Math Mammoth Fractions 1

Copyright 2012 Taina Maria Miller. EDITION 2.2 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.

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Visit www.MathMammoth.com for more information about ebooks and books by Maria Miller. Visit www.HomeschoolMath.net/worksheets/ to create free math worksheets. Visit also my blog at Homeschoolmath.blogspot.com - math teaching tips, articles, news, reviews.

2

Math Mammoth Fractions 1

Contents Introduction ................................................ ........................

4

Fraction Term inology .... ...................................................

8

Understanding Fractions .................................................. Mixed Numbers ................................................................. Part of a Whole Group 1 ................................................... Part of a Whole Group 2 ................................................... Part of a Whole Group 3 ...................................................

9 12 16 19 21

Adding and Subtracting Like Fractions .......................... Review: Mixed Numbers...................................................

23 27

Adding Mixed Numbers 1 ................................................ 31 Subtracting Mixed Numbers 1 ......................................... 34 Subtracting Mixed Numbers 2 — Renaming ................. 37 Subtracting M ixed Num bers — Extra Pra ctice ... .......... 40 Equivalent Fractions ......................................................... 42 Equivalent Fractions 2 ...................................................... Adding Unlike Fractions 1 ................................................

46 48

Add Unlike Fractions 2: Finding the Common Denominator.................................

51

Add and Su btract Un like Fract ions .... ............................ Mixed Numbers with Unlike Fractional Parts ............... Add and Subtract Several Unlike Fractions ..................

54 56 60

Measuring in Inches ......................................................... Comparing Fractions 1 .....................................................

62 66

Comparing Fractions 2 ..................................................... Fraction Problems ............................................................ Review ...............................................................................

68 71 72

Answers .............................................................................

75

Fractions Cutouts .............................................................

99

More from Math Mammoth ............................................ 111

3

Introduction Math Mammoth Fractions 1 is the first book of two that cover all aspects of fractions in arithmetic. This book covers the concepts of fraction and mixed numbers, equivalent fractions, adding and subtracting like and unlike fractions, adding and subtracting mixed numbers, and comparing fractions. The book Fractions 2 covers simplifying fractions and multiplication and division of fractions.

I have made a set of videos to match many of the lessons in this book. You can access them at http://www.mathmammoth.com/videos/fractions_1.php Studying fractions involves lots of rules, and many students learn them only mechanically, not really understanding the underlying concepts and principles. Then they end up making lots of mistakes because they confuse the different rules and either apply the wrong one or apply the right rule but don't remember it quite right. All this can make students even fear fractions in math. To avoid that, this book uses the visual model of a pie divided into slices all the way through the book. It is a very natural model, because it uses a circle that can be divided into any number of circle sectors (slices). When students work with this model from lesson to lesson, they will eventually be able to “see” these pies in their mind. This, in turn, gives them the ability to do many of the easier fraction calculations mentally. It also enables students to really UNDERSTAND these concepts, and not just learn mechanical rules. You are welcome to use manipulatives alongside the book; however the visual pie model is probably sufficient for most students in 5th grade level. I have also included (in the appendix) printable cutouts for fractions from halves to twelfths. You can use them to make your own fraction manipulatives. To make the manipulatives sturdier, glue the printed pages on cardboard, and cut the parts only after gluing. The whole circle is there to illustrate “one whole” - needed when studying mixed numbers. You will probably need to print at least two copies of each cutout page. You can use the white cutout fractions if you need to save on ink and let children color them. Just use consistent colors so that thirds are always the same color, fourths are the same color, etc. Fractions, the student learns to draw pie models of certain common In the firstThis lesson, fractions. skillUnderstanding is needed later on in various exercises through the book.

The lesson Mixed Numbers teaches the concept with pictures. The child also writes mixed numbers as fractions. The next lessons, titled Part of a Whole Group 1, 2, and 3 have to do with finding a part of a certain number of objects, and of course has lots of practical applications. It ties the concept of a fraction with division of whole numbers. Next we study adding and subtracting like fractions, which is an easy topic. Next is lesson reviews mixed numbers and further practices changing mixed numbers to fractions and vice versa. Next we add and subtract mixed numbers with like fractional parts. Then, it is time to study equivalent fractions, as a prerequisite for adding unlike fractions. Equivalent fractions are presented as parts that have been split further. The rule is to multiply both the numerator and the denominator by the same number, but try to emphasize the terminology of “splitting the existing parts into so-and-so many pieces” or something similar. That should help students to understand the concept instead of memorizing a mechanical rule. Add Unlike Fractions 1 is an introductory lesson in the sense that the student is not yet introduced to the rule for finding the common denominator. In this lesson, the common denominator is either given, or the student figures it out using pictures.

4

Add Unlike Fractions 2 emphasizes the idea that we need to find a common denominator, and then convert the fractions to like fractions before adding. Many textbooks introduce here the concept of Least Common Denominator, LCD, which is the best common denominator to use since it is the smallest. That concept tends to be only memorized and poorly understood, so the lesson here does not overtly emphasize that one always needs to find the LCD. Your student will encounter the concept of LCD again in 6th and 7th grade.

Finally we also add and subtract mixed numbers with unlike fractional parts, and add & subtract several unlike fractions. The last new concept in this book is that of comparing fractions. Once the student has mastered converting two fractions to equivalent, like fractions, this should be fairly easy. Answers are in the end of the book. I wish you success in your math teaching! Maria Miller, the author

Helpful Resources on the Internet Use these free online resources to supplement the “bookwork” as you see fit. You can access an up-to-date online version of this list at www.mathmammoth.com/weblinks/fractions_1.htm

Fraction Videos for Math Mammoth Fractions 1 book A set of videos by the author that tie in with the lessons in this book. http://www.mathmammoth.com/videos/fractions_1.php

Fractions and Mixed Numbers Clara Fraction's Ice Cream Shop A game in which you convert improper fractions to mixed numbers and scoop the right amount of ice cream flavors on the cone. www.mrnussbaum.com/icecream/index.html Equivalent Fractions Equivalent Fractions from National Library of Virtual Manipulatives (NLVM) See the equivalency of two fractions as the applet divides the whole into more pieces. http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html Equivalent Fractions Draw two other, equivalent fractions to the given fraction. Choose either square or circle for the shape. http://illuminations.nctm.org/ActivityDetail.aspx?ID=80 Fraction Frenzy

Click on pairs of equivalent fractions, as fast as you can. See how many levels you can get! www.learningplanet.com/sam/ff/index.asp

5

Fresh Baked Fractions Practice equivalent fractions by clicking on a fraction that is not equal to others. www.funbrain.com/fract/index.html ddition and Subtraction MathSplat Click on the right answer to addition problems (like fractions) or the bug splats on your windshield! http://fen.com/studentactivities/MathSplat/mathsplat.htm Adding fractions Illustrates with pictures finding the common denominator. http://matti.usu.edu/nlvm/nav/frames_asid_106_g_2_t_1.html Old Egyptian Fractions Puzzles to solve: add fractions like a true Old Egyptian Math Cat! www.mathcats.com/explore/oldegyptianfractions.html Fraction Bars Blackjack Computer deals you two fraction cards. You have the option of getting more or “holding”. The object is to get as close as possible to 2, without going over, by adding the fractions on your cards. http://fractionbars.com/Fraction_Bars_Black_Jack/ Action Fraction A racing game with several levels where you answer questions about adding and subtraction fractions. The levels advance from using like fractions to using unlike fractions and eventually subtraction. http://funschool.kaboose.com/formula-fusion/number-fun/games/game_action_fraction.html Fishy Fractions Select the correct answer and the pelican catches the fish. Options for fraction addition or subtraction, like or unlike denominators, simplifying, comparing, and more. www.iknowthat.com/com/L3?Area=FractionGame Comparing Fractions Comparison Shoot Out Choose level 2 or 3 to compare fractions and shoot the soccer ball to the goal. www.fuelthebrain.com/Game/play.php?ID=47 Order Fractions On each round, you drag five given fractions in the correct order. http://www.bbc.co.uk/schools/ks2bitesize/maths/activities/fractions.shtml Comparing Fractions - XP Math Simple timed practice with comparing two fractions. http://xpmath.com/forums/arcade.php?do=play&gameid=8 Fractional Hi Lo Computer has selected a fraction. You make guesses and it tells you if your guess was too high or too low. www.theproblemsite.com/games/hilo.asp

6

ll Aspects Visual Fractions Great site for studying all aspects of fractions: identifying, renaming, comparing, addition, subtraction, multiplication, division. Each topic is illustrated by either a number line or a circle with a Java applet. Also couple of games, for example: make cookies for Grampy. www.visualfractions.com Conceptua Math Conceptua Math has free, interactive fraction tools and activities that are very well made. The activities include identifying fractions, adding and subtracting, estimating, finding common denominators and more. Each activity uses several fraction models such as fraction circles, horizontal and vertical bars, number lines, etc. that allow students to develop conceptual understanding of fractions. www.conceptuamath.com Who Wants Pizza? Explains the concept of fraction, addition and multiplication with a pizza example, then has some interactive exercises. http://math.rice.edu/~lanius/fractions/index.html Fraction lessons at MathExpression.com Tutorials, examples, and videos explaining all the basic fraction math topics. Look for the lesson links in the left sidebar. www.mathexpression.com/understanding-fractions.html Visual Math Learning Free tutorials with some interactivity about all the fraction operations. Emphasizes visual models and lets student interact with those. www.visualmathlearning.com/pre_algebra/chapter_9/chap_9.html Fractioncity Make “fraction streets” and help kids with comparing fractions, equivalent fractions, addition of fractions of like and unlike denominators while they drive toy cars on the streets. This is not an online activity but has instructions of how to do it at home or at school. www.teachnet.com/lesson/math/fractioncity.html Online Fraction Calculator Add, subtract, multiply or divide fractions and mixed numbers. www.homeschoolmath.net/worksheets/fraction_calculator.php Fraction Worksheets: Addition, Subtraction, Multiplication, and Division Create custom-made worksheets for the four operations with fractions and mixed numbers. www.homeschoolmath.net/worksheets/fraction.php Fraction Worksheets: Equivalent Fractions, Simplifying, Convert to Mixed Numbers Create custom-made worksheets for some other fraction operations. www.homeschoolmath.net/worksheets/fraction-b.php

7

Fraction Terminology As we study fractions and their operations, it’s important that you understand the terms, or words, that we use. This page is a reference. You can even post it on your wall or make your own fraction poster based on it. 3 11

The top number is the numerator. It enumerates, or numbers (counts), how many pieces there are. The bottom number is the denominator. It denominates, or names, what kind of parts they are.

A mixed number has two parts: a whole-number part and a fractional part. 3

3

For example, 2 7 is a mixed number. Its whole-number part is 2, and its fractional part is 7 . The mixed number 2

3 7

actually means 2 +

3 7

.

Like fractions have the same denominator. They have the same kind of parts.

It is easy to add and subtract like fractions, because all you have to do is look at how many of that kind of part there are.

2

7

and

9

are like fractions.

9

Unlike fractions have a different denominator. They have different kinds of parts.

It is a little more complicated to add and subtract unlike fractions. You need to first change them into like fractions. Then you can add or subtract them.

2 9

3

and

are unlike fractions.

4

A proper fraction is a fraction that is less than 1 (less than a whole pie). 2/9 is a proper fraction.

11 9

An improper fraction is more than 1 (more than a whole pie). Since it’s called a fraction, it’s written as a fraction and not as a mixed number.

But 1

Equivalent fractions are equal in value. If you think in terms of pies, they have the same amount of “stuff to eat,” but they’re written using different denominators, or are “cut into different kinds of slices.”

3 9

and

2 9

is an improper fraction.

is a mixed number.

1 3

are equivalent fractions.

Simplifying a fraction means that, for a given fraction, you find an equivalent fraction that

has a “simpler,” or smaller, numerator and denominator. (It has fewer but bigger slices.)

9 12

8

simplifies to

3 4

.

Math Mammoth Fractions 1 (Blue Series)

Understanding Fractions Fractions are PARTS of a WHOLE. The WHOLE is always divided into EQUAL parts. One part is colored; two equal parts; one half.

Two parts are colored; two equal parts; two halves OR one whole.

1 2

2 2

=1

One part is colored;

1

Two parts are colored;

2

four equal parts; one fourth.

4

five equal parts, two fifths.

5

Three colored parts; seven equal parts; three sevenths.

3

Can you tell what fraction this is?

7

The number ABOVE the line tells HOW MANY PARTS are colored. It enumerates or counts the colored parts. 3

NUMERATOR

8

DENOMINATOR

The number BELOW the line tells WHAT KIND OF PARTS the whole is divided into. It denominates or names the parts. We use ordinal numbers to name the fractional parts.

“three eighths”

1. Color the parts to illustrate the fraction.

f.

b.

a.

c.

d.

e.

7

6

4

4

2

4

8

10

6

5

4

7

g.

h.

i.

l.

k.

j.

2

11

5

1

9

2

6

12

9

5

10

7

9


2. Write the fractions, and read them aloud.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

How to draw pie models

Halves: split the circle with a straight line.

Thirds: draw lines at 12 o'clock, 4 o'clock, and 8 o'clock.

Fourths: First draw halves, then split those like a cross pattern.

Fifths: Draw as a man doing jumping jacks.

Sixths: First draw thirds, then split those

Eighths: First draw fourths, then split those.

3. Draw the pie models and color the parts to illustrate the fractions.

a.

e.

2 3

4 5

b.

f.

2

c.

5

3

g.

8

10

1 6

1 3

d.

h.

6 8

4 4


4. Draw the fractions. Then compare: which is more pie? Write > , < , or = between the fractions.

a.

b.

c.

2

1

1

4

1

3

3

3

5

5

6

6

d.

e.

f.

6

7

3

1

2

4

8

8

8

8

4

4

g.

h.

i.

1

5

5

3

6

7

9

9

12

12

10

10

What can you notice about comparing two fractions when the denominators are the same?

j.

k.

l.

1

1

1

1

1

1

2

3

5

8

6

2

m.

n.

o.

1

1

1

2

4

4

6

8

2

3

8

5

p.

q.

r.

2

2

3

3

1

2

4

6

8

6

2

4

What can you notice about comparing two fractions when the numerators are the same?

11


Mixed Numbers Mixed numbers have two parts: a part that tells you the whole number, and a fractional part.

1

1

2

3

“One and one-third”

3 4

“Two and three-fourths”

1. Write what mixed numbers the pictures illustrate.

a.

b.

c.

d.

e.

f.

g.

h.

i.

