____ mm

40 cm 8 mm = _____ mm

55 cm = ____ mm

12 cm 8 mm = _____ mm

100 cm = _____ mm

6. Change between millimeters and centimeters. a.

b.

c.

70 mm = ____ cm

21 mm = ____ cm ____ mm

453 mm = ____ cm ____ mm

430 mm = ____ cm

78 mm = ____ cm ____ mm

390 mm = ____ cm ____ mm

1,200 mm = ____ cm

109 mm = ____ cm ____ mm

5,000 mm = ____ cm ____ mm

7. Draw here a triangle so that its one side measures 4 1/4 in. and another side 3 3/8 in. Measure its third side. Find the perimeter.

160

Inches, Feet, Yards, and Miles Remember? 12 inches equal 1 foot. 12 in = 1 ft.

Three feet make up one yard. 3 ft = 1 yd.

1. Draw a long line on the yard and mark on it 1 ft, 2 ft, 3 ft, etc. marks up until at least 20 ft. Walk along your line. First, try to take 1-foot steps. Then, try to take 2-foot steps. Then, try to take 1-yard steps. Which kind of steps were the most comfortable and easiest steps for you to take? After practicing the 2-foot steps, measure some distances using your steps. For example, measure how wide a street is, or how long a room is. Count your steps, and then figure out the distance in feet.

2. Measure out the length and width of two rooms in your house using feet and inches.

3. Convert. You will need your multiplication skills! a. 6 ft = ____ in. 11 ft = ____ in. d. 36 in. = ____ ft 50 in. = ____ ft ____ in. g. 6 yd = ____ ft 13 yd = ____ ft

b. 2 ft 5 in. = ____ in. 7 ft 8 in. = ____ in. e. 27 in. = ____ ft 100 in. = ____ ft ____ in.

c. 13 ft 7 in. = ____ in. 24 ft = ____ in. f. 64 in. = ____ ft 85 in. = ____ ft ____ in.

h. 2 yd 2 ft = ____ ft

i. 24 ft = ____ yd

5 yd 1 ft = ____ ft

42 ft = ____ yd

j. 13 ft = ____ yd ____ ft

k. 22 ft = ____ yd ____ ft

l. 32 ft = ____ yd ____ ft

17 ft = ____ yd ____ ft

29 ft = ____ yd ____ ft

40 ft = ____ yd ____ ft

161

How much is 3 × 2 ft 7 in? Multiply the feet and the inches separately: 3 × 2 ft = 6 ft and 3 × 7 in = 21 in. Then add those. But first you need to convert the 21 inches into 1 ft 9 in: 6 ft + 1 ft 9 in = 7 ft 9 in.

4. Multiply. a. 7 × 5 in = ____ ft ____ in

c. 8 × 3 ft 5 in = ____ ft ____ in

b. 4 × 4 ft 4 in = ____ ft ____ in

d. 7 × 2 ft 9 in = ____ ft ____ in

5. Solve the problems about the perimeter (P).

a. The long sides are 6 ft 4 in, The short sides are 2 ft 10 in. P = ___ ft ___ in

b. Each side is 8 in. P = ___ ft ___ in

c. Each side is 1 ft 8 in. P = ___ft ___ in

d. A rectangle's long sides are 5 ft 6 in, and its perimeter 16 ft 10 in. How long are the shorter sides?

A mile is used to measure long distances. 1 mile = 5,280 feet. Mile originates from the Roman measure “mille passus”, or thousand paces. (A pace is a double-step.) The Roman mile was exactly 5,000 Roman feet. Read here how the 5,000-foot mile became a 5,280-foot mile around the year 1300: http://www.sizes.com/units/mile.htm and http://en.wikipedia.org/wiki/Furlong

6. a. Which is a longer distance, 2 mi 300 ft or 13,000 ft? b. An airplane flies at the height of 21,000 feet. About how many miles is that? c. How many feet is four miles? d. About how many miles tall is Mt. Everest (elevation 29,029 ft)? e. Andrew can walk 300 feet in one minute. How many whole miles can he walk in an hour? 162

Metric Units for Measuring Length The basic unit in the metric system is the meter. All of the other metric units for measuring length have the word “meter” in them.

Units of length in the metric system 10

The conversion factors in the metric system are based on 10. That is why you will use either 10, 100, or 1,000 when changing one metric unit of length to another.

