MathMammoth_Grade3A

Copyright 2007-2010 Taina Maria Miller. EDITION 1.3 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/

2

Contents Foreword ...........................................................................

5

Chapter 1: Addition and Subtraction Strategies Introduction ......................................................................

6

Addition Review ...............................................................

8

Addition Terminology and Practice ..............................

11

Ordinal Numbers and Roman Numerals ......................

15

Add in Columns ................................................................ 18 Subtraction Review .......................................................... 20 Subtraction Strategies and Terminology ......................

23

Subtracting in Columns ................................................... 27 Addition/Subtraction Connection ..................................

31

Borrowing Over Zero Tens ............................................

34

Mileage Chart .................................................................

37

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

38

Graphs .............................................................................

40

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

42

Chapter 2: Multiplication Concept Introduction ....................................................................

44

Many Times the Same Group ......................................

46

Multiplication and Addition ...........................................

47

Multiplication as an Array .............................................

50

Multiplying on a Number Line ......................................

52

Multiplication in Two Ways ...........................................

55

Multiplying by Zero ........................................................

59

Word Problems ...............................................................

61

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

63

Understanding Word Problems .....................................

65

Practice with Parts ..........................................................

68

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

71

3

Chapter 3: Multiplication Tables Introduction ....................................................................

72

Multiplication Table of 2 ...............................................

76

Multiplication Table of 4 ...............................................

78

Multiplication Table of 10 .............................................

80

Multiplication Table of 5 ...............................................

83

More Practice and Review (Tables of 2, 4, 5, and 10) ...............................................

86

Multiplication Table of 3 ................................................ 89 Multiplication Table of 6 ................................................ 92 Multiplication Table of 11 .............................................. 94 Multiplication Table of 9 ................................................ 97 Multiplication Table of 7 ................................................ 101 Multiplication Table of 8 ................................................ 103 Multiplication Table of 12 .............................................. 106 Review ............................................................................. 109

Chapter 4: Telling Time Introduction ..................................................................... 112 Review: Reading the Clock ............................................ 114 Reading the Clock to the Minute ................................

116

Elapsed Time ................................................................... 119 More on Elapsed Time ................................................... 121 Using the Calendar ......................................................... 125 Changing Time Units ...................................................... 127 Review ............................................................................. 131

Chapter 5: Money Introduction .................................................................... 132 Using the Half-Dollar ..................................................... 134 Dollars ............................................................................. 136 Making Change .............................................................. 139 Mental Math and Money Problems .............................. 143 Solving Money Problems ................................................ 146 Review ............................................................................. 150

4

Foreword Math Mammoth Grade 3-A and Grade 3-B worktexts comprise a complete math curriculum for the third grade mathematics studies. Third grade is a time for learning and mastering two (mostly new) operations: multiplication and division with single-digit numbers. The student also deepens his understanding of addition and subtraction, and uses those in many different contexts, such as with money, time, and measuring. The first chapter in this book deals with addition and subtraction strategies. The student does a lot of mental math, learns addition and subtraction terminology, touches on algebraic problems in the lesson about addition/subtraction connection, practices borrowing, and more. Then we tackle the multiplication concept in chapter 2. After that come multiplication tables in chapter 3, so multiplication does take a big part of book A. Then comes a chapter about reading the clock time (chapter 4) and a chapter about money (chapter 5). In part B, we study place value with thousands (chapter 6), then measuring and geometry (chapters 7 and 8), followed by division in chapter 9. In chapter 10, we study a little about multiplying bigger numbers, and finally in chapter 11, it is time for some introductory fraction and decimal topics. 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. Chapter 1 should be studied before chapter 8 (place value). Multiplication chapters need to be studied before the division chapter, and all of those need to be studied before the chapter about all four operations (chapter 10). You can go through the chapters about clock, money, geometry, measuring, and fractions/decimals in some other order, if you desire. This curriculum aims to concentrate on a few major topics at a time and study them in depth. It is for this reason that you will not see some topics that might be present in other third grade books, such as long division, or the standard way of multiplying vertically. I wanted the student to get a very good foundation in basic multiplication and basic division by single-digit numbers. I did not want to hurry through measuring topics, yet I didn't want to make a 500-600 page book either. There is plenty of time in grades 4, 5, and 6 to master division and multiplication with bigger numbers. This is totally opposite to the continually spiraling step-by-step curricula in which each lesson typically is about a different topic from the previous or next lesson, and includes a lot of review problems from past topics. This does not mean that your child wouldn't need occasional review. However, when each major topic is presented in its own chapter, this gives you more freedom to plan the course of study and choose the review times yourself. In fact, I totally encourage you to plan your mathematics school year as a set of certain topics, instead of a certain book or certain pages from a book. For review, I have included an html page called Make_extra_worksheets_grade3.htm that you can use to make additional worksheets for computation or for number charts. You can also always simply reprint some already studied pages. I wish you success in your math teaching! Maria Miller, the author 5

Chapter 1: Addition and Subtraction Introduction This first chapter of Math Mammoth Grade 3-A Complete Worktext covers various addition and subtraction topics, ordinal numbers, and Roman numerals. This chapter includes a lot of mental adding and subtracting. Some of it is review from second grade, but some lessons probably contain ideas and strategies that will be new to the student. Adding or subtracting in parts is a strategy that is emphasized a lot. With subtraction, the strategy of “adding up” is taught again. Children also get to notice how the sum or difference changes when the numbers in the problems change and how that can be used to solve problems mentally. The connection between addition and subtraction should already be a familiar topic, but the lesson Addition/Subtraction Connection practices the conversions with bigger numbers. This lesson also aims to help children think algebraically. We also practice subtracting in columns, borrowing from both hundreds and tens, and borrowing “over zeros.” The lessons illustrate this last process with the help of pictures that relate to the hundreds, tens, and ones place values. The idea stressed is that a “borrowed” unit gets broken down into 10 smaller units, a hundred into 10 tens or a ten into 10 ones, and that is what lets you subtract. Make sure the student masters this topic. You can make more practice sheets using the “Make More Worksheets” file on the accompanying CD. This chapter also introduces parentheses and order of operations. Students get to practice their adding and subtracting skills in a practical way through reading a mileage chart and other types of graphs

The Lessons in Chapter 1 page

span

8

3 pages

Addition Terminology and Practice ................... 11

4 pages

Ordinal Numbers and Roman Numerals ............ 15

3 pages

Add in Columns .................................................

18

2 pages

Subtraction Review ............................................ 20

3 pages

Subtraction Strategies and Terminology ...........

23

4 pages

Subtracting in Columns .....................................

27

4 pages

Addition/Subtraction Connection ......................

31

3 pages

Borrowing Over Zero Tens ...............................

34

3 pages

Mileage Chart ....................................................

37

1 pages

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

38

2 pages

Graphs ................................................................ 40

2 pages

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

2 pages

Addition Review ................................................

42

6

Helpful Resources on the Internet Use these free online resources to supplement the “bookwork” as you see fit. Number Puzzles Place the numbers to the puzzle so that each side adds up to a given sum. Practices mental addition and logical thinking. http://nlvm.usu.edu/en/nav/frames_asid_157_g_2_t_1.html Speedy Sums Click on numbers that add to the target sum. The more numbers you use, the more you score. http://www.mathplayground.com/speedy_sums.html Thinking Blocks Thinking Blocks is an interactive math tool that lets students build diagrams similar to the bar diagrams used in this chapter. Choose the Addition and Subtraction section. http://www.mathplayground.com/thinkingblocks.html Callum's Addition Pyramid Add the pairs of numbers to get a number on the next level and finally the top number. Three difficulty levels. http://www.amblesideprimary.com/ambleweb/mentalmaths/pyramid.html MathBlox Click on two falling blocks that add up to the given number and they disappear. Try some of the harder levels, such as addition to 50. http://www.iknowthat.com/com/L3?Area=Mathblox

7

Addition Review Breaking numbers into their parts often makes adding easier:

30 + 28

8+6

/ \

12 + 60

/ \

8 + 2 + 4 = ____

/ \

30 + 20 + 8 = ____

2 + 10 + 60 = ____

1. Let's review addition! a.

b.

c.

d.

8+7=

23 + 7 =

6+6=

45 + 5 =

18 + 7 =

23 + 6 =

7+7=

45 + 8 =

18 + 8 =

23 + 9 =

8+8=

45 + 6 =

2. Break one of the numbers into two parts to make the adding easier: a. 50 + 14 =

b. 80 + 11=

c. 50 + 39 =

d. 43 + 20 =

e. 35 + 60 =

f. 22 + 50 =

g. 29 + 40 + 30 =

h. 10 + 5 + 21 =

3. Add a number between 1 and 10 so that the sum (the answer) ends in 1. a.

b.

c.

d.

28 + ___ = 31

76 + ___ = ____

83 + ___ = ____

64 + ___ = ____

45 + ___ = 51

59 + ___ = ____

66 + ___ = ____

83 + ___ = ____

8

4. Add the same number repeatedly. You can add in parts (part-by-part). a. 15

b. 25

c. 40

d. 9

+ 15 30

+ 25 ____

+ 40 ____

+ 9 ____

+ 15 ____

+ 25 ____

+ 40 ____

+ 9 ____

+ 15 ____

+ 25 ____

+ 40 ____

+ 9 ____

+ 15 ____

+ 25 ____

+ 40 ____

+ 9 ____

+ 15 ____

+ 25 ____

+ 40 ____

+ 9 ____

+ 15 ____

+ 25 ____

+ 40 ____

+ 9 ____

If the number you add changes, the sum (answer) changes in the same way!

56 + 4 = 60 56 + 5 = 61

17 + 100 = 117 17 + 99 = 116

1 more

15 + 15 = 30 15+ 17 = 32

1 less

2 more

5. Compare each pair of problems as you go: a.

b.

c.

d.

48 + 20 = ____

28 + 100 = ____

25 + 25 = ____

15 + 15 = ____

48 + 21 = ____

28 + 99 = ____

25 + 27 = ____

18 + 15 = ____

e.

f.

g.

h.

200 + 36 = ____

36 + 40 = ____

8 + 8 = ____

46 + 50 = ____

199 + 36 = ____

36 + 39 = ____

8 + 9 = ____

46 + 47 = ____

i.

j.

k.

l.

220 + 50 = ____

530 + 80 = ____

270 + 30 = ____

670 + 20 = ____

227 + 50 = ____

532 + 82 = ____

276 + 32 = ____

669 + 19 = ____

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6. Add these part-by-part. First add what is inside the parentheses ( ). a.

b.

(20 + 40) + (2 + 7) ____ + ____ = ____

c.

(30 + 50) + (8 + 2)

(40 + 60) + (4 + 3)

____ + ____ = ____

____ + ____ = ____

7. Add these in the easiest order. You can break numbers into their parts and add part-by-part. a. 10 + 12 + 7 =

b. 50 + 4 + 30 + 7 =

c. 78 + 10 + 2 + 20 =

8. Draw a line to connect each problem to its answer.

29 + ___ = 36

86 + ___ = 96

7

66 + ___ = 76

46 + ___ = 56

48 + ___ = 56

10

57 + ___ = 66

50 + ___ = 56

9

38 + ___ = 46

87 + ___ = 96

89 + ___ = 96

6

70 + ___ = 76

39 + ___ = 46 8

68 + ___ = 76

77 + ___ = 86

Solve the mystery numbers a.

+

!

b.

c.

+

= 22

+

–

=4

+

+ 10 = 34

= _____

and

= _____ = _____

10

= 22 = 36

= _____ = _____

Addition Terminology and Practice The numbers you add are all called addends. The answer is called the sum. It is called a sum even when you haven’t yet calculated it. So 13 + 7 is the sum of 13 and 7. The whole “thing” is an addition sentence.

1. Write 20, 100, 500, and 138 as sums in three different ways. a. 20 = 14 + 6

b. 100 =

c. 500 =

d. 138 =

20 =

100 =

500 =

138 =

20 =

100 =

500 =

138 =

How many different answers are there? ________________________ 2. Fill in each blank, and then write an equivalent addition sentence. a. Two of the addends are 8 and 7, and the sum is 20.

The third addend is ______.

____ + ____+ ____ = ____

b. The addends are 50, 60, and 70.

The sum is ______. c. The sum is 80, and three of the addends are

22, 20, and 10. The fourth addend is ______. 3. Complete the next whole hundred. a. 40 + ____ = 100

b. 96 + ____ = 100

c. 60 + ____ = 100

80 + ____ = 100

196 + ____ = ____

360 + ____ = ____

180 + ____ = 200

496 + ____ = ____

960 + ____ = ____

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If an addend changes, then the sum changes the same way!

90 + 30 = 120 95 + 30 = 125

200 + 40 = 240 199 + 40 = 239

40 + 70 = 110 38 + 69 = 107

You start out with 5 more, so the answer is 5 more.

You start out with 1 less, so the answer is 1 less.

You start out with 3 less, so the answer is 3 less.

4. Add and compare the problems. a. 7 + 8 =

b. 9 + 9 =

c. 5 + 6 =

d. 8 + 5 =

70 + 80 =

90 + 90 =

50 + 60 =

180 + 50 =

70 + 82 =

90 + 95 =

54 + 60 =

180 + 57 =

5. The addends change; think carefully how the sum changes. a. 100 + 60 =

99 + 60 =

b. 30 + 140 =

c. 500 + 60 =

d. 110 + 80 =

29 + 139 =

499 + 63 =

108 + 79 =

6. Can you think of an easy “trick” to find these sums? a. 99 + 99

b. 499 + 299

c. 199 + 198 + 197

7. In each top problem, complete the next hundred. Use the top problem to help you solve the bottom one. a.

640 + 60 = 700

b.

640 + 70 = 710 d.

350 + ___ = 400

592 + ___ = 592 + ___= 622

c.

180 + ___ = 210 e.

350 + ___ = 420 g.

180 + ___ = 200

230 + ___ =

850 + ___ = 910 f.

230 + ___ = 330 h.

420 + ___ = 420 + ___ = 530

12

850 + ___ = 900

660 + ___ = 660 + ___ = 740

i.

770 + ___ = 770 + ___ = 850

8. Add. Compare the problems in each set. Follow the pattern to write the third problems for sets (g) and (h) yourself. a. 5 + 6 =

b. 7 + 5 =

c. 5 + 7 =

d. 8 + 7 =

35 + 6 =

77 + 5 =

35 + 7 =

18 + 7 =

350 + 60 =

770 + 50 =

350 + 70 =

180 + 70 =

e. 9 + 8 =

f. 6 + 9 =

69 + 8 =

76 + 9 =

690 + 80 =

760 + 90 =

g. 4 + 8 =

h. 9 + 9 =

34 + 8 =

49 + 9 =

+

=

9. Add these in parts. a. 200 + 50 + 4

b. 40 + 500 + 9

c. 300 + 20 + 400

d. 4 + 9 + 20 + 800

e. 18 + 700 + 40

f. 29 + 40 + 30 + 6 + 600

g. 400 + 506

h. 144 + 50

i. 24 + 512

10. Carmen had 23 pretty stones, Jane had 18, and Julie had 30. a. How many stones do the three girls have all together? b. Julie gave five of her stones to Jane. How many more does Julie have than Jane now? c. Now how many stones do the three girls have all together?

11. Continue the sequences. a. 6, 27, 48, 69, _____, _____, _____, _____, _____, _____, _____, _____. b. 14, 34, 24, 44, 34, 54, _____, _____, _____, _____, _____, _____, _____. c. 250, 305, 360, 415, _____, _____, _____, _____, _____, _____, _____.

13

12. Add the tens and the ones separately. a.

36 + 22 = (30 + 20) + (6 + 2) = + =

b.

72 + 18 = (70 + 10) + (2 + 8) = + =

c.

54 + 37 = (50 + 30) + (4 + 7) = + =

d.

24 + 55 = (__ + __) + (_ + _) = + =

e.

36 + 36 = (__ + __) + (_ + _) = + =

f.

42 + 68 = (__ + __) + (_ + _) = + =

13. Add mentally. a. 14 + 14 =

b. 23 + 23 =

c. 35 + 35 =

d. 17 + 17 =

16 + 16 =

24 + 24 =

38 + 38 =

27 + 27 =

18 + 18 =

25 + 25 =

32 + 32 =

37 + 37 =

e. 36 + 38 =

f. 45 + 46 =

g. 39 + 56 =

h. 47 + 34 =

23 + 57 =

14 + 28 =

16 + 78 =

27 + 24 =

27 + 41 =

28 + 13 =

37 + 33 =

72 + 19 =

Multiplication means adding the same number multiple times.

2 × 14

3 × 12

14 + 14 = 28

12 + 12 + 12 = 36

14. Solve these by adding. a. 2 × 15 =

b. 2 × 23 =

c. 2 × 150 =

2 × 16 =

2 × 32 =

2 × 64 =

2 × 408 =

2 × 19 =

2 × 37 =

3 × 11 =

3 × 31 =

15. A bundle of two towels costs $13. How much would four towels cost?

14

d. 2 × 214 =

Ordinal Numbers and Roman Numerals Ordinal numbers are used when you put things in order—in other words, when the order is important.

TIMBUKTU

TIMBUKTU

The third letter from the left is M.

The first three letters from the left are TIM.

Here is a short list of some ordinal numbers and how they are abbreviated: first - 1st second - 2nd third - 3rd fourth - 4th fifth - 5th sixth - 6th ninth - 9th

tenth - 10th eleventh - 11th twelfth - 12th thirteenth - 13th fifteenth - 15th sixteenth - 16th eighteenth - 18th

twentieth - 20th twenty-first - 21st twenty-second - 22nd twenty-fifth - 25th twenty-ninth - 29th

thirtieth - 30th fortieth - 40th fiftieth - 50th hundredth - 100th hundred first - 101st hundred twelfth - 112th two hundred twentythird - 223rd

Most of the time, you just add “-th” to the normal (cardinal) number. Some exceptions: z z z

If the number ends in “y,” such as “twenty,” change the “y” to “ie.” “Five” changes to “fifth” and “twelve” changes to “twelfth” (“-ve” changes to “f”) “Nine” and “twelve” drop the final “e”: “ninth,” “twelfth.”

For 1 and any number ending in 1, use “first”; for example, thirty-first. For 2 and any number ending in 2, use “second”; for example, fifty-second. For 3 and any number ending in 3, use “third”; for example, one hundred and sixty-third.

1. a) Color the first four people from the right.

b) Color the fourth person from the right.

2. Write the ordinal number. a. 31

e. 99

b. 9

f. 52

c. 12

g. 61

d. 57

h. 43

15

3. Number the people with ordinal numbers starting from the left. Use the picture to help answer the questions. a. How many people are there between the third person and the sixth person?

Note: Do not count the 3rd and the 6th persons themselves. b. How many people are there between the first person and the seventh person? c. Jeff is the ninth in the line. Jane is ahead of him,

and there are three people between Jeff and Jane. What is Jane’s position in the line? d. Jack is the tenth in the line. Mark is ahead of him,

and there are seven people between them. What is Mark’s position in the line? e. The first three people in line are wearing black.

The fifth person after those three is wearing blue. What is that person’s position in the line?

Roman Numerals In ancient Rome, people wrote numbers using letters such as I, V, X, L and C.

I is 1

V is 5

X is 10 L is 50

C is 100

Using I, X, and C more than once just means you add their values:

II 2

III 3

XX 20

XXX 30

CC 200

When bigger symbols come before smaller ones, add the values.

VI 6

XI 11

LX 60

XXVI 26

CXXV 125

There are more rules too: for example, 5 is not IIIII, but let’s practice these first. 4. Write the Roman numerals using normal numbers. a. II

b. XV

c. XXXVIII

d. LXIII

e. LXXXIII

VII

XXI

LIII

LXV

CX

VIII

XXII

LVI

LXXX

CVII

XII

XXXV

LXI

LXXVII

CLXXX

16

If a smaller unit comes before a bigger unit, you subtract the smaller from the bigger.

IV is 4

IX is 9

1 before 5

1 before 10

5−1

10 − 1

50 − 10

100 − 10

You can combine IV, IX , XL and XC symbols with others, and add their values:

XIV = 14 10 and 4

XXIX = 29 20 and 9

XLV = 45 40 and 5

XCIX = 99 90 and 9

NOTE: For 49 don’t use IL, but XLIX.