2. Draw pictures to illustrate these mixed numbers.

a. 1

d. 1

1 2

5 6

b. 2

e. 3

2

c. 2

3

1

f. 3

3

12

3 5

5 8


1

You can count all of the same kind of fractional parts from a mixed number, and then write the mixed number as a fraction.

1

2

3

If I count all of the thirds, 4 I get four thirds: 3

3 4

If I count all of the fourths, 11 I get eleven fourths: 4

3. Find the matching pairs.

1

2

1

3

1

2

3

9

4

4

1

9

4

8

1

12

8

5

1

11

5

8

3

7

8

4

2

16

5

5

4. Fill in the blanks.

1= a.

3

1=

3

b.

One whole is three thirds.

2= d.

3

1 =

6

c.

.

2= e.

2 = f.

13


5. Write these both as mixed numbers AND fractions.

b.

a.

d.

c.

f.

e.

6. Draw enough “pies” so you can color all of the parts. Then write the fractions as mixed numbers.

a.

d.

g.

5 4

14 6

14 9

=1

1 4

b.

=

e.

=

h.

7 3

14 5

21 6

=

c.

=

f.

=

i.

14

8 5

15 8

11 4

=

=

=


7. The number line from 0 to 1, and from 1 to 2, is divided into parts. Label the tick marks between 0 and 1 with fractions. Label the tick marks between 1 and 2 with mixed numbers.

15


Part of a Whole Group 1 Four of the ten parts are colored. 4 10

Three of the nine apples are dark. 3

of the parts are colored.

9

of the apples are dark.

1. In each group are two different kinds of things. Write each part as fractions of the whole group. b.

a.

of the objects are carrots.

of the objects are stars.

of them are apples.

of them are hearts.

c.

d.

of the shapes are rectangles.

of the trees are dark.

of them are parallelograms.

of them are not dark.

2. Fill in using fractions. (Hint: you need to think how many are in the whole group.) a. There are 5 girls and 6 boys.

b. There are 4 apples, 5 bananas, and 3 mangoes

of all the students are boys.

of all the fruit are apples.

of all the students are girls.

of all the fruit are mangoes.

3. An egg carton contains a dozen eggs. Write what fraction of the carton of eggs is

6eggs

4eggs

8eggs

3eggs

2eggs

10eggs

4. Write as a fraction what part of the day is... a. 12hours

b. 6hours

c.3

hours

16

d.9 hours

e. 16 hours


12 flowers are divided into three groups. Each group is a third of the whole. 1 12 ÷ 3 = 4

3

Twelve fish are divided into two groups. Each group is a half of the whole. 1

of 12 is 4.

2

12 ÷ 2 = 6

of 12 is 6.

5. Circle to make groups.

a. Make 3 groups.

1

of 9 apples is ___ apples.

3

3

d.

1 4

1 of __ carrots is __ carrots.

1 6

5

of __ fish is ___ fish.

f.

e.

of 12 flowers is __ flowers.

c. Make 5 groups.

b. Make 3 groups.

1

of __ flowers is __ flowers.

1 5

of __ berries is __ berries.

6. Draw sticks. Make groups. Write a “part” sentence, and a division sentence.

a.

1

of 18 is _____

3 18 ÷ ____ = _____

b.

1

of 20 is _____

5

c.

1

of 24 is _____

4

______ ÷ ____ = _____

17

______÷ ____ = _____


1

One group is 1/5 of the total.

Look again. 10 fish are divided into 5 groups.

5 2

Two groups is 2/5 of the total.

5 3

Three groups is 3/5 of the total.

5 4

Four groups is 4/5 of the total.

5 5

Five groups is 5/5 of the total.

5

of 10 is 2. of 10 is 4. of 10 is 6. of 10 is 8. of 10 is 10.

7. Fill in the blanks.

c. Make 5 groups.

1 b. Make 3 groups.

a. Make 3 groups.

1 3 2 3 3

of 9 apples is ____ apples. of 9 apples is ____ apples.

3 of 9 apples is ____ apples.

1 3 2 3 3

4 2 4 3 4 4 4


5

e.

1 of 12 flowers is __ flowers.

of __ carrots is ___ carrots.

5 3

3 of __ carrots is ___ carrots.

d.

1

of __ carrots is ___ carrots.

5 2

6 2 6 3


6 4


6

of __ fish is ___ fish. of __ fish is ___ fish. of __ fish is ___ fish.

4 of __ fish is ___ fish. 5

f.

of __ flowers is __ flowers. 1 of __ flowers is __ flowers.

5 2

of __ berries is __ berries. of __ berries is __ berries.


5


3 of __ berries is __ berries. 5

18


Part of a Whole Group 2 One half of twelve fish is six fish.

One third of twelve flowers is four flowers. 1 3

1

of 12 is 4.

2

To find a third part, you can use division: 12 ÷ 3 = 4.

To find To find

1 2 1 4

of 12 is 6.

To find a half, you can use division: 12 ÷ 2 = 6. 1 3

of a number, divide by ______. To find

1

of a number, divide by ______. To find

5

of a number, divide by ______. of a number, divide by ______.

1.Figure out these parts using division.

a.

1 3

of 18 is _____.

b.

18 ÷ 3 = ____ d.

1 3

1 5

of 10 is _____

10 ÷ 5 = ____

of 15 is _____

e.

___ ÷ ___ = ____

1 2

of 14 is _____

___ ÷ ___ = ____

We can see that We also see that Similarly,

3 5

1 5

c.

1 5

of 25 is _____

25 ÷ ___ = ____ f.

1 3

of 21 is _____

___ ÷ ___ = ____

of 20 berries is 4 berries. 2 5

of 20 berries is double that, or 8 berries.

of 20 is three times 4, or 12.

2. Fill in the missing parts. Use the top problem to help you solve the bottom one. a. 1 of 12 is _____. 4

3 4

of 12 is _____.

b. 1 of 12 is _____. 6

2 6

c. 1 of 10 is _____. 5

3

of 12 is _____.

5

19

of 10 is _____.

d. 1 of 10 is _____. 2

3 2

of 10 is _____.


3. Fill in the missing parts.

a.

1 7 3 7

of 28 is _____.

1

b.

9 2

of 28 is _____.

9

of 18 is _____.

c.

of 18 is _____.

1 11 3 11

of 55 is _____.

d.

of 55 is _____.

1 8 7 8

of 56 is _____. of 56 is _____.

4. Continue the patterns! 1

1

1

1

7 of 21 is 3.

5 of 20 is 4.

2 of 10 is _____.

3 of 21 is _____.

2

2

1

1

7 3 7

of 21 is _____. of 21 is _____.

5 3 5

of 20 is _____.

2 1

of 20 is _____.

2

of 12 is _____. of 14 is _____.

3 1 3

of 24 is _____. of 27 is _____.

of 21 is _____.

of 20 is _____.

of ____ is _____.

of ___ is _____.

of 21 is _____.

of 20 is _____.

of ____ is _____.

of ___ is _____.

of 21 is _____.

of ____ is _____.

of ___ is _____.

of 21 is _____.

of ____ is _____.

of ___ is _____.

5. Solve how much is left or how much there was in the beginning! The bar models can help. a. There were 16 apples. Rhonda ate 1/4 of them. How many are left? b. Mom used 1/3 of the 12 eggs. How many are left? c. Tom ate 4 hazelnuts, which was 1/8 part of the nuts. How many nuts were there to start with? d. Lisa used $5 to buy a gift, which took 1/3 of her savings. How much did she have in the beginning? How much does she have left now? e. Mom used 2/3 of the eggs she had, and now she has 2 eggs left. How many did she have srcinally?

20


Part of a Whole Group 3 1. Solve these like you learned in the previous lesson, but the “helping problem” is not written. a.

3 7

of 21 is _____

b.

2 9

of 36 is _____

c.

of 90 is _____

g.

4 5

of 30 is _____

d.

of 54 is _____

h.

7 8

of 64 is _____

Find first 1/7 of 21.

e.

5 6

of 42 is _____

f.

9 10

7 9

2 5

of 45 is _____

2. The bar model can help you. Answer the questions. a. 1/3 of the 33 club members are boys.

b. 1/6 of the team, which is 3, are adults.

What part (what fraction) are girls?

What part of the team are not adults?

How many are girls?

How many people on the team are not adults?

3. Half of the cookies is 20 cookies.

Part

1

1/2

1/4

3/4

1/5

3/5

3/8

Fill in the rest of the chart. 20

Amount

4. Solve the problems. a. Of the 25 members of the stamp club, 2/5 are girls. How many boys are there?

b. The air conditioner is on for 5/8 of the day. How many hours is it not on?

c. Jennie has 50 apples. Fill the table.

d. The store had 48 bananas, and sold 2/3 of them. How many are left?

Color

Part Amount

Red

1/5

Yellow 1/2 Green

3/10

21


5. The word quarter means 1/4. Find where all the word “quarter” is used! a. What common expressions about time use the word “quarter”?

How many minutes is three-quarters of an hour? b. How many cents is a quarter dollar? c. How many feet is a quarter mile (a mile is 5280 feet)? d. What do we call a quarter of a gallon?

6. Jerry has saved 2/5 of the price of a $625 notebook. How much does he still need to save?

7. Jamie started watching a 2-hour video at 4:15 pm but he only watched 2/5 of it. At what time did he stop?

8. Maria has $50 in her savings. She used 1/5 of it for apples. Then, of the remaining money she used 3/8 for some other fruit. How many dollars does she have left?

9. Can you figure these out? It requires some 'reverse thinking'! a.

e.

1 2 9 10

of ___ is 6.

b.

of ___ is 45.

f.

1 3 2 3

of _____ is 18.

c.

of _____ is 12.

g.

1 5 3 5

of _____ is 10.

d.

of _____ is 15.

h.

1 4 3 4

of _____ is 8.

of _____ is 33.

How many cookies? a. Mom baked cookies, and gave 1/2 of them to Beth, and of the remaining ones, she gave 2/3 to Seth. Then she ate the rest herself, which was 4 cookies.

How many did she bake srcinally? b. Mom baked cookies again. She gave 1/2 of them to Beth, and 1/3 of the remaining ones to Seth. Then there were 12 cookies left.

How many did Mom bake srcinally?

22


Adding and Subtracting Like Fractions Fractions that have the same denominator are called like fractions. Fractions that have a different denominator are called unlike fractions. Write the answers to the pictures. Check with your teacher. +

=

1 4

1 4

+

2 4

+ 2 9

=

and

2 4

are like fractions since they have

the same denominator, 4. In other words, they are the same kind of parts - fourth parts.

= 4

+

9

+ 3

=

+

8

= 5

=

8

=1

Adding the same kind of parts is easy - just add the __________________.

1. Write the addition sentence and the answer. Give your final answer as a mixed number if possible. Shade parts in the answer picture.

+

=

a.

+

=

+

=

+

=

b.

+

=

c.

d.

+

=

e.

f.

23


+

=

g.

+

=

+

+

h.

+

=

i.

=

j.

2. Shade parts and write an addition sentence for each problem.

a. Shade

c. Shade

e. Shade

g. Shade

5 8

6 10

6 14

5 4

. Shade another

. Shade another

. Shade another

. Shade another

7 8

7 10

6 14

6 4

b. Shade

.

.

d. Shade

.

f. Shade

h. Shade

.

24

3

. Shade another

5

2

. Shade another

7

3 4

. Shade another

9 10

. Shade another

1

.

5

6

.

7

2 4

.

3 10

.


3. Fill in the missing fractions. Write your answer as a mixed number if possible.

a.

d.

g.

j.

3 2 2 14 10 20 1 13

1

+

=

2 6

+

14

b.

=

e.

19

+

=

+

= 1

h.

20

k.

4 7 3 4 1 3 5 6

+

4 7

=

+

c.

=

+

= 1

+

= 1

6

f.

4 2

i.

3 2

l.

6

5

2

4

=

=

+

5

8

10

+

13

7

6

+

10

8 13

= 1

+

= 1

1 8

4. Shade parts and cross out parts, and write a subtraction sentence for each problem.

a. Shade 1

c. Shade 1

e. Shade

5 8

5 10

10 14

g. Shade 2

. Cross out

1 4

. Cross out

. Cross out

7 8

.

b. Shade

.

d. Shade 1

2 10

3 14

. Cross out 1

.

2 4

f. Shade 1

h. Shade 2

.

25

5 5

. Cross out

3 7

2 4

3 8

. Cross out

. Cross out

. Cross out

1

.

5

6

.

7

3 4

.

11 8

.


5. Subtract like fractions. The principle is the same as in adding: you just subtract the numerators. a.

e.

i. 1

3 5

1 

=

5

8 4

=



1

=



6

b.

4

6 4

f.

4 

7

=

7

12

j. 1

=



15

c.

=



8

g.

15

2

9 10

10

16 20

k. 1

3

6 

=

=



=



12

d.

11 20

5

h.

7

3

l. 1

9

6 

14

14

=

= 1



=



1 3

5 8

6. Make your own fraction problems! Make two different problems for each answer.

9

+

=

+

=

+

9

5 4 5 4

= 1

9 + 9 = 1

20

+

=

+

=

+

20

5 5 5 5

= 1

20 + 20 = 1

+

=

+

=

+

4

13 20 13 20

= 1

4

6

6 1 4

6

+

+

+

= 1

6

= 1

6

= 1

6

5 6

1

5

4 + 4 = 1 4

6 + 6 = 1 6

7. Add. Write a subtraction problem for each addition. Use mixed numbers when possible. a.

1 3

+

2 3 2



3

=

b.

6 11

+

6 11 6

=



11

e.

4

2

5 + 5 =

c.

2

=

10

=

6

+

10

8

=



f.

7

=

d.

9

+

2 8

=



g.

6

6

10 + 10 =

7

7 + 7

26

h.

3 =

=

7

8 + 8 =


Review: Mixed Numbers This lesson should be mostly review. But please don’t go on to the lessons about adding and subtracting mixed numbers until you understand well the concepts in this lesson.

Mixed numbers as pictures 1. Write the mixed numbers that these pictures illustrate.

a. b.

c. d.

2. Draw pictures of “pies” that illustrate these mixed numbers.

a. 4

c. 3

e. 6

2

b. 2

3

2

d. 4

6

3 5

7 8

8 10

27


Mixed numbers on a number line 3. Write the mixed number that is illustrated by each number line.

a.

b.

4. Write the fractions and mixed numbers that the arrows indicate.

a.

b.

c.

