Exploring Fractions: Mastering Fractional Concepts and Operations (Middle-Upper Grades) 9781580374477, 9781580377096, 1580374476

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About the Authors Karise Mace is the founder and president of Mathematical Expressions, a company dedicated to providing support to mathematics educational companies in the areas of writing, editing, curriculum development, project management, and textbook alignment. Mace has a Bachelor’s Degree in mathematics from Greenville College in Greenville, Illinois, and a Master’s Degree in secondary mathematics education from the University of Kentucky in Lexington, Kentucky. She is a certified high school mathematics educator in Pennsylvania. She has four years teaching experience and over four years experience in mathematics text and software publishing. Amy Doverspike earned a Bachelor’s Degree in secondary mathematics education from Indiana University of Pennsylvania in Indiana, Pennsylvania, and a Master’s Degree in curriculum and instruction from Gannon University in Erie, Pennsylvania. Doverspike has been teaching high school mathematics in Pennsylvania for over six years. In addition to teaching, she has over four years experience writing and editing mathematics texts and software.

Exploring Fractions: Mastering Fractional Concepts & Operations By KARISE MACE & AMY DOVERSPIKE

COPYRIGHT © 2008 Mark Twain Media, Inc. ISBN 978-1-58037-447-7 978-1-58037-709-6 CD-404088-EB Printing No. CD-404088

Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing Company, Inc.

The purchase of this book entitles the buyer to reproduce the student pages for classroom use only. Other permissions may be obtained by writing Mark Twain Media, Inc., Publishers.

All rights reserved. Printed in the United States of America.

Exploring Fractions

Table of Contents

Table of Contents Adding and Subtracting Mixed Numbers With Unlike Denominators Let’s EXPLORE!...........................................43 Let’s EXERCISE!..........................................48 Show Your EXPERTISE!.............................. 49

Letter From the Authors...............................................iii Alignment to NCTM Number and Operation Standard..................................................................... iv Unit 1: The Basics.....................................................1 Teacher Notes for Unit 1.......................................1 Understanding Fractions Let’s EXPLORE!.............................................3 Let’s EXERCISE!............................................5 Show Your EXPERTISE!................................ 6 Equivalent Fractions Let’s EXPLORE!.............................................7 Let’s EXERCISE!............................................9 Show Your EXPERTISE!.............................. 10 Simplifying Fractions Let’s EXPLORE!........................................... 11 Let’s EXERCISE!..........................................13 Show Your EXPERTISE!.............................. 14 Comparing and Ordering Fractions Let’s EXPLORE!...........................................15 Let’s EXERCISE!..........................................17 Show Your EXPERTISE!.............................. 18

Unit 4: Mixed Numbers and Improper Fractions..................................................................50 Teacher Notes for Unit 4.....................................50 Mixed Numbers and Improper Fractions Let’s EXPLORE!...........................................51 Let’s EXERCISE!..........................................54 Show Your EXPERTISE!.............................. 55 Adding Mixed Numbers With Regrouping Let’s EXPLORE!...........................................56 Let’s EXERCISE!..........................................58 Show Your EXPERTISE!.............................. 59 Subtracting Mixed Numbers With Regrouping Let’s EXPLORE!...........................................60 Let’s EXERCISE!..........................................63 Show Your EXPERTISE!.............................. 64 Unit 5: Multiplying and Dividing Fractions and Mixed Numbers........................................................65 Teacher Notes for Unit 5.....................................65 Multiplying Fractions Let’s EXPLORE!...........................................66 Let’s EXERCISE!..........................................68 Show Your EXPERTISE!.............................. 69 Dividing Fractions Let’s EXPLORE!...........................................70 Let’s EXERCISE!..........................................72 Show Your EXPERTISE!.............................. 73 Multiplying Mixed Numbers Let’s EXPLORE!...........................................74 Let’s EXERCISE!..........................................76 Show Your EXPERTISE!.............................. 77 Dividing Mixed Numbers Let’s EXPLORE!...........................................78 Let’s EXERCISE!..........................................80 Show Your EXPERTISE!.............................. 81