XL is 40

XC is 90

10 before 50 10 before 100

For 99, don’t use IC, but XCIX.

5. Write the Roman numerals using normal numbers. a. IV

b. XXIV

c. XXIX

d. XL

e. XLI

f. XLIX

g. XLIV

h. XCIII

i. LXXIV

j. LIX

k. LXXXV

l. LXXXIX

m. LIV

n. LVI

o. CCIX

p. XCIV

6. Write using Roman numerals. a. 15

b. 31

c. 42

d. 50

16

32

43

51

17

33

44

52

e. 62

f. 75

g. 69

h. 97

63

76

70

98

64

77

71

99

7. Add and subtract using Roman numerals. Write your answer as a Roman numeral. a. IV + VI

b. XI + IX

c. XX + LX

d. XII + XVI

e. XL + LIX

f. XXXIX + L

g. LX − XXX

h. XC − XL

i. LXXIV − IV

j. C − VI

k. XLIX − XXIII

l. LXXX − XXVI

17

Add in Columns 3 5 6 + 5

3 6 + 7

1 5 6 + 9

31 55 86 + 29

45 42 75 + 57

First add the numbers that make a ten.

1. Add. a.

b.

34 212 258 + 56

182 527 159 + 43

c.

280 149 154 + 276

d.

138 364 265 + 182

e.

56 229 119 + 454

2. Add in columns. Write the hundreds, tens, and ones neatly in the correct columns. a. 524 + 68

b. 56 + 309 + 162

c. 435 + 79 + 8

1

5 +

2 6

4 8

+

+

2 d. 17 + 8 + 340

+

e. 222 + 38 + 159

+

f. 135 + 235 + 96

+

18

3. Solve the word problems. a. One night Dad came home with 24 new

crayons for Louise. She had 36 crayons before. How many does she have now?

b. Dad drove 35 kilometers to a farm and picked

60 pounds of strawberries. Then he drove 19 km to a library, and from there 22 km home. How many kilometers did Dad drive?

c. Jim mixed together pieces from two 192-piece

puzzles and from one 48-piece puzzle. How many pieces were in the pile?

d. Candles are packaged in boxes of 300. Mary has three

boxes. She took 12 candles out of one of the boxes. How many candles are in the three boxes now?

e. Mr. Jackson put two 50-kg sacks of potatoes,

two 35-kg sacks of carrots, and three crates of tomatoes weighing 25 kg each onto his pickup truck. Then two workmen weighing 78 kg and 92 kg climbed onto the truck. Find the total weight of the load on the truck.

19

Subtraction Review 1. Let’s review some easy subtraction problems! a.

b.

c.

d.

10 − 7 = _____

70 − 5 = _____

50 − 2 = _____

12 − 2 − 5 = _____

10 − 4 = _____

70 − 2 − 2 = _____

50 − 5 − 2 = _____

25 − 5 − 8 = _____

20 − 4 = _____

70 − 8 = _____

50 − 7 = _____

73 − 3 − 5 = _____

e.

f.

g.

h.

70 − _____= 63

46 − 6 − 6 = _____

90 − 8 = _____

14 − ____ − 5 = 5

40 − _____ = 35

89 − 9 − 9 = _____

100 − 8 = _____

18 − ____ − 4 = 9

100 − _____ = 91

77 − 7 − 7 = _____

110 − 8 = _____

15 − ____ − 6 = 7

2. Subtract whole tens. The box with a “T” is a ten. a.

c.

b.

d.

84 − 40 = ______

______ − 10 = 57

35 − 20 = ______

______ − 10 = 83

54 − 10 = ______

67 − 20 = ______

51 − 30 = ______

______ − 20 = 17

54 − 30 = ______

67 − 50 = ______

62 − 30 = ______

______ − 40 = 43

54 − 20 = ______

67 − 60 = ______

87 − 50 = ______

______ − 60 = 3

3. Test yourself! – 30

– 4

100 _____

– 5

____

– 20

____

– 1

____

20

– 9

____

– 10

____

– 21

____

____

4. Subtraction is used: a. to find what’s left

b. to find a difference

c. to find a part of the whole

Debbie had $45, and she spent $10. How much does she have left?

A certain lawnmower costs $340 in one store and $360 in another. How much more does it cost in the second store?

In a package of 100 buttons, 50 of them are white, 25 are blue, and the rest are red. How many are red?

$360 − $340 = ________

100 − 50 − 25 = _______

It costs $________ more.

________ buttons are red.

$45 − $10 = _______. She has ________ left.

5. Fill in the missing numbers. a. Ben has saved 22 dollars. He still

b. Jill earned $5 for raking the yard

needs $_______ to buy a bicycle that costs $30.

and another $5 for weeding. She had already saved $20, so now she has $________.

c. Mom bought 28 bushes and planted

d. The Smiths drank seven bottles

eight of them. She still needs to plant ________ bushes.

of water. The Burns drank ________. All together they consumed 21 bottles.

e. Ann bought 18 candles and Jill bought 5.

f. Dad ate 12 cookies, Mom ate five,

Ann has _______ more candles than Jill, and together they have _______ candles.

little sister ate two, and there are still nine left. So originally the family had _______ cookies.

g. Meredith had $20, and she bought a gift

h. A box has 35 thumbtacks, and another

for $________. Now she has $13 left.

has 42. The latter box has _______ more.

6. Subtract part-by-part: first to the previous whole ten, and then the rest. a. 64 − 7

b. 72 − 8

c. 54 − 8

d. 45 − 9

f. 27 − 9

g. 43 − 5

h. 51 − 5

64 − 4 − 3 = _____ e. 75 − 7

75 − 5 − 2 = _____

21

7. Basic subtraction facts might need practice! Point to the problems, think of the answer, and drill them. a.

b.

c.

d.

e.

12 − 5

13 − 8

14 − 5

15 − 6

16 − 7

12 − 7

13 − 4

14 − 7

15 − 8

16 − 9

12 − 8

13 − 5

14 − 9

15 − 9

16 − 8

12 − 6

13 − 6

14 − 6

15 − 7

12 − 4

13 − 9

14 − 8

12 − 9

13 − 7

f.

17 − 8 17 − 9

12 − 3 8. Subtract and compare the results! What pattern do you notice? a. 14 − 7 = _____

b. 12 − 8 = _____

c. 16 − 7 = _____

d. 15 − 7 = _____

34 − 7 = _____

42 − 8 = _____

56 − 7 = _____

75 − 7 = _____

64 − 7 = _____

82 − 8 = _____

156 − 7 = _____

105 − 7 = _____

9. Subtract in parts: Break the second number into tens and ones. a.

26 89 – 20 – 6 79 – 6 = _____ 89 –

b.

35 56 – – _____ – ___ = _____ 56 –

d.

c.

51 75 – – _____ – ___ = _____ 75 –

e.

f.

69 –

19 69 – – _____ – ___ = _____

67 – 36 =

64 – 33 =

g. 97 – 64 =

h. 55 –

i. 56 – 23 =

34 =

22

Subtraction Strategies and Terminology The number you subtract from is called the minuend. The number you subtract from the minuend is called the subtrahend. Mnemonic: minuend comes first, subtrahend after, just like m is before s in the alphabet. When you subtract two numbers, the answer is called the difference. It is called a difference even when you haven't yet calculated it. So 12 – 6 is the difference of 12 and 6.

1. Write 1, 10, and 62 as differences in many different ways. a. ____ – ____ = 1

b. ____ – ____ = 10

c. ____ – ____ = 62

____ – ____ = 1

____ – ____ = 10

____ – ____ = 62

____ – ____ = 1

____ – ____ = 10

____ – ____ = 62

Think carefully: How many different answers are there in (a)? In (b)? In (c)? 2. Write a subtraction sentence and fill in the blanks. a. The subtrahend is 15 and the difference is 4.

The minuend is ________. b. The minuend is 49 and the subtrahend is 23.

The difference is ________. c. The minuend is 38 and the difference is 19.

_____ – _____ = _____ _____ – _____ = _____ _____ – _____ = _____

The subtrahend is _________.

3. Think carefully: how can you find the missing minuend no matter what the numbers are? a. _____ – 4 = 5

_____ – 10 = 20

b. ______ – 15 = 30

______ – 200 = 5

23

c. ______ – 23 = 800

______ – 30 – 22 = 10

If the minuend changes, the difference changes the same way!

90 – 30 = 60 95 – 30 = 65

75 – 25 = 50 74 – 25 = 49

240 – 165 = 75 238 – 165 = 73

If you start out with 5 more; the answer is 5 more.

If you start out with 1 less; the answer is 1 less.

If you start out with 2 less; the answer is 2 less.

If the subtrahend changes, the difference changes in the opposite way!

56 – 30 = 26 56 – 29 = 27 If you subtract 1 less; the answer is 1 more.

650 – 100 = 550 650 – 99 = 551 1 less; 1 more

72 – 12 = 60 72 – 15 = 57 3 more; 3 less

4. The subtrahend or minuend changes; think carefully how the difference changes. a. 95 – 66 = 29

b. 900 – 240 = 660

c. 504 – 37 = 467

95 – 67 = _______

900 – 239 = _______

507 – 37 = _______

95 – 68 = _______

900 – 243 = _______

500 – 37 = _______

d. 340 – 100 = 240

e. 67 – 50 = 17

f. 600 – 28 = 572

340 – 99 = _______

67 – 49 = _______

598 – 28 = _______

340 – 102 = _______

67 – 47 = _______

605 – 28 = _______

5. Compare these expressions without actually calculating. Write or = . a. 60 – 28

60 – 25

b. 90 – 25

90

c. 43 – 8

43 – 18

d. 75 + 24

75 + 36

e. 97 – 32

90 – 32

f. 43 – 28

67 – 28

g. 89 + 32

50 + 89

h. 45 + 27

27 + 44

i. 65 – 28

43 – 28

j. 65 + 13

13 + 65

k. 52 – 25

92 – 25

l. 27 + 27

47

24

Strategy: Add up to find the difference. To find the difference, start at the smaller number, and add up till you get to the bigger number. When adding up, first complete the ten, then add whole tens, then ones again.

84 – 37 = ?

92 – 35 = ?

37 + 3 = 40 40 + 40 = 80 80 + 4 = 84

35 + 5 = 40 40 + 50 = 90 90 + 2 = 92

I added 3, 40, and 4 - total 47. 84 – 37 = 47.

I added total 57. 92 – 35 = 57.

6. Add up to find the difference. a.

+

+

b.

+

65 – 26

+

+

+

83 – 35 26

= _____

30

60

65

35

= _____

40

80

83

c.

d.

e.

f.

56 – 28 = ______

72 – 18 = ______

54 – 37 = _______

74 – 55 = _______

55 – 24 = ______

82 – 46 = ______

91 – 57 = _______

63 – 34 = _______

7. Find missing addends. The same method works here. Think: First, add up to the next whole ten, and then see how much more you need. a.

b.

c.

d.

13 + _____ = 30

25 + _____ = 50

43 + _____ = 70

36 + _____ = 60

37 + _____ = 70

25 + _____ = 54

43 + _____ = 72

36 + _____ = 64

28 + _____ = 90

25 + _____ = 60

54 + _____ = 90

65 + _____ = 80

54 + _____ = 80

25 + _____ = 61

54 + _____ = 93

65 + _____ = 83

25

Strategy: Subtract an easy number that is close, and then correct the answer.

74 – 39 74 – 40 = 34 34 + 1 = 35

81 – 57 81 – 60 = 21 21 + 3 = 24

Subtract 40 since it is close to 39. You subtracted one too much, so add one back.

First subtract 60. Then add 3 back.

8. Subtract mentally. a.

c.

b.

d.

34 – 18 = ______

65 – 27 = ______

97 – 49 = ______

65 – 29 = ______

42 – 29 = ______

55 – 38 = ______

62 – 19 = ______

83 – 38 = ______

e.

f.

g.

h.

66 – 38 = ______

55 – 46 = ______

89 – 56 = ______

52 – 36 = ______

93 – 57 = ______

48 – 13 = ______

57 – 33 = ______

66 – 37 = ______

Solve the mystery numbers

a.

–

= 14

+

= 30

and

!

b.

c.

–7= 99 –

–

–

= 36

–

–

=4

= _______ = _______

= _______

= _______ = _______

26

Subtracting in Columns 1. Remember borrowing? If you can’t subtract the ones or the tens, you need to borrow one unit from the next bigger place value. It is also called regrouping, because you take a bigger unit (a ten or a hundred) and group it with the smaller units (the ones or tens). Remember also to check each subtraction result by adding. Check: a.

762 – 156

Check: b.

+ 156

c.

580 – 341

529 – 357

749 –376

+

Check: d.

Check:

Check: e.

Check: f.

630 – 217

+

465 – 283

2. Solve the word problems. Write an addition or subtraction sentence (equation) for each problem. Check your subtraction by adding. a. Mark has saved $327. The computer that he wants to buy costs $495.

How much more does he need to save in order to buy the computer?

b. Mark’s grandpa gives him $50 for his birthday.

How many dollars does Mark still need to save?

c. Jack’s family is driving from Easttown to Middletown, a distance

of 149 km. The car’s odometer shows they have already driven 67 kilometers. How far do they still have to go?

d. A store sold 178 blue balls and 149 red balls in one day.

At the end of the day, there were 210 blue balls and 239 red balls left. How many blue and red balls did the store start with at the beginning of the day?

27

Borrowing twice: 325 – 169 First break one ten into 10 ones. Now you can cross out the 9 ones, but you still can’t subtract the 6 tens.

There are not enough ones to subtract 9.

Next break one of the hundreds into 10 tens. Now you can cross out the 6 tens, 9 ones, and one hundred. What is left? ______

→

→

325

3 2 5 – 1 6 9

3 hundreds + 1 ten + 15 ones 1

→

15

2 hundreds + 11 tens + 15 ones 2

→

3 2 5 – 1 6 9

11 1 15

3 2 5 – 1 6 9

6

6

First borrow a ten. Then borrow a hundred.

3. Subtract in columns. Draw pictures to illustrate the process. H is hundreds, T is tens, and O is ones.

→ break a 10

a. 221

→

→ break a 100

→

1 H; 11 T; 11 O Now cross out 97.

→ break a 100

→ break a 10

b. 341

→

2 H; 1 T; 11 O

2 2 1 – 9 7

→

__H; ___T; ___O

28

3 4 1 – 1 7 5 __H; ___T; ___O Now cross out 175.

→ break a 100

→ break a 10

c. 350

→

__H; ___T; ___O

→

__H; ___T; ___O Now cross out 287.

→ break a 100

→ break a 10

d. 423

→

3 5 0 – 2 8 7

→

__H; ___T; ___O

4 2 3 – 1 5 6 __H; ___T; ___O Now cross out 156.

4. It’s time for practice. Subtract in columns. Find the answers in the number queue below.

a.

616 – 469

b.

734 – 265

c.

421 – 326

d.

743 – 578

e.

747 – 269

f.

921 – 145

g.

354 – 157

h.

765 – 379

i.

614 – 426

j.

850 – 262

k.

811 – 156

l.

643 – 277

m.

746 – 378

n.

916 – 249

o.

933 – 388

The number queue: 197395265591654783867761881474695883686675458366 29

5. Solve the word problems. Write an addition or subtraction sentence, or several, for each problem. Check your subtractions by adding. a. During a certain year, Littletown had 168 rainy days.

How many days did it not rain that year?

b. Uptown School had 267 students, and Downtown

School had 650. Then 125 students transferred from Downtown School to Uptown School. How many students does Uptown School have now? How about Downtown School? Which school has more students now? How many more?

c. A shipment contains shirts that are either light blue,

white, or striped. 250 shirts are white, and 180 are light blue. Altogether, the shipment contains 775 shirts. How many striped shirts are there in the shipment?

What numbers are missing from the subtractions?

6 0 –

8 3

3 1 5

–

1

6 3 6

–

5 6 4

5 7 2 4 4

30

0

– 4

2 0 7

Addition / Subtraction Connection When two parts make a total, you can: z z

z

add the parts to get the total; subtract the first part from the total to get the second part; subtract the second part from the total to get the first part.

300 + 531 = 831 831 – 300 = 531 831 − 531 = 300

1. For each addition, write two subtraction sentences. Fill in the missing numbers. |-------- total _____ ---------| 670

|-------- total _____ ---------|

|---------- total 390 ----------|

120

a. _____+ _____ = _____

200 b. _____+ _____ = _____

c. _____+ _____ = _____

_____ – _____ = _____

99 – 65 = _____

_____ – _____ = _____

_____ – _____ = _____

_____ – _____ = _____

_____ – _____ = _____

d. _____ + _____ = _____

e. _____ + _____ = _____

400 – 199 = _____ _____ – _____ = _____

_____ – _____ = _____ 95 – _____ = 28

f.

291 + ____ = 400 _____ − ____ = _____ _____ − ____ = _____

2. Find the difference of: a. 100 and 499

b. 405 and 704

3. Ellie has saved $190. She wants a computer that costs $429. How many more dollars does she need to buy it?

4. Peter has saved $49. He wants to buy a cell phone for $80 and the cell phone service for $42. How much does he still need to save?

31

c. 200 and 650

Sometimes you know the total and one part, but you don't know the other part. In this case, you can write both a “missing addend” sentence and a subtraction sentence.

5. Write a missing addend problem and a subtraction problem using the given numbers. |-------- total _____ ---------| 560

|-------- total _____ ---------|

|-------- total _____ ---------|

b. 200 + ____ = 900

c.

100

a. 560 + 100 = 660

750 – 150 = _____

660 – 560 = 100 d. 609 + _____ = 809

f. 700 + _____ = 830

e.

965 – 400 = _____ 6. Solve the problems. Write both a missing addend sentence and a subtraction sentence. a. Ann needs 56 pins for a sewing project. She only has 41.

How many more does she need?

b. You are on page 224 of a book that has 380 pages.

How many pages are left to read?

7. Write a subtraction problem with the given numbers so that the numbers in the boxes are the same. a.

199 +

= 234

____ – ____ = d.

15 +

= 153

____ – ____ =

b.

17 +

= 85

____ – ____ = e.

307 +

= 449

____ – ____ =

32

c.

44 +

= 93

____ – ____ = f.

101 +

= 155

____ – ____ =

8. Solve the problems. a. A cook needs 84 eggs. Eggs are sold in packages of 30.

How many packages does the cook need to buy? How many eggs will be left over? b. The temperature outside is 25 degrees Fahrenheit,

and inside it is 74 degrees. What is the difference in temperature? c. Jack had 83 tennis balls, and Robert had 45.

How many do they have together? Jack lost 11 of his. How many more does Jack have now than Robert? d. A diving suit costs 66 dollars. John has saved $37,

and his grandma gives him $15 more. How much more money does he still need to buy it?

9. Here three parts make up a whole. Write both an addition and a subtraction sentence and solve them. |--------------- total 130 ------------------| 20 b.

a. 560 + 100 + _____ = 960

960 – 560 – 100 = ______ |-------------- total 99 --------------| ?

28

20

c.

d.

33

40

?

Borrowing Over Zero Tens →

→

300

Can't cross out 4 ones - there are none. Can't borrow a ten - there are none!

→

2 hundreds + 10 tens

First break down one hundred into 10 tens. Still can't subtract 4 ones.

→

Then break down one ten-pillar into 10 ones. That leaves only 9 tens. Now, you can subtract. Cross out 1 hundred, 7 tens, and 4 ones. What is left? _______

2 10

3 0 0 – 1 7 4

2 hundreds + 9 tens + 10 ones

2

3 0 0 – 1 7 4

9 10 10

3 0 0 – 1 7 4 6

→

204 Can't cross out five ones since there are only 4.

→

→ 1 hundred + 10 tens + 4 ones →

Break down one hundred into 10 tens.

Break one ten down into 10 ones. Now cross out 6 tens and 5 ones. What is left? _______

1 10

2 0 4 – 6 5

1 hundred + 9 tens + 14 ones

1

2 0 4 – 6 5

9 10 14

2 0 4 – 6 5

First borrow a hundred to make 10 tens, and then borrow one ten to make 10 ones.