5. Draw arrows to indicate where these mixed numbers occur on the number line.

6. a. Indicate 2

c. Indicate 5

3 5

1 5

d.

a. 4

2 6

b.

on the number line. b. Write the mixed number that is

on the number line. d. Write the mixed number that is

28

2 6

4 5

3 5

c. 5

3 6

d. 1

1 6

to its right: _______

to its left: _______


Changing mixed numbers to fractions To write 3   

3 4

as a fraction, count how many fourths there are:

Each pie has four fourths, so the three complete pies have 3 × 4 = 12 fourths. Additionally, the incomplete pie has three fourths. The total is 15 fourths or 15/4. Numerator: 3 × 4 + 3 = 15

Shortcut:

=

Denominator: 4

15 4

Multiply theofwhole number the denominator, then add numerator.will Theremain result gives you the number fourths, or thetimes numerator, for the fraction. Thethe denominator the same. 7. Write as mixed numbers and as fractions.

a. 1

2 5

=

5

=

d.

8. Mae changed 5

9 13

b.

=

c.

=

e.

=

f.

=

to a fraction, and explained how the shortcut works. Fill in the blanks.

There are ____ whole pies, and each pie has _____ slices. So ____ × ____ tells us the number of slices in the whole pies. Then the fractional part 9/13 means that we add _____ slices to that. In total we get ____ slices, each one a 13th part. So the fraction is

.

9. Write as fractions. Think of the shortcut. a. 7 e. 2

1 2 5 11

b. 6 f. 8

2 3 1 12

c. 8 g. 2

29

3 9 5 16

d. 6 h. 4

6 10 7 8


Changing fractions to mixed numbers To write a fraction, such as  



7

, as a mixed number, you need to figure out:

How many whole “pies” there are, and How many slices are left over.

In the case of 

58

58 7

, each whole “pie” will have 7 sevenths. (How do you know?) So we ask:

How many 7s are there in 58? (Those make the whole pies!) After those 7s are gone, how many are left over?

All that is solved by the division 58 ÷ 7! That division tells you how many 7s there are in 58. Now, 58 ÷ 7 = 8 R2. So you get 8 whole pies, with 2 slices or 2 sevenths left over. To write that as a fraction, we get Example:

45 4

58

=8

7

2 7

.

is the same as 45 ÷ 4, and 45 ÷ 4 = 11 R1. So, we get 11 whole pies and

1 fourth-part or slice left over. Writing that as a mixed number,

45 4

= 11

1

.

4

The Shortcut: Think of the fraction bar as a division symbol, then DIVIDE. The quotient tells you the whole number part, and the remainder tells you the numerator of the fractional part.

10. Rewrite the “division problems with remainders” as problems of “changing fractions to mixed numbers.” b. 35 ÷ 8 = 4 R3 a. 47 ÷ 4 = 11 R3 47 3 = 11 4 4

c. 19 ÷ 2 = ___ R ___

=

d. 35 ÷ 6 = ___ R ___

=

e. 72 ÷ 10 = ___ R ___

=

f. 22 ÷ 7 = ___ R ___

=

=

11. Write these fractions as mixed numbers (or as whole numbers, if you can). a.

62 8

b.

7 e.

i.

2 24 11

16

c.

3 25

f.

j.

27 5

d.

50 g.

4 39

k.

3

30

6 57 8

32 9 32

h.

l.

5 87 9


Adding Mixed Numbers 1 1. Can you figure out these addition problems without reading the instructions in the blue box below? First write an addition sentence for each problem. Then afterwards read the instructions below.

+

+

a.

b.

Adding mixed numbers You can simply add the whole numbers and fractional parts separately: 1

1

+ 5

7

3 7

= 6

4

1

1 7

+ 5

3 7

6

4 7

or in columns →

7

However, many times the sum of the fractional parts goes over one whole pie. + 1

3

+

= 1

6

4

=

7

2

6

6

→

3

1 6

So first, simply add the fractional parts as usual. Then, change the fraction that is more than one pie into one or more whole pies and a fractional part that is less than one pie.

2. These mixed numbers have a fractional part that is more than one “pie.” Change them so that the fractional part is less than one. The first one is done for you. a. 3

3 2

→4

1

b. 1

2

11

c. 3

9

5

d. 6

3

7

e. 3

4

13 8

3. Write the addition sentences that the pictures illustrate and then add.

+

+

+

a.

b.

31


4. Write the addition sentences that the pictures illustrate and then add.

+

a.

c.

+

e.

+

b.

+

d.

+

+

5. Add. a. 3

c. 6

2 3 8 9

+ 8 + 1

1 3 2 9

=

b. 4

=

d. 3

4 5 6 7

+ 1 + 2

3 5 4 7

= =

6. Add. 3 7 5 + 5 7 a. 4

9

3 5 4 3 5

6 9 7 + 2 9

b. 3

+

8 1 → 10 7 7

c. 4

6 8 7 2 8

d. 7

+

→

7. Find the missing addend. Imagine drawing the rest of the picture.

a. 1

c. 1

1 2

1 4

+

=3

b. 2

+

=5

d. 2

32

2 3

3 4

+

=5

+

=8


Sometimes the sum of the fractional parts can be two or more whole pies. Just figure out how many whole pies you can make from the fractional part and add them to the whole number part. +

= 1

5

+ 1

6

3

1

5 6

1

3 6

+ 1

5 6

3

13 6

+

+ 1

6

5

= 3

6

13

→

1

5

6

→ 5

1 6

6

8. Convert these mixed numbers so that the fractional part is less than one. a. 3

13

3

→ 5

5

b. 1

5

11

c. 3

4

11

d. 6

3

13

e. 3

4

19 8

9. Add these mixed numbers. a. 3 c. 6

1

5

+ 2

6 4

+ 1

8

6 6

= + 1

8

4

b. 4 7 8

=

d. 3

+ 1

5 6

+ 3

10

2 5 8 10

+ 5 +

2 5 9 10

= =

10. Tom has one string that is 7 3/8 inches long and another that is 5 7/8 inches long. He tied them together. In making the knot, he lost 1 4/8 inches from the total length. How long is the combined string now? 11. Add the mixed numbers. 7 9 5 2 9 8 3 9

5 11 9 3 11 8 + 2 11

a. 10

+

15

5 6 4 5 6 3 + 2 6

b. 1

20 → 9

7 10 9 10 6 + 10 10

c. 2

→

d.

1

→

→

12. Find the missing addends. a. 2

1 4

+1

1 4

+

= 5

b. 3

2 5

+2

2 5

+

33

= 8

c. 2

1 3

+

2 3

+

= 4

1 3


Subtracting Mixed Numbers 1 1. Warm-up: see if you can subtract mixed numbers!

3 4

Shade 2

. Cross out 1

1 4

a. 2 34 − 1 14 =

3 7

Shade 2 c. 2

3 − 7

1

. Cross out

5 8

b. 2 18 − 58 =

. Cross out 1 6 7

1 8

Shade 2

Shade 2

6 7

1 2

. Cross out 1

2 3

.

(Hint: Change the pictures into sixths.)

=

d. Challenge. 2

Strategy 1: Subtract whole-number parts and the fractional parts separately. This is natural to do, and it works well when the subtrahend (the number that you subtract) has a smaller fractional part than the minuend (the number that you start with).

1 − 2

6 − 3

3

1

2 3

=

7 8 2 8 5 8

2. Subtract. 1 − 1

3 4 1 4

11 c. b. 2

4 − 7

1

3 7

f. 6

2 7

− 1

2 7

− 4

7 9 3 9

8 d. − 1

4 5 1 5

=

a. Cross out 1 1/4. e. 5

6 − 11

3

2 11

=

34

=

g. 7

9 − 10

1

1 − 10

4

6 10

=


Strategy 2: Subtract in Parts First subtract what you can from the fractional part of the minuend. Then subtract the rest from one of the whole pies. The examples show two slightly different ways to understand this.

2

1 5 − 8 8

=?

3

We can’t subtract 5/8 from 1/8. So ... First subtract only 1/8 of the 5/8, which leaves 2 whole pies. The rest (4/8) of the 5/8 is subtracted from one of those whole pies. 2

1 − 8

5 8

=

2

1 − 8

=

1 8

−

2

−

4 8 4 8

2 − 9

= 1

7 9

=?

We can’t subtract 7/9 from 2/9. So, first subtract 2 2/9, which leaves 1 whole pie, and the rest (5/9) is subtracted from the last whole pie. 3

=

2

2 − 9

2

7 9

=

3

2 − 9

=

4 8

2

2 9

1

− −

5 9 5 9

= =

4 9

2 5

.

3. Subtract in parts. Remember: You can add to check a subtraction problem. 2 6

First subtract

a. 2

2 5 − 6 6

=

c. 3

1 − 8

1

7 8

2

5

e. 5 9 − 9

, then

=

3 6

.

First subtract 2, then

b. 3

1 − 5

2

3 5

=

d. 3

2 − 7

2

6 7

=

1 5

, then

8

− 1 9 =

35


4. Subtract in two parts. Write a subtraction sentence.

a.

c.

Cross out 1

Cross out

3

2 . 3

b.

.

d.

Cross out

4 . 5

5

Cross out 1

4

7

e.

Cross out 1

7 . 8

f.

Cross out 1

g.

Cross out 1

3 . 5

h.

Cross out

i.

Cross out

j.

Cross out 1

5 . 6

Cross out 3

k.

36

5 6

6 9

7 9

4 . 5


Subtracting Mixed Numbers 2 — Renaming Strategy 3: Rename the mixed number. In this method you divide one whole pie into “slices” and join these slices from the whole pie to the existing slices. After that, you can subtract. It’s just like “borrowing” in the subtraction of whole numbers. Examples will make this strategy clear. Rename 3

2 8 5 as 2 , and then subtract 1 6 6 6

Borrow 1 whole pie.

.

Borrow 1 whole pie from

the whole pies’ column into the fractional parts’column. The one whole pie is 6 sixths. There were already 2 sixths in the fractional parts’ column, so that is why it becomes 8/6 after borrowing. Now you can subtract the 5/6.

At first we have three uncut pies and 2/6 more. Then we cut one of the whole pies into 6 sixth parts. We end up with only two whole (uncut) pies and 8 sixth parts. We say that 3 2/6 has been renamed as 2 8/6. Now we can subtract 1 5/6 easily. 3

2 − 6

Rename 2

1

5 6

= 2

8 − 6

1

5 6

= 1

3 6

1 9 5 as 1 , and then subtract 8 8 8

.

Borrow 1 whole pie.

Borrow 1 whole pie from the whole pies’ column into the fractional parts’ column.

The one whole pie is 8 eighths. Since there was already one eighth in the fractional parts’ column, it becomes 9/8 after borrowing. Now you can subtract 5/8.

At first we have two uncut pies and 1/8 more. Then we cut one of the whole pies into 8 eighth parts. We end up with only one whole (uncut) pie and 8 eighth parts. We say that 2 1/8 has been renamed as 1 9/8. Then we can subtract 5/8 easily. 2

1 5 − 8 8

= 1

9 5 − 8 8

=1

4 8

1. Do not subtract anything. Just cut up one whole pie into fractional parts, and rename the mixed number.

a. 2

1 6

is renamed as

b. 3

1 8

is renamed as

37

c. 2

2 9

is renamed as


2. Do not subtract anything. Just cut up one whole pie into fractional parts, and rename the mixed number.

a. 2

3 5

is renamed as

b. 3

3 10

is renamed as

c. 2

1 4

is renamed as

3. Rename, then subtract. Be careful. Use the pie pictures to check your calculation.

4 8 − 9 9

a. 3

↓

↓

13 8 = 2 = − 9 9

c. 3

7 9 − 1 10 10 ↓

=

3 7 − 2 12 12

b. 3

−

= 2

7 10 9 −1 10

3

d. 2

12

−

2

7 = 12

3 8 7 − 1 8

3 7 − 1 8 8

2

↓

1 9 =

=

10

−

17 = 8

4. Subtract. You can draw a picture to help you. a. 4

2 5 = − 1 6 6

b. 3

2 7 = − 1 9 9

c. 4

5 9 = − 2 12 12

d. 3

1 5 = − 1 8 8

38


5. Rename (if needed) and subtract.

a.

1 7 3 − 1 7

3

b.

7 9 5 − 3 9

14

e.

f.

4 9 7 − 2 9

7

9 12 11 − 6 12

12

c.

5 21 15 − 7 21

11

d.

4 19 15 − 14 19

26

g.

h.

3 14 9 − 5 14

8

3 20 7 − 5 20

10

6. Figure out what was subtracted. You can use the circle pictures to help you.

a. 3 −

=1

1

b. 4 −

=2

4

d. 4 −

=1

5 6

3

c. 4 −

=1

5

e. 5 −

=1

7 12

1 8

f. 4 −

=

5 6

7. Subtract. a.

5

b.

9

3 7 = −1 8 8 2 15

c.

7

− 5

8

=

d.

4

e. 13

15

11 9 = − 4 17 17

5 7 = − 2 8 8 2 5

f. 16

39

− 4

2

=

5

5 11 = − 4 12 12


Subtracting Mixed Numbers - Extra Practice 1. Subtract. You can draw a pie picture to help you.

a. 4

1 2 = − 2 3 3

b. 3

2 4 = − 1 5 5

c. 4

1 3 − 2 = 4 4

d. 3

2 5 − 1 = 6 6

e. 3

2 5 = − 1 8 8

f. 4

1 5 = − 2 12 12

g. 7

3 7 3 = − 2 − 1 8 8 8

h. 7

1 3 3 = − 2 − 2 4 4 4

i. 6 5 − 2 9 − 2 5 = 12 12 12

40


2. Add and subtract. a. 2

1 3 2 + 5 = − 3 4 4 4

b. 4

5 3 4 + 6 = − 1 6 6 6

c. 9

3 7 6 + 3 = − 2 8 8 8

d. 7

7 11 2 + 3 = − 1 12 12 12

3. Find the missing minuend or subtrahend. a. 5 −

e.

= 2

−

2 3

b. 7 −

4 3 = 3 8 8

f.

− 1

= 6

1 5

c. 6 −

2 2 = 3 3

= 4

− 2

g.

1 6

d. 2 −

5 1 = 1 6 6

− 5

h.