Unit 2: Adding and Subtracting Fractions............ 19 Teacher Notes for Unit 2.....................................19 Adding and Subtracting Fractions With Like Denominators Let’s EXPLORE!...........................................20 Let’s EXERCISE!..........................................23 Show Your EXPERTISE!.............................. 24 Finding Common Multiples Let’s EXPLORE!...........................................26 Let’s EXERCISE!..........................................28 Show Your EXPERTISE!.............................. 29 Adding and Subtracting Fractions With Unlike Denominators Let’s EXPLORE!...........................................30 Let’s EXERCISE!..........................................33 Show Your EXPERTISE!.............................. 34 Unit 3: Mixed Numbers............................................36 Teacher Notes for Unit 3.....................................36 Adding and Subtracting Mixed Numbers With Like Denominators Let’s EXPLORE!...........................................37 Let’s EXERCISE!..........................................41 Show Your EXPERTISE!.............................. 42

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Answer Keys.............................................................82

ii

Exploring Fractions

Letter From the Authors

Letter From the Authors We are so glad that you have purchased Exploring Fractions: Mastering Fractional Concepts and Operations. This book is divided into five units covering fractional concepts from the basics to operations with fractions and mixed numbers. A unit is divided into several lessons, each of which covers one or two concepts. You will notice that each lesson is made up of three parts. The first part, called “Let’s EXPLORE!”, is a teacher-guided exploration that is hands-on. The goal of this part of the lesson is to help students understand the why of the particular fraction concept being covered by using real-life context. The second part, called “Let’s EXERCISE!”, is a set of exercises where students can practice the concepts they learned in the exploration. The third part, called “Show Your EXPERTISE!”, is an activity, small project, or set of critical-thinking questions that asks the students to apply what they have learned about the concepts covered in the lesson. This part of the lesson could be used as an assessment to help the teacher evaluate what the students have learned. Each unit also contains a page or two of teacher notes. In these pages, we have made suggestions on how to make the explorations run smoothly as well as paper and pencil alternatives for the activities that require extra materials. We have also provided an alignment to show you how this book addresses the Number and Operation Standard of the NCTM standards. We hope that you and your students will have fun exploring fractions!

Mathematically yours,



Karise Mace and Amy Doverspike

© Mark Twain Media, Inc., Publishers

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Exploring Fractions

Alignment to NCTM Number and Operation Standard

Alignment to NCTM Number and Operation Standard Expectations



Unit 1 Unit 2 Unit 3 Unit 4 Unit 5

Number and Operation Standard Students should understand numbers, ways of representing numbers, relationships among numbers, and number systems. Students should work flexibly with fractions, decimals, and percents to solve problems.

• •

Students should compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line.

•

•

•

•

Students should develop meaning for percents greater than 100 and less than 1. Students should understand and use ratios and proportions to represent quantitative relationships. Students should develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation. Students should use factors, multiples, prime factorization, and relatively prime numbers to solve problems.



•

Students should develop meaning for integers and represent and compare quantities with them.

Students should understand meanings of operations and how they relate to one another. Students should understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.



•

•

•

•

Students should select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods.



•

•

•

•

Students should develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.



•

•

•

•

Students should use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals. Students should understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

Students should compute fluently and make reasonable estimates.

Students should develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results. Students should develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Reprinted with permission from Principles and Standards for School Mathematics, copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved. Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.

© Mark Twain Media, Inc., Publishers

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Exploring Fractions

Unit 1: The Basics

Unit 1: The Basics Teacher Notes Understanding Fractions Let’s EXPLORE! A paper-pencil alternative for using real fruit is to use the pictures of fruit on the following page. Allowing students to color their fruit will engage them more in the lesson.