34

1. Follow the example. Break down the minuend (the number you subtract from) as needed. Draw squares for hundreds, sticks for tens, and dots for ones. Cross out what is needed. a. Cross out 167 from 300

First break one hundred down into 10 tens. Then break down one ten into 10 ones.

→

300 – 167

→

300

__ hundreds __ tens

__hundreds __tens __ones

b. Cross out 238 from 403 First break one hundred down into 10 tens. Then break down one ten into 10 ones.

→

403

403 – 238

→

__ hundreds __ tens __ ones

__ hundreds __ tens __ ones

c. Cross out 222 from 400

→

400

400 – 222

→

__hundreds __tens

__ hundreds __ tens __ ones

d. Cross out 176 from 305

→

305

305 – 176

→

__ hundreds __ tens __ ones

35

__ hundreds __ tens __ ones

2. Subtract. a.

906 – 469

b.

500 – 225

c.

404 – 326

d.

703 – 536

e.

407 – 269

f.

900 – 111

g.

300 – 172

h.

707 – 309

i.

606 – 448

j.

801 – 232

k.

701 – 156

l.

500 – 341

m.

700 – 376

n.

903 – 646

o.

900 – 611

3. For each problem, write a companion subtraction or addition sentence, and solve. a. ___ + 23 = 66

b. 19 + ___ = 53

c.

66 – 23 = __

d.

___ – 14 = 26

80 – ___= 35

4. Same as above, but the numbers are more difficult. a.

b.

c.

d.

___ + 140 = 660

___ + 111 = 532

770 – ___ = 505

____ – 99 = 365

e.

f.

g.

h.

500 + ___ = 760

567 + ___ = 800

___ – 105 = 230

___ – 102 = 237

36

Mileage Chart To find the distance between Brigham City and Emery, look at the column downwards from Brigham City, and at the row across from Emery. The number at the intersection of that column and row tells you the distance in miles. The distance between Brigham City and Emery is 230 miles. 1. How many miles is it from Fillmore to Emery? 2. How many miles is it from Delta to Brigham City? 3. How many miles is it from Bryce Canyon to Echo Junction?

4. A family made a trip from Emery to Delta, to Cedar Breaks, to Canyonlands National Park, and back home to Emery. What was their total mileage? 5. a. On his way from Cedar Breaks to Brigham City, Dad stopped to refuel after driving 35 miles. How many miles did he still have to go?

b. After another 80 miles he has to stop again.

Now how far does he still have to go?

6. How much longer is the drive from Echo Junction to Canyonlands National Park than to Cedar Breaks?

7. Suppose that Dad averages 45 miles in one hour. Can he then drive from Delta to Capitol Reef National Park in four hours?

37

Order of Operations Addition is an operation, subtraction is another operation, and multiplication is another. If you have more than one operation, there are rules that tell you which one to do first. First calculate the operations enclosed inside parentheses ( ) .

15 – (2 + 3)

(6 + 7) – (4 – 3)

15 – 5 = 10

13

1 = 12

90 – 60 – 20 + 4

80 + 20 – 30

If there are no parentheses, then add and subtract from left to right.

–

\

/

30 – 20 + 4 100 – 30 = 70

\

10

/

+ 4 = 14

1. Calculate. a. 20 – 6 – 2

20 – (6 – 2)

b. 20 + (6 + 2)

c. 20 – 6 + 2

(20 + 6) + 2

20 – (6 + 2)

d. 20 + 6 – 2

20 + (6 – 2)

2. Write an expression to match each instruction. a. “First add 40 and 50, then subtract that sum from 120.”

b. “First add 40 and 50, then subtract 120 from that sum.”

3. Calculate. a. (13 – 6) – (2 + 5)

b. (50 + 8) + (7 – 4)

c. (200 – 40) – (90 – 70)

13 – 6 – 2 + 5

50 + 8 + 7 – 4

200 – 40 – 90 – 70

d. 25 – 10 – 4 + 6

25 – (10 – 4) + 6

e. 700 – (200 + 30 – 1)

700 – 200 + (30 – 1)

f. (100 – 50) – (20 – 10)

100 – (50 – 20 – 10)

4. Add parentheses to each equation to make it true. a. 10 – 5 – 2 = 7

b. 20 – 5 – 2 – 1 = 16

c. 15 – 5 + 2 – 1 = 9

10 – 5 + 2 = 3

20 – 5 – 2 – 1 = 18

15 – 5 + 2 – 1 = 7

38

5. Remember to do addition and subtraction from left to right. For the following problems, use the extra space to do the calculations. a. 234 + 567 – 135 = ___

b.

505 – 317 + 195 = ____

Add these first. Then from the sum subtract 135.

234 + 567

c.

– 135

–

364 + 409 – 238 = ____

+

d.

–

735 – 218 + 450 = ____

–

6. Solve the word problems. a. Julie earned money from picking strawberries as follows:

The first week she earned $178, the second week $215, the third week $230 and the last week $212. The people who owned the farm where she worked subtracted $88 for the cost of her food and lodging. How much did Julie bring home from her job?

b. Ken earns $90 each week. After four weeks, he

paid his parents $125 to help on the cost of food. How much does he have left for himself?

39

+

+

Graphs 1. The graph shows the number of books some children read during a vacation reading assignment. Read the graph and fill in the blanks. ________________ read the most books. ________________ read the fewest books. All together, the four girls read _____ books. All together, the four boys read _____ books. The smallest number of books read was _____ books, and the biggest number was _____ books.

The difference between the most books read and the fewest books read was _____ books. The difference between ______________ and ______________, who read the least amount of books, was only ___ book. 2. Below you see a pictogram that shows how many vegetables were used in certain places. It is called a pictogram because it uses a picture to represent a certain amount. In this case, each picture of a carrot represents 5 kilograms. Read the pictogram to answer the questions. Restaurant B used _____ kg of vegetables. Vegetable use in one month

The Millers used _____ kg. The Jacksons used _____ kg.

Jacksons

The school cafe used _____ kg.

Joneses

Restaurant B used _____ kg more than the Joneses.

Millers School cafe

The Millers used _____ kg fewer vegetables than the school cafe.

Restaurant A

All together, the Jacksons, Joneses, and Millers used _____ kg of vegetables.

Restaurant B

The two restaurants used a total of _____ kg of vegetables.

= 5 kilograms of vegetables

40

3. On Monday a small art museum had 29 adult and 14 child visitors; on Tuesday it had 23 adult and 10 child visitors; on Wednesday 34 adult and 18 child visitors; and on Thursday 38 adult and 19 child visitors. Use that data to fill in the table. Also calculate the total visitor counts.

Museum visitors Day

Adults Children

Total Visitors

Monday Tuesday Wednesday

a. The busiest day was _______________.

The least busy day was _______________. b. What was the difference in the total visitor

count between those two days?

Thursday Friday

35

19

Saturday

57

25

Sunday

63

31

Totals c. The total visitor count for the whole week

(both adults and children) was ______ visitors. d. During the whole week, how many more

adults than children visited the museum? e. Make a double-bar graph of this data. For each day, make a bar for the number of adults

and another bar for the number of children. The bars for Monday and Tuesday are already made.

41

Review 1. Use the solution for the top problem to help you solve the bottom problem mentally. a. 300 + 50 =

b. 60 + 70 =

c. 500 – 60 =

d. 990 – 400 =

299 + 50 =

59 + 68 =

501 – 60 =

990 – 402 =

2. Write the numbers as hundreds, tens, and ones. Then add in parts. a. 44 + 503

b. 643 + 52

3. Write the Roman numerals using normal numbers. a. VI

b. LVI

c. LXV

d. XLVIII

4. Add up to find the difference. +

a. 71 – 26 =

26

+

30

+

70

71

b. 63 – 27 =

d. 94 – 58 =

c. 82 – 15 =

e. 101 – 27 =

5. Subtract. a.

405 – 266

b.

510 – 216

c.

807 – 429

d.

503 – 126

6. Calculate. a. 50 – 20 – 5 + 6

d. (500 – 50) + (70 – 10)

b. 50 – (20 – 5) + 6

e. 500 – (50 + 70 – 10)

c. 50 – (20 – 5 + 6)

f. 500 – 50 + (70 – 10)

42

e.

415 – 249

7. Solve: ___ – 318 = 467

8. Solve: 693 + 183 – (800 – 34).

9. A store sells CDs in boxes of 100. Ann bought three full boxes and one box from which 14 CDs had been removed earlier. How many CDs did she get?

10. There are 800 beads in a bag. Some are yellow, some are red, and some are blue. If there are 270 red and 270 blue beads, find how many yellow beads are in the bag.

11. a. Which day was the busiest? b. Which day was the least busy? c. These numbers are visitors counts for adults for the days shown on the chart, but they are not in the right order. Match each visitor count with the right day. 109

67

47

144

54

d. How many adults all totaled visited the park during the week?

43

132

34

Chapter 2: Multiplication Concept Introduction The second chapter of Math Mammoth Grade 3-A Complete Worktext covers the concept of multiplication. (However, memorizing and drilling “times tables” is postponed until chapter 3.) The first lessons introduce the concept of multiplication as repeated addition of groups of the same size. Then the lesson Multiplication as an Array shows a different model for multiplication: objects arranged in rows and columns. This lesson teaches the student to think of the rows as groups, showing the fundamental unity of the two models. The whole lesson is presented in pictures. Multiplication on a Number Line illustrates repeated addition as consecutive jumps or skips on a number line. The student learns to connect skip-counting with multiplication. Multiplication in Two Ways concentrates on the fact that it does not matter in which order the factors appear (the commutative property of multiplication). Objects in an array illustrate this fact nicely: either the row or the column can be taken as the group being multiplied. This lesson also deals with jumping on the number line. Multiplying By Zero is illustrated both with the group model (either several groups of zero size or zero groups of any size) and with the jump-on-a-number-line model (either several jumps of zero distance or zero jumps of any distance). Understanding Word Problems shows how problems, including multiplication, have the idea of “each,” “every,” or “all.” For example: each item does or has the same number of something. If students find these problems difficult, they can draw pictures to help, such as drawing flowers in pots, slices of pizza, etc. The lesson Order of Operations teaches that multiplication is to be done before addition or subtraction and that addition and subtraction are to be done from left to right. Understanding Word Problems, Part 2 gives more challenging problems. The word problems in traditional school texts are often so easy that students learn just to take the numbers in the problem and mechanically apply the operation that the lesson is about without really understanding what they’re doing. If this lesson is too difficult, skip it for the time being and come back to it later. You can help your student to draw a picture for each problem.

The Lessons in Chapter 2 page

span

46

1 page

Multiplication and Addition ................................ 47

3 pages

Multiplication as an Array ..................................

50

2 pages

Multiplication on a Number Line .......................

52

3 pages

Multiplication in Two Ways ...............................

55

4 pages

Many Times the Same Group .............................

44

Multiplication by Zero ........................................

59

2 pages

Word Problems ...................................................

61

2 pages

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

63

2 pages

Understanding Word Problems ...........................

65

3 pages

Practice with Parts ..............................................

68

3 pages

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

71

1 pages

Helpful Resources on the Internet Use these free online resources to supplement the “bookwork” as you see fit. Math Dice Game for Addition and Multiplication Instructions for three simple games with dice: one to learn the concept of multiplication, another to practice the times tables, and one more for addition facts. http://www.teachingwithtlc.blogspot.com/2007/09/math-dice-games-for-addition-and.html Explore the Multiplication Table This applet visualizes multiplication as a rectangle. http://www.mathcats.com/explore/multiplicationtable.html Multiplication Number Lines First choose a tile from the 10×10 grid to pose a problem, then you will see it illustrated on a number line. http://www.ictgames.com/multinumberlines.html Multiplication Memory Game Click on corresponding pairs (the problem and its answer). http://www.dositey.com/2008/addsub/memorymult.html Multiplication Mystery Drag the answer tiles to right places in the grid as they are given, and a picture is revealed http://www.harcourtschool.com/activity/mult/mult.html Multiplication.com Interactive Games A bunch of online games just for the times tables. http://www.multiplication.com/interactive_games.htm

45

Many Times the Same Group This means “3 times a group of 4.”

This means “2 times a group of 6.”

It is called multiplication.

You multiply 2 times 6.

3 × 4

2 × 6

1. Write the multiplication.

a.

2

×

4

d. ____ × ____

b. ____ × ____

c. ____ × ____

e. ____ × ____

f. ____ × ____

2. Now it’s your turn to draw! Notice the symbol × which is read “times.”

a. 2 times 4

b. 3 times 6

c. 1 times 7

2×4

3×6

1×7

d. 6 × 1

e. 4 × 0

f. 2 × 2

46

Multiplication and Addition The symbol × indicates multiplication. Multiplication means that you have a certain number of groups of the same size.

Here we have five groups, and each group has two elephants.

5 × 2 = 10

how many groups

how many in each group

5

×

“Five

2

times two elephants

We can solve it by adding:

=

2 + 2 + 2 + 2 + 2 = 10

is

ten elephants.”

Here there are three groups, and each group has four dogs.

3 × 4 = 12

how many groups

how many in each group

We can solve it by adding:

3

×

4

=

4 + 4 + 4 = 12

“Three

times

four dogs

is

twelve dogs.”

1. Draw dots in groups to match the multiplications.

a. 2 × 6

b. 4 × 2

47

2. Fill in the missing parts. a.

b.

____ groups, ____ scissors in each.

____ groups, ____ rams in each.

____ × ____ scissors = ____ scissors

____ × ____ rams = ____ rams

_____+ _____ + _____ + _____

_____+ _____ + _____

c. d.

____ groups, ____ apples in each. ____ group, ____ carrots in it.

____ × ____ apples = ____ apples

1 × ___ carrots = ___ carrots _____+ _____ + _____

3. Write an addition and a multiplication sentence for each picture. a.

b.

___ + ___ + ___ + ___ = ___

___ + ___ + ___ + ___ + ___ = ___

____ × ____ = ____

____ × ____ = ____

c.

d.

___ + ___ + ___ + ___ + ___ = ___

___ + ___ + ___ + ___ = ___

____ × ____ = ____

____ × ____ = ____

48

4. Now it is your turn to draw. Draw balls or sticks. Write the multiplication sentence. b. Draw 2 groups of eight balls.

a. Draw 3 groups of seven sticks.

____ × ____ = ____ c. Draw 4 groups of four balls.

d. Draw 5 groups of two balls.

5. Draw a picture and solve the problems.

IIII IIII IIII IIII IIII a. 5 × 4 = _____

b. 4 × 6 = _____

c. 3 × 8 = _____

d. 2 × 10 = _____

49

Multiplication as an Array An array is an orderly arrangement of things in rows and columns. When things are aligned in an array, we can treat the rows like groups, so an array still pictures multiplication as repeated addition.

3 rows, 6 crosses in each row.

4 rows, 8 camels in each row.

6+6+6=

8+8+8+8=

3 × 6 = 18

4 × 8 = 32

1. Fill in the missing numbers.

5 + 5

+

10 a. ____ rows, ____ carrots in each row.

b. ____ rows, ____ rams in each row.

_____+ _____ carrots

_____ + _____ + _____ rams

_____ × _____ carrots = _____

_____ ×_____ rams = _____

+

+

c. ____ rows, ____ bear in each row. d. ____ rows, ____ lightbulbs in each row.

_____+ _____ bears

_____+ _____+ _____ bulbs

_____ × _____ bears = _____

_____ × _____ lightbulbs = _____

50

2. Write the addition and multiplication facts that the pictures are illustrating. The box with a “T” is a ten. a.

b.

4+4= 2 × 4 = ____

c.

d.

f.

e.

g.

h.

i.

j.

51

IIIIIIIIII IIIIIIIIII IIIIIIIIII

Multiplying on a Number Line Five jumps, each jump is two steps.

5 × 2 = 10.

Four jumps, each jump is three steps.

4 × 3 = 12.

1. Write the multiplication sentence that the jumps on the number line illustrate.

` a. _____ × ______ = ______

b. _____ × ______ = ______

c. _____ × ______ = ______

d. _____ × ______ = ______

52

2. Draw more “skips” of three.

3. Multiply times 3. Use the skips above to help. a. 5 × 3 =

b. 8 × 3 =

c. 6 × 3 =

d. 2 × 3 =

4×3=

7×3=

3×3=

9×3=

4. How many skips of three are needed? Use the number line above to help. a. ____ × 3 = 24

____ × 3 = 9

b. ____ × 3 = 18

c. ____ × 3 = 21

d. ____ × 3 = 6

____ × 3 = 15

____ × 3 = 12

____ × 3 = 3

5. Draw more “skips” of four.

6. Multiply times 4. Use the skips above to help. a. 2 × 4 =

b. 6 × 4 =

c. 8 × 4 =

d. 5 × 4 =

4×4=

7×4=

3×4=

1×4=

7. How many skips of four are needed? Use the number line above to help. a. ____ × 4 = 24

____ × 4 = 8

b. ____ × 4 = 0

____ × 4 = 12

53

c. ____ × 4 = 16

d. ____ × 4 = 20

____ × 4 = 8

____ × 4 = 4

8. Continue and draw jumps to fit the multiplication problem.

a. 6 × 4 =

b. 5 × 5 =

c. 6 × 5 =

d. 7 × 4 =

e. 3 × 10 =

9. Add repeatedly (or skip-count) to multiply. You can use the number line to help.

a. 3 × 2 =

b. 5 × 2 =

c. 5 × 6 =

d. 3 × 10 =

6×3=

7×4=

3×9=

2 × 11 =

4×5=

3×8=

4 × 10 =

3×7=

54

Multiplication in Two Ways Compare the two pictures: 4 4 +4 3 + 3 + 3 + 3 = 12

12 Three rows; four dogs in each row.

Four columns; three dogs in each column.

3 × 4 = 12

4 × 3 = 12

Five rows; each row has two rams.

Two columns; each column has five rams.

___+___+___ +___+___ rams

___ + ___ rams

5 × 2 = _____

2 × 5 = _____

One row; it has five giraffes.

Five columns; each column has one giraffe.

_____ giraffes

___ + ___ + ___ + ___ + ___ giraffes.

1×5=5

5 × 1 = _____

You can do any multiplication in two different ways, but the result is the same. The order of the numbers to be multiplied does not matter. (In other words, multiplication is commutative.)

55

1. Group the animals in two different ways, and write the addition and multiplication facts that go with the pictures. When do you get the same fact either way? a.

b.

4+4

2+2+2+2

____________

____________

___ × ___ = ___

___ × ___ = ___

___×___ = ___

___×___ = ___

c.

d.

_____________

____________

_____________

_____________

___×___ = ___

___×___ = ___

___ × ___ = ___

___ ×___ = ___

2. Draw X’s and group them in two ways to illustrate the two ways to multiply.

a. 9 × 2 = _____

2 × 9 = _____

b. 4 × 10 = _____

10 × 4 = _____

56

Multiplying in two ways on the number line

5 × 2 = 10 2 × 5 = 10

7 × 2 = 14 2 × 7 = 14

3. For each number line, write the two multiplication sentences that the arrows portray.

a.

b.

c.

d.

57

4. Write down the multiplication sentence. Then write the multiplication the other way, and draw the arrows for that multiplication sentence.

a.

b.

c.

5. Skip-count to fill in the multiplication table of 3. How does the picture relate to it?

1×3=

4×3=

7×3=

10 × 3 =

2×3=

5×3=

8×3=

11 × 3 =

3×3=

6×3=

9×3=

12 × 3 =

6. Skip-count and fill in the multiplication table of 4. Draw arrows on the number line to illustrate the skip counting.

1×4=

4×4=

7×4=

10 × 4 =

2×4=

5×4=

8×4=

11 × 4 =

3×4=

6×4=

9×4=

12 × 4 =

58

Multiplying by Zero

Five groups, each has zero giraffes.

Zero groups or NO groups of five giraffes.

5 × 0 = _____

0×5=0

Take three jumps of zero steps. Where do you end up? 3 × 0 = 0.

Take ZERO or no jumps of three steps: 0 × 3 = 0.