= 1

1 9

7 11 = 1 12 12

4. Subtract. Color the answer squares as indicated. a. (yellow) 5

2 9

b. (blue)

7

8 − 15

c.(blue)

5

11 −

− 2

4

6

7 9

j. (yellow) 8

1 8

11 15

k.(blue)

7

1 − 11

l.(blue)

9

2

3 11

1 9

− 2

3 9

e. (green) 17

2 9

− 4

5 9

4

11 12

f. (green)

1 − 11

3

9 11

3

7 11

1 − 12

4

7 12

d.(yellow)

4

5

g.(green) 10

h. (yellow) 4

3

− 2

10

i(green)

5

1 − 10

m. (yellow) 15

3

5 8 5 11

7

1

8

− 3 8

3 − 12

2 10

10

4 12

12

6 9

2

4 9

2

4 11

6

6 8

2 12 2 15

3

17 9

10

3

1

− 2

13 11

5 6 12

54 8

9 10

41


Equivalent Fractions This is an important topic. You need to understand equivalent fractions well before learning how to add unlike fractions.

= 1 3

=

2

These two fractions are equivalent fractions because they picture the same amount. You could say that you get to “eat” the same amount of “pie” either way. In the second picture each slice has just been split into two pieces.

6

Splitting the pieces The arrows above and below the equivalent fractions show into how many new pieces each old piece was split. 

Each slice has been split in two. =

3

BEFORE: There are 3 colored pieces out of a total of 4. AFTER: 6 colored pieces out of 8.

4

2

= 

2



3

6 8

Each slice has been split into three. 1

=

BEFORE: 1 colored piece, 3 total AFTER: 3 colored pieces, 9 total

3

Each slice has been split into four. =

BEFORE: 1 colored piece, 2 total AFTER: 4 colored pieces, 8 total (The whole number does not change.)

1

1 2

3

= 

3



4

= 1 

4



2

9

4 8

Each slice has been split in two. =

1

BEFORE: 1 colored piece, ___ total AFTER: ___ colored pieces, ___ total

= 

2

When all of the pieces are split the same way, both the number of colored pieces (the numerator) and the total number of pieces (the denominator) get multiplied by the same number.

42


1. Connect the pictures that show equivalent fractions. Write the name of each fraction beside its picture. 1 2

2. Split the pieces by drawing the new pieces into the right-hand picture. Write down the equivalent fractions. If you get confused, you can also draw in the “helping arrows” above and below the fractions. a. Split each piece in two.

b. Split each piece into three.

= 

2 5



=

2

=

2

d. Split each piece in two.

c. Split each piece in two.



1 2

3

=



=

3

e. Split each piece into three.



2 3

2

=



2

f. Split each piece in two.

=

=

=

=

=

=

g. Split each piece in two.

h. Split each piece in two.

i. Split each piece into five.

=

=

=

=

=

=

43


3. Write the equivalent fraction. Imagine or draw the helping arrows. a. Split each piece into four.

3 4

b. Split each piece in two.

5

=

8

f. Split each piece into three.

c. Split each piece into six.

1

=

2 = 7

2

=

2

g. Split each piece into ten.

d. Split each piece into four.

7

h. Split each piece into eight.

5 = 8

1

=

4

i. Split each piece into seven.

1 = 2

e. Split each piece into five.

=

j. Split each piece into eight.

3 = 5

3 = 7

4. Make chains of equivalent fractions according to the model. Pay attention to the patterns formed by the numerators and the denominators.

= 1

a.

1

=

=

=

=

=

=

=

=

=

=

=

=

=

=

2

=

2

= 1

b.

1

c.

d.

e.

2 3 5 2

=

f.

3 4

6 10

=

7

2

=

3

6

=

= 14

=

15

=

=

=

=

4

5

=

4

5

=

=

=

=

=

6

=

=

=

6

=

5

=

6

=

=

=

=

=

=

=

=

=

21 =

=

=

=

=

=

44

=


5. Figure out how many ways the pieces were split and write the missing numerator or denominator. a. Pieces were split into three. 

4



7 10

d. Pieces were split into ____ .

e. Pieces were split into ____ .









4 21

5

3

=

=

1 20

14

5 9



=

15

3

=

1

6

4 9



2 11

=

6

2 14

=

7



j.

8 54

11

56

=

44



o.

1

=

8



n.

4

=

3



m.

9

=



i.

=

8

l.

10

7





h.



k.

18





g.

6

=

6





f.

c. Pieces were split into ____ .

3

=

7

b. Pieces were split into ____ .

6

=

7 54

8

=

64

6. The three number lines have been divided into twelfths (12th parts), thirds (3rd parts), and twenty-fourths (24th parts). a. Find the fraction that is

equivalent to

2 3

on the 12th

parts number line and on the 24th parts number line. Mark all three fractions on the number lines: 2 = = 3 12 24 b. Do the same with

1 3

. Mark the equivalent fractions on the three number lines:

c. Find and mark the fraction that is equivalent to

7 12

1 3

=

12

=

on the 24th parts number line. What is it?

24

24

d. Find and mark a fraction on the 12th parts number line that does not have an equivalent fraction on the 3rd parts number line. Write it here → e. Find and mark a fraction on the 24th parts number line that does not have an equivalent fraction on the 12th parts number line. Write it here →

45


Equivalent Fractions 2 These are mixed numbers: There is a whole-number part and a fractional part (some whole pies and some cut-up slices).

=

1

2

1

=

3

2

6

Here you see almost the same thing. But this time the whole pies are cut up into slices, so we can write the whole amount as a single (improper) fraction.

= 7

14

=

3

2

The fractional parts and are 3 6 equivalent fractions. Note that the whole number 2 does not change.

2

7

6

3

14

and

are equivalent fractions.

6

We use equivalent fractions both with mixed numbers and with improper fractions.

1. These are improper fractions. Split the slices in the right-hand picture. Write the equivalent fractions. a. Split each slice into three.

b. Split each slice in two.



7 =



5

=

4

= 

c. Split each slice in two.



d. Split each slice into four.



12 =

=

3

3

=

5



2

=

= 



2. Fill in the missing numbers in the equivalent fractions. a. 1

3

= 1

12

b. 5

4 f.

7 2

=

7

=

28

c.

10 21

g. 6

2 9

9

=

36

d. 3

4 =

12

h.

7 1

46

2

=

12

e.

3 =

6

i. 5

7 10

8

=

3 =

80

j.

9 3

15 =

18


3. Each column contains a set of equivalent fractions (or mixed numbers). Fill in the missing numbers. a.

b.

5

11

3

5

9

c.

d.

e.

f.

3

2

9

10

1

6

10

40

2

12

24

12

15

50

3

24

48

21

25

80

4

36

96

30

50

100

5

54

100

2

2

5

5

3 4

4. If you can find an equivalent fraction, then write it. If you can’t, then cross the whole problem out. a. The pieces were split into ____ .

5 7

2 =

5

28

f. The pieces were split into ____ .

1 6

b. The pieces were split into ____ .

1 18

g. The pieces were split into ____ .

1 =

=

6

28

=

c. The pieces were split into ____ .

4

2 14

h. The pieces were split into ____ .

5 24

=

4

=

d. The pieces were split into ____ .

3

=

5 12

6

i. The pieces were split into ____ .

1 32

3

=

e. The pieces were split into ____ .

8

j. The pieces were split into ____ .

3

5

=

8

=

8

5. Explain in your own words how you can know when an equivalent fractions problem is not possible to do. You can use an example problem or problems in your explanation.

6. Write the number 3 as a fraction using... wholepies

halves

thirds

fourths

fifths

tenths

hundredths

3 1

2

47


Adding Unlike Fractions 1 Cover the page below the black line. Then try to figure out additions below.

+ 1

= 1

+

3

=

2

+ 1

What fraction would this be?

3

+

= 1

=

4

What fraction would this be?

Did you solve the problems above? Study the pictures below for solutions. Discuss them with your teacher.

+ 1

+

3 ↓

+ 1

1

2

3

↓

↓

+ 2

= 3

+

6

+

=

6

+ 1

1

4

6

↓

↓

+ 5

4

6

12

12

=

1 2 ↓

= 3

+

+

7 12

+

=

+

=

To add unlike fractions, first convert them to ____________ fractions. Then add. 1. Write the fractions, convert them into equivalent fractions, and then add them. Color the missing parts.

+

+

+

+

+

+

↓

a.

↓

↓

+

=

+

=

b.

↓

↓

+

=

+

=

48

c.

↓

+

=

+

=


Let’s write the addition all on the same line now (horizontally).

+=+= 3 10

2

+

5

3

=

10

4

+

=

10

10

2. Change these to equivalent fractions first and then add them. Each box below has TWO problems. In the bottom problem, you need to figure out what kind of pieces to use, but the top problem will help you do that!

a.

+ 1 2

+

= 1 6

=

b.

+

6

=

+

+=+=

1 6

=

+

+=+=

+

=

+

=

+

+=+=

=

+

=

+

c.

+=+=

=

+

=

+

+=+=

+

=

d.

+=+=

+

=

=

+

=

+

=

+=+=

=

+

=

+

=

3. Look at the problems above in exercise (2). What kind of parts did the fractions have? What kind of parts did you use in the final addition? Fill in the table. Types of parts to add: a. 2nd parts + 6th parts

Converted to:

Types of parts to add:

6th parts

Converted to:

b. 4th parts + 6th parts ___ parts

c. 8th parts + 4th parts ___ parts

d. 2nd parts + 8th parts ___ parts

49


4. In the problems below, split the parts (as in the example at the right) so that both fractions will have the same kind of parts.

These pictures change into...

...these.

Example:

Write an addition sentence. 1 4

+

3

2

8

8

+

a.

+

=

b.

+

=

c.

+

=

d.

+

=

e.

+

=

f.

+

=

3 8

=

5 8

5. Fill in the table based on the problems above in exercise (4). Types of parts to add:

Converted to:

Types of parts to add:

Converted to:

a. 2nd parts + 8th parts ___ parts

d. 2nd parts + 5th parts

___ parts

b. 2nd parts + 4th parts ___ parts

e. 3rd parts + 5th parts

___ parts

c. 3rd parts + 6th parts ___ parts

f. 3rd parts + 2nd parts ___ parts

6. Now think: How can you know into what kind of parts to convert the fractions that you are adding? Can you see any patterns or rules in the table?

7. Challenge: If youworry thinkifyou kind to convert will thesestudy fractions to, then try these problems. Don’t youknow don’twhat know howoftoparts do them—we this more in the next lesson. a.

1 2

+

2 3

=

b.

2 5

+

2 3

=

50

c.

1 3

+

1 4

=


Adding Unlike Fractions 2: Finding the Common Denominator How do we add unlike fractions? The principle is very simple:

First convert the unlike fractions to like fractions. Then add. In this lesson we study how to do this conversion.

Like fractions havetothe samewhat kind kinds of parts: they have theconvert same denominator. add unlike fractions, we need decide of parts we can them into so When that thewe converted fractions have the same denominator. We call this same denominator the common denominator because all of the converted fractions that we add together will have this same denominator in common. After you know what denominator to use (the kind of parts to convert to), you can use the principles of equivalent fractions to do the actual conversion.

1. This table gives you some example additions and tells you what denominator to convert to. Convert the fractions using the rule for writing equivalent fractions. Then add them. Note: sometimes you need to convert only one fraction, not both. fractions to add a.

b.

c.

d.

1 3 1 3 1 8 1 2

+

+

+

+

denominator 1 2 1 4 1 4 1 6

2 1 e. 7 + 2

6

fractions to add

addition sentence 2 6

+

3 6

=

5

f.

6

1 3 5

+

denominator 3 5 1

12

+

=

g.

8

+

=

h.

6

+

=

i.

14

+

=

2 1 j. 5 + 4

51

9 2 5 3 7

+

+

+

3 1 2 1 3

addition sentence

15

+

=

9

+

=

10

+

=

21

+

=

20

+

=


The rule for finding a common denominator The common denominator has to be a multiple of each of the denominators.

In other words, the common denominator has to be in the multiplication table of the individual denominators. Or, the individual denominators have to “go into” the common denominator, just like 5 goes into 30. Examples:

2

1

+

3 3

1

+

8

7

=

6

3

+

8

=

5

+

24

7

=

4

+

15

8

+

15

24

The common denominator must be a multiple of 5 and also a multiple of 3. Fifteen will work because 5 goes into 15, and 3 goes into 15. We need to find a number that is both a multiple of 8 and a multiple of 6. Let’s check the multiples of 8: They are 8, 16, 24, 32, etc., and we notice that 24 is the smallest one that is also a multiple of 6, so 24 works. We need a number that 4 can “go into” and that 8 can “go into.” Actually, the smallest such number is 8 itself. So in this case, the 7/8 does not need to be converted at all; you just convert the 3/4 into 8th parts.

8

2. Find a common denominator (c.d.) that will work if you are adding these kinds of fractions. fractionstoadd

c.d.

fractionstoadd

c.d.

a. 4th parts

and

5th parts

e. 2nd parts

and

7th parts

b. 3rd parts c.10th parts

and and

7th parts 2nd parts

f. 5th parts g. 4th parts

and and

10th parts 6th parts

d.3rd parts

and

12th parts

h. 9th parts

and

6th parts

3. Find a common denominator (c.d.) that will work for adding these fractions. Fractions c.d. a.

4 5

and

Fractions c.d.

1

b.

4

1 9

and

Fractions c.d.

1

c.

2

3 4

and

1 12

4. Now let’s add the fractions in Exercise #3. Use the common denominators you found above.

a.

4

+

1

b.

1

+

1

c.

3

+

1

5

4

9

2

4

12

↓

↓

↓

↓

↓

↓

20

+

20

=

+

=

52

+

=


You can always multiply the deno minators to get a common denominator . That is one possibility. But you can often find a common denominator that is smaller than the one you get by multiplying the denominators. Just compare the multiplication tables of the denominators. Fractions 2

and

3 7

and

10 2

and

7

Common denominator

5

You could use 3 × 18 = 54, but actually 18 works as well, and is smaller!

18 1

You could use 10 × 15 = 150, but actually 30 works as well, and is smaller!

15 1

Use 7 × 6 = 42. There are no smaller numbers that would work.

6

5. Try to find the smallest common denominator (c.d.) that will work for adding these fractions. Fractions 2

a.

3 7

d.

12 3

g.

5

and and and

c .d .

Fractions

5

4

b.

9 1

7 5

e.

6 1

12 3

h.

2

4

and and and

c.d.

Fractions

3

7

c.

2 1

4 1

f.

2 1

6 1

i.

6

12

and and and

c.d. 9 11 7 9 1 9

6. Add the fractions in the above exercise. Use the common denominators you found above. a.

2 3

+

↓

5 ↓

+

d.

7 12

b.

9

+

↓

= 1

+

5 12

=

c.

2 ↓

+

↓

3 1 g. 5 + 2

3

+

e.

6

+

=

+

9 11 ↓

+

1

f.