Equivalent Fractions Let’s EXPLORE! A paper-pencil alternative for using the candy bar is to use the picture of the bar on the following page.

Simplifying Fractions Let’s EXPLORE! A paper-pencil alternative for using the fish crackers is to use the pictures of the fish on the following page.

Comparing and Ordering Fractions Let’s EXPLORE! A paper-pencil alternative for using the fruity cereal is to use the pictures on the next page. Make sure to copy them onto colored paper. If there is no colored paper to copy on, you may want the students to color the cereal pieces.

© Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Grapes

Orange slices

Candy Bar

Fish Crackers

Fruity Cereal

© Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Understanding Fractions Let’s EXPLORE! Materials: q small bunch of grapes (at least 10 grapes on the bunch) q 1 citrus fruit, such as an orange, tangerine, or Clementine Vocabulary: ƒ fraction ƒ numerator ƒ denominator A fraction is defined as a part of a group or a part of a whole. Fractions are written as one number over another number. The top number is known as the numerator, and the bottom number is known as the denominator. What does all of this mean exactly? Let’s go on an exploration and figure it out! First, we’ll explore fractions as a part of a group. 1. How many grapes are in your bunch?

2. Pull several grapes off the bunch to eat. How many did you pull off? 3. Use your answers to Questions 1 and 2 to write the fraction of grapes you pulled off to eat. Then eat the grapes you pulled off.

number of grapes to eat total grapes

=

4. How many grapes do you have left? 5. Use your answers to Questions 1 and 4 to write the fraction of grapes you have left. number of grapes left total grapes

=

© Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Understanding Fractions (cont.) Now, let’s explore fractions as a part of a whole. 6. Peel your citrus fruit and divide it into segments. 7. How many segments do you have? 8. Put aside several segments to eat. How many did you put aside? 9. Use your answers to Questions 7 and 8 to write the fraction of your fruit you put aside to eat. Then eat these segments. number of segments to eat total number of segments

=

10. How many segments do you have left? 11. Use your answers to Questions 7 and 10 to write the fraction of your fruit that you have left. number of segments left total number of segments

=

Now, enjoy eating your fruit while you answer the following questions. 12. Why were the grapes used to show a fraction as part of a group? Use complete sentences in your answer. 13. Why was the citrus fruit used to show a fraction as part of a whole? Use complete sentences in your answer. © Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Understanding Fractions (cont.) Let’s EXERCISE! Fractions as Parts of a Group Write a fraction that names how many in each set are circled. 1.



2.

3.



4.

Fractions as Part of a Whole Write a fraction that names the shaded part of each picture. 5.



6.

7.



8.

Extend Your Knowledge! 9. Draw a picture that shows

#g as parts of a group.

10. Draw a picture that shows

#g as part of a whole.

© Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Understanding Fractions (cont.) Show your EXPERTISE! Fractions as Parts of a Group You are going to make a fruit salad. Use the fruit salad to answer each question. 1. You would like 15 pieces of fruit in your bowl. Determine the number of pieces of each fruit below that you are going to put in your bowl so that the total is 15. Then draw the fruit in your bowl.

strawberries



chunks of pineapple



cherries 2. What fraction of your fruit salad is strawberries? 3. What fraction of your fruit salad is chunks of pineapple? 4. What fraction of your fruit salad is cherries? 5. What fraction of your fruit salad is strawberries and chunks of pineapple? 6. What fraction of your fruit salad is strawberries and cherries?

Fractions as Part of a Whole You are going to make a paper fruit pizza. Use the pizza to answer each question. 7. Draw slices of kiwi on

How many pieces have kiwi slices on them? 8. Draw blueberries on



#k of your pizza.

&k of your pizza.

How many pieces have blueberries on them? 9. Draw slices of peaches on



$k of your pizza.

How many pieces have peaches on them?

10. What fraction of your pizza has slices of kiwi and peaches? 11. What fraction of your pizza has slices of peaches and blueberries? 12. What fraction of your pizza has all three toppings?

© Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Equivalent Fractions Let’s EXPLORE! Materials: q one 12-piece chocolate bar Vocabulary: ƒ equivalent Equivalent is another word for equal. Let’s talk chocolate! Work through the questions before your chocolate melts! 1. How many small rectangles make up the chocolate bar? 2. Break the chocolate bar into halves, making two equal parts. How many small rectangles are in one of the halves? 3. Use your answers from Questions 1 and 2 to write a fraction that is equivalent to

!s of the

chocolate bar. number of small rectangles in one half total of small rectangles in the whole bar

=

Lay the pieces of the chocolate bar side by side so it is whole again to complete Questions 4 and 5. 4. Break the chocolate bar into thirds by making three equal parts. How many small rectangles are in one of the thirds? 5. Use your answers from Questions 1 and 4 to write a fraction that is equivalent to chocolate bar. number of small rectangles in one third total of small rectangles in the whole bar

© Mark Twain Media, Inc., Publishers

=



!d of the

Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Equivalent Fractions (cont.) Lay the pieces of the chocolate bar side by side so it is whole again to complete Questions 6 and 7. 6. Break the chocolate bar into fourths by making four equal parts. How many small rectangles are in one of the four parts? 7. Use your answers from Questions 1 and 6 to write a fraction that is equivalent to

!f of the

chocolate bar. number of small rectangles in one fourth total of small rectangles in the whole bar

=

Now, have some chocolate while you answer the following questions. 8. How would you define what equivalent fractions are using the information from our exploration? Use complete sentences in your answer. 9. How did breaking the chocolate bar into fractional pieces help us find an equivalent fraction? Use complete sentences in your answer.

© Mark Twain Media, Inc., Publishers



Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Equivalent Fractions (cont.) Let’s EXERCISE! Writing Equivalent Fractions Write two equivalent fractions that represent the shaded part of each picture. 1.





2.



3.





4.



Draw a picture with rectangles to represent the fraction, and then write an equivalent fraction. 5.

aR;



6.

aTg

7.

aWs



8.

aRh

Extend Your Knowledge! 9. How many equivalent fractions can be written for sentences. © Mark Twain Media, Inc., Publishers



!a ? Explain why, using complete

Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Equivalent Fractions (cont.) Show Your EXPERTISE! Writing Equivalent Fractions You are going to share three candy bars and one roll of caramels with your friends. Use the candy bars and caramels to answer each question. 1. You and a friend are going to share a candy bar that has 4 total pieces. To be fair, you and your friend should have the same number of pieces. Draw the candy bar with your share shaded in one color and your friend’s share in another color.

2. What two equivalent fractions of the candy bar did you share with your friend?

3. You and two friends are going to share a candy bar that has 6 total pieces. To be fair, everybody should have the same number of pieces. Draw the candy bar, shading everybody’s share differently.

4. What two equivalent fractions of the candy bar did each person get? 5. You and a friend are going to share an 8-piece roll of caramels. Your friend would like 2 caramels. Draw the roll of caramels with your pieces shaded.

6. What two equivalent fractions represent your pieces?

© Mark Twain Media, Inc., Publishers

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Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Simplifying Fractions Let’s EXPLORE! Materials: q 36 fish crackers Vocabulary: ƒ factor ƒ common factor ƒ simplify When two numbers are multiplied together, the result is a product. The two numbers that were used to find the product are called factors. A common factor is a factor that two numbers share. To simplify a fraction, you must divide out the common factors from the numerator and the denominator. Let’s swim through this exploration to learn about simplifying fractions. 1. How many total fish crackers do you have? 2. Divide your fish crackers in half by making 2 equal piles. How many are in each half? 3. Use your answers from Questions 1 and 2 to write a fraction that represents the number of fish crackers in one of the 2 piles. number of fish crackers in one of the 2 piles total number of fish crackers

=

4. How does the fraction you made in Question 3 compare to the fraction of sentences in your answer. 5. What number was divided out of the numerator in Question 3 to get 1? © Mark Twain Media, Inc., Publishers

11

!s ? Use complete

Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Simplifying Fractions (cont.) 6. What number was divided out of the denominator in Question 3 to get 2? 7. What is the common factor that was divided out of the fraction from Question 3 to simplify the fraction to

!s ?