10

10 millimeters makes 1 centimeter. 10 centimeters makes 1 decimeter. 10 decimeters makes 1 meter. And so on.

10

10 10 10

kilometer km 1,000 meters hectometer

hm

(not used)

decameter

dam

(not used)

meter decimeter

m the basic unit dm

(not used much)

centimeter cm look at your ruler! millimeter mm look at your ruler!

Remember that 1 meter is very close to 1 yard. 1 meter is a tiny bit longer than 1 yard.

1. Outside, or in a long corridor or room, draw two lines that start at the same place. a. Using a measuring tape, mark on the one line 1 m, 2 m, 3 m, and 4 m. Can you take “hops” 1 meter long?

1 meter

1 meter

1 meter

1 meter

b. Mark on the second line marks from 1 foot to 13 feet. Make 1-yard hops. Compare: do the two kinds of hops feel about the same?

2. Measure how tall you and other people are in centimeters. Write it also using whole meters and centimeters. Name

How tall _____ cm = 1 m ____ cm.

163

Conversions between units Remember what millimeters look like on your ruler. 10 mm = 1 cm. Decimeters aren't usually marked on rulers.10 centimeters make 1 decimeter. 10 decimeters end up being 100 centimeters, and that is 1 meter. 1 km = 1,000 m

1 m = 100 cm

1 cm = 10 mm

3. Convert between meters, centimeters, and millimeters. a. 5 m = _______ cm

b. 4 m 6 cm = _______ cm

c. 800 cm = _______ m

12 m = _______ cm

10 m 80 cm = _______ cm

239 cm = ___ m ____ cm

6 m 20 cm = _______ cm

9 m 9 cm = _______ cm

407 cm = ___ m ____ cm

d. 58 mm = ___ cm ___ mm

e. 5 km = ________ m

f. 2 km 800 m = _______ m

78 cm = _____ mm

57 km = _________ m

6 km 50 m = _______ m

234 mm = ____cm ___ mm

5,000 m = _____ km

60,000 m = _______ km

4. Calculate. Give your answer using kilometers and meters. a. 5 km 200 m + 8 km 900 m

b. 3 × 2 km 800 m

c. 1,500 m + 2 km

d. 6 × 700 m

5. Solve the problems. a. How many millimeters are in a meter? b. Mary can walk 1 km in 10 minutes. How far can she walk in 34 minutes? c. John jogs through a track 1 km 800 m long twice a day, five days a week. How long a distance does he jog in a week? d. A 10-meter wall is divided into five segments (not of equal length). Four of the segments are 1 m 20 cm each; how long is the fifth segment? e. Kathy's wallpaper has butterflies that are 80 mm wide. She will put the wallpaper in her room. How many complete butterflies can she have on a wall 3 meters long?

164

Measuring Weight Units of weight in the customary system 2,000

(short) ton

16

T

for very heavy things

pound

lb for medium-heavy things

ounce

oz to measure light things

1. Choose the right weight for each thing. Sometimes there are two possibilities. a. a sparrow 10 oz

b. a book

1 oz 16 oz

1 lb

d. a car 2T

2 oz

c. a 3-year old boy

20 oz

22 lb

e. a magazine

3,500 lb 300 lb

5 oz

2 lb

f. a healthy woman

1 lb

1 lb = 16 oz

44 lb 66 lb

80 lb

130 lb 60 lb

1 T = 2,000 lb

6 lb 4 oz = 6 × 16 oz + 4 oz = 100 oz

17 T = 17 × 2,000 lb = 34,000 lb

4 lb 9 oz + 1 lb 7 oz = 5 lb 16 oz = 6 lb

4 T 800 lb + 1,500 lb = 4 T 2,300 lb = 5 T 300 lb

2. Fill in the tables. Pounds

1

2 1/2

4

10

Ounces

Tons

15

80

3

5 1/2

480

10

Pounds

14,000

25

41

40,000

3. Convert pounds and ounces. a. 7 lb 7 oz = ______ oz

b. 33 oz = ____ lb _____ oz

2 lb 14 oz = ______ oz

c. 7 lb 7 oz = ______ oz 42 lb 14 oz = ______ oz

52 oz = ____ lb _____ oz

4. Circle the heavier amount. a. 1500 lb

1 1/2 T

b.

3T

4000 lb

165

c.