2

+

1 6

+

↓

=

= 7 9 ↓

+

=

1 1 i. 12 + 9

↓

+

4

=

↓

↓

7 ↓

3 1 h. 4 + 6

↓

+

7 ↓

↓

↓

4

↓

=

↓

+

53

=


Add and Subtract Unlike Fractions Whether you add or subtract unlike fractions, the principle is the same:

To add or subtract unlike fractions, we convert them to like fractions first. To do the conversion, find a common denominator. Then convert the fractions. Then add or subtract. Once you learn this well, you will have mastered the hardest part of fraction math! Multiplication, division, and simplification of fractions are much easier than adding unlike fractions.

1. In the problems below and on the following page, find the letters that match these answers. Write them in the boxes to solve the riddle.

Why did the banana go to the doctor? 23

5

7

3

1

4

5

23

7

8

31

17

36

6

15

10

10

15

6

30

45

15

30

28

17

9

9

83

11

11

27

23

5

1

1

104

7

13

21

10

35

72

24

24

40

30

9

6

14

63

6

20

!

L

1 2 6

E

2 6

+ +

+

2

6

1

W

3 =

1

2 3

=

A

1 10

+

1

1

15

+

5

2 

15 =



G

5

5 6

=

2 

3 =



13

U

30 +

2

=

1 

3 =

S

3



I

5

+

15

C

2

1

+

5

3 5

=

1 

2 

54

=


These problems belong to the riddle on the previous page, as well. E

1 3

+

1

5

T

8

9

2 

N

5

2 3

1

+

7

+ =

E

5 6

3 

1

O

8

2

=

L

4 5

3 

6 7

=

T

6 7



3

1 5

2 3



1

E

9

+

10 7

+

B

4

+

Hint: If the answer fraction's denominator is 15, think what the denominators of the two addend fractions could have been.

1 

2 =

2 9

5

+

12

+

5

P

6

=

11 8

=

2 

9 =



2

L

9 =

7 8

1 

5 =



+ Find the fractions that can go into the puzzles.

7

=

+

=

4



1 

+

=

N

W

=



A

5

4 10

+

S

20

+

=

=



=

13 15

+

+

=

+

= 72

17

= 12

= 5

= 9

= 7

= 16

6

20

24

63

55

42

+

5

+

13

+


Mixed Numbers with Unlike Fractional Parts Adding and subtracting mixed numbers when the fractional parts are unlike

This is easy: First convert the unlike fractional parts into like fractions. Then add or subtract the mixed numbers. You already know how to do both steps. 1 2 7 +1 8

Split the half into eighths.

2

1 2

+ 1

7 8

= 2

4 8

+ 1

7 8

4 8 7 +1 8 11 3 8

2

+

= 3

11 8

= 4

2

3 8

4

3 8

Convert 1/2 into 4/8 and add. Then, change the improper fraction 11/8 into 1 3/8.

Split the pieces so you have sixth parts.

2

1 − 2

1

2 3

= 2 = 1

3 − 6 9 − 6

1 1

4 6 4 6

5 6

=

First convert 1/2 and 2/3 into like fractions. But before you can subtract, you also need to rename 2 3/6 as 1 9/6 (that is, to borrow 1 whole pie).

1. Split the pieces in such a way that you can cross out what is indicated. Write a subtraction sentence.

a.

c.

Cross out 1

Cross out 1

3 . 8

7 . 10

b.

d.

56

Cross out 1

1 . 3

Cross out 1

4 . 9


2. Split the pieces so you have like fractional parts. Write an addition sentence.

+

a.

+

c.

+

b.

+

d.

3. First convert the fractional parts into like fractions, then add or subtract. Before subtracting, you may need to borrow. After adding, you may need to change the mixed number so that its fractional part is a proper fraction.

6

2 3

6

b. 10

1 8

c. 17

1 16

+ 3

4 5

+ 3

+ 3

2 5

+ 3

3 8

a.

5

1 2

15

4 8

− 2

4 5

− 8

5 6

g. 4

1 6

h. 11

1 12

− 2

3 5

+ 3

1 4

d.

e.

16

5 9

− 10

1 2

f.

8

2 9

− 2

3 4

i.

57


We can also subtract in parts. But we still need to convert unlike fractions into like fractions: Split the fourth and one whole pie into twelfths. Cross out 2 5/6 = 2 10/12. 5

1 − 4

2

5 6

= 5 = 5

3 − 12 3 − 12

=

2 2

10 12 3 12

3

First convert into like fractions. −

7 12

−

7 12

Break 2 10/12 into two parts. = 2

5 12

Finish the subtraction.

4. First convert into like fractions. Then subtract in parts. Follow the model.

a. 5

3 7 = 5 − 1 4 8 = 5

− 1

−

b. 8

(like fractions)

= 8

−

=

2 1 c. 3 9 − 1 3 = 3 = 3

− 5

−

(like fractions)

−

=

− 1

−

2 1 d. 7 7 − 2 2 = 7

(like fractions)

= 7

−

=

e. 8

9 4 = 8 − 5 15 5

− 2

−

(like fractions)

−

=

3 4 − 2 = 10 5

f. 6

58

2 1 − 1 = 3 7


5. a. Sally needs 3 1/4 feet of material to make a blouse and 4 1/2 feet to make a skirt. How many feet of material should she buy?

b. Betty uses 3 1/4 feet of material to make one shirt. She has one piece that is 5 1/2 feet and another piece that is 4 1/2 feet. She made one shirt from each piece of material. Now, if she combines the remnants left over from both pieces, will she have enough to make a third shirt?

c. Hal wants to make a picture frame out of wood. He has a 24-inch strip of wood of the correct width. The frame needs to be 4 1/4 in. high and 2 3/4 in. wide on both sides. How long a piece of wood will be left after he finishes?

6. Lily's notebook measures 3 1/4 in. wide and 6 1/8 in. long. She wants to glue a picture on the front so that the margins on all sides are 3/4 in. What size should her picture be (width and length)?

7. A picture that is 10 inches high and 4 1/2 inches wide needs a frame. The frame is made out of wood that is 1/2 inch wide. Mark the measurements in the picture. The gray outline shows how the wood needs to be cut. How long a piece of wood is needed to make the frame?

8. Cindy needs to make two cakes, one batch of pancakes, and some sauce. She needs 3 1/2 dl of flour for a cake, 5 dl of flour for a batch of pancakes, and 3/4 dl of flour for the sauce. (The abbreviation dl means deciliter.) A 1 kg bag of flour is about 15 dl. Will that one bag of flour be enough for her?

59


Add and Subtract Several Unlike Fractions (This lesson is challenging. Students who are struggling to add or subtract just two fractions may skip it.) When we add or subtract more than two unlike fractions, the principles remain the same: First, find a common denominator for all of the fractions. This common denominator has to be a multiple of each of the individual denominators. Next, convert all the fractions to like fractions. Then add or subtract.

Study these examples carefully:

2

1

+

3

5

The common denominator must be a multiple of 5, a multiple of 3, and a multiple of 2. The “obvious” one is obtained by multiplying all three: 3 × 5 × 2 = 30. None of the smaller numbers, such as 5, 10, 15, 20, or 25 work.

1

+

2

↓

↓

↓

20

6

15

+

30

1

30

7

+

2

+

30

↓

12

21

20

24

= 1

11

Notice that we give the answer as a mixed number. The answer 41/30 is not wrong, but it’s customary to give the final answer as a mixed number when possible.

30

6

↓

+

30

The first “candidate” that might come to mind is 16, but 6 does not go into 16. So let’s keep checking the multiples of 8. Indeed, 24 will work because 2, 8, and 6 all go into 24 evenly.

↓

24

41

5

−

8

=

−

24

=

13

Notice that we add the first two fractions, then subtract the third.

24

1. Find a common denominator (c.d.) that will work to add or subtract these fractions. fractions a. 2

3

1 

6

+

c.d.

fractions

1

b. 1

2

6

+

1 4

+

c.d.

fractions

2

c. 1

3

2

1 

10

+

c.d.

2 3

2. Add or subtract the fractions from Exercise #1. Use the common denominators that you found.

a.

2 3

1 

6

+

1 2

b.

1 6

+

1 4

+

60

2 3

c.

1 2

1 

10

+

2 3


3. Add or subtract. a.

7 8

1 

4

7 d.

1 

1

b.

3

1

1 5

1

4 − 5 − 2

e.

+

1 10

3

+

2

2

c.

3

1

9

5

7 + 4 − 2

f.

+

1 2

1

+

2 3

1

6 + 9 + 2

4. Solve the equations for x. Remember, the connection between addition and subtraction works even when the numbers are fractions. a.

x +

c.

x−

e.

x +

1 4

5 6

1 3

=

=

+

7

b.

8

3 8

2 7

= 4

61

9 10

d.

x−

f.

2

+ x = 2

7 11

1 6

−

2 3

− x = 1

1 5

=

1 2

1 2


Measuring in Inches Here are four rulers that all measure in inches. They are NOT to scale. Instead, they are “blown up” to be “bigger” than the actual rulers, so you can see the divisions better. The tickmarks are: every 1/2-inch: every 1/4-inch:

every 1/8-inch:

every 1/16-inch:

1. Find the

2. Find the the 2

3 4

1 2 1 4

-inch mark, 1

1 2

-inch mark, the

-inch mark, and 2 3 4

-inch mark, and the 3

1 2

-inch mark, the 1 1 4

-inch mark on all of the rulers above. Label them. 1 4

-inch mark, the 1

3 4

-inch mark, the 2

1 4

-inch mark,

-inch mark on the bottom three rulers above. Label them.

3. Look at the ruler that measures in 1/8-inches. On that ruler find and label tick marks for these points: The 1/8-inch point, the 5/8-inch point, the 7/8-inch point, the 1 5/8-inch point, and the 2 3/8-inch point. Find these same points also on the ruler that measures in 1/16-inches.

4. Look at the ruler that measures in 1/16-inches. On that ruler find tick marks for these points: the 1/16-inch point, the 3/16-inch point, the 5/16-inch point, the 7/16-inch point, the 9/16-inch point, the 11/16-inch point, the 13/16-inch point, and the 15/16-inch point.

62


5. Measure the following colored lines with the rulers given. If the end of the line does not fall exactly on a tick mark, then read the mark that is CLOSEST to the end of the line.

a.

d.

b.

e.

c.

f.

g.

h.

i.

.

k.

l.

63


6. Measure the following lines using different rulers. Cut out the rulers from the bottom of this page. a.

b.

Using the 1/4-inch ruler: __________ in.






c.

d.







e.

f.







You may cut out the following rulers:

64


7. Find six items in your home that you can measure with your ruler and measure them. a. __________________________

_______ in.

b. __________________________

_______ in.

c. __________________________

_______ in.

d. __________________________

_______ in.

e. __________________________

_______ in.

f. __________________________

_______ in.

8. It’s time to put your fraction skills to use adding and subtracting fractions of inches! a. Carefully measure the sides of the purple quadrilateral at the right and find its perimeter.

b. A company packs little jars of skin salve in boxes. Each jar is 1 3/8 inches high. How many of those jars can be stacked on top of each other in a box that is 6 inches tall? 10 inches tall?

c. A small rectangular bulletin board measures 15 3/4 in. by 9 1/8 in.

What is its perimeter? d. Draw below any triangle, and find its perimeter.

65


Comparing Fractions 1 Sometimes it is easy to know which fraction is greater.

With like fractions, all you have to do is to check which fraction has more “slices,” and that fraction is greater.

7 9

3 9

2 9

If both thepieces same number of pieces, then thefractions one withhave bigger is greater.

3

<

>

8

Sometimes you can see that one fraction is less than 1/2 and the other is more than 1/2. Here, 4/7 is clearly more than 1/2, and 5/12 is clearly less than 1/2.

4 7

6

2 5

12

Any fraction that is bigger than one must also be bigger than any fraction that is less than one. Here, 6/5 is more than 1, and 9/10 is less than 1.

9

>

5

5

>

10

If you can imagine the pie pictures in your mind, then you can “see” which fraction is bigger. For example, it’s easy to see that 2/5 is more than 1/4.

1

>

4

1. These are like fractions. Compare them, and write > or < . a.

8

4

11

11

b.

21

25

16

16

c.

4

2

20

20

d.

49

61

100

100

2. These fractions have the same number of pieces. Compare them, and write > or <. c.

a.

1

1

8

10

b.

3

3

9

7

66

d.

2

2

11

5

5

5

14

9

e.

f.

7

7

4

6

1

1

20

8


3. Compare these fractions to one half. Then write >, <, or =.

a.

e.

4

1

9

2

1

3

2

4

b.

f.

1

4

2

7

3

1

6

2

c.

7

1

12

2

g.

1

5

2

8

d.

6

1

10

2

5

1

11

2

h.

4. Compare each fraction to one. Then write >, <, or = in the box. a.

8

3

7

3

b.

4

9

4

11

c.

6

3

5

4

d.

7

8

8

7

5

7

6

8

e.

12

8

9

11

5

3

8

4

5. Compare these fractions by imagining the pies in your mind. a.

3

5

4

6

b.

1

2

3

8

c.

1

3

3

9

d.

e.

6. Here are three number lines that are divided respectively into halves, thirds, and fifths. Use them to help you put the given fractions in order, from the least to the greatest.

a.

1 3

,

2 5

,

2 3

,

1 5

,

1

b.

2

___< ___< ___ < ___ < ___

7 5

,

3 2

,

4 3

,

6 5

,

2 2

___< ___< ___< ___ < ___

7. For each pair of fractions, find one that is between them. Any such fraction will do! (Hint: You can visualize pies in your mind, or convert the fractions into like fractions.)

a.

1 6

<

<

1 3

b.

2 3

<

<

67

7 8

c.

3 8

<

<

1 2


Comparing Fractions 2 Comparing unlike fractions

Sometimes none of the “tricks” explained in the previous lesson work. But we do have one more up our sleeve!

3

5

Convert both fractions into like fractions. Then compare.

5

9

In the picture on the right, it’s hard to be sure if 3/5 is really more than 5/9. Convert both into 45th parts, and then it is easy

↓

↓

27

25

to see that 27/45 is more than 25/45. Not by much, though!

45

>

45

1. Convert the fractions into like fractions, and then compare them.

a.

e.

2

5

3

8

↓

↓

b.

5

7

8

12

↓

↓

f.

5

7

6

8

↓

↓

c.

3

4

8

10

↓

↓

g.

1

3

3

10

↓

↓

6

8

9

12

↓

↓

d.

h.

8

7

12

10

↓

↓

1

2

5

9

↓

↓

2. Convert the fractions into like fractions and compare them. a.

7

5

10

7

↓

↓

b.