(no jumps)

Remember, multiplication means you have so many same-size groups. How many groups

4

how many in each group

× 0 =

How many groups

0+0+0+0 = 0

0

How many in each group

× 4

(empty groups)

=

0 (nothing)

Don't forget about multiplying by one, either.

Four groups, each has one giraffe.

One group has four giraffes.

4 × 1 = 1 + 1 + 1 + 1 = _____

1 × 4 = _____

59

1. Fill in the easy multiplication tables of zero and one.

Zeroes Table

Ones Table

1 × 0 = _____

7 × 0 = _____

1 × 1 = _____

7 × 1 = _____

2 × 0 = _____

8 × 0 = _____

2 × 1 = _____

8 × 1 = _____

3 × 0 = _____

9 × 0 = _____

3 × 1 = _____

9 × 1 = _____

4 × 0 = _____

10 × 0 = _____

4 × 1 = _____

10 × 1 = _____

5 × 0 = _____

11 × 0 = _____

5 × 1 = _____

11 × 1 = _____

6 × 0 = _____

12 × 0 = _____

6 × 1 = _____

12 × 1 = _____

2. Write each multiplication as an addition. (One you cannot do since there are no groups.) a. 5 × 1 = 1 + 1 + 1 + 1 + 1 = 5

f. 6 × 1 =

b. 2 × 3 =

g. 2 × 6 =

c. 6 × 0 =

h. 0 × 8 =

d. 4 × 5 =

i. 3 × 8 =

e. 1 × 4 =

j. 4 × 0 =

3. Multiply. a. 35 × 1 = ____

4. Fill in the multiplication table:

×

b. 6 × 5 = ____

1 × 1 = ____

1 × 0 = ____

10 × 3 = ____

67 × 1 = ____

0 1 2

c. 1 × 45 = ____

d. 7 × 2 = ____

0 × 1 = ____

0 × 0 = ____

0 × 99 = ____

0 × 10 = ____

3 4 5

60

0

1

2

3

4

5

Word Problems 7

7

7

7 5

There are seven rocks in each box. That is a total of 4 × 7 = 28 rocks.

5

5

Each house has five people living in it. That is a total of 3 × 5 = 15 people.

The SAME amount of something IN EACH thing is solved with multiplication.

1. Write a multiplication sentence for each problem and solve it. You can draw pictures to help. a. Four children are playing tennis together.

They each brought six balls. How many tennis balls do they have all together? b. There are five people in the Smith family. Each person

keeps a hand towel and a bath towel in the bathroom. How many towels are there hanging in their bathroom? c. The Jones family ordered four pizzas.

Each pizza was sliced into four pieces. How many slices of pizza were there? d. A certain town has three post offices. Each post

office has five workers. How many postal workers do the post offices have all together? e. Mrs. Anderson has four flower pots, and in

each pot there are five flowers. How many flowers does she have? f. An egg carton holds six eggs. A bottle of juice

has two liters of juice in it. Mommy bought three cartons of eggs, and three bottles of juice. How many eggs did she get? How many liters of juice did she buy?

61

g. Mary has a jar of marbles, a jar of pretty stones,

and a jar of pine cones. Each jar has 20 items in it. What is the total number of items that Mary has? h. Twelve children each wore a pair of socks.

What was the total number of socks worn? How many toes do these children have all together?

i. Angela needs to memorize Spanish words. The five

word boxes on the page all have six words in them. In each box, two of the words are in bold. How many words does she need to learn? How many words are in bold?

j. Jerry bought two tangerines, five boxes of oats,

and seven bananas. Each box of oats weighs 2 pounds. What is the total number of pieces of fruit that he bought? What was the total weight of the boxes of oats?

3. Fill in the multiplication table:

2. Multiply.

a. 4 × 5 = ___

×

b. 10 × 0 = ___

0 × 4 = ___

6 × 3 = ___

10 × 3 = ___

1 × 78 = ___

0 1 2

c. 25 × 1 = ___

d. 0 × 49 = ___

2 × 4 = ___

10 × 1 = ___

2 × 7 = ___

2 × 6 = ___

3 4 5

62

0

1

2

3

4

5

Order of Operations Do MULTIPLICATION first, before addition or subtraction. Then ADD and SUBTRACT from left to right.

1. Add and subtract from left to right. The operation to be done FIRST is marked. a.

15 + 3 – 7 18 – 7 =

b.

15 – 3 + 7 12 + 7 =

d.

22 – 4 + 3

e.

100 – 60 + 12

60 + 30 – 50 90 – 50 =

c.

f.

50 + 40 – 40

2. Multiply first. Then add and subtract from left to right. a.

3×2 +2 6 +2=

b.

5+ 4×2 5+ 8 =

c.

4×4 –2 16 – 2 =

d.

5×3 +6 ___ + 6 =

e.

25 – 5 × 2 ___ – ___ =

f.

3 × 6 – 11 ___ – ___ =

6×3 – 2×5 ___ – ___ =

i.

15 – 3 × 3 15 – ___ =

g.

j.

3×5 – 2×4 15 – 8 =

h.

k.

2×5 + 4 – 5 ___ + 4 – 5 =

l.

2 × 4 + 3 × 10 + 8 ___ +

___

10 + 2 × 4 – 7 10 + ___ – 7 =

m.

+ 8=

50 – 7 + 2 + 10 × 4 50 – 7 + 2 + ___ =

63

3. Do the multiplication first. a.

15 – 2 × 3

b.

2×5+4

c.

16 – 3 × 5

d.

0×7+2

e.

5×1+3

f.

5 × 4 + 10

g.

30 – 5 × 4

h.

55 + 0 × 3

i.

8 × 2 – 12

4. Now watch carefully! You will need more steps. a.

3×4–2×3

b.

6+7×2–4

c.

2×5+4×5

d.

30 – 2 – 7 × 2

e.

6×2–1×5

f.

20 – 5 + 1 × 3

Which operations will make the number sentences true?

4

5

1 = 21

3

7

7 = 14

3

4

5

6 = 29

16

1

1 = 15

10

5

2 = 20

9

3

5

2 = 17

35

5

4 = 15

5

7

6 = 41

3

1

2

2=7

You can make a game out of this. Just make problems beforehand, and use a board game, the rule being that you can only roll the die if you first answer the question right. (It's a lot more fun if you include division, too.)

64

Understanding Word Problems Mr. Johnson usually eats three meals a day. How many meals does he eat in a normal week? Again we have a situation where the SAME thing happens EACH DAY. 7 days × 3 meals a day = _______ meals in a normal week. One Friday he skipped breakfast. How many meals did he eat during that week? Now one day is different. It is only ONE day though, so we just subtract the one meal from the total. days times meals take away

7

×

3

the skipped breakfast

–

1

= _______

The next week he ate normally, and additionally he had some ice cream on Saturday. How many meals did he eat during that week? days times meals and the ice cream

7

×

3

+

1

= _______

During the following week, he ate three times on Monday, Tuesday, and Friday, and four times on the rest of the days of the week. How many meals did he eat during that week? Now we have the one situation three times, and the other situation four times. We calculate each situation separately, then add. days times meals and

3

×

3

+

the rest of times meals the days

4

×

4 = _______

During a certain busy week, he ate twice on Monday, Tuesday, Thursday, and Friday; three times on Sunday; and four times on the rest of the days. How many meals did he eat during that week? The one situation (two meals a day) happens four times, there are three meals on Sunday, and the other situation (four meals a day) happens twice. We calculate each situation separately, and then add. days times meals and

4

×

2

+

the rest of times meals add Sunday’s the days

2

×

65

4

+

3

= _______

1. Fill in the numbers in the number sentences for each problem and solve it. For the last problems, write the number sentence yourself. You can write words above the numbers to describe the numbers. You can also draw pictures to help you! a. Mommy bought four cartons of eggs, and each carton had six eggs.

Two of the eggs were bad. How many good eggs did Mommy get? egg eggs in take times bad ones cartons each one away

×

–

=

b. The Johnsons ordered 4 pizzas again, sliced into four pieces each.

This time the dog ate one piece. How many pieces did the people eat? number of times pizzas

pieces in each

take what the away dog ate

×

–

=

c. Joe has three friends who each have five toy cars, and also two friends

who have only two toy cars. How many cars do Joe’s friends have? friends friends who have times 5 cars and who have times 2 cars 5 cars two cars

×

×

+

=

d. Grandpa worked six days to make wooden toys for his grandchildren.

The first three days he made 4 toys each day, and the last three days he made 5 toys each day. How many toys did he make in total? days when he days when he times 4 toys and times 5 toys made 4 toys made 5 toys

×

×

+

=

e. Grandma has to take four pills each day, but on Sundays she takes none.

How many pills does she take during one week? days when takes pills

number of pills

×

66

=

f. During the first three weeks of December, Sally borrowed 4 books from the library

each week, but during the last week she borrowed 5 books. How many books did she borrow in total?

g. On the family dinner table there are two plates for everybody except little Hannah.

Hannah gets only one plate. Hannah and ten other people came to dinner. How many plates were on the table?

h. There are four horses and three people. How many legs are there in total?

i. In a little restaurant, there are five tables for two people and four tables

for four people. How many people can the restaurant seat?

j. The neighbor’s cat eats two cans of cat food each day. Last week it was sick

and didn’t eat any food for two days. How many cans of cat food did it eat during that week?

67

Practice with Parts 1. Make three different problems from the same picture. The box with a “T” is a ten.

c. ____ + ____ + ____ = ____

a. ____ + ____ = ____

b. ____ + ____ = ____

____ × ____ = ____

____ × ____ = ____

3 × ____ = ____

1 of ____ is ____. 2

1 of ____ is ____. 2

1 of 12 is ____. 3

d. ____ + ____ + ____ + ____ = ____

e. ____ + ___ + ____ = ____

____ × ____ = ____

____ × ____ = ____

1 of _____ is ____. 4

1 of ____ is ____. 3

2. Draw a picture to match the problems. Fill in the missing numbers.

b. ____ + ____ + a. ___ + ___ + ___ = 15

___ + ___ = ____

c. ___ + ___ = ____

____ × ____ = 15

4 × 10 = ____

___ × 12 = ____

1 of 15 is 5. 3

1 of ____ is 10. 4

1 of ____ is 12. 2

68

3. Draw a picture to match the problems. Fill in the missing numbers.

a. ___ + ___ + ___ = 21

____ × ____ = 21

b. 6 × 10 = ___

1 of 21 is ____. 3

c. 5 × 8 = ___

1 of ____ is 10. 6

1 of ____ is ____. 5

4. Notice something interesting in the pattern below.

2 × 5 = ____

1 2

of 10 is 5 .

2 × 25 = ____

1 2

of 50 is ____.

2 × 10 = ____

1 2

of 20 is 10 .

2 × 30 = ____

1 2

of ____ is ____.

2 × 15 = ____

1 2

of ____ is ____.

2 × 35 = ____

1 2

of ____ is ____.

2 × 20 = ____

1 2

of ____ is ____.

2 × 40 = ____

1 2

of ____ is ____.

5. Here’s some more about finding half of a number. a.

b.

c.

1 2

of 90 is ____.

Finding

1 2

of 100 is ____.

Find

1 2

of 60.

Find

1 2

1 2

of 110 is ____.

Find

1 2

of 2.

Find

1 2

1 2

of 120 is ____.

1 2

of 130 is ____.

1 2

of 62:

d. 1 2

of 32 is ____.

of 50.

1 2

of 38 is ____.

of 4.

1 2

of 46 is ____. of 52 is ____. of 78 is ____.

Finding

1 2

of 54:

Add those.

Add those.

1 2

I get _____.

I get _____.

1 2

69

6. Jane has 16 dolls and Marie has 6. Jane gave half of her dolls to Marie. How many dolls do Jane and Marie have now?

7. Mom gave half of her flowers to Betty, and Betty now has 18 flowers. How many did Mom have initially?

8. Andy gave half of his marbles to Ben. Now, Andy has 12 and Ben has 20. How many did Andy and Ben have initially?

9. Find a path from the top to the bottom. Each answer number on your path must be bigger than the previous one. You can only travel up, down, right, or left at each turn. 1 2

of 22

17 + 17

45 – 9

78 + 60

13 + 13

100 – 83

50 – 25

2×6

50 – 30 – 9

210 + 30

1 × 10

8×0

14 + 25

1 2

23 – 9

2 × 15

43 – 6 – 5

of 68

83 – 60

40 – 20 – 7

5 + 47

100 – 65 + 12

35 + 15 +7

120 – 50

230 – 50

98 – 78

2 × 200

300 + 500

2 × 30

210 – 90

1 × 92

60 + 80 + 80

400 – 90

2 × 70

400 – 80 – 140

60 + 60 + 30

36 + 81 1 2

of 200

170 + 59

1 2

of 100

1 2

70

of 34

152 – 20 1 2

of 160

Review 1. Multiply. a. 2 × 2 = ___

b. 2 × 10 = ___

c. 2 × 40 = ___

d. 1 × 8 = ___

1 × 4 = ___

3 × 3 = ___

3 × 30 = ___

12 × 0 = ___

0 × 5 = ___

2 × 7 = ___

2 × 400 = ___

12 × 1 = ___

2.Write the multiplication as an addition. a. 3 × 7 b. 5 × 1 3. Draw a picture to illustrate the problems.

a. 4 × 5

b.

1 of 18 is 6. 3

4. Write a mathematical sentence (equation) for each problem and solve them. a. If each bag holds five balls, then

how many balls are in four bags?

b. During one week, Karen read three books each day,

except on Thursday, when she read only one book. How many books did she read in total that week?

5. Calculate. a. 3 + 2 × 5

b. 2 × 10 – 1 × 3

c. 12 + 3 × 5 – 4

f. 0 + 7 × 2 – 4

71

Chapter 3: Multiplication Tables Introduction In the third chapter we concentrate on memorizing the times tables.

How to Do Effective Oral Drilling When you are doing memorization drills, be sure to explain to the student that the goal is to memorize the facts—to recall them from memory—and not to get the answers by counting or any other method. Just like your child has probably already memorized your address and phone number, now she or he is going to memorize some math facts. You can easily see if the student is trying to count because producing the answer by counting takes much more time. You should expect the child to answer immediately when you are drilling. If he or she doesn't know the answer by heart (from memory), then tell him or her the right answer. Short drill sessions are usually best. For example, you might drill for five or ten minutes at a time, depending on the attention span of the child. However, try to have at least two sessions during the day as your schedule permits. Research on how the brain learns has shown that new memories are forgotten soon and that new information is best retained when it is reviewed within 4-6 hours of the time it is initially learned. (By the way, this principle applies to anything new a person is learning.) Pencil and paper activities that the student completes alone don’t work really well for memorizing facts because the child can get the answers by counting and not from memory. Proper drill requires an investment in time from the instructor. If you can, utilize older siblings, too, in the task of drilling. Moreover, computers are great drillmasters since they never get tired or bored and since you can usually choose a timed session in which the child must produce the answers quickly. Computer-based drilling can be very rewarding to children when they notice that they are truly learning the facts and are able to complete the drills successfully. They can actually come to enjoy the process of memorization. I’ve included a list of free online multiplication activities at the end of this introduction. Here’s a five-step method for memorization. Normally only a few of the steps would be included in any one session, depending on the child's concentration and ability.

Memorizing the table of 3 — in steps Have the table to be learned already written on paper. Here we will use the table of three as an example. You can view a short video explaining the main points of the drill here: http://www.youtube.com/watch?v=sZlBtMPrMyk

1. The first task is to memorize the list of answers. Have your child study the first half of the skip-counting list (3, 6, 9, 12, 15, 18), saying the numbers aloud while pointing to the answers one by one with a finger or a pen. You may also use a number line. This technique uses the senses of seeing, hearing, and touch simultaneously to fix the information in the brain. After he has gone through the list a few times, ask him to repeat it from memory. Expect your child to answer, and don’t give her the answers too easily, because ONLY by putting forth an effort will she memorize the facts. Just like the muscles, the mind needs exercise to become stronger. Require her to memorize the skip-counting list both forwards and backwards. Keep practicing until she can “rattle off” the first list of 3, 6, 9, 12, 15, 18. With some tables, like the tables of 2, 5, and 10, it helps to point out the pattern in them. The pattern in table of 9 is more subtle but still useful.

1x3=3 2x3=6 3x3=9 4 x 3 = 12 5 x 3 = 15 6 x 3 = 18 7 x 3 = 21 8 x 3 = 24 9 x 3 = 27 10 x 3 = 30 11 x 3 = 33 12 x 3 = 36

2. Tackle the last half of the list: 21, 24, 27, 30, 33, 36. Do the same things you did with the first half of the list. 72

3. Next, work with the whole list of answers. Practice the list going up and down until it goes smoothly and easily. These steps may be enough for one day. But be sure to review again later in the day.

4. Next, practice individual problems randomly. You can ask orally (“What is 5 times 3?”), point to the problems on the paper, or use flashcards. However, I would recommend reading the question aloud while simultaneously pointing to the problem or showing the flashcard because, again, using multiple senses helps to fix the information in the mind better. The goal at this stage is to associate each answer 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, with a certain multiplication fact (such as 7 x 3). You can also mix earlier tables that she already knows with these new problems, and drill both with flashcards.

5. The last step is to do this the other way around. Now you say the answer (“21”), and the student has to produce the problem (“3 × 7”). Keep the table handy, hide the problems, and point to the answers in random order. This technique can also work the other way around, where the student says the answers, and you produce the problems. Give wrong answers sometimes, too, to check her out. As an extension, you can say answers from several tables that you've studied, and the student gives the corresponding problem. Sometimes there are several answers. For example, 36, 30, 24, and 20 are in several different times tables. This is an especially good exercise as it prepares for the concepts of division and factoring. The memorization probably won't happen overnight. On subsequent days, you can mix drills 1-5 (hopefully you won’t need to concentrate on steps 1 and 2). This kind of drilling takes a little time and effort from the teacher, but it can be very effective. Homeschoolers can obviously do some of it while going about other tasks, while traveling in the car, etc. While you are doing this table by table, you can also try to teach the process to your child, so that she will learn how to do the memorization herself. She can hide the answers and try to reproduce the list in her mind.

Other helpful ideas z

Hang a poster with the 12×12 or 10×10 table on the wall. Remind your child to glance at it a few times a day. It can work wonders for visual learners!

z

Hang beside it another poster, with an empty grid, in which the child fills in those facts he has mastered.

z

Recite the skip-counting lists or multiplication facts aloud just before going to bed. This can turn them into mastered facts by the next morning.

Are timed drills necessary? I feel that timed drills are a tool among many, when it comes to learning math facts. Some kids will “thrive” on them; in other words learn quickly when they are used. Perhaps they like racing against the clock or like the challenge. There exist timed computer games that can work very well for drilling facts. For example, Math Magician games has a simple 1-minute countdown, and if you answer 20 questions in that time, you get an award. http://www.oswego.org/ocsd-web/games/Mathmagician/cathymath.html Some of the games at the link below don't time you but give you more points the faster you go. That site is actually filled with several types of games just for math facts practice. http://www.sheppardsoftware.com/math.htm Yet for other kids timed drills may be counterproductive and end up in tears and frustration. The proof is in the pudding: just try it and see how it goes.

73

The Lessons in Chapter 3 page

span

Multiplication Table of 2 ..................................... 76

2 pages

Multiplication Table of 4 ..................................... 78

2 pages

Multiplication Table of 10 ................................... 80

3 pages

Multiplication Table of 5 ..................................... 83

3 pages

More Practice and Review (Tables of 2, 4, 5, and 10) ...................................