64 oz

3 lb 6 oz

5. Make a line graph of baby's weight.

Week Weight 0

6 lb 14 oz

1

6 lb 12 oz

2

6 lb 14 oz

3

7 lb

4

7 lb 2 oz

5

7 lb 4 oz

6

7lb 6oz

7

7lb 7oz

You can add pounds to pounds and ounces to ounces. Afterwards, figure out if the total ounces actually make some whole pounds. In the example on the right, 33 oz needs changed to 2 lb 1 oz before giving the final answer 5 lb 1 oz.

6. Solve the problems. a. Jose's packages and letters for the week weighed 2 oz, 6 oz, 5 oz, 1 lb 1 oz, and 1 lb 4 oz. What was their total weight? b. The cocoa powder on the scales weighs 2 lb 5 oz. Janet puts some in a bag, and then the scales shows 1 lb 8 oz. How much cocoa powder did she put in the bag? c. How many more bags can Janet make of the same size from the 1 lb 8 oz she still has unbagged? d. Farmer Smith's red apples weigh about 4 oz each. How many apples are in 5 lb of apples? e. An ounce of rice costs $0.09. Find the price for 2 pounds of rice. f. A bag of split peas weighs 1 lb 4 oz. How much do ten bags weigh?

166

9 oz 1 lb 11 oz + 2 lb 13 oz 3 lb 33 oz = 5 lb 1 oz

Measuring Weight in the Metric System Units of weight in the metric system 10 1 kg = 1,000 g

10 10

kilogram

kg to measure heavy things

hectogram

hg

decagram

dcg (not used)

gram

g

(not used)

to measure light things

1. Fill in the table. kilograms

2

2 1/2

3 1/2

10

grams

7,000

9,500

20,000

2. Convert. a. 5 kg 400 g = ______ g

b. 2,500 g = ____ kg ______ g

c. 60 kg = ______ g

32 kg 40 g = ______ g

20,250 g = ____ kg ______ g

8 1/2 kg = ______ g

5000 3650 + 490 can change them all to grams first. 9140

You can also add kilograms to kilograms and grams to grams. Remember that 1,000 grams makes a kilogram.

5 kg + 3 kg 650 g + 490 g

4 kg 250 g + 5 kg 800 g

When some weights are given in kilograms and some in grams, you

= 5,000 g + 3,650 g + 490 g

Answer: 9,140 g or 9 kg 140 g

= 9 kg 1,050 g = 10 kg 50 g

3. a. Jeremy received in the mail, packages that weighed 700 g, 350 g, 4 kg 400 g, and 1 kg 900 g. What was the total weight of the packages? b. Angi bought three 1 1/2 kg packages and seven 400-gram packages of buckwheat flour. What is the weight of the flour she got? c. You need 2 kg of flour to make bread. The scale shows you already have 1,050 g. How many more grams of flour do you need? d. A 200-gram bag of millet costs $1.69. How many bags do you need for 1 kg of millet? What is the total cost? 167

4. Circle the heaviest amount. a. 3 kg 300 g

3,030 g

b. 6 kg 400 g

640 g

c. 10 kg

5,000 g

5. a. Fill in the table how much weight Greg gained during each year. b. When did he grow the fastest? c. How can you see the 'fast' growth periods on the chart?

AGE WEIGHT Weight gain from (yrs) (kg) previous year 0

3 kg 300 g

1

10 kg 200 g 6 kg 900 g

2

12 kg 300g

3

14 kg 600 g

4

16 kg 700 g

5

18 kg 700 g

6

20 kg 700 g

7

22 kg 900 g

8

25 kg 300 g

9

28 kg 100 g

AGE WEIGHT Weight gain from (yrs) (kg) previous year

-

168

10

31 kg 400 g

11

32 kg 200 g

12

37 kg

13

40 kg 900 g

14

47 kg

15

52 kg 600 g

16

58 kg

17

62 kg 700 g

18

65 kg

Customary Units of Volume Units of volume in the customary system gallon for large amounts of liquid (gal) 4

quart for medium-size amounts of liquid (qt)

2

pint

for medium-size amounts of liquid (pt)

2

cup

for small amounts of liquid (C)

8

ounce for small amounts of liquid (oz.)

1. Fill in the tables. cups

1

2

3 1/2

ounces gallons

4

6

24

96

1

4 1/2

quarts

8

6 22

Cups

64

160

2. More, less, or the same amount? Write >,