Put the fish crackers back into one pile for Questions 8–13. 8. Divide up your fish crackers into thirds by making 3 equal piles. How many are in each pile? 9. Use your answers from Questions 1 and 8 to write a fraction that represents the number of fish crackers in one of the 3 piles. number of fish crackers in one of the 3 piles total number of fish crackers

=

10. How does the fraction you made in Question 9 compare to the fraction of

!d ? Use complete

sentences in your answer. 11. What number was divided out of the numerator in Question 9 to get 1? 12. What number was divided out of the denominator in Question 9 to get 3? 13. What is the common factor that was divided out of the fraction in Question 9 to simply the fraction down to

!d ?

Help yourself to some fish crackers while you come up for air to answer the following questions. Write the answers on your own paper, and use complete sentences in your answers. 14. How are equivalent fractions related to simplifying fractions? 15. How do you find the largest common factor to divide out of the numerator and the denominator? © Mark Twain Media, Inc., Publishers

12

Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Simplifying Fractions (cont.) Let’s EXERCISE! Finding the Greatest Common Factor Find the greatest common factor between the two given numbers. 1. 4 and 12



2. 2 and 18

3. 10 and 25



4. 9 and 21

Simplifying Fractions Find the fraction that represents the shaded region in the pictures. Write the fraction in simplified form. 5.



6.

7.



8.

Extend Your Knowledge! 9. How do you know when a fraction is as simplified as possible? Use complete sentences in your answer. © Mark Twain Media, Inc., Publishers

13

Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Simplifying Fractions (cont.) Show Your EXPERTISE! Simplifying Fractions You are going to have a party where you serve rainbow fish crackers and pretzel fish crackers as snacks. There are going to be big bowls of fish crackers to share and gift bags filled with fish crackers for your guests to take with them at the end of the party. One of the bowls is going to have a mixture of fish crackers. There are going to be 15 pretzel, 5 red, 8 purple, 10 orange, 5 yellow, and 7 green fish crackers in the bowl. Use the numbers of fish crackers to write the fractions in simplest form.

1. What fraction of the fish crackers in the bowl are pretzel? 2. What fraction of the fish crackers in the bowl are red? 3. What fraction of the fish crackers in the bowl are purple? 4. What fraction of the fish crackers in the bowl are orange? 5. What fraction of the fish crackers in the bowl are yellow? 6. What fraction of the fish crackers in the bowl are green? The gift bags for your guests are going to have a mixture of fish crackers as well. There are going to be 5 pretzel, 2 red, 2 purple, 3 yellow, and 8 green fish crackers in each bag. Use the number of fish crackers to write the fractions in simplest form. 7. What fraction of the fish crackers in the bag are pretzel? 8. What fraction of the fish crackers in the bag are red? 9. What fraction of the fish crackers in the bag are purple? 10. What fraction of the fish crackers in the bag are yellow? 11. What fraction of the fish crackers in the bag are green?  

© Mark Twain Media, Inc., Publishers

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Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Comparing and Ordering Fractions Let’s EXPLORE! Materials: q fruity cereal pieces: 2 yellow, 10 blue, 5 orange, 3 green Vocabulary: ƒ descending order ƒ ascending order ƒ common denominator Descending order is arranged from biggest to smallest. Ascending order is arranged from smallest to biggest. A common denominator is a multiple of all the denominators. Let’s pick through the exploration to learn about simplifying fractions! 1. How many pieces of each color do you have?