86

3 pages

Multiplication Table of 3 ..................................... 89

3 pages

Multiplication Table of 6 ..................................... 92

2 pages

Multiplication Table of 11 ................................... 94

3 pages

Multiplication Table of 9 ..................................... 97

4 pages

Multiplication Table of 7 ..................................... 101

2 pages

Multiplication Table of 8 ..................................... 103

3 pages

Multiplication Table of 12 ................................... 106

3 pages

Review ................................................................. 110

3 pages

Helpful Resources on the Internet You can use these free online resources to supplement the “bookwork” as you see fit. As you can see, there are many resources available for drilling and practicing the tables online. Megamaths Tables Explore the facts of each table, the patterns in each table, or check your skills in a nice timed game. http://www.bbc.co.uk/education/megamaths/tables.html Multiplication Grid Drag the scrambled answer tiles into the right places in the grid as fast as you can! http://www.mathcats.com/microworlds/multiplication_grid.html Multiplication.com Interactive Games A bunch of online games just for the times tables. http://www.multiplication.com/interactive_games.htm The Times Tables at Resourceroom.net Fill in the multiplication chart—part of it or the whole thing—or take quizzes and get graded. http://www.resourceroom.net/Math/1timestables.asp Math Trainer - Multiplication Multiplication table training online that responds to your answers and will train your weaknesses. http://www.mathsisfun.com/games/math-trainer-multiply.html

74

Table Mountain Climb the mountain with 20 questions from a selected table. http://www.teachingtables.co.uk/tm/tmgame/tgame2.html Multiplication Table Challenge 100 questions, timed. http://www.programmingart.com/free/games/multiply/ Multiplication Mystery Drag the answer tiles to right places in the grid as they are given, and a picture is revealed http://www.harcourtschool.com/activity/mult/mult.html Mr. Taylor's Multiplication Facts Drill Simple practice (click on the right answer) for the easy ones, the hard ones, the monsters, or all of them. http://www.geocities.com/multiplicationfacts Multiplication Memory Game Click on corresponding pairs (problem-answer). http://www.dositey.com/2008/addsub/memorymult.html Quiz Hub - Multiplication game Click on corresponding pairs (problem-answer). http://quizhub.com/quiz/f-multiplication.cfm Times tables from BBC Skillswise Has printable factsheets, online quizzes, two grid games, and five printable worksheets. http://www.bbc.co.uk/skillswise/numbers/wholenumbers/multiplication/timestables/index.shtml Math Dice Game for Addition and Multiplication Instructions for three simple games with dice; one to learn multiplication concept, another to practice the times tables, and one more for addition facts. http://www.teachingwithtlc.blogspot.com/2007/09/math-dice-games-for-addition-and.html Product Game A fun, interactive two-player game that exercises your skill with factors and multiples. http://illuminations.nctm.org/ActivityDetail.aspx?ID=29 Two Minute Warning Solve as many problems as you can in two minutes. http://www.primarygames.com/flashcards/multiplication/start.htm Button Beach Challenge Figure out what number the various colored buttons represent. http://www.amblesideprimary.com/ambleweb/mentalmaths/buttons.html

75

Multiplication Table of 2 1. Skip-count by twos. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 2, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 24 2. a. Fill in the table of 2. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 2 till 6 × 2, and the rest at a later time, such as the next day. a.

1 × 2 = ____

7 × 2 = ____

2 × 2 = ____

b.

____ × 2 = 2

____ × 2 = 14

8 × 2 = ____

____ × 2 = 4

____ × 2 = 16

3 × 2 = ____

9 × 2 = ____

____ × 2 = 6

____ × 2 = 18

4 × 2 = ____

10 × 2 = ____

____ × 2 = 8

____ × 2 = 20

5 × 2 = ____

11 × 2 = ____

____ × 2 = 10

____ × 2 = 22

6 × 2 = ____

12 × 2 = ____

____ × 2 = 12

____ × 2 = 24

3. Don't write the answers down. Use these problems for random drill practice.

6×2

7×2

2×3

2×7

2×8

9×2

2×2

2 × 11

2×4

3×2

4×2

8×2

2×9

2×6

2×5

2×1

12 × 2

2 × 12

8×2

10 × 2

4. Don't write the answers down. Use these problems for random drill practice.

× 2 = 14

× 2 = 12

×2 =6

× 2 = 12

× 2 = 22

× 2 = 18

× 2 = 16

× 2 = 18

×2 =8

× 2 = 10

×2=8

× 2 = 24

× 2 = 14

× 2 = 20

× 2 = 24

× 2 = 16

×2=2

× 2 = 22

×2 =4

×2 =6

76

5. Solve the word problems. a. There were seven birds in each of the two trees.

Then three of them flew away. How many birds stayed in the trees?

b. John gets a weekly allowance of two dollars.

After saving money for six weeks, John bought a book that cost eight dollars. How much money did he have left?

c. Lisa also saved her weekly allowance of $2

for six weeks. Then, she earned $4 for helping the neighbor who was sick. How much money does she have now?

d. Right now Fred has $11 in his piggy bank.

He is going to save his weekly allowance of $2 for eight weeks. Then, he will have just enough money to buy an expensive model airplane. How much does it cost?

6. Multiply. a.

b.

c.

d.

2 × 12 = ____

4 × 6 = ____

9 × 2 = ____

8 × 2 = ____

7 × 1 = ____

2 × 5 = ____

3 × 0 = ____

10 × 2 = ____

1 × 8 = ____

6 × 2 = ____

1 × 2 = ____

0 × 7 = ____

e.

f.

g.

h.

2 × 1 = ____

8 × 0 = ____

2 × 11 = ____

5 × 3 = ____

2 × 5 = ____

1 × 9 = ____

0 × 0 = ____

2 × 8 = ____

7 × 2 = ____

1 × 0 = ____

2 × 4 = ____

3 × 8 = ____

77

Multiplication Table of 4 1. Skip-count by fours. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 4, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 48

2. a. Fill in the table of 4. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 4 till 6 × 4, and the rest at a later time, such as the next day. a.

1 × 4 = ____

7 × 4 = ____

2 × 4 = ____

b.

____ × 4 = 4

____ × 4 = 28

8 × 4 = ____

____ × 4 = 8

____ × 4 = 32

3 × 4 = ____

9 × 4 = ____

____ × 4 = 12

____ × 4 = 36

4 × 4 = ____

10 × 4 = ____

____ × 4 = 16

____ × 4 = 40

5 × 4 = ____

11 × 4 = ____

____ × 4 = 20

____ × 4 = 44

6 × 4 = ____

12 × 4 = ____

____ × 4 = 24

____ × 4 = 48

You can find these facts by doubling the facts from the table of 2! 6 × 4 is double 6 × 2. 3. Don't write the answers down. Use these problems for random drill practice.

6×4

7×4

4×3

4×7

4×8

9×4

2×4

4 × 11

4×4

3×4

4×4

8×4

4×9

4×6

4×5

4×1

12 × 4

4 × 12

8×4

10 × 4

4. Don't write the answers down. Use these problems for random drill practice.

× 4 = 44

× 4 = 12

× 4 = 28

× 4 = 48

× 4 = 24

× 4 = 32

× 4 = 36

× 4 = 44

×4 =4

× 4 = 16

×4=8

× 4 = 24

× 4 = 20

× 4 = 40

× 4 = 48

78

5. Circle the numbers (products) that appear in both lists. Then fill in the table below. Table of 2: 0, 2, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Table of 4: 0, 4, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Products in both tables of 2 and 4

Using 2 as a factor

Using 4 as a factor

0

= 0 × 2

=

0 ×4

___

= ___ × 2

___ ___

Products in both tables of 2 and 4

Using 2 as a factor

Using 4 as a factor

___

= ___ × 2

= ___ × 4

= ___ × 4

___

= ___ × 2

= ___ × 4

= ___ × 2

= ___ × 4

___

= ___ × 2

= ___ × 4

= ___ × 2

= ___ × 4

6. Solve the problems. Remember to write number sentence(s) for each. a. The cheap socks cost 1 dollar for each pair, and the expensive ones cost 3 dollars

for each pair. Which takes less money: to buy 10 pairs of cheap socks or to buy four pairs of expensive socks?

b. How many pairs of cheap socks can you buy with 15 dollars?

How many pairs of expensive socks can you buy with 15 dollars?

c. Liz bought three pairs of cheap socks and two pairs of expensive socks.

How much money did she spend?

d. How many legs do seven goats have altogether?

How about three cats and seven chickens?

79

Multiplication Table of 10 1. Skip-count by tens. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 10, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 120 2. a. Fill in the table of 10. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 10 till 6 × 10, and the rest at a later time, such as the next day. a.

1 × 10 = ____

7 × 10 = ____

2 × 10 = ____

b.

____ × 10 = 10

____ × 10 = 70

8 × 10 = ____

____ × 10 = 20

____ × 10 = 80

3 × 10 = ____

9 × 10 = ____

____ × 10 = 30

____ × 10 = 90

4 × 10 = ____

10 × 10 = ____

____ × 10 = 40

____ × 10 = 100

5 × 10 = ____

11 × 10 = ____

____ × 10 = 50

____ × 10 = 110

6 × 10 = ____

12 × 10 = ____

____ × 10 = 60

____ × 10 = 120

What same multiplication fact is...

...both in the table of two and the table of ten? ...both in the table of four and the table of ten?

3. Multiply. Don't write the answers down. Use these problems for random drill practice.

5 × 10

6 × 10

10 × 8

10 × 7

2×5

12 × 10

9 × 10

10 × 4

10 × 10

10 × 3

7 × 10

11 × 10

10 × 12

10 × 11

10 × 6

4. Don't write the answers down. Use these problems for random drill practice.

× 10 = 30

× 10 = 20

× 10 = 80

× 10 = 40

× 10 = 90

× 10 = 10

× 10 = 40

× 10 = 90

× 10 = 110

× 10 = 30

× 10 = 60

× 10 = 50

× 10 = 100

× 10 = 70

× 10 = 120

80

5. a. You see chickens and cats walking in the yard and they have total of 22 legs. How many cats and how many chickens are there?

b. Find two other solutions to the previous problem.

6. Multiply. a.

b.

12 × 4

c.

8 2

×

d.

7 × 1

×

e.

9 0

×

5 4

10

11

f.

7. Fill in parts of the multiplication table that we have studied. ×

0

1

2

3

4

5

6

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

81

7

8

9

12

11 × 4

One centimeter (cm) has ten millimeters (mm). 1 cm = 10 mm.

8. Change centimeters into millimeters and vice versa! a. 2 cm = ___ mm

e. 7 cm = ___ mm .

i. ___ cm = 20 mm

b. 5 cm = ___ mm

f. 9 cm = ___ mm

j. ___ cm = 80 mm

c. 11 cm = ___ mm

g. 6 cm = ___ mm

k. ___ cm = 120 mm

d. ___ cm = 30 mm

h. ___ cm = 100 mm

l. ___ cm = 40 mm

9. Do the conversions according to the example. a. 2 cm 2 mm = 22 mm

f. ___ cm ___ mm = 37 mm

b. 5 cm 4 mm = ___ mm

g. ___ cm ___ mm = 89 mm

c. 8 cm 8 mm = ___ mm

h. ___ cm ___ mm = 45 mm

d. 11 cm 1 mm = ___ mm

i. ___ cm ___ mm = 29 mm

e. 10 cm 6 mm = ___ mm

j. ___ cm ___ mm = 103 mm

10. Measure a spoon, a pencil, a pen, a nail, and a safety-pin in centimeters and millimeters. You can also measure other items that you find. spoon:

___ cm ___ mm

nail:

___ cm ___ mm

pencil:

___ cm ___ mm

safety-pin:

___ cm ___ mm

pen:

___ cm ___ mm

82

Multiplication Table of 5 1. Skip-count by fives. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 5, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 60 2. a. Fill in the table of 5. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 5 till 6 × 5, and the rest at a later time, such as the next day. a.

1 × 5 = ____

7 × 5 = ____

2 × 5 = ____

b.

____ × 5 = 5

____ × 5 = 35

8 × 5 = ____

____ × 5 = 10

____ × 5 = 40

3 × 5 = ____

9 × 5 = ____

____ × 5 = 15

____ × 5 = 45

4 × 5 = ____

10 × 5 = ____

____ × 5 = 20

____ × 5 = 50

5 × 5 = ____

11 × 5 = ____

____ × 5 = 25

____ × 5 = 55

6 × 5 = ____

12 × 5 = ____

____ × 5 = 30

____ × 5 = 60

...both in the table of five and the table of two? What same multiplication fact is... ...both in the table of five and the table of four? ...both in the table of five and the table of ten? 3. Don't write the answers down. Use these problems for random drill practice.

6×5

7×5

5×3

5×7

5 × 10

9×5

12 × 5

5 × 11

5×4

3×5

4×5

8×5

5×9

5×6

5×5

4. Don't write the answers down. Use these problems for random drill practice.

× 5 = 35

× 5 = 20

× 5 = 55

× 5 = 40

× 5 = 55

×5=5

× 5 = 45

× 5 = 25

× 5 = 50

× 5 = 30

× 5 = 60

× 5 = 10

× 5 = 35

× 5 = 60

× 5 = 15

83

5. Find the numbers (products) that appear in both lists. Then fill in the table. Table of 5: 0, 5, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Table of 10: 0, 10, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Products in both tables of 5 and 10

Using 5 as a factor

Using 10 as a factor

= 0×5

= 0 × 10

___

= ___ × 5 = ___ × 10

___

= ___ × 5

= ___ × 10

___

= ___ × 5 = ___ × 10

20

= ___ × 5

= ___ × 10

___

= ___ × 5 = ___ × 10

0

Products in both tables of 5 and 10

Using 5 as a factor

Using 10 as a factor

6. Continue the patterns:

2 × 3 = __

10 × 3 + 1 = 31

5 × 1 + 1 = __

3 × 4 = __

10 × 4 + 2 = 42

5 × 2 + 2 = __

4 × 5 = ___

__ × __ + __ = 53

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ + ___ = ___

84

7. Fill in parts of the multiplication table that we have studied. ×

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

What numbers can go into the puzzles?

× ×

= 20 ×

× = 10

× ×

= 20

= 12 ×

×

= 40

= 6

85

= 12 = 24

More Practice and Review (tables of 2, 4, 5, and 10) 1. Review the tables of two and four. Then check yourself with these problems. a.

b.

c.

d.

9 × 2 = ____

5 × 2 = ____

7 × 2 = ____

4 × 4 = ____

7 × 4 = ____

3 × 4 = ____

9 × 4 = ____

12 × 4 = ____

10 × 2 = ____

2 × 4 = ____

4 × 10 = ____

3 × 2 = ____

e.

f.

g.

h.

6 × 4 = ____

4 × 2 = ____

6 × 2 = ____

8 × 4 = ____

8 × 2 = ____

12 × 2 = ____

4 × 11 = ____

2 × 2 = ____

11 × 2 = ____

2 × 1 = ____

5 × 2 = ____

1 × 4 = ____

2. Write a mathematical sentence to each problem and solve them. You can draw pictures to help you solve the problems! a. Mom does laundry three days a week, and each time she uses two cupfuls of detergent. How much detergent does she use within a week?

b. The big egg cartons have one dozen eggs each. Mom bought two cartons, and has now used four eggs. How many eggs does Mom have left?

c. Eleven shops in a shopping mall have three workers, and two shops have 9 workers. How many workers are there total?

d. Of Marie's relatives, 7 families have two children, 8 families have three children. All of the children are Marie's cousins. How many cousins does Marie have?

86

3. Review the table of five and the table of ten. Check yourself with these problems. a.

b.

c.

d.

5 × 9 = ____

10 × 8 = ____

6 × 5 = ____

7 × 10 = ____

7 × 5 = ____

4 × 10 = ____

2 × 5 = ____

10 × 1 = ____

5 × 10 = ____

10 × 10 = ____

11 × 5 = ____

11 × 10 = ____

e.

f.

g.

h.

6 × 10 = ____

8 × 5 = ____

5 × 10 = ____

5 × 5 = ____

12 × 10 = ____

12 × 5 = ____

10 × 9 = ____

5 × 4 = ____

10 × 2 = ____

5 × 1 = ____

3 × 10 = ____

5 × 3 = ____

4. Write a mathematical sentence to each problem and solve them. a. There are 10 millimeters in each centimeter. How many millimeters are in five centimeters?

b. There are twelve inches in each foot. How many inches are there in five feet?

c. Karen is five feet and five inches tall. How many inches tall is she total?

d. Each minibus holds ten passengers. There are six full minibuses, and one with one seat is empty. How many passengers are there total?

e. Smiths takes three bags with three bottles each and two large bags with five bottles in each bag; how many bottles are there total?

87

5. What was the order of operations?

Do ________________________________ first. Do _____________________ and ____________________ from left to right. 6. Calculate. a. 3 + 7 × 5

b. 10 × 6 – 10 × 3

c. 10 + 5 × 5 – 4

d. 17 + 2 × 3

e. 5 × 4 + 12 × 4

f. 0 + 7 × 2 – 4

a. 3 × 10 + 3 × 2 =

b. 1 × 1 + 1 =

c. 100 – 1 × 1 =

4 × 10 + 4 × 2 =

2×2+2=

100 – 2 × 2 =

5 × 10 + 5 × 2 =

3×3+3=

100 – 3 × 3 =

7. Continue the patterns.

is a certain number, and is another number. Solve what they are in each case. a.

× +

b.

c.

= 42

×

= 24

×

= 24

= 13

−

= 10

+

= 10

88

Multiplication Table of 3 1. Skip-count by threes. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 3, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 36 2. a. Fill in the table of 3. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 3 till 6 × 3, and the rest at a later time, such as the next day. a.

1 × 3 = ____

7 × 3 = ____

2 × 3 = ____

b.

____ × 3 = 3

____ × 3 = 21

8 × 3 = ____

____ × 3 = 6

____ × 3 = 24

3 × 3 = ____

9 × 3 = ____

____ × 3 = 9

____ × 3 = 27

4 × 3 = ____

10 × 3 = ____

____ × 3 = 12

____ × 3 = 30

5 × 3 = ____

11 × 3 = ____

____ × 3 = 15

____ × 3 = 33

6 × 3 = ____

12 × 3 = ____

____ × 3 = 18

____ × 3 = 36

Note: the fact 2 × 3 = 6 or 3 × 2 = 6 is both in the table of three and the table of two. 3. Don't write the answers down. Use these problems for random drill practice.

6×3

7×3

3×3

3×7

3×8

9×3

2×3

3 × 11

3×4

3×3

4×3

8×3

3×9

3×6

3×5

3×1

12 × 3

3 × 12

8×3

10 × 3

4. Don't write the answers down. Use these problems for random drill practice.

× 3 = 15

× 3 = 12

× 3 = 27

× 3 = 36

× 3 = 30

× 3 = 33

× 3 = 36

× 3 = 33

×3 =3

×3=6

×3=9

× 3 = 24

× 3 = 27

× 3 = 18

× 3 = 21

89

5. Continue the patterns.

12 × 2 = ___

1 × 2 + 2 = 4

1 × 2 – 1 = ___

13 × 2 = ___

2 × 2 + 4 = ___

2 × 2 – 2 = ___

14 × 2 = ___

3 × 2 + 6 = ___

3 × 2 – 3 = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

___ × ___ = ___

___ × ___ + ___ = ___

___ × ___ – ___ = ___

6. Solve the word problems. a. John gets $3 a week as an allowance. He wants to buy a toy train

that costs $14. How many weeks will he have to save for that? The train is discounted and now costs only $11. How many weeks will he have to save for it now?

b. John saved money for five weeks and then his Grandpa gave him

five dollars as a present. How much money does John have now? If he buys the 11-dollar train, how much will he have left?

c. John wants to buy a book about nesting birds that costs $18.

He earned $5 for yard work and plans to save the rest from his weekly allowance of $3. How many weeks will he need to save his allowance money to buy the book?

90

d. Roses are sold in bunches of three. Dad bought eleven bunches

and one extra rose for Mom's birthday - a rose for each year. How old is mom?

e. How many bunches of roses and extra roses would you need to

buy if mom was 31 years old?

f. How about your mom? How many rose bunches and extra

roses would you need?

7. Fill in the parts of the multiplication table that we have studied. ×

0

1

2

3

4

5

6

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

91

7

8

9

10

11

12

Multiplication Table of 6 1. Skip-count by sixes. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 6, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 72 2. a. Fill in the table of 6. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 6 till 6 × 6, and the rest at a later time, such as the next day. a.

1 × 6 = ____

7 × 6 = ____

2 × 6 = ____

b.