4

3

9

7

↓

↓

c.

68

7

6

8

7

↓

↓

d.

7

2

10

3

↓

↓


3. A certain coat costs $40. Which is a bigger discount: 1/4 off the normal price, or 3/10 off the normal price? Does your answer change if the srcinal price of the coat was $60 instead? Why or why not?

4. Compare the fractions using any method. a.

e.

5

3

12

8

4

1

f.

3

15

i.

b.

3

4

4

11

j.

5

4

12

11

5

11

6

16

13

9

10

8

c.

3

1

10

5

7

10

6

8

2

1

13

5

g.

k.

d.

h.

l.

3

4

8

7

5

5

12

8

1

1

10

11

5. Find the equivalent fraction when the denominator is 100. a.

e.

1

=

2 1

=

5

3

b.

100

f.

100

10 2 10

=

=

c.

100

6

g.

100

=

10 3

=

4

6. Compare these fractions with hundredth parts. Write a.

1

40

2

100

b.

6

42

10

100

c.

>

d.

100

h.

100

1 4 2 5

=

=

100

100

or <.

7

75

10

100

d.

1

23

4

100

e.

1

9

10

100

7. Write the three fractions in order. a.

7 8

,

9 10

,

7 9

___< ___< ___

b.

69

1 3

,

4 10

,

2 9

___< ___< ___


8. Rebecca took a survey of a group of 600 women. She found that 1/3 of them never exercised, that 22/100 of them swam regularly, 1/5 of them jogged regularly, and the rest did other sports. Which was a bigger group, the women who jogged or the women who swam? How many women in her survey sample do exercise?

9. The number lines below are divided into eighths, tenths, and sixths. Use the number lines to put the given fractions in order.

a.

5 6

,

8 10

,

7 8

,

9 10

,

7

b.

10

___< ___< ___ < ___ < ___

9 8

,

11 10

,

7 6

,

12 10

,

10 8

___< ___< ___< ___ < ___

The seven dwarfs could not divide a pizza into 7 equal slices. The oldest suggested, “Let’s cut it into 8 slices, let each dwarf have one piece, and give the last piece to the dog.” But then another dwarf said, “No! Let’s cut it into 12 slices instead, and give each of us 1 1/2 of those pieces, and the dog gets the 1 1/2 pieces left over.”

Which suggestion would give more pizza to the dog?

70


Fraction Problems 1. Angela weighs 25 lb, which is 1/5 of her mother's weight. What does her mother weigh? 2. Julie cut off 1/4 of a 1-m ribbon. What part is left? How long is it?

3. A store discounted a $45-jacket by 1/5. How much does it cost now?

4. What fractional part of a 36-sq. ft. floor does a 4 ft x 3 ft carpet cover?

5. Shirley's skirt was 2 ft. 8 in. long. She decided to cut off 1/4 of her skirt's length. How long will her skirt be after she shortens it?

6. 2/9 of the library's 5,400 books need protective covering. a. How many books is that?

b. If it takes one person 2 minutes to cover a book, how much time does it take for one person to cover those books?

7. Mom needs to give her child 3/4 of the adult dose (which is 600 mg) of medicine. How much is that?

8. A company divided a project so that Mark would do 1/10 of it, Leslie would do 1/2 of it, and Jerry the rest. What part was left for Jerry?

9. A family shared a 4x8-chocolate bar. Dad got 1/4, each of the four kids got 1/8, and Mom the rest. What part did Mom get?

71


6. Subtract.

a.

5 – 1

d.

6 – 2

1 3 5

–

12

3 4 8 9

9

b.

5

9

e.

–

2

3

–

5

1 6 3

5

8

f.

–

5

7

12

c.

10 2

5

20 3 4

1 2 3 8

7. Solve the problems. a. Baby has gotten in 2/5 of her 20 baby teeth. How many are still to come?

b. A picture is 5 1/4 inches wide, and 3 1/8 inches tall. What is its perimeter?

c. The bus can seat 44 passengers. It is now 3/4 full. How many seats are still empty?

d. The Hill family spends about 1/10 of the year in Florida. Is that more or less than one month?

e. The school has budgeted $7,560 to be used for school lunches during the nine school months. How much of that money is meant for January and February?

f. 3/10 of the 50 states have implemented the newest education policy. How many have not?

g. Jane slept 5/12 of the day's hours, worked for 1/3 of the day's hours, did housework for 1/8 of the day's hours. How many hours total did she spend on those?

73


Math Mammoth Fractions 1 Answers Note: The answers are not simplified to the lowest terms since that is only studied in Math Mammoth Fractions 2. If your student gives the answer in lowest terms, that is of course still a right answer.

Understanding Fractions, p. 9 Teaching box: Can you tell what fraction this is? 7/10. 1. a.

b.

c.

d.

1 2. a. one-third 3

1 b. one-fifth 5

3.a.

c.

e.

f.

g.

h.

i.

3 2 2 c. three-fourths 4 d. two-eights 8 e. two-fifths 5 5 3 7 3 g. five-tenths h. three-twelfths i. seven-twelfths j. three-sixths 10 12 12 6 4 7 k. four-eighteenths l. seven-eighths 18 8

b.

d.

e.

f.

g.

j.

k.

l.

4 f. four-sevenths 7

h.

4. The student might color different pieces than what is shown below. a.

d.

g.

2 3 6 8 1 9

>

<

<

1 3 7 8 5 9

b.

e.

h.

1 5 3 8 5 12

<

>

>

4 5 1 8 3 12

c.

f.

i.

1 6 2 4 6 10

<

<

<

3 6 4 4 7 10

What can you notice about comparing two fractions when the denominators are the same? You can just compare the numerators. In other words, the numerators tell you which is more. j.

m.

p.

1 2 1 6 2 4

>

>

>

1 3 1 8 2 6

k.

n.

q.

1 5 1 2 3 8

>

<

<

1 8 2 3 3 6

l.

o.

r.

1 6 4 8 1 2

<

<

=

1 2 4 5 2 4

What can you notice about comparing two fractions when the numerators are the same? The fraction with the bigger size pieces is more. In other words, the fraction with the smaller denominator is more.

75


Mixed Numbers, p. 12 1. a. 1 2/5

b. 1 4/6

c. 1 1/4

d. 2 3/8

e. 1 1/3

f. 2 1/5

g. 2 3/4

h. 1 2/9

i. 1 5/12

2.

1

a. 1

d. 1

b. 2

2

5

e. 3

6

2

3

c. 2

3

1

f. 3

3

5

5 8

3.

4. b. 6/6 c. 5/5

d. 6/3 e. 12/6

f. 10/5

5. a. 3 1/2 = 7/2

b. 1 2/10 = 12/10

c. 2 3/4 = 11/4

d. 3 2/3 = 11/3

e. 2 1/6 = 13/6

f. 4 5/10 = 45/10.

6.

a.

d.

g.

5 4

14 6

14 9

=1

=2

1 4

2 6

= 1 5/9

b.

e.

h.

7 3

14 5

21 6

=2

=2

=3

1 3

4 5

3 6

c.

f.

i.

8 5

15 8

11 4

=1

=1

=2

3 5

7 8

3 4

76


7.

Part of a Whole Group 1, p. 16 1. a. 3/7; 4/7

b. 4/9; 5/9

c. 1/6; 5/6

d 3/8; 5/8

2. a. 6/11 of all students are boys; 5/11 of all students are girls.

b. 4/12 of all fruit are apples; 3/12 of all fruit are mangoes.

3. Two kinds of answers are given below: the non-simplified ones and simplified ones. Accept either answer from the student. 6eggs

4eggs

6 1 or 12 2

4 1 or 12 3

8eggs

3eggs

8 2 or 12 3

2eggs

3 1 or 12 4

10eggs

2 1 or 12 6

10 5 or 12 6

4. Two kinds of answers are given below: the non-simplified ones and simplified ones. Accept either answer from the student. 12 1 6 1 3 1 9 3 16 2 a. or b. or c. or d. or e. or 24 2 24 4 24 8 24 8 24 3 5.

a. Make 3 groups. 1 3

of 9 apples is 3 apples.

b. Make 3 groups. 1 3

d. 1 4

of 12 flowers is 3 flowers.

c. Make 5 groups. 1

of 15 carrots is 5 carrots.

5

f.

e. 1 6

of 10 fish is 2 fish.


77

1 5

of 20 berries is 4 berries.


6. III III III

III III III III

III III III

a.

1 3

of 18 is 6

18 ÷ 3 = 6

II II II II II

b.

1 5

III III III III

of 10 is 2

10 ÷ 5 = 2

c.

1 4

of 24 is 6.

24 ÷ 4 = 6

7.

c. Make 5 groups. 1 a. Make 3 groups. 1 3 2 3 3 3

of 9 apples is 3 apples. of 9 apples is 6 apples. of 9 apples is 9 apples.

1 3 2 3 3 3

d. 1 4 2 4 3 4 4 4

6 2

of 12 flowers is 6 flowers. of 12 flowers is 9 flowers.

6 3 6 4


5


2 5


3


5 4 5

e. 1


b. Make 3 groups.

6


1 2 5


3 5


4 5

78

of 15 fish is 6 fish. of 15 fish is 9 fish. of 15 fish is 12 fish.

f.

5 of 12 flowers is 4 flowers.

of 15 fish is 3 fish.

of 20 berries is 4 berries. of 20 berries is 8 berries. of 20 berries is 12 berries. of 20 berries is 16 berries.


Part of a Whole Group 2, p. 19 1

To find

2 1

To find

4

of a number, divide by 2 . To find of a number, divide by 4. To find

1 3 1 5

of a number, divide by 3. of a number, divide by 5.

1. a.

1 3

of 18 is 6.

b.

18 ÷ 3 = 6 d.

1

of 10 is 2

5

c.

10 ÷ 5 = 2

1 of 15 is 5 3

e.

15 ÷ 3 = 5

1 5

of 25 is 5

25 ÷ 5 = 5

1 of 14 is 7 2

f.

14 ÷ 2 = 7

1 of 21 is 7 3 21 ÷ 3 = 7

2. a.

1 4 3 4

of 12 is 3.

b.

1 6 2

of 12 is 9.

6

of 12 is 2.

c.

1 5 3

of 12 is 4.

5

of 10 is 2.

d.

1 2 3

of 10 is 6.

2

of 10 is 5. of 10 is 15.

3. a.

1 7 3 7

of 28 is 4.

b.

of 28 is 12.

1 9 2 9

of 18 is 2.

c.

1 11 3

of 18 is 4.

11

of 55 is 5. of 55 is 15.

d.

1 8 7 8

of 56 is 7. of 56 is 49.

4. 1 7 2 7 3 7 4 7 5 7 6 7

of 21 is 3. of 21 is 6. of 21 is 9. of 21 is 12. of 21 is 15. of 21 is 18.

1 5 2 5 3 5 4 5 5 5

of 20 is 4. of 20 is 8. of 20 is 12. of 20 is 16. of 20 is 20.

1 2 1 2 1 2 1 2 1 2 1 2

of 10 is 5. of 12 is 6. of 14 is 7. of 16 is 8. of 18 is 9. of 20 is 10.

1 3 1 3 1 3 1 3 1 3 1 3

of 21 is 7. of 24 is 8. of 27 is 9. of 30 is 10. of 33 is 11 of 36 is 12.

7

1

1

7 of 21 is 21.

2 of 22 is 11.

3 of 39 is 13.

5. a. 12 apples are left b. 8 eggs are left c. 32 hazelnuts to start with d. She had $15; now she has $10. e. She srcinally had 6 eggs.

79


Part of a Whole Group 3, p. 21 1. a. 9

b. 8

c. 24

d. 56

e. 35

2. a. 2/3 are girls; 22 are girls

f. 81

g. 42

h. 18

b. 5/6 are not adults; 15 are not adults.

3. Part

1

1/2

1/4

3/4

1/5

3/5

3/8

Amount

40

20

10

30

8

24

15

4. a. There are 15 boys c. Color R ed

b. The air conditioner is not on for 9 hours each day.

Part Amount 1/5

ellow Y 1/2 Green

d. 16 bananas are left.

10 25

3/10

15

5. a. a quarter till or a quarter after; 45 minutes

b. 25 cents

c. 1,320 feet

d. one quart

6. Jerry needs to save $375. 7. He stopped watching at 5:03. 8. She has $25 left. (She used $10 for apples and had $40 left. 3/8 of $40 is $15, and $40 − $15 = $25.) 9. To solve for example (g), 3/5 of what number is 15, notice that if 3/5 of a number is 15, then 1/5 of a number is a third of 15, or 5. Since 5 is 1/5 of this number, the number is 5 × 5 = 25. a.

e.

1 2 9 10

of 12 is 6.

b.

of 50 is 45. f.

1 3 2 3

of 54 is 18. c.

of 18 is 12. g.

1 5 3 5

of 50 is 10. d.

of 25 is 15. h.

1 4 3 4

of 32 is 8.

of 44 is 33.

Puzzle corner: a. She baked 24 cookies. This can be solved by thinking from the end to the beginning: 4 cookies is 1/3 of what was left after giving some to Seth. So after giving some to Beth, she had 12 cookies. Those 12 were half of the srcinal batch; so srcinally she had 24. b. Mom baked 36 cookies. 12 cookies is 2/3 of what was left after giving some to Seth - which makes 18 cookies left after giving some to Beth. Since 18 was half the batch, she srcinally baked 36 cookies.

80


Adding and Subtracting Like Fractions, p. 23 Fractions that have the same denominator are called like fractions. Fractions that have a different denominator are called unlike fractions. Write the answers to the pictures. Check with your teacher. + 1

2

+

4

4

+

=

3

2

and

4

are like fractions since they have the

4

same denominator 4. In other words they are same kind of parts - fourth parts.

4

=

2

+

4 +

9

1

=

9

=

6

3

9

8

= 5

+

8 = 8 =1

8

Adding the same kind of parts is easy - just add the numerators. 2

1. a.

4

e.

1

2. a. e.

5 6

2

8 6

+

=

2

7

=

=1

6 3

=1

2 4

= 1

1

b.

5

=

6

7

+

8

3

+

2

4

=

5

+

6

i.

2

+

5

1 6 1 2

10 5

f.

12 3

j.

8 3

b.

8

12

5 3

f.

14 14 14 3. a. 2 b. 1 1/7 c. 1 1/10 d. 8/14 4. a. 1

e.

5 8

10 14

5. a. 2/5

−

−

7 8 3 14

b. 2/7

=

=

6 8 7

14

c. 3/10

b.