Yellow



Blue

Orange



Green

2. How many total pieces do you have? 3. Use your answers from Questions 1 and 2 to write fractions that represent how many pieces of each color there are in your group. The second fraction should be simplified. total number total number of yellow pieces of blue pieces = = total number total number of pieces of pieces

=

=

total number total number of orange pieces of green pieces = = total number total number of pieces of pieces

=

=

© Mark Twain Media, Inc., Publishers

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Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Comparing and Ordering Fractions (cont.) 4. Which color is most plentiful, and what two equivalent fractions represent that color? 5. Which color is least plentiful, and what two equivalent fractions represent that color? 6. Write two inequalities that compare the fractions of the most plentiful color and the least plentiful color. Use < or > in your answers. 7. Which set of fractions is easier to compare, the original or simplified fractions? Use complete sentences to explain why in your answer. 8. Using the information you know about common denominators, write the simplified fractions of pieces in ascending order. Pick some pieces of fruit cereal for yourself while you answer the following questions. 9. Why is it important to have common denominators when comparing or ordering fractions? Use complete sentences in your answer. 10. How can you get common denominators in fractions when you compare or order them? Use complete sentences in your answer.

© Mark Twain Media, Inc., Publishers

16

Exploring Fractions

Unit 1: The Basics

Name:

Date:

Unit 1: The Basics—Comparing and Ordering Fractions (cont.) Let’s EXERCISE! Comparing Fractions Write the fraction that represents each picture, and then write the fractions with common denominators. Use >, , sW;; !s > aQ; 3. yellow:

7. The original fractions are easier to compare because when the denominators are the same, you only have to compare the numerators. 8.

aQ;, sE;, !f, !s



9–10. Answers may vary.

© Mark Twain Media, Inc., Publishers

83

Exploring Fractions

Answer Keys

Let’s Exercise! (page 17)

#h @d #h < $h

1. ; ; 7. > 12.



8. >

%k #f %k < ^k 9. aYs, aRs, aQs 2. ; ;

3. <

4. >

!d aTk, @l

5. <

6. =

!j $j ^j

10. ,

11. , ,

sTf, #k, %h

13. Multiply each fraction’s numerator and denominator by the denominator of the other fraction. Show Your Expertise! (page 18) 1. green 2. red 3. no 4. orange 6. blue 7. green 8. blue, orange, brown, green

5. purple

Unit 2: Adding and Subtracting Fractions Adding and Subtracting Fractions With Like Denominators Let’s Explore! (pages 20–22) 1. See student’s work. 2. 28 3. Addition

tQ Up 5. Qt Qp 6. Qt Up + Qt Qp = Wt Ip 10. gO; 11. gU; 12. gO; – gU; = gW; 4.

wQ Rt 13. sQg 7.

8. 2

9. Subtraction

14. The numerators changed. The numerators were added or subtracted. 15. The denominators stayed the same. The denominators represented the total, so they did not need to change. Let’s Exercise! (page 23)

!f + @f = #f 6. &l – $l = #l = !d 1.

#g + @g = %g = 1 3. #g 7. %h – $h = !h 8. #g 2.

%k 9. aRa 4.

wQ Eq 10. aQj 11. a negative one 5.

Show Your Expertise! (pages 24–25)

dUh 2. Qe Py 3. dIh 7. dOh + dUh = Qe Yy = $l 10. dOh – dUh = dWh = aQk 1.

© Mark Twain Media, Inc., Publishers

dOh 5. dWh 8. dUh + Qe Py = Qe Uy 11. Qe Py – dOh = dQh

4.

84

eQ Py + dIh = Qe Iy = !s 9. dIh – dWh = dYh = !h

6.