____ × 6 = 6

____ × 6 = 42

8 × 6 = ____

____ × 6 = 12

____ × 6 = 48

3 × 6 = ____

9 × 6 = ____

____ × 6 = 18

____ × 6 = 54

4 × 6 = ____

10 × 6 = ____

____ × 6 = 24

____ × 6 = 60

5 × 6 = ____

11 × 6 = ____

____ × 6 = 30

____ × 6 = 66

6 × 6 = ____

12 × 6 = ____

____ × 6 = 36

____ × 6 = 72

You can find these facts by doubling the facts from the table of 3: 8 × 6 is double 8 × 3! 3. Don't write the answers down. Use these problems for random drill practice.

9×6

8×6

6×8

6×5

3×6

2×6

10 × 6

6 × 12

6×7

6×6

4×6

3×6

6×9

6×2

6×4

11 × 6

12 × 6

6×4

6×6

7×6

4. Don't write the answers down. Use these problems for random drill practice.

× 6 = 72

× 6 = 18

× 6 = 54

× 6 = 42

× 6 = 54

×6=6

× 6 = 48

× 6 = 24

× 6 = 36

× 6 = 30

× 6 = 60

× 6 = 12

× 6 = 42

× 6 = 66

× 6 = 72

92

5. Circle the numbers (products) that appear in both lists. There are seven of them. Then fill in the table below. Table of 3: 0, 3, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Table of 6: 0, 6, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Products in both tables of 3 and 6

0

Using 3 as a factor

Using 6 as a factor

=

=

0 ×3

Products in both tables of 3 and 6

Using 3 as a factor

Using 6 as a factor

0 ×6

___

= ___ × 3

= ___ × 6

___

= ___ × 3

= ___ × 6

___

= ___ × 3

= ___ × 6

___

= ___ × 3

= ___ × 6

___

= ___ × 3

= ___ × 6

___

= ___ × 3

= ___ × 6

6. Fill in parts of the multiplication table that we have studied. ×

0

1

2

3

4

5

6

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

93

7

8

9

10

11

12

Multiplication Table of 11 1. Skip-count by elevens. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 11, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 132 2. a. Fill in the table of 11. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 11 till 6 × 11, and the rest at a later time, such as the next day. a.

1 × 11 = ____

7 × 11 = ____

2 × 11 = ____

b.

____ × 11 = 11

____ × 11 = 77

8 × 11 = ____

____ × 11 = 22

____ × 11 = 88

3 × 11 = ____

9 × 11 = ____

____ × 11 = 33

____ × 11 = 99

4 × 11 = ____

10 × 11 = ____

____ × 11 = 44

____ × 11 = 110

5 × 11 = ____

11 × 11 = ____

____ × 11 = 55

____ × 11 = 121

6 × 11 = ____

12 × 11 = ____

____ × 11 = 66

____ × 11= 132

3. Don't write the answers down. Use these problems for random drill practice.

5 × 11

2 × 11

11 × 7

11 × 3

11 × 5

12 × 11

8 × 11

11 × 12

11 × 10

11 × 11

9 × 11

7 × 11

11 × 4

11 × 4

11 × 9

3 × 11

6 × 11

11 × 11

11 × 8

11 × 6

4. Don't write the answers down. Use these problems for random drill practice.

× 11 = 88

× 11 = 77

× 11 = 55

× 11 = 66

× 11 = 11

× 11 = 132

× 11 = 121

× 11 = 33

× 11 = 22

× 11 = 44

× 11 = 110

× 11 = 99

× 11 = 132

× 11 = 121

× 11 = 110

94

5. Continue the patterns. a.

b.

c.

1 × 11 + 1 =

1 × 10 + 1 × 5 =

12 × 10 – 2 × 3 =

2 × 11 + 2 =

2 × 10 + 2 × 5 =

11 × 10 – 3 × 3 =

3 × 11 + ___ =

3 × 10 + 3 × 5 =

10 × 10 – 4 × 3 =

6. Write different problems for these answers. You can also use 1 as a factor! a.

b.

c.

d.

___× ___ = 20

___× ___ = 18

___× ___ = 40

___× ___ = 36

___× ___ = 20

___× ___ = 18

___× ___ = 40

___× ___ = 36

___× ___ = 20

___× ___ = 18

___× ___ = 40

___× ___ = 36

e.

f.

g.

h.

___× ___ = 30

___× ___ = 12

___× ___ = 24

___× ___ = 100

___× ___ = 30

___× ___ = 12

___× ___ = 24

___× ___ = 100

___× ___ = 30

___× ___ = 12

___× ___ = 24

___× ___ = 100

95

7. Fill in parts of the multiplication table that we have studied. ×

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

Can you figure out what these “MYSTERY NUMBERS” are? a. I am in the table of four but not in the table of three. If you add two to me,

the new number is in the table of ten, in the table of five, and in the table of six. b.. I'm in the table of 11! If you take away one from me, you'll get a number

that is in the table of ten. c. I'm in the table of five but not in the table of ten. Adding my digits you get seven.

96

Multiplication Table of 9 1. Skip-count by nines. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 9, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 108 2. a. Fill in the table of 9. b. Fill in the missing factors. Then cover the answers. Choose problems in random order and practice. You may first practice only the part from 1 × 9 till 6 × 9, and the rest at a later time, such as the next day. a.

1 × 9 = ____

7 × 9 = ____

2 × 9 = ____

b.

____ × 9 = 9

____ × 9 = 63

8 × 9 = ____

____ × 9 = 18

____ × 9 = 72

3 × 9 = ____

9 × 9 = ____

____ × 9 = 27

____ × 9 = 81

4 × 9 = ____

10 × 9 = ____

____ × 9 = 36

____ × 9 = 90

5 × 9 = ____

11 × 9 = ____

____ × 9 = 45

____ × 9 = 99

6 × 9 = ____

12 × 9 = ____

____ × 9 = 54

____ × 9 = 108

What same multiplication fact is both in...

... the table of nine and the table of two?____________________ ... the table of nine and the table of five? ___________________ ... the table of nine and the table of three? __________________ ... the table of nine and the table of ten? ____________________ ... the table of nine and the table of four? ____________________ ... the table of nine and the table of eleven? ____________________

3. Multiply. Don't write the answers down. Use these problems for random drill practice.

5×9

8×9

9 × 10

9×5

9×8

11 × 9

9×9

10 × 9

9×3

9×7

1×9

9×2

12 × 9

6×9

9×1

9×4

9×6

9×9

97

4. Don't write the answers down. Use these problems for random drill practice.

× 9 = 18

× 9 = 36

× 9 = 72

× 9 = 108

× 9 = 81

× 9 = 45

×9=9

× 9 = 90

× 9 = 99

× 9 = 72

× 9 = 27

× 9 = 72

× 9 = 81

× 9 = 63

× 9 = 54

There are some special things in the table of nine. Add up the DIGITS of the answers:

1 × 9 = 9;

9=9

2 × 9 = 18;

1+8=

3 × 9 = 27;

2+7=

4 × 9 = ___

___ + ___ =

5 × 9 = ___

___ + ___ =

6 × 9 = ___;

___ + ___ =

7 × 9 = ___;

___ + ___ =

8 × 9 = ___;

___ + ___ =

9 × 9 = ___;

___ + ___ =

10 × 9 = ___

___ + ___ =

11 × 9 = ___

9 + 9 = 18; 1+8=

12 × 9 = ___;

1+0+8=

What do you notice??

1 × 9 = 09 2 × 9 = 18 3 × 9 = 27 4 × 9 = ____ 5 × 9 = ____ 6 × 9 = ____ 7 × 9 = ____ 8 × 9 = ____ 9 × 9 = ____ 10 × 9 = ____

For each answer, color the FIRST digit in yellow and the SECOND digit in blue. Then look at the line of yellow numbers, and the line of the blue numbers from top to bottom!

98

5. Circle the products that appear in both lists. There are five of them. Then fill in the table below. Table of 3: 0, 3, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Table of 9: 0, 9, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Products in both tables of 3 and 9

Using 3 as a factor

Using 9 as a factor

0

= ___× 3

= ___ × 9

___

= ___ × 3

= ___ × 9

___

= ___ × 3

= ___ × 9

Products in both tables of 3 and 9

Using 3 as a factor

Using 9 as a factor

___

= ___ × 3

= ___ × 9

___

= ___ × 3

= ___ × 9

Now make a longer list for the table of 3 and compare that with the table of nine. What can you notice? 0, 3, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, Table of 3: 36, 39, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Table of 9:

0, 9, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ___

6. Look further into the special thing about the table of nine! Continue the table of nine and add the digits of the products.

Multiplication facts:

Multiplication facts:

add digits:

10 × 9 = ____

16 × 9 = ____

11 × 9 = ____

17 × 9 = ____

12 × 9 = ____

18 × 9 = ____

13 × 9 = 117

1+1+7=9

add digits:

19 × 9 = ____

14 × 9 = ____

20 × 9 = ____

15 × 9 = ____

21 × 9 = 189

99

1 + 8 + 9 = 18; 1 + 8 = 9

7. Fill in the parts of the multiplication table that we have studied. ×

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

represents a mystery number. It is more than 0 and less than 100. See if you can compare the expressions even when you don't know the mystery number. Write , or = in the boxes. There is one comparison you can't do! Hint: Go back to exercise 5 for some ideas.

10 ×

9× ×8

×4

×8

×5

×5 4× ×2

100

×4

×0

3×6

×8

×1

10 × 7

+

×3

+

Multiplication Table of 7 1. Practice the skip-count pattern until you can say it from memory. Also practice it backwards. You may practice one-half of it at first, and the other half later.

0, 7, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 84 2. Fill in. Then cover the answers. Choose problems in random order and practice. a.

1 × 7 = ____

7 × 7 = ____

2 × 7 = ____

b.

____ × 7 = 7

____ × 7 = 49

8 × 7 = ____

____ × 7 = 14

____ × 7 = 56

3 × 7 = ____

9 × 7 = ____

____ × 7 = 21

____ × 7 = 63

4 × 7 = ____

10 × 7 = ____

____ × 7 = 28

____ × 7 = 70

5 × 7 = ____

11 × 7 = ____

____ × 7 = 35

____ × 7 = 77

6 × 7 = ____

12 × 7 = ____

____ × 7 = 42

____ × 7 = 84

What same multiplication fact is both... ...in the table of 7 and the table of 2? ...in the table of 7 and the table of 3? ...in the table of 7 and the table of 4? ...in the table of 7 and the table of 5?

...in the table of 7 and the table of 6? ...in the table of 7 and the table of 9? ...in the table of 7 and the table of 10? ...in the table of 7 and the table of 11?

Therefore, the only new facts to study from the table of seven are:

7 × 7 = 49

8 × 7 = 56

12 × 7 = 84

3. Don't write the answers down. Use these problems for random drill practice.

9×6

8×6

6×8

6×5

3×6

4×6

10 × 6

6 × 12

6×7

6×6

11 × 6

6×6

6×9

6×2

4×6

× 7 = 35

× 7 = 70

× 7 = 42

× 7 = 28

× 7 = 56

× 7 = 77

× 7 = 21

× 7 = 56

× 7 = 84

× 7 = 49

× 7 = 42

× 7 = 14

× 7 = 35

× 7 = 35

× 7 = 63

101

4. Solve the word problems. a. Jenny has seven little boxes and each has seven pretty stones

in them, except one only has six stones. Ann has eight boxes and each has six stones. Who has more stones? b. Tom has twelve pairs of socks. How many individual socks does he have?

c. Mom bought three dozen eggs, and has already used 8 of them.

How many are left? d. Each table can seat six people. How many people do seven tables seat?

How many tables do you need if you are serving 30 people at a dinner?

5. Fill in the parts of the multiplication table that we have studied. ×

0

1

2

3

4

5

6

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

102

7

8

9

10

11

12

Multiplication Table of 8 1. Skip-count by eights. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 8, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 96 2. Fill in. Then cover the answers. Choose problems in random order and practice. a.

1 × 8 = ____

7 × 8 = ____

2 × 8 = ____

b.

____ × 8 = 8

____ × 8 = 56

8 × 8 = ____

____ × 8 = 16

____ × 8 = 64

3 × 8 = ____

9 × 8 = ____

____ × 8 = 24

____ × 8 = 72

4 × 8 = ____

10 × 8 = ____

____ × 8 = 32

____ × 8 = 80

5 × 8 = ____

11 × 8 = ____

____ × 8 = 40

____ × 8 = 88

6 × 8 = ____

12 × 8 = ____

____ × 8 = 48

____ × 8 = 96

The only new facts to study from the table of eight are:

8 × 8 = 64

12 × 8 = 96

3. Don't write the answers down. Use these problems for random drill practice.

8×8

9×8

8×4

8×5

8×8

8×6

8 × 11

8 × 12

7×8

8 × 10

3×8

8×6

2×8

8×9

8×6

4. Don't write the answers down. Use these problems for random drill practice.

× 8 = 32

× 8 = 24

× 8 = 88

× 8 = 40

× 8 = 64

×8=8

× 8 = 48

× 8 = 72

× 8 = 56

× 8 = 96

× 8 = 64

× 8 = 16

× 8 = 80

× 8 = 48

× 8 = 88

103

5. Circle the numbers that appear in both lists. There are seven of them. Then fill in the table below. Table of 4: 0, 4, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Table of 8: 0, 8, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____ Products in both tables of 4 and 8

0

Using 4 as a factor

Using 8 as a factor

Products in both tables of 4 and 8

Using 4 as a factor

Using 8 as a factor

= ___ × 4 = 0 × 8

___

= ___ × 4

= ___ × 8

___

= ___ × 4

= ___ × 8

___

= ___ × 4

= ___ × 8

___

= ___ × 4

= ___ × 8

___

= ___ × 4

= ___ × 8

___

= ___ × 4

= ___ × 8

Make a longer list for the table of 4 and compare it with the table of 8. What do you notice? 0, 4, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, Table of 4: 48, 52, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ___ Table of 8:

0, 8, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ___

6. a. Mommy bought two dozen eggs. Aunt Kathy bought five cartons of eggs with six eggs in each. Who had the most eggs? b. Each of Grandma's chickens lays approximately six eggs in

a week. She has 9 chickens. Approximately how many eggs does Grandma get in a week? c. One of the chickens is kind of old and now lays only four

eggs a week. How many eggs is Grandma getting now? d. Brenda's family eats two kilograms of beans weekly.

How many kilograms of beans do they eat in nine weeks? How many weeks does it take for them to eat 10 kilograms of beans?

104

7. Fill in the parts of the multiplication table that we have studied. ×

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

represents a number, and represents another number. Solve what they are in each case, if a.

b.

c.

×

= 48

×

= 48

×

= 36

+

= 14

+

= 16

+

= 24

= ____ = ____

= ____ = ____

105

= ____ = ____

Multiplication Table of 12 1. Skip-count by twelve. Practice this pattern until you can say it from memory. Also practice it backwards (up-down). You may practice one-half of it at first, and the other half later.

0, 12, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, 144 2. Fill in. Then cover the answers. Choose problems in random order and practice. a.

1 × 12 = ____

7 × 12 = ____

2 × 12 = ____

b.

____ × 12 = 12

____ × 12 = 84

8 × 12 = ____

____ × 12 = 24

____ × 12 = 96

3 × 12 = ____

9 × 12 = ____

____ × 12 = 36

____ × 12 = 108

4 × 12 = ____

10 × 12 = ____

____ × 12 = 48

____ × 12 = 120

5 × 12 = ____

11 × 12 = ____

____ × 12 = 60

____ × 12 = 132

6 × 12 = ____

12 × 12 = ____

____ × 12 = 72

____ × 12 = 144

The only new fact to study from the table of twelve is

12 × 12 = 144

3. Don't write the answers down. Use these problems for random drill practice.

3 × 12

9 × 12

12 × 4

12 × 1

7 × 12

2 × 12

10 × 12

12 × 5

12 × 7

12 × 3

1 × 12

6 × 12

12 × 8

12 × 9

4 × 12

8 × 12

12 × 12

12 × 11

12 × 6

12 × 2

4. Don't write the answers down. Use these problems for random drill practice.

× 12 = 36

× 12 = 24

× 12 = 84

× 12 = 72

× 12 = 144

× 12 = 12

× 12 = 48

× 12 = 144

× 12 = 120

× 12 = 132

× 12 = 72

× 12 = 60

× 12 = 96

× 12 = 60

× 12 = 108

106

Find a tape measure and look at it. It has INCHES and FEET. Each foot is 12 inches:

In. is short for inches, and ft. is short for feet. For example, 24 inches is 2 feet and 36 inches = 3 feet. You need the table of 12 here! Then 37 inches is 3 feet 1 inch, and 40 inches is 3 feet 4 inches.

5. Measure with a ruler. a. side of a table

___ feet ___ inches

d. the width of the room

___ feet ___ inches

b. your height

___ feet ___ inches

e. the length of the room

___ feet ___ inches

c. a jump rope

___ feet ___ inches

f. _________________

___ feet ___ inches

6. Convert between feet and inches: 2 feet = 24 inches

10 ft = ___ in

___ ft = 60 inches

4 feet = ___ inches

7 ft = ___ in

___ ft = 96 inches

6 feet = ___ inches

3 ft = ___ in

___ ft = 72 inches

10 feet 1 inch = ___ inches

5 ft 5 in = ___ in

___ ft = 24 in

3 feet 2 inches = ___ inches

2 ft 8 in = ___ in

5 ft = ___ in

6 feet 7 inches = ___ inches

1 ft 10 in = ___ in

11 ft = ___ in

__ ft ___ in = 16 in

__ ft ___ in = 27 in

4 ft 4 in = ___ in

__ ft ___ in = 20 in

__ ft ___ in = 31 in

5 ft 8 in = ___ in

107

7. Fill in the table - for the last time. ×

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

After completing the table, take each number in the table of 12 and color the squares that have the same number. z z z z

color yellow all squares with 12 color red all squares with 24 color blue all squares with 36 color pink all squares with 48 etc.

You can choose your own colors. This should make a pretty chart!

108

Review ×

0

1

2

3

4

5

6

7

8

9

10

11

0 1 2 3 4

1. Fill in the complete multiplication table!

5 6 7 8 9 10 11 12

2. Fill in the table of 9. Write down the SPECIAL THINGS about the table of nine.

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

If you add up the digits, _____________________________ ________________________________________________ If you look at the first digits of the answers from 9 till 90, ________________________________________________ If you look at the last digits of the answers from 9 till 90, ________________________________________________

109

12

3. Fill in the tables:

1 × 3 = __

7 × 3 = __

1 × 6 = __

7 × 6 = __

1 × 9 = __

7 × 9 = __

2 × 3 = __

8 × 3 = __

2 × 6 = __

8 × 6 = __

2 × 9 = __

8 × 9 = __

3 × 3 = __

9 × 3 = __

3 × 6 = __

9 × 6 = __

3 × 9 = __

9 × 9 = __

4 × 3 = __ 10 × 3 = __

4 × 6 = __ 10 × 6 = __

4 × 9 = __ 10 × 9 = __

5 × 3 = __ 11 × 3 = __

5 × 6 = __ 11 × 6 = __

5 × 9 = __ 11 × 9 = __

6 × 3 = __ 12 × 3 = __

6 × 6 = __ 12 × 6 = __

6 × 9 = __ 12 × 9 = __

Then do some coloring in the answers above: color the ANSWER z z

yellow, if it is in the tables of 3 and 6 orange, if it is in the tables of 3 and 9

z z

blue, if it is in the tables of 6 and 9 red, if it is in ALL three tables.

What numbers were “all colored”? _____________________ 4. Compare the expressions and write < , > , or = . a. 9 × 8

10 × 8

b. 9 × 5

11 × 4

c. 9 × 2

d. 9 × 8

9×4

e. 4 × 4

2×8

f. 10 × 11

g. 10 × 8

10 × 5

h. 9 × 2

4×5

i. 9 × 8

3×6 10 × 7 9×6

5. Solve the problems. a. The class has eleven girls. They each have seven schoolbooks. Three of the girls have a cat,

two of the girls each have 5 goldfish, and six of them have no pets. How many schoolbooks do the girls have altogether? How many pets do they have altogether?

b. A zoo has five

s, three s, and 20 s. How many feet do those animals have altogether?