+

+

+

+ +

3

=

10 8

=

12 4

+

8 1

=

5 2

4

13

=1

12 3

=

8

10 8

1

= 1

f. 1

1

−

5 2 4

d. 6/14 e. 2/4

−

3 4

c. 1

c. 1 3

g. 2

4 g. 5/20

5

+ +

4 h. 1 1/3

5

f. 4/15

6 10

g.

4

=

+

8 10

=

=

7

d.

10 12 10

=1

2 10

h.

1 6 2 3

+

+

5

=

6 2

=

3

6

=1

6 4

= 1

3

1 3

8

5

=

5

4 10

2 10

2

4 4 4 e. 3/4 f. 6/13 g. 9/20 5

+

10

g.

12

4

= 1

5

c.

10

h. 1/3

5 10 1 4

7 10 6

− 1

i. 3/4

2 10 2 4

d.

10

= 2

4 i. 1/5 −

3

=1

3

h.

4 j. 12/13 = 1

=

j. 1/3

3 10

3

2 7 9

k. 2/9

+

6

d. 1

3 7 3 8

1

= 1

7 3

7 2

= 1

10 10 k. 3/6 l. 2/8

h. 2

4

+

−

−

10

6 7 11 8

=

4 7

= 1

l. 3/8

6. Answers will vary. Check the student's work.

81


7. a. 1 3

+

1−

2 3 2 3

= =

3

b. 6

= 1

3 1

11 1

3

+ 1

11

6

=

11 −

6 11

12

e. 4 5 1

+ 1 5

2 5 −

= 2 5

6 5 =

2

11

6

10 8

11

10

= 1

11 =

c. 1

+ −

6

=

10 6

=

10

9

10 2

8

1

7

5

10

4

1

5

+ 3

10

6

=

10 −

6 10

+ 3 8

2

=

8 −

2

11

= 1

10

3

6

10

7

7

1

10

+ 6 7

7

=

7 −

6 7

13 7

= 1

= 1

8

8

3 8

1

= 1

g.

13

=

1

10

f.

= 1

d.

8

8

h. 6

3

7

8

= 1

1

+ 2 8

7

=

8 −

7 8

10

= 1

8 =

2 8

3 8

Review: Mixed Numbers, p. 27 1. a. 1 1/3

b. 2 2/6

c. 3 3/5

d. 6 4/12

2. a.

b.

c.

d.

e. 3. a. 2 2/5 4. a. 2/4

b. 1 6/7 b. 1 1/4

c. 3 3/4

d. 4 2/4

5.

6. a. & c. see image:

b. 3 2/5

d. 4 3/5

7. a. 1 2/5 = 7/5 b. 2 4/6 = 16/6

c. 2 3/8 = 19/8

d. 4 5/12 = 53/12

e. 3 1/4 = 13/4

f. 5 2/9 = 47/9

8. There are 5 whole pies, and each pie has 13 slices. So 5 × 13 tells us the number of slices in the whole pies. Then the fractional part 9/13 means that we

74 add 9 slices to that. All total we get 74 slices, and each one is 13th part. So the fraction is 13 .

82


9. a. 15/2

b. 20/3

c. 75/9

d. 66/10

e. 27/11 f. 97/12

g. 37/16

h. 39/8

10. a. 47 ÷ 4 = 11 R3 47 4

= 11

b. 35 ÷ 8 = 4 R3

3

35

4

8

d. 35 ÷ 6 = 5 R 5 35 6

= 5

11. a. 7 6/8

=4

c. 19 ÷ 2 = 9 R 1

3

19

8

2

e. 72 ÷ 10 = 7 R 2

5

72

6

10

b. 5 1/3

= 7

c. 5 2/5

2

f. 22 ÷ 7 = 3 R 1

2

22

10

7

d. 3 5/9

1

= 9

e. 3 1/2

1

= 3

f. 6 1/4

7 g. 8 2/6

h. 6 2/5

i. 2 2/11

j. 13

k. 7 1/8

l. 9 6/9

13

5

Adding Mixed Numbers 1, p. 31 1. a. 1

2. b. 1

3. a. 1

4. a. 2 e. 2

1

11 9 1

1

+

8 3 10

= 3

3

→ 2

+ 1

2

5. a. 12

1

+ 2

3

+ 1

1

+

= 2 5 10

b. 6 2/5

= 3

2

10

c. 8 1/9

3 7 5 5 7

= 5

8. b. 3 3/4

c. 6 2/3

9. a. 6

b. 11 3/5

2

d. 9 1/4 c. 10 1/8

3

+ 1

2 3

3 5 4 3 5 6

c. 3 3/4

7

d. 6

3

4

= 3

1

c. 2

3

5 8 7 8

6 → 7

+ 1

+

3 8

3

e. 3

4 5 8

= 4

= 3

2 8

8

→ 4

8

2 8

d. 1

3 4

+ 1

3 4

= 3

2 4

d. 6 3/7

+

b. 2 1/3

2

2

10

+

7. a. 1 1/2

= 4

6

4

b. 3

8 1 → 10 7 7

5

+ 2

b. 2

b. 1

8 6

→ 4

6

2

4

+ 1

3

3

1

6. a. 4

9

5

c. 3

9

2

8

b. 1

3

2

1

3

2

6 9 7 2 9

c. 4 +

7 2 → 7 5 5

6

13 4 → 7 9 9

6 8 7 + 2 8 d. 7

9

13 5 → 10 8 8

d. 5 1/4 e. 5 3/8 d. 8 3/10

10. 7 3/8 + 5 7/8 − 1 4/8 − 1 4/8 = 10 2/8. The combined string is 10 2/8 inches long now.

83


11. a. 10

7 9

b. 1

5 11

2

5 9

3

9 11

3

8 9

+ 2

8 11

15

20 2 → 17 9 9

+

12. a. 1 2/4

b. 2 1/5

5 6

c. 2

+

5

4 6

2

3 6

9

12 → 11 6

22 → 8 11

6

7 10

d. 1

9 10 6 10

+ 10 11

22 2 → 13 10 10

c. 1 1/3

Subtracting Mixed Numbers 1, p. 34 1. a. 1 2/4 2. a. 2/4 3. a. 1 3/6

b. 1 4/8

c. 4/7

d. 5/6

b. 1 1/7

c. 7 4/9

d. 7 3/5

b. 3/5

c. 1 2/8

d. 3/7

e. 2 4/11 f. 5 e. 2 7/9

4. a. 2

1 2 − 1 3 3

=

2 3

b. 1

3 4 − 5 5

e. 2

3 7 − 1 8 8

=

4 8

f. 2

1 5 − 1 6 6

=

i. 2

2 5 − 6 6

j. 2

5 7 − 1 9 9

=

= 1

3 6

g. 2 2/10

4 5

=

c. 2

1 3 − 4 4

2 6

g. 2

2 3 − 1 5 5

=

7 9

k. 5

1 4 − 3 5 5

= 1

=1

2 4 4 5

d. 2

3 5 − 1 7 7

=

5 7

h. 2

2 6 − 9 9

= 1

5 9

2 5

Subtracting Mixed Numbers 2 — Renaming, p. 37 1. a. 1 7/6

b. 2 9/8

2. a. 1 8/5 13 3. a. 2 9

b. 2 13/10

c. 1 5/4 15 b. 2 12

8 9

−

2

−

5 9

2

7 12

c. 2

17 10

d

1

11 8

1

9 10

−

1

7 8

1

8 10

−

8 12

4. a. 2 3/6

b. 1 4/9

5. a. 1 5/7

b. 4 6/9

6. a. 1 3/4 b. 1 2/5 7. a. 3 4/8

c. 1 11/9

b. 3 9/15

c. 1 8/12 c. 5 10/12 c. 2 5/12 c. 3 2/17

4 8

d. 1 4/8 d. 2 8/14 e. 11 2/9 d. 2 1/6

e. 3 7/8

d. 1 6/8 e. 9

f. 3 11/21

g. 11 8/19

h. 4 16/20

f. 3 1/6

f. 11 6/12

84


Subtracting Mixed Numbers - Extra Practice, p. 40 1. a. 1 2/3 d. 1 3/6 g. 3 1/8

b. 1 3/5 e. 1 5/8 h. 1 3/4

c. 1 2/4 f. 1 8/12 i. 1 3/12

2. a. 4 2/4

b. 9 4/6

c. 10 2/8

3. a. 2 1/3 e. 3 7/8

b. 4/5 f. 2 1/3

4. m. 4

11 2 6 i. 1 e. 12 12 10 9

a. 2

4 9

k. 3

7 11

c. 2

4 11

b. 2

12 15

l. 6

6 8

d. 10 4/12

c. 1 5/6 d. 8/9 g. 4 h. 7 6/12

h. 2

7 d. 1 9

6

4

f. 1 11 g. 5 12

3

j. 5 8

Equivalent Fractions 1, p. 42 1.

2. a. Split each piece in two.

b. Split each piece into three.





2 2 5

=



1 4

4

1

10

2

2







2

3

8

3

2

1 2



2

6

3



2

3

4

8



4 6

2

f. Split each piece in two.

3



9

1

9

5

3

2

=



2 10

2

i. Split each piece into five.

2

=

=



h. Split each piece in two.

2

=

2

3

=



g. Split each piece in two. 

=

2 3

e. Split each piece into three.

2

=



3

d. Split each piece in two. 

c. Split each piece in two.



6

1

16

2

2

=



85

5 5 10

5


3. a. 3 12 = 4 16

4. a.

d.

e.

f.

2

=

1 1

=

2 6

2

10

c. 1

16

2

=

2

=

3 1

=

8

1

b.

c.

b. 5

=

3 3

= 3

3 9 =

=

= 4

=

6

d. 2

12

7

4

=

4 4 12 =

= 5

5

=

5 5 15 =

=

= 6

8

e. 1

28

4

6

6 18 =

2

4

6

8

10

12

14

3

6

9

12

15

18

21

5 2 7 3 4

=

=

=

10 4 14 6 8

=

=

=

15 6 21 9 12

=

=

=

20 8 28 12 16

=

=

=

25 10 35 15 20

=

=

=

30 12 42 18 24

=

=

=

35 14 49 21 28

f. 2

20

7

7 21

=

=

6

g. 5

21

8

=

50

h. 1

80

2

=

8

i. 3

16

5

=

21

j. 3

35

7

=

24 56

8

=

7

=

7

5

7

=

6

=

8

=

8 24

8 16

=

=

=

24 40 16 56 24 32

=

=

=

27 45 18 63 27 36

5. a. 4/7 = 12/21 b. four; 4/5 = 16/20. c. three; 1/6 = 3/18. d. two; 6/7 = 12/14. e. four; 2/3 = 8/12. f. two; 7/10 = 14/20. g. three; 5/9 = 15/27. h. six; 1/8 = 6/48. i. six; 4/9 = 24/54. j. four; 8/11 = 32/44. k. three; 3/10 = 9/30. l. three; 2/11 = 6/33. m. eight; 4/7 = 32/56. n. nine; 1/6 = 9/54. o. eight; 7/8 = 56/64. 6. a. 2/3 = 8/12 = 16/24 b. 1/3 = 4/12 = 8/24 c. 14/24 d. Answers vary; any of the following will do: 1/12, 2/12, 3/12, 5/12, 6/12, 7/12, 9/12, 10/12, or 11/12. e. Answers vary; any of the following will do: 1/24, 3/24, 5/24, 7/24, 9/24, 11/24, 13/24, 15/24, 17/24, 19/24, 21/24, or 23/24.

86


Equivalent Fractions 2, p. 46 1. a. Split each slice into three.

3



7

c. Split each slice in two.



2



12

b. 5 28/40

c. 36/16

d. 3 12/18



=

2



2.a. 1 12/16

d. Split each slice into four.

24 = 10

5

f. 21/6

g. 6 12/54

h. 42/6

i. 5 56/80

6

2 4

3

12

2 =

8



e. 40/15

10

=

3

=

3



=

5

12

2



21

=

4

=

b. Split each slice in two.

4

j. 54/18

3. a.

b.

c.

d.

e.

5

11

3

2

9

10

1

6

12

4

18

40

2

12

7

5

3 4

3

5

15

22

9

10

20

33

15

6

36

12

15

50

3

24

48

35

55

54

25

2 24 80

8

21

4

36

5 72 96

50

110

30

10

81

30

50

100

5

54

4. a. four 5

2

f.

=

20

2

2

2

b. NOT POSSIBLE

c. NOT POSSIBLE

d. four 2

28

f. NOT POSSIBLE

3 g. four 1 6

=

h. eight 4

5

24

4

=

=

1

32

3

=

5

5

18 24 36

75 100

e. NOT POSSIBLE

8 12

i. five 40

5

j. NOT POSSIBLE

5 15

5. Answers vary. If we know the new numerator is not divisible by the old one, or if the new denominator is not divisible by the old one, then the conversion is not possible. In other words, if the numerator does not “go into” or divide into the new numerator and similarly with the denominators, then we cannot find an equivalent fraction. 6. 3

halves

thirds

fourths

fifths

6

9

12

15

tenths 30

300

2

3

4

5

10

100

87

hundredths


Adding Unlike Fractions 1, p. 48 + 1

+

+

3 ↓

1

1

2

3

↓

↓

+ 2

= 3

+

6

+

1

4

6

↓

↓

+

=

6

+ 1

5

4

6

12

+

= 3

1 2 ↓

+ 1

7 12

=

12

+

6

+

= 3

4

=

6

6

To add unlike fractions, first convert them to like fractions. Then add.

1. a.

1

+

2 ↓

b.

4 ↓

2

+

4

1

2. a.

2 1 2

c.

1

1 8 3 8

+

+

1 4 +

+

1 4 1 4

2

1

+

5 ↓

= 1 6 2 6

=

=

4

4

10

=

1 8 3 8

3 6 3 6

+

+

+

+

2 8 2 8

1 6 2 6

=

=

=

=

3 8

4 6

9

↓

3

=

3

c.

2

+

5 10

b.

1 4

5

1

6

4

d.

1 2

5

1

8

2

1

+

3

↓

=

+

+

+

+

9

3

10

9

1 6 4 6 1 8 3 8

=

=

=

=

3 12 3 12 4

+

+

+

8 4

+

8

2 12 8 12 1 8 3 8

=

=

=

=

↓

3

+

9

=

6 9

5 12 11 12 5 8 7 8

3. b. 12th parts c. 8th parts d. 8th parts

4. a.

b.

4 8

+

5 8

=

c.

9

2

8

4

d.

+

3 4

=

5

2

4

6

e.

5 + 10

4 = 9 10 10

5. a. 8th parts

b. 4th parts

5 6

7

=

6

f.