Exploring Fractions

Answer Keys

Finding Common Multiples Let’s Explore! (pages 26–27) 1. Days 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 2. Days 5, 10, 15, 20 3. Days 10 and 20 4. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 5. The multiples of 2 are the same as the days the map quiz is given. 6. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 7. The first four multiples of 5 are the same as the days the state capital quiz is given. 8. 10, 20 9. 10 10. The least common multiple of 2 and 5 is the same as the first day that both quizzes were given. The second common multiple is the second day both quizzes were given. 11. Yes; Answers may vary. Let’s Exercise! (page 28) 1. 3, 6, 9, 12, 15, 18, 21, 24 3. 10, 20, 30, 40, 50, 60, 70, 80 5. 2, 4, 6, 8, 10, 12, 14, 16 7. 12 8. 70 9. 63 10. 90 11. 28 13. Answers may vary but could be 40 and 80. 14. Answers may vary but could be 33 and 66. 15. The other number; Answers may vary. Show Your Expertise! (page 29) 1. 6, 12, 18, 24, 30 2. 6 5. 6, 12, 18, 24, 30 6. 6

2. 4, 8, 12, 16, 20, 24, 28, 32 4. 12, 24, 36, 48, 60, 72, 84, 96 6. 9, 18, 27, 36, 45, 54, 63, 72 12. 100

3. 6, 12, 18, 24, 30 7. snake 8. 30

4. 6

Adding and Subtracting Fractions With Unlike Denominators Let’s Explore! (pages 30–32) 1. See student’s work. 2. 2 3. 7 4. Addition 5. 36 6. 3

aWs x #d = dYh 8. 2 12. Qe Ry – dYh = dIh = @l 7.

9.

aUk x @s = Qe Ry

10.

dYh + Qe Ry = We Py



11.

%l



Let’s Exercise! (page 33)

@f + @k; $k + @k = ^k; #f 4. sRf + sYf = Qw Pr ; aTs 7. aOf – aYf = aEf 1.

#h + !d; #h + @h = %h 5. %l – !d; %l – #l = @l 8. Qq Pw – aOs = aQs 2.

aRk + aTk = aOk; !s 6. $h – Qw Qr ; Qw Yr – Qw Qr = sTf 3.

9. The least common denominator would be the product of the denominators.

© Mark Twain Media, Inc., Publishers

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Exploring Fractions

Answer Keys

Show Your Expertise! (pages 34–35) 1–4. See student’s work; Answers will vary. Unit 3: Mixed Numbers Adding and Subtracting Mixed Numbers With Like Denominators Let’s EXPLORE! (pages 37–40)

!g 3. 2 !g 5. @g 10. 4 #g 11. 2 !g + 2 @g = 4 #g 18. 2 $g – 1 #g = 1 !g 2.

6. 2

@g

8. 4

$g

14. 2

13.

9. 3

$g

!g

17. 1

Let’s EXERCISE! (page 41)

#k + 3 $k = 5 &k 5. 6 !s 6. 7 @d 11. 8 @g 12. 15 !s 1. 2

%h – 1 $h = 1 !h 3. 4 #g 7. 6 @d 8. 3 aQ; 9. 1 @d 13. 1 !j 14. 1 !l 2. 2

!g 10. 5 !j

4. 5

Show Your EXPERTISE! (page 42) 1–2.

3. 1

Answers will vary.

#f inches

4. 1

!k inches

Adding and Subtracting Mixed Numbers With Unlike Denominators Let’s EXPLORE! (pages 43–47) 1–4. 9. 2

See student’s work.

!f + 1 !a!h = 3 !a%h

5. 3

6.

aRh

7.

!a%h

8. 3

10. See student’s work. Two whole acres of pumpkins and two whole acres of corn. 11. 0

12.

@k

13.

#k

14.