110

6. Find the missing factors. a.

b.

c.

d.

____× 8 = 24

6 × ____ = 18

7 × ____ = 49

____× 5 = 25

____ × 8 = 64

6 × ____ = 66

____ × 7 = 56

____ × 5 = 45

____ × 8 = 40

6 × ____ = 12

7 × ____ = 63

____ × 5 = 35

e.

f.

g.

h.

____× 4 = 16

____× 3 = 36

____× 8 = 48

____× 12 = 60

____ × 4 = 28

____ × 3 = 21

____ × 8 = 32

____ × 12 = 84

4 × ____ = 36

3 × ____ = 27

8 × ____ = 72

12 × ____ = 108

MYSTERY NUMBERS 

(all numbers are less than 100)

b. I am more than 15. I am in the table of

a. You can find me both in the table of

two, the table of three, and the table of four!

eleven and in the table of four. I am _______.

I am _______.

c. I am between 15 and 35. The number

d. I am both in the table of four and in

one more than me is in the table of five. The number one less than me is in the table of four.

the table of three, and if you add one to me, I am in the table of five. I am _______.

I am _______. e. I am in the table of 11. The number that

f. I am less than 22 but more than 9, and

is one more than me, is in the table of five, but not in the table of ten.

I am in the table of four. If you exchange my digits, I am in the table of three! I am _______.

I am _______.

111

Chapter 4: Telling Time Introduction This chapter of Math Mammoth Grade 3-A Complete Worktext covers reading the clock to the minute, finding time intervals (elapsed time), using the calendar, and making simple conversions between units of time. The lesson “Reading the Clock to the Minute” completes the topic, begun in earlier grades, of reading the clock because the student will now be able to tell the complete time. From that point on, the focus switches to finding time intervals and other time-related calculations. The lessons about calculating elapsed time emphasize dividing the time interval into easily-calculated parts: For example, to find the time elapsed from 10:30 AM to 7:00 PM, the student learns to find the elapsed time from 10:30 AM to 12:00 noon and then from 12:00 noon to 7 PM. The same principle is followed when the time-interval looks more complex. This chapter does not yet introduce the idea of adding or subtracting hours and minutes vertically in columns.

The Lessons in Chapter 4 page

span

Review: Reading the Clock ............................. 114

2 pages

Reading the Clock to the Minute ..................... 116

3 pages

Elapsed Time ................................................... 119

2 pages

More on Elapsed Time .................................... 121

4 pages

Using the Calendar .......................................... 125

2 pages

Changing Time Units ...................................... 127

4 pages

Review ............................................................ 131

1 pages

Helpful Resources on the Internet Use these free online resources to supplement the “bookwork” as you see fit.

Analog and Digital Clocks These clocks show you the current time, side by side. Useful for illustration. http://nlvm.usu.edu/en/nav/frames_asid_316_g_2_t_4.html What Time Will it Be? Move the hands on the clock to show what time it will be after certain amount of minutes. http://nlvm.usu.edu/en/nav/frames_asid_318_g_2_t_4.html

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Match Clocks Make the digital clock to show the time given with the analog clock. http://nlvm.usu.edu/en/nav/frames_asid_317_g_2_t_4.html Flashcard Clock Read the analog and type in the time in digital. Very clear clock and good fast response! http://www.teachingtreasures.com.au/maths/FlashcardClock/flashcard_clock.htm Telling Time Practice Interactive online practice: you drag the hands of the clock to show the correct time. http://www.worsleyschool.net/socialarts/telling/time.html

Teaching Time Analogue/digital clock games and worksheets. Also an interactive “class clock” to demonstrate time. http://www.teachingtime.co.uk/ Time-for-time Resource site to learn about time: worksheets, games, quizzes, time zones. http://www.time-for-time.com/default.htm A Matter of Time Lesson plans for telling time, interactive activities, and some materials to print. http://www.fi.edu/time/Journey/JustInTime/contents.html Clockwise Plug in a time, and the clock runs till it, or clock runs to a time and you type it in. http://www.shodor.org/interactivate/activities/clock2/index.html The Right Time A couple of interactive exercises about reading the clock. http://www.pitara.com/activities/math/time/time.asp?QNum=3 What Time Is It? Look at the analog clock and pick the digital clock that shows the same time. http://www.primarygames.com/time/start.htm Calculating Time from BBC SkillsWise Factsheets, worksheets, and an online game to practice time calculations. http://www.bbc.co.uk/skillswise/numbers/measuring/time/calculatingtime/

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Review: Reading the Clock People use several expressions in telling time. We can say the hours, and then the minutes. We can use “o'clock” to mean the whole hour, and “half past” to mean the half hour. We can say how many minutes it is past the whole hour, or how many minutes it is till the next whole hour.

25 past 2

25 till 7

2:25

6:35

Imagine the clock face divided into four parts, or quarters. The word quarter means a fourth. Each quarter of an hour is 15 minutes. “A quarter past 7” means fifteen minutes past 7. “A quarter till 2” means fifteen minutes till 2.

It is a quarter past 7. It is a quarter till 2.

1. Write the time the clock shows, and the time 15 minutes later.

a. ______ : ______

b. ______ : ______

c. ______ : ______

d. ______ : ______

e. ______ : ______

f. ______ : ______

g. ______ : ______

h. ______ : ______

15 min. later →

15 min. later →

114

2. Write the time the clock shows using words “past”, “till”, and “quarter”.

a. a quarter till _______

e.

b.

c.

d.

_________________

_________________

_________________

_________________

_________________

_________________

f.

g.

h.

_________________

_________________

_________________

_________________

_________________

_________________

_________________

_________________

3. Write the time using the hours:minutes way. a. a quarter past 8

b. 10 till 7

c. 25 past 3

d. half-past 5

______ : ______

______ : ______

______ : ______

______ : ______

e. a quarter till 5

f. 25 till 6

g. 5 till 12

h. 15 till 1

______ : ______

______ : ______

______ : ______

______ : ______

i. a quarter till 12

j. a quarter past 6

k. 5 till 11

l. 20 till 9

______ : ______

______ : ______

______ : ______

______ : ______

4. Write the time using the expressions “quarter past”, “quarter till”, “till”, and “past”. a. 5:45

b. 2:15

c. 3:55

d. 8:25

e. 11:55

f. 2:45

g. 6:35

h. 10:15

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Reading the Clock to the Minute The minute hand goes through 60 minutes, or 60 steps, each time it goes around the clock. You can see some numbers for those “steps” marked outside this clock face. (They’re the green numbers 5, 10, 15, and so on). On this clock we have marked the steps between 10 and 15 with dots. Those dots show 11, 12, 13, and 14 minutes. The minute hand is pointing to 12 minutes, so the time is 3:12, which is read, “Twelve past three.” Some clocks have dots like these between the numbers; others don't. But even if the clock does not have the dots (or lines), you need to remember the little steps between the marked numbers that the minute hand passes. These clocks have the little lines to tell the minutes. On the first clock, the minute hand is pointing to the first line past the 15-minute mark (which is at the number 3). So the time is sixteen minutes past 1:00, or 1:16. On the second clock, the minute hand points to the third line after the 45-minute mark. So the time is 2:48.

1:16

2:48

1. Write the time the clock shows.

a.

____:____

b. ____:____

c. ____:____

116

d. ____:____

2. Write the time the clock shows.

a.

____:____

b. ____:____

c. ____:____

d. ____:____

e.

____:____

f. ____:____

g. ____:____

h. ____:____

i.

____:____

j. ____:____

k. ____:____

l. ____:____

m.

____:____

n. ____:____

o. ____:____

p. ____:____

q.

____:____

r. ____:____

s. ____:____

t. ____:____

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John started jumping rope at 7:23 and jumped for 10 minutes. What time did he stop? Add the elapsed minutes to the minutes on the clock time. John stopped jumping at 7:33. The time is now 3:49. What time will it be in 20 minutes? If you add the elapsed minutes to the time, you get 3 hours, 69 minutes. Since 1 hour is 60 minutes, 69 minutes is 1 hour plus 9 minutes more. So the time will be 9 minutes past the next whole hour, or 4:09. 3. Write the time the clock shows on the line below it. Then in the box below that, write the time the given number of minutes later.

a.

____:____

b. ____:____

c. ____:____

d. ____:____

e.

____:____

f. ____:____

g. ____:____

h. ____:____

i.

____:____

j. ____:____

k. ____:____

l. ____:____

10 min. later

20 min. later

15 min. later 118

Elapsed Time How many minutes is it till the next whole hour? It is 4:38. The minute hand needs to go 2 minutes till the 40-minute point (number 8), and then 20 more minutes till the next whole hour. So it is 22 minutes till 5 o'clock. Or, you can subtract 38 minutes from 60 minutes: 60 − 38 = 22. Remember, a complete hour is 60 minutes. It is 2:34. How many minutes is it till 2:50? The hour is the same (2 hours) in both times, you can simply subtract the minutes: 50 − 34 = 16 minutes. Or, add up from 34 till 50: 34 + 6 = 40 40 + 10 = 50. You added 16 minutes. Or, imagine the minute hand moving on the clock face: it moves 1 minute, and then another 15 minutes — total 16 minutes.

1. How many minutes is it till the next whole hour?

a. _____ minutes

b. _____ minutes

c. _____ minutes

d. _____ minutes

e. _____ minutes

f. _____ minutes

g. _____ minutes

h. _____ minutes

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2. How many minutes is it from the time on the clock face until the given time?

till 12:40

till 7:30

till 10:45

till 3:58

a. _____ minutes

b. _____ minutes

c. _____ minutes

d. _____ minutes

till 1:00

till 5:55

till 12:50

till 4:55

e. _____ minutes

f. _____ minutes

g. _____ minutes

h. _____ minutes

3. How many minutes is it? a. From 5:06 till 5:28

b. From 2:05 till 2:54

c. From 3:12 till 3:47

d. From 12:11 till 12:55

e. From 7:27 till 7:48

f. From 9:06 till 10:00

4. a. The pie needs to bake half an hour. Marsha's clock shows 4:22 when she puts it into the oven. When should she take it out? b. Juan notices, “In 14 minutes the lesson will end.” If the lesson ends at 2 PM, what time is it now? c. The sun rises at 5:49 AM. Marge wants to wake up 15 minutes before that. When should she wake up? d. Edward was 8 minutes late for math class, and came at 1:53 PM. When did the class start?

120

More on Elapsed Time How many minutes is it from 1:47 till 2:10? Notice the hour changes from 1 to 2. We need to figure this out carefully, but it is easy when you do it in parts: From 1:47 to 2:00 is 13 minutes. From 2:00 to 2:10 is 10 minutes. The total is 23 minutes.

How many minutes is it from 4:28 till 5:15? Again the hour changes. We do this in two parts: from 4:28 till 5:00 is 32 minutes. From 5 till 5:15 is 15 minutes. Total: 32 + 15 = 47 minutes.

1. How many minutes is it from the time on the clock face until the given time?

till 3:15

till 11:25

till 4:05

till 1:30

a. _____ minutes

b. _____ minutes

c. _____ minutes

d. _____ minutes

till 5:05

till 4:23

till 10:18

till 12:10

e. _____ minutes

f. _____ minutes

g. _____ minutes

h. _____ minutes

121

How much time passes from 5:38 till 8:38? The minute amounts are the same (38), so the minute hand has made some full rounds - full hours - and come back to the same place. So, you only need to look at the difference in the hours: from 5 to 8 is 3 hours. Three hours have passed. How much time passes from 11:30 till 4:30? Again the minute hand has made several full rounds. From 11 to 4 is five hours. Check by turning the hands on your practice clock! You can also figure the passed time in parts: From 11:30 till 12:00 is half an hour. From 12:00 till 4:00 is four hours. From 4:00 till 4:30 is another half an hour. The total is four hours and two half-hours, but two half hours makes a whole hour. The total is five hours. The passing whole hours and half hours — figure it out in parts. How much time passes from 5:00 till 7:30?

How much time passes from 11:30 AM till 5:00 PM?

From 5:00 till 7:00 is two whole hours. From 7:00 till 7:30 is half an hour. Total: two and a half hours.

From 11:30 AM till 12:00 noon is half an hour. From 12:00 till 5:00 PM is five hours. Total: five and a half hours.

2. How much time passes? a. From 2:06 till 10:06

b. From 8:25 till 12:25

c. From 3:30 till 6:00

d. From 7:30 AM

e. From 10:00 AM

f. From 9:49 till 1:49

till 1:30 PM

till 3:30 PM

g. From 5 AM till 5 PM

h. From 11 PM till 12 noon

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i. From 6 AM till 4 PM

3. How much time passes? Figure it out in parts. a. From 1:40 till 2:30

b. From 7:30 AM till 3:10 PM

From 7:30 till 8:00 ____ h ____ min.

From 1:40 till 2:00 ______ minutes

From 8:00 till 12:00 ____ h ____ min.

From 2:00 till 2:30 ______ minutes

From 12:00 till 3:00 ____ h ____ min. Total ______ minutes

From 3:00 till 3:10 ____ h ____ min. Total ____ h ____ min.

c. From 2:35 till 8:15

d. From 6:40 till 4:15

4. Write the ending time or starting time. Imagine turning the minute hand on a clock, or use your practice clock. a. 6:15 → _______

b. 2:03 → _______

c. 11:30 → _______

40 minutes

25 minutes

35 minutes

d. _______ → 5:50

e. _______ → 7:00

f. _______ → 12:10

35 minutes

45 minutes

20 minutes

5. Solve the problems. Imagine turning the minute hand on a clock. a. The science class starts at 10:55 and ends 50 minutes later. When does it end? b. The airplane took off at 3:35 PM and landed at 7:10 PM. How long was the flight? c. Eddie played with building blocks for 2 hours 40 minutes, starting at 3:30. When did he stop? d. Jane played for 45 minutes, starting at 2:30. When did she stop?

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6. Write what time it is in the story as it goes along. Mr. Toad was sitting in his favorite chair at home, excited. It was 4:20, and in fifteen minutes he was going to go out with his friend to do something special. Soon he heard the doorbell ring. Mr. Rat was at the door and it was time to go. (time ________) It was a 20-minute walk to Mr. Mole's house, and the two friends were not planning to be late. But, after walking for 10 minutes (time ________), they heard a familiar voice behind their backs, “Hey, where are you going with such brisk steps?” It was Mr. Duck, wandering around without any particular place to go in mind. The three friends chatted for five minutes, and then (time ________) decided to go together to surprise Mr. Mole. Soon they arrived there (time ________), and knocked on the door. Mr. Mole had no idea they'd be coming, and was very happy seeing his friends. After chatting for ten minutes, they headed for the table to sip some cranberry juice (time ________). 7. Answer the questions about the television programs. Channel 1 4:30 Nature film: Whales 5:30 Children's Story Time 6:05 Early News 6:35 Shopping Spree Show 7:05 The Week in Politics

Channel 2 4:30 Cooking Class 5:05 Kids TV 5:55 Quick News 6:20 Nature Film: The Antarctica 7:25 Current Trends

Channel 3 4:45 Afternoon Bits 5:15 Nature film: Bees and Honey 6:20 Flash News 6:40 The Silly Faces Show 7:20 Arnold's Kitchen

a. All three channels have a nature film. List here how long each nature film lasts.

b. Which is a longer program, “Children's Story Time” or “Kids TV”? How many minutes longer? c. Which channel has the longest news? Which one has the shortest news? What is the difference in minutes? d. Megan does some channel surfing this way: From 4:30 till 5:15 Channel 1 From 5:15 till 6:20 Channel 3 From 6:20 till 7:25 Channel 2 Which programs did Megan see either wholly or partially?

124

Using the Calendar On a calendar, the weekdays from Sunday through Saturday (or sometimes from Monday through Sunday) are written in a row above the numbers.

October 2010 Sun Mon Tue Wed Thu

Fri

Sat

1

2

3

4

5

6

7

8

9

A week has 7 days.

10

11

12

13

14

15

16

The week from the 12th through the 18th is highlighted. It is 7 days.

17

18

19

20

21

22

23

24

25

26

27

28

29

30

What day is one week after October 20th?

31

It is the day directly under it on the calendar, October 27th. You could also add 7 to the day number: 20 + 7 = 27. What day is one week after October 25th? It is November 1st, because it would be directly under the number 25, if we continued writing the days of November in the boxes. What day is three weeks after October 25th? You need to “jump” down three rows. First go from October 25th to November 1st on the next calendar, and then two weeks more from there. November 15th is exactly three weeks later than October 25th. What day is six weeks after October 14th? Go down six rows, but skip from October 28th directly to the first Thursday of November, and then on from there. It is November 25th. What day is three weeks before November 10th? November 3rd is one week before that. Go “up” one more week - to October 27th, and one more, to October 20th. What day is two months after March 10th? Just change the name of the month: it is May 10th.

125

1. Find out the later or earlier dates using the calendar. November 2010

a. November 6th

Sun Mon Tue Wed Thu

4 days earlier ______________ b. November 30th

5 days later ______________ c. November 21st

Fri

Sat

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Fri

Sat

1 week earlier _____________ December 2010

d. November 22nd

Sun Mon Tue Wed Thu

2 weeks later ______________ e. November 19th

4 weeks later ______________ f. December 6th

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

2 weeks earlier _____________

January 2011 Sun Mon Tue Wed Thu

g. December 26th

Fri

Sat 1

4 weeks later _____________ h. January 14th

5 weeks earlier ____________ i. January 18th

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

27

26

27

28

29

30

31

8 weeks earlier _____________ 2. A summer camp was scheduled to start on July 15th and run for 5 days, but was delayed by one week. When did it start? What was the last day of the camp? 3. Jen's dentist appointment was scheduled for the 23rd of December, but was postponed by 2 weeks because she was sick. What was the new date? 4. It is November 2nd. How many days is it till Jack's birthday on November 27th? Till mom's birthday on December 5th? 5. Mary borrowed a book from her friend on February 18th, and she said she needs to return it in 2 months. When should Mary return it at the latest?

126

Changing Time Units Some important numbers to remember! z

1 hour = 60 minutes

z

1/2 hour = 30 minutes

1 hour 15 minutes is 60 + 15 = 75 minutes. 78 minutes is 60 + 18 minutes, so it is 1 hour 18 minutes.

1. Fill in the table. a. Notice the pattern!

b. Compare to the ones on the left!

1 hour is _____ minutes

1 hour 10 minutes is _______ minutes

2 hours is _____ minutes

2 hours 10 minutes is _______ minutes

3 hours is _____minutes

3 hours 20 minutes is _______ minutes

4 hours is _____ minutes

4 hours 50 minutes is _______ minutes

5 hours is _____minutes

5 hours 30 minutes is _______ minutes

6 hours is _____minutes

6 hours 40 minutes is _______ minutes

c. Practice some more.

d. Explain how to convert 2 hours 50 minutes to minutes.

1 h 4 min is _____ minutes 1 h 28 min is _____ minutes 2 h 12 min is _____ minutes 2 h 30 min is _____ minutes 3 h 50 min is _____ minutes 3 h 11 min is _____ minutes

127

2. Are the times less than 1 hour? Between 1 and 2 hours? Or more than 2 hours? Connect with a line. Between 1 and 2 hours

Less than 1 hour

56 minutes

80 minutes

66 minutes

78 minutes

123 minutes

110 minutes

95 minutes

118 minutes

136 minutes

104 minutes

43 minutes

55 minutes

More than 2 hours

3. Now to do the same thing the other way around: convert minutes to hours and minutes. a. More than one hour.

b. More than one hour.

60 minutes is __ hour

60 minutes is ___ hour

70 minutes is ___ hour ____ minutes

75 minutes is ___ h min

80 minutes is ___ hour ____ minutes

105 minutes is ___ h ____ min

100 minutes is ___ hour ____ minutes

108 minutes is ___h ____ min

90 minutes is ___ hour ____ minutes

111 minutes is ___ h ____ min

110 minutes is ___ hour ____ minutes

96 minutes is ___h ____ min

c. More than two hours.

d. More than three hours.