5 + 15 c. 6th parts

+

d. 10th parts

6 = 11 15 15 e. 15th parts

88

2 6

+

3 = 6

5 6

f. 6th parts


6. The two denominators always “go into” the number that tells us what kind of parts we are converting to. In other words, we need to find a number that is divisible by the two denominators, or in yet other words, a number that is a multiple of both of the denominators. 7. a. 7/6

b. 16/15

c. 7/12

Adding Unlike Fractions 2: Finding the Common Denominator, p. 51 1. fractions to add

denominator

1 1 + a. 3 2

6

1

b.

3 1

c.

8 1

d.

2 2

e.

7

+

+

+

+

1

12

4 1

8

4 1

6

6 1

14

2

addition sentence 2 6 4 12 1 8 3 6 4 14

+

+

+

+

+

3 6 3 12 2 8 1 6 7 14

=

=

=

=

=

fractions to add 5

f.

6 7

g.

12 3

h.

8 4

i.

6 11

j.

14

1 3 5 9 2 5 3 7 2 5

+

+

+

+

+

addition sentence

denominator

3

15

5 1

5

9

3 1

9

10

2 1

21

3 1

20

4

5 15

4 10 9 21 8 20

+

+

+

+

+

9 15 3 9 5 10 7 21 5 20

=

=

=

=

=

14 15 8 9 9 10 16 21 13 20

2. fractionstoadd

c.d.

fractionstoadd

c.d.

a. 4th parts

and

5th parts

20

e. 2nd parts

and

7th parts

14

b. 3rd parts

and

7th parts

21

f. 5th parts

and

10th parts

10

c. 10th parts

and

2nd parts

10

g. 4th parts

and

6th parts

12

d. 3rd parts

and

12th parts

12

h. 9th parts

and

6th parts

18

3. Fractions a.

4 5

and

1 4

Fractions

c.d 20

b.

1 9

and

1 2

Fractions

c.d c.

18

89

3 4

and

1 12

c.d 12


4. a.

4 5

1

+

↓

16 20

b.

=

4

9

+

↓

↓

+

1

5 20

=

21

2

20

18

1 2

=

3

c.

4

↓

+

9 18

+

↓

=

11

9

18

12

1 12

=

↓

+

1 12

=

10 12

5. Fractions a.

2

c.d 5

and

3

d. g.

9

9

7

Fractions 4

b.

and

12 3

5

6 1

and

5

and

7

1

2

12

e.

10

h.

c.d 3

Fractions

14

2

c.

1

12 3

and and

4

7 4

c.d

and

1

2 1 6

12

f.

12

i.

6 1 12

9 11

44

7 and and

9 1 9

18 36

6. a.

2 3

5 ↓

6

5

7 12

+

+

=

9

↓

9 d.

+

=

9 1 6

↓

↓

7

2

b.

3 5

+

1 2

↓

↓

6

5

10

+

10

7

+

3 2

↓

↓

11

8

21

9

14

=

e.

9

12 + 12 = 12 g.

4

=

=

5 12

+

+

14 1 2

↓

↓

5

6

=

=

c.

3 4

+

1 6

↓

↓

11

9

2

10

12

+

12

4

+

↓

↓

77

36

14

44

=

f.

11

1 6

+

+

44 7 9

↓

↓

3

14

=

=

113 44

=

17

18 + 18 = 18 i.

=

1 12

+

↓

=

9 11

29

12 + 12 = 12 h.

7

11

3

12

36

90

1 9

=

↓

+

4 36

=

7 36


Add and Subtract Unlike Fractions, p. 54 1. 23

5

7

3

1

4

5

23

7

8

31

17

36

6

15

10

10

15

6

30

45

15

30

28

A

USE

BEC

I

T W

AS

17

9

9

83

11

11

27

23

5

1

1

104

7

13

21

10

35

72

24

24

40

30

9

6

14

63

6

20

N

O

T

P

E

E

L

I

N

G

W

E

L

L

Puzzle Corner

2 + 1 = 13 3 5 15 +

1 + 1 = 13 6 7 42

+ 1 6

+ +

= 5 6

1 4

=

!

+

5

1

12

8

= 9 20

+

= 7 24

1 9

=

17 72

= 16 63

Mixed Numbers with Unlike Fractional Parts, p. 56 1. a. 2

3 3 6 3 3 − 1 = 2 − 1 = 1 4 8 8 8 8

b. 2

1 1 3 2 1 − 1 = 2 − 1 = 1 2 3 6 6 6

c. 2

1 7 12 7 5 − 1 = 1 − 1 = 5 10 10 10 10

d. 3

1 3

2. a. 2

c. 1

1 4 1 2

+ 1

+ 1

1 3 3 5

= 2

= 1

3 12 5 10

+ 1

+ 1

4 12 6 10

= 3

= 3

7 12 1 10

b. 2

d. 1

91

1 3 2 3

− 1

+ 1

+ 2

4 24 8 16 = 2 − 1 = 1 9 18 18 18 7 9 1 5

= 2

= 1

3 9 10 15

+ 1

+ 2

7 9 3 15

= 3

= 3

10 9

= 4

1 9

13 15


3. 2 3 4 + 3 5

a.

10 15 12 + 3 15

6

d. −

1 8 2 + 3 5

b. 10

6

9

22 7 = 10 15 15

5 40 16 + 3 40 13

21 40

1 16 6 + 3 16 17

7 16

20

5

1 2

4

15 10

e. 15

4 8

14

36 24

f. 16

5 9

16

10 18

2

4 5

− 2

8 10

− 8

5 6

− 8

20 24

− 10

1 2

− 10

9 18

2 7 10

6 16 24

4

1 6

3

35 30

h. 11

1 12

− 2

3 5

− 2

18 30

+ 3

1 4

1

17 30

g.

1 16 3 + 3 8

c. 17

10

6 1 18

11

1 12

i. 8

2 9

3

3 12

− 2

3 4

14

4 12

+

−

7

44 36

2

27 36

5

17 36

4. a. 5

3 7 6 7 − 1 − 1 = 5 (like fractions) 4 8 8 8 = 5

=

c. 3

3

−

=

2 2 − 1 9 9

1

9 4 9 12 − 5 − 5 = 8 (like fractions) 15 5 15 15

1 8

= 8

7 8

=

2 1 2 3 − 1 = 3 − 1 (like fractions) 9 3 9 9 = 3

e. 8

6 6 − 1 8 8

b. 8

−

d. 7

= 7

8 9

=

f. 6 3 3 = 8 − 2 − 10 10

5 10

12 15

4 4 −2 − 14 14

4

3 14

11 14

2 1 14 3 − 1 = 6 − 1 (like fractions) 3 7 21 21

= =

2

3 15

2 1 4 7 − 2 = 7 − 2 (like fractions) 7 2 14 14

1 9

3 4 3 8 − 2 = 8 − 2 (like fractions) 10 5 10 10

9 9 − 5 − 15 15

5 5 10

92

5 11 21


5. a. Sally needs 7 3/4 feet of material. She could probably buy 8 feet, just to make sure she has enough after cutting. b. Yes. She will have left 5 1/2 − 3 1/4 = 2 1/4 feet from one piece, and 4 1/2 − 3 1/4 = 1 1/4 feet from the other piece. Combining both of those gives 2 1/4 + 1 1/4 = 3 1/2 feet, which should be enough even with material allowed for hems. c. 2 3/4 + 2 3/4 + 4 1/4 + 4 1/4 = 14 inches. He will have 24 − 14 = 10 inches of wood left. 6. The margins are a total of 3/4 in. + 3/4 in. = 1 1/2 inches. Subtract that from the width and height of the notebook. The width of the picture will be 3 1/4 in. − 1 1/2 in. = 1 3/4 in. and the height will be 6 1/8 in. − 1 1/2 in. = 4 5/8 in. 7. The total height is 10 + 1/2 + 1/2 = 11 in. The total width = 4 1/2 + 1/2 + 1/2 = 5 1/2 inches. You need 11 + 11 + 5 1/2 + 5 1/2 = 33 inches of wood.

8. She needs 3 1/2 + 3 1/2 + 5 + 3/4 = 12 3/4 dl of flour. Since 1 kg is 15 dl, it is enough.

Add and Subtract Several Unlike Fractions, p. 60 1. a. 6

b. 12

c. 30

2. a. 1 2/6

b. 1 1/12

3. a. 7/24

b. 29/30

4. a. x = 5/8

c. 1 2/30 c. 1 7/18

b. x = 1 3/10

d. 1 1/20 c. x = 1 5/24

e. 11/28

f. 1 8/18

d. x = 1 2/6

e. x = 3 8/21

93

f. x = 1 3/22


Measuring in Inches, p. 62 1.

2.

3.

4.

5. a. 1 1/4 in g. 2 3/4 in.

b. either 1 2/8 in. or 1 3/8 in. c. 1 5/16 in. d. either 1/2 in. or 3/4 in. h. 2 5/8 in. i. 2 11/16 in. j. 2 1/4 in. k. 2 1/8 in. l. 2 3/16 in.

6. a. either 3 inches or 3 1/4 inches; 3 1/8 in.; 3 1/8 in. c. 2 3/4 in. on all rulers

d. 1 in; 1 in; 15/16 in.

e. 5/8 in.

f. 10/16 in.

b. 3/4 in.; either 5/8 or 6/8 in.; 11/16 in. e. 1 1/4 in.; 1 3/8 in.; 1 5/16 in.

f. 1 3/4 in. on all rulers.

7. Answers will vary. Please check the student’s work. 8. a. The sides measure: 1 1/2 in., 3 15/16 in., 1 11/16 in., and 3 1/4 in. The perimeter is 10 3/8 in. b. Four jars can be stacked on top of each other in a 6 inch high box, seven in a 10-inch high box. c. 49 3/4 in. d. Answers will vary. Check the student’s work.

94


Comparing Fractions 1, p. 66 1. a. >

b. <

c. >

d. <

2. a. >

b. <

c. <

d. <

e. >

f. <

3. a. <

b. <

c. >

d. >

e. <

f. =

4. a. >

b. >

c. >

d. <

e. >

5. a. <

b. >

c. =

d. < e. <

1

6. a.

<

5

1 3

<

2 5

1

<

2

<

2

g. <

2

b.

3

h. <

<

2

6 5

4

<

7

<

3

3

<

5

2

7. Answers will vary. a. For example 1/4, 1/5, 2/9 or their equivalent fractions. b. For example, 4/5, 5/6, 5/7, 6/7, 3/4, 17/24, 19/24

c. For example 3/7, 4/9, 4/10, 5/11, 5/12, 7/16.

Comparing Fractions 2, p. 68 1. a.

e.

2. a.

16

>

24 15

>

24 49 70

<

15

b.

24 14

f.

24 50

b.

70

20 24 15 40 28 63

<

<

>

21

c.

24 16

10 30

24

g.

40 27

c.

63

>

36 49 56

=

>

9

d.

30 24

h.

36 48

d.

56

40 60

<

9 45 21 30

42 60

<

>

10 45

20 30

3. 1/4 of $40 is $10, and 3/10 of $40 is $12, so 3/10 is the bigger discount. The answer does not change at $60, because the fraction 3/10 is bigger than 1/4. 4. a. >

b. >

5. a. 50/100 6. a. > 7. a.

c. >

b. 30/100

b. > 7 9

<

d. <

c. < 7 8

<

9 10

e. <

f. >

c. 60/100

d. >

g. <

h. <

d. 25/100

i. >

e. 20/100

j. >

k. <

f. 20/100

l. >

g. 75/100

h. 40/100

e. > b.

2 9

<

1 3

<

4 10

8. The women who swam were the bigger group. 400 women of that group do exercise. 7 8 5 7 9 11 9 7 12 10 9. a. < < < < b. < < < < 10 10 6 8 10 10 8 6 10 8 Puzzle corner. Either way, the dog would get 3/24 (or 1/8) of the pizza.

95


Fraction Problems, p. 71 1. Angela's mother weighs 125 pounds. 2. 3/4 of the ribbon is left. It is 75 centimeters long. 3. The jacket costs $36 after the $9 discount. 4. The carpet covers 1/3 of the floor. (Its area is 12 sq. ft.) 5. The skirt will be 2 ft or 24 inches long after she shortens it. 6. a. 1,200 books need covered. b. It would take one person 40 hours to cover all of the books that need covered. 7. She needs to give the child 450 mg of the medicine. 8. 2/5 or 4/10 was left for Jerry to do. 9. Mom got 1/4 of the chocolate bar.

Review, p. 72 1. a. white - 5/7, pink - 2/7 c. white - 4/5, pink - 1/5

b. white - 1/2, pink - 1/2 (or 3/6 and 3/6) d. white - 3/8, pink - 5/8

2.a. 4/12or1/3

b. 8/12or2/3

c. 3/12or 1/4

3. a. 7; 14

c. 4; 14

e. 140; 420

b. 24; 20

d. 25; 200

d. 9/12or3/4

e.10/12or5/6

4. 6

x

10

y

x+y

5. a.

7 10

1

1 4

4

3

10

4

1

2

b. 1

6. a. 3 11/12

5

2

7

5 8

3

7

3

10

10

4

2

5

2

10

10

4

5 10

c. 1

1 2 6

b. 3 9/10

2 10

1

d. 1

1

6

1

2

1

7 12

y

5 2

8

4

x 1

5

e. 5

c. 6 12/20

3 10

f. 6

2

d. 3 31/36

2 7

7

1

2

7

7

12

1

3

12 5

x−y

5

3

6

2

g. 9

7 7

12

e. 6 17/30

5

3

3

h. 4

5 12 5 12 2

5

1

4

5 8

1

1 6

1

3 7

1

4

4

8

6

7

4

3

6

8

6

7

86 100

f. 3 1/8

7. a. 12 teeth b. 5 1/4 + 3 1/8 + 5 1/4 + 3 1/8 = 16 3/4 inches. c. 11 seats are still empty. d. 1/10 of a year is 36.5 or 36 1/2 days, which IS more than one month. e. The budget has 7,560 ÷ 9 = $840 for each month; for two months the budgeted amount is $1,680. f. 35 have not. g. Jane slept 10 hours, worked 8 hours, and did housework for 3 hours; a total of 21 hours. 8. a. <

b. =

c. >

d. >

e. >

f. >

9. a. 1/4 < 2/7 < 1/3 < 3/7 < 2/4

g. >

h. <

i. <

j. <

b. 3/4 < 6/7 < 3/3 < 8/7 < 5/4

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99


100


101


102


103


104


105


106


107


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109


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