120 minutes is ____ hours

180 minutes is ____ hours

130 minutes is ____ h _____ min

189 minutes is ____ h _____ min

150 minutes is ____ h _____ min

190 minutes is ____ h _____ min

144 minutes is ____ h _____ min

204 minutes is ____ h _____ min

156 minutes is ____ h _____ min

215 minutes is ____ h _____ min

165 minutes is ____ h _____ min

222 minutes is ____ h _____ min

e. Explain how to convert 88 minutes to hours and minutes.

128

4. Solve the problems. Give the answers as hours and minutes, not just plain minutes. a. Lisa rode on her horse 45 minutes in the morning, and 65 minutes in the evening. What was the total amount of that time she rode on her horse? b. It takes 30 minutes to mix the cake, and 40 minutes to bake it. What is the total time? c. It takes Mom 10 minutes to ride her bike to a store, and 12 minutes to ride back home. She does this three times a week. How much time does she spend bicycling in a week? d. A doctor tells Mom she needs more exercise, so Mom decides to start shopping in another store that is further away. Now it takes her 18 minutes to get there, and 22 to come back. How much time does she now spend bicycling in a week? e. See John's jogging schedule below. How much time does he spend jogging in a whole week? Monday 45 minutes

Tuesday Wednesday Thursday rest

45 minutes

rest

Friday

Saturday

Sunday

30 minutes 30 minutes

rest

Some important numbers to remember! z z

1 day = 24 hours 1 week = 7 days

z z z

1 year = 12 months 1 year = 52 weeks 1 year = 365 days

5. Fill in the empty lines. a. Weeks to days! Follow the pattern.

b. Weeks and days!

1 week = ____ days

1 week 2 days = ____ days

2 weeks = ____ days

2 weeks 4 days = ____ days

3 weeks = ____ days

17 days = ___ weeks ___ days

4 weeks = ____ days

24 days = ___ weeks ___ days

Your last answer should be very close to one month, which usually is 30 or 31 days.

15 days = ___ weeks ___ days

129

6. a. The doctor is going to take the cast off of Andy's leg in 4 weeks and 3 days. How many days is that? b. Matthew is choosing between three different summer math programs. Figure out how long they are, in weeks and days. Duration (weeks / days) Super Summer Math Camp Duration: 24 days

____ weeks _____ days

Boost Your Grades Course Duration: 20 days

____ weeks _____ days

Summer School Math Program ____ weeks _____ days Duration: daily from July 9th till July 31st 7. Change between the different time units. Fill in the empty lines. a. Years and months!

b. Days and hours!

1 year = ____ months

1 day = ____ hours

2 years = ____ months

2 days = ____ hours

1 year 5 months = ____ months

3 days 3 hours = ____ hours

19 months = ___ year ___ months

27 hours = ___ days ___ hours

25 months = ___ years ___ months

42 hours = ___ days ___ hours

8. a. Marsha's baby is 23 months old, and Brenda's baby is 2 years and 2 months old. Which baby is older? How many months older? b. After an operation, a patient can't eat solid food for 48 hours. How many whole days and hours is that? c. It is 7 AM now. What time will it be 12 hours later? 24 hours later? 36 hours later?

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Review 1. Write the time the clock shows, and the time given minutes later.

a.

____:____

b. ____:____

c. ____:____

d. ____:____

10 min. later 2. a. - b. How many minutes is it from the time on the clock face until the given time? c. - g. How much many minutes is between the given times? c. from 4:08 till 10:08 d. from 3 AM till 5 PM e. from 8:23 till 8:41

till 7:55

till 11:23

a. _____ minutes

b. _____ minutes

f. from 5:15 till 6:06 g. from 3:37 till 4:17

3. Solve the problems. Imagine turning the minute hand on a clock. a. The music class starts at 1:45 and ends 50 minutes later. When does it end? b. The train left at 11:10 and arrived at 12:20 PM. How long was the trip? 4. Convert between the time units. a. 2 h 40 min is ____ minutes

d. 1 year 5 months = ____ months

b. 110 minutes is __ hour ____ minutes

e. 27 hours = ___ days ___ hours

c. 2 weeks 4 days = ____ days

f. 3 days 1 hour = ____ hours

131

Chapter 5: Money Introduction This chapter of Math Mammoth Grade 3-A Complete Worktext teaches counting coins, making change, and solving simple problems about money. The first lesson, Using the Half-Dollar, reviews counting coins, including half-dollars. In the lesson Dollars, the student writes dollar amounts using the “$” symbol and the decimal point. The lesson Making Change explains two basic ways of making change: (1) counting up and (2) subtracting (finding the difference). This is all done with mental math. The following lesson, Mental Math and Money Problems, also uses mental math, this time in solving simple money problems. The lesson Solving Money Problems introduces the concept of adding and subtracting amounts of money vertically in columns.

The Lessons in Chapter 5 page

span

Using the Half-Dollar ........................................ 134

2 pages

Dollars .............................................................

136

3 pages

Making Change ...............................................

139

4 pages

Mental Math and Money Problems ................

143

3 pages

Solving Money Problems ................................

146

4 pages

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

150

1 pages

Helpful Resources on the Internet Use these free online resources to supplement the “bookwork” as you see fit. 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 Using Money Drag the right amount of coins and bills (US) to the answer space to match a given amount. The pictures look a little fuzzy. http://www.mathcats.com/microworlds/usingmoney.html Counting Money Activity from Harcourt Count the coin value, type it in the box, and click ‘Check’ to verify your answer. http://www.hbschool.com/activity/counting_money/

132

Cash Out Make the correct change by clicking on the bills and coins. http://www.mrnussbaum.com/cashd.htm Piggy bank When coins fall from the top of the screen, choose those that add up to the given amount to fill up the piggy bank. http://fen.com/studentactivities/Piggybank/piggybank.html Money Instructor Exercises and worksheets for checkbook math. Includes a checkbook to print, a worksheet for writing dollars and cents, checking account deposits, checkbook transactions, and word problems. http://www.moneyinstructor.com/checks.asp

Money Activities at the National Library of Virtual Manipulatives Count the money shown, or make the given change, or make one dollar. http://nlvm.usu.edu/en/nav/frames_asid_325_g_2_t_1.html Making Change Game at MathPlayground.com An interactive game where you figure out the change and then make it using the fewest possible bills and coins. http://www.mathplayground.com/making_change.html

133

Using the Half-Dollar

This is a half-dollar. It is worth 50 cents.

a quarter

a dime

a nickel

a penny

____ ¢

____ ¢

____ ¢

____ ¢

= A half-dollar is worth two quarters, because 50 = 25 + 25.

A half-dollar and a quarter is 75 cents.

A half-dollar and two quarters make $1.

1. Half-dollars and quarters. Write the total amount in cents.

a.

b.

d.

e.

c.

2. Write how many half-dollars and how many quarters you need to make these amounts. a. 150 cents

b. 200 cents

c. 150 cents

d. 75 cents

____ half-dollars

____ half-dollars

____ quarters

____ quarters

e. 175 cents

f. 225 cents

____ half-dollars and ____ quarter(s)

____ half-dollars and____ quarter(s)

134

Count up, starting with the coin(s) with the most value.

100 ¢

125 ¢

127 ¢

50 ¢

100 ¢

3. How much money? Write down the amount in cents.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

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105 ¢

You may count two quarters as fifty.

Dollars 2 One dollar.

Five dollars. $5 or $5.00.

$1 or $1.00

Write “$” symbol in front of dollar amounts. Write first the dollars, then a decimal point, and then the cents.

$1.51

1. How much money? Write the amount.

a. $________

b. $________

c. $________

d. $________

e. $________

f. $________

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$5.30

If you have more than 100 cents, those 100 make a dollar.

100 ¢ = $1

100 ¢ = $1 100 ¢ = $1 Total $2.10

Total $2.32

2. How much money? Write the amount.

a. $________

b. $________

c. $________

Remember to put 0 into the dollar's place if you have a total cent amount that is less than 100. 40 cents = $0.40

82 cents = $0.82

9 cents = $0.09

3. Write as dollar amounts.

three nickels and a dime a. $________

b. $________

c. $________

eight dimes

seven pennies and a nickel

three quarters and two dimes

d. $________

e. $________

f. $________

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4. Write the cent amounts as dollar amounts, and vice versa. a. 56 ¢ = $______

b. 6 ¢ = $______

c. 425 ¢ = $______

d. 209 ¢ = $______

e. _____ ¢ = $5.69

f. _____ ¢ = $0.30

g. _____ ¢ = $3.06

h. _____ ¢ = $0.79

5. The picture shows how much money you have. Write how much you will have left if you buy the items listed. If I buy:

I will have left:

a. a puzzle for $5.20

$

b. a book for $7.35

$

c. a pineapple for $3.52 $

6. Add the money amounts. You can add the cents and dollars separately in your head. a. $0.37 + $0.40 = $______

b. $1.25 + $4.00 = $______

c. $5.43 + $1.20 = $______

7. The picture shows how much money you have. Write how much you will have left if you buy the items listed. If I buy: a. a book for $4.20 and

a magazine for $1.50

I will have left: $

b. two brushes for $3.35 each $ c. candles for $4.09 and

paper cups for $2.07

If I buy:

$

I will have left:

d. a pen ($0.60) and an eraser ($0.50) $ e. three pencils for $0.40 each f. a notebook for $1.12 and

loose paper for $0.90

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$ $

Making Change 1. To give change, or to check the change you are given, you can count up from the price of the item until you reach the amount the customer gives. First count up to the next whole dollar. Then use 1-dollar or 5-dollar bills.

a. Price: $0.76 Customer gives $1

The change is Count up →

$0.80

$1.00

b. Price: $8.90

The change is

Count Customer gives $10 up →

$9.00

$10.00

c. Price: $2.35 Customer gives $5

$______

The change is Count up →

$______

d. Price: $4.18 Customer gives $10

$______

The change is Count up →

$______

e. Price: $3.04

The change is Customer gives $10

Count up →

$______

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2. Figure out the change. You can draw coins or use real money to help.

a. Price: $3.55

The change is

Customer gives $5

$______

b. Price: $8.60

The change is

Customer gives $10

$______

c. Price: $4.70

The change is

Customer gives $10

$______

d. Price: $7.99

The change is

Customer gives $10

$______

e. Price: $3.25

The change is

Customer gives $5

$______

f. Price: $4.15

The change is

Customer gives $10

$______

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Finding change is finding the difference.

Example:

You can also find the change by subtracting the item price from the money amount the customer gives. You are just finding the difference between the price and the money given.

A book costs $6. You give $10. Your change: $10 − $6 = $4.

You can add up to find the change.

A toy costs $3.30. You give $10.

Another method is to first add up to the next whole dollar to find the cents. Then find the dollar-amount by subtracting.

First find how many cents there are to the next whole dollar: $3.30 + $0.70 = $4.

Again, you are finding the difference between the price and the money given, but you're finding that in two parts.

Then find the difference between $4 and $10, which is $6. The total change is $6.70.

3. Find the change. a. A book costs $7.

b. A basket costs $4.

c. A train costs $5.50.

You give $10.

You give $20.

You give $10.

Change: $_______

Change: $_______

Change: $_______

d. A magazine costs $2.40.

e. A meal costs $7.60.

f. A drink costs $1.30.

You give $10.

You give $10.

You give $5.

Change: $_______

Change: $_______

Change: $_______

g. Crayons cost $3.80.

h. Staples cost $1.40.

i. Paper costs $7.20.

You give $5.

You give $2.

You give $10.

Change: $_______

Change: $_______

Change: $_______

4. Did these people receive the correct change? If not, correct it. a. Margie bought a few items that cost $7.86. She paid with a $10-bill. She got back two dollars, two dimes, and four pennies. b. Fred bought a toy car for $2.76 and gave $5 for it. The clerk handed back to him a quarter and two dollars.

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Here's a little trick for finding two 2-digit numbers that add up to 100: The ones add up to 10. The tens add up to 9... ...plus there is one ten that is “carried” from the ones — total 10 tens or a hundred.

5. Try it yourself! Find the two-digit number so the sum is 100. a.

56 + 100

b.

c.

19 + 100

72 + 100

d.

44 + 100

e.

34 + 100

6. Fill in the missing cent-amount. You can use the “trick” explained above. a. 54¢ + _____¢ = 100¢

b. 38¢ + _____¢ = $1

c. 33¢ + _____¢ = $1

76¢ + _____¢ = 100¢

$1.13 + _____¢ = $2

$4.39 + _____¢ = $5

27¢ + _____¢ = 100¢

$3.86 + _____¢ = $4

$9.37 + _____¢ = $10

7. Find the change. Find also what coins and bills that could be used to make the change. a. A book costs $3.55. You give $5.

b. Pencils cost $2.88. You give $5.

Change: $1.45. Use a quarter, two dimes, and a dollar bill.

c. A shirt costs $7.76. You give $10.

d. Sunglasses cost $8.95. You give $10.

e. A sandwich costs $4.26. You give $5.

f. Flowers cost $6.28. You give $10.

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Mental Math and Money Problems You can add money amounts in your mind, as well. Add the dollars and the cents separately. If you get more than 100 cents, then those make another dollar.

$1.20 + $1.50

$0.14 + $1.20

= $2.70

= $1.34

$0.70 + $0.70

$0.99 + $0.06

= 140 cents = $1.40

= 105 cents = $1.05

1. Find the total cost of buying the things listed. Add mentally if you can.

$3.10

$1.50

$1.00

$1.90

55 ¢

$2.20

80 ¢

50 ¢

$20 $35

a. scissors and pencils

b. pen and glue

c. crayons, glue, and pencils

d. eraser and calculator

e. microscope and scissors

f. book bag, pen,

and crayons

g. stapler and glue

h. glue and eraser

i. scissors and stapler

j. pen, pencil, and crayons

k. calculator, pen,

l. scissors and eraser

and microscope

143

2. Add up to the next whole dollar. a. $0.30 + ____ = $1.00

b. $3.30 = ____ = $4.00

c. $1.10 = ____ = $2.00

50 ¢ + ____ = $1.00

$2.20 = ____ = $3.00

$1.05 = ____ = $2.00

70 ¢ + ____ = $1.00

$5.60 = ____ = $6.00

$1.15 = ____ = $2.00

Price: $1.20. Customer gave $5.

Add up to find the change

$1.20 $2.00 $5.00

To find the change, find the difference between the price and the money given.

differences →

Start from the price and add till you reach the amount the customer gives.

80 ¢ $3

Change: $3.80 Price: $3.37. Customer gave $5.

First add up to the next whole ten cents.

$3.37 $3.40 $4.00 $5.00

Then add up to the next whole dollar (if need be).

differences →

Lastly add all the differences to find the total change.

3 ¢ 60 ¢

$1

Change: $1.63

3. Find the total change. a. Price: $1.80. Customer gave $5.

b. Price: $3.26. Customer gave $4.

$1.80 $2.00 $5.00

$3.26 $3.30 $4.00

Change: $_______

Change: $_______

c. Price: $2.19. Customer gave $5.

d. Price: $0.82. Customer gave $5.

$2.19 $2.20 $3.00 $5.00

$0.82 $0.90 $1.00 $5.00

Change: $_______

Change: $_______

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4. Find the change. a. Price: $0.45. Customer gave $1.

b. Price: $2.40. Customer gave $5.

Change: $_______

Change: $_______

c. Price: $3.15. Customer gave $3.50.

d. Price: $4.36. Customer gave $5.

Change: $_______

Change: $_______

e. Price: $0.28. Customer gave $0.50.

f. Price: $1.34. Customer gave $5.

Change: $_______

Change: $_______

g. Price: $2.29. Customer gave $2.50.

h. Price: $3.58. Customer gave $3.75.

Change: $_______

Change: $_______

5. Solve the word problems. a. Mary bought ice cream for $2.20 and

water for $0.70. Find her total bill and her change from $3.

b. John bought three slices of pizza for

$1.15 each. Find his total bill and his change from $5.

c. If you have $3, can you buy two boxes

of crayons for $1.40 each? If not, find how much more you would need. If yes, find your change if you buy them.

d. If you have $5, can you buy a calculator,

a stapler, and a pen (see problem 1)? If not, find how much more you would need. If yes, find your change if you buy them.

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Solving Money Problems dollars cents 1 1 1

Add dollar and cent amounts in columns the same way as any other numbers. You can imagine that the decimal point is not there while calculating. Just remember to put it in the answer!

$ 1 4. 0 5 2. 1 1 + 5 4. 9 5

Use the dollar symbol ($) in the first item and in the answer, when adding in columns.

$ 7 1.1 1

1. Add the dollar amounts. a.

b.

$2.24 + 4.69

c.

$ 5.69 7.50 + 22.25

$ 2 . 9 9 d. 5.79 1.40 + 6.72

$20.46 2.79 5.62 + 6.68

e.

$12.99 25.59 41.80 + 26.70

2. Find the total cost of buying the items listed.

$3.10

$11.45

a. a skirt and a bookbag

$15.99

$1.50

b. a teddy bear, crayons,

scissors, and two pens

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$4.87 c. a pen and three

pairs of scissors

$1.99

Subtract or add up to find the change. To find the change, you find the difference between the price and the money given. To find any difference, you can: z

subtract the price from the money given, or

z

add up from the price to the money given.

Subtracting to find the change often involves borrowing over many zeros. A bag costs $11.28. A customer pays with $20. What is his change? Add up: + $0.72

The price was $5.65. A customer paid with $20 and got back $14.55. Was that correct change?

Subtract:

+ $8

9 1 10

$11.28 $12.00 $20.00

Add the price and the change:

9 10 10

1 1

$1 4 . 5 5 + 5.6 5

$2 0 . 0 0 − 1 1.2 8 $

The difference is $8.72.

1

$2 0 . 2 0

8.7 2

No, it was 20 cents too much.

3. Find the difference by counting up. +

a.

+

+

b.

$10 – $2.66 =

+

$20 – $7.52 = $2.66 $3.00 $10.00 +

c.

$7.52 $8.00 $20.00

+

+

d.

$20 – $14.47 =

+

$50 – $28.33 = _____ _____ _____

_____ _____ _____

4. Subtract the dollar amounts. a.

$5.50 – 2.39

b.

$ 10 . 9 0 – 4.45

c.

$20.00 – 7.29

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d.

$10.00 – 6.44

e.

$50.00 – 34.56

5. Solve the problems.

$6.90

$6.75

$3.48

a. Mark bought two computer mice and paid

$15.99

$35.90

b. Judy bought a book and a bookbag.

with a $20-dollar bill. What was his change?

She paid with $30. How many dollars and cents did she receive in change?

c. Mark bought a microscope and paid

d. Mark has saved $5.50, and he wants to

with a $50-dollar bill. He received $14.10 as change. Was that correct?

buy a calculator and a book. What is their total cost?

e. How many calculators can Ernest

How much more money does Mark need to buy them?

buy with $10?

What will his change be from the purchase?

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6. Solve the word problems. a. Dad bought a meal for $15.55 and a

b. Dad paid with a $50 bill.

drink for $2.39 at a restaurant. What was his bill?

What was his change?

c. You have saved $15, and you want to buy

d. Melissa bought a book for $4.55,

a toy for $22.95. How much do you still need to save?

a magazine for $2.30, and a pencil for $0.85. Find her total bill.

What is her change from $10?

e. John buys two ice creams, a fruit juice, and a sandwich.

What is his total bill?

What is John's change from $20?

f. Can Mom buy a jacket for $14.55 and a blouse for $23.95 with $40?

If yes, what is her change from that? If no, how much is she missing?

149

Ice cream $2.15 Fruit juice $1.45 Soda pop $1.56 Sandwich $3.98 Coffee $1.55

Review 1. How much money? Write the amount.

a. $________

b. $________

2. Write as dollar amounts.

five dimes and a quarter

three half-dollars, three nickels, and 8 pennies

three quarters, two dimes, and a half-dollar

a. $________

b. $________

c. $________

3. Solve the problems. a. Maria has saved $23, and she wants to buy

b. Arnold bought a sandwich for $2.55,

a game for $42.95. How much does she still need to save?

soup for $2.30, and a juice for $1.85. Find his total bill.

What is his change from $10?

4. Solve using mental math. a. You buy stickers for $2.35 and a notebook for $1.20. What is your total bill? b. What is your change from $5?

150