#k

15. 2

%k – 2 !f = #k

16–17. Answers will vary. Let’s EXERCISE! (page 48)

!f + 3 !h = 5 aTs 5. 4 dU; 6. 12 Ww Pq 11. 7 Qw Er 12. 7 Ry Qp 1. 2



© Mark Twain Media, Inc., Publishers

$g – 1 !d = 1 aUg 3. 6 aO; 7. 3 Qe Oy 8. 2 dU; 9. 3 aTs 13. 3 Qw Er 14. 9 Wr Ei 2. 2

86

!k 10. 5 aQk

4. 2

!a%h

Exploring Fractions

Answer Keys

Show Your EXPERTISE! (page 49) 1. Pumpkin 3 and Pumpkin 4; 18 3. 1

fU;

sO;

2. Pumpkin 1 and Pumpkin 5; 20

Qw Ur

4. He forgot to find a common denominator before subtracting.

Unit 4: Mixed Numbers and Improper Fractions Mixed Numbers and Improper Fractions Let’s EXPLORE! (pages 51–53) 3. 2

4. 8 divided by 4 is equal to 2.

7. 2

@f 10. 2 @f 8.

9. 10 divided by 4 is equal to 2 with a remainder of 2. 11. Divide the numerator by the denominator to get the whole number part of the mixed number. Write the remainder over the denominator to get the fractional part of the mixed number. 13. 6 14. 15. 2 times 3 is equal to 6. 17. 8

^d

*d

18. 19. 2 times 3 plus 2 is equal to 8. 20. multiplication 21. Multiply the denominator by the whole number and then add the numerator. Write the total over the denominator. Let’s EXERCISE! (page 54)

&k 6. 6 @d 11. AtA 16. DiG 1. 1

!d 7. 5 #f 12. SuH 17. Qq Up 2. 3

$j 8. 8 !s 13. AoJ 18. Wq Ep 3. 3

%l 9. 9 !s 14. AyJ 19. AwA 4. 3

Show Your EXPERTISE! (page 55) 1. Amount Ingredient

#f cups 4 !s teaspoons 1 !k teaspoons s! teaspoon s! cup 3

all-purpose flour baking powder salt baking soda shortening

© Mark Twain Media, Inc., Publishers

87

#g 10. 4 aE; 15. ArD 20. StD 5. 3

Improper Fraction

ArG cups (s teaspoons k( teaspoons s! teaspoon s! cup

Exploring Fractions

Answer Keys

!f cups shredded cheddar cheese (f cups 1 #f cups buttermilk &f cups 2. 1 !d cups; Answers should include that you divided the numerator by the denominator to

2

get the whole number and then wrote the remainder over 3 to get the fractional part of the mixed number.

Adding Mixed Numbers With Regrouping Let’s EXPLORE! (pages 56–57) 3. 4

!d

$d

4. 1

5. 4

6. 3

9. They are the same.

12. 4

13. 1

!f

16. 1

!f

17. 4

!d 14. 4 !f 7. 1

!d 15. 3 %f 8. 4

18. They are the same.

19. Use regrouping when the sum contains an improper fraction.

Ai:. Use regrouping to rewrite Ai: as 1 @k. Then add this to 6 to get a sum of 7 @k, which can be simplified to 7 !f.

20. When you add, the sum is 6

Let’s EXERCISE! (page 58)

@j 6. 9 #g 11. 6 !f 16. 14 aUg 1. 7

@g 7. 10 #g 12. 15 !s 17. 20 !s

!s 8. 7 !g 13. 10 sTf 18. 13 Ey Qp

2. 5

3. 8

!d 9. 10 !s 14. 12 dTh 19. 16 !a!k 4. 11

Show Your EXPERTISE! (page 59) Ingredient Amount Water

10 cups

Honey

9

Salt

!d tablespoons or 28 teaspoons 10 !d teaspoons

Bread flour

25 cups

Active dry yeast

18

Margarine

10 tablespoons

Dry milk powder

8

aOf teaspoons

!s tablespoons

© Mark Twain Media, Inc., Publishers

88

@d 10. 13 !f 15. 8 aTf 20. 13 Qw Qq 5. 10

Exploring Fractions

Answer Keys

Subtracting Mixed Numbers With Regrouping Let’s EXPLORE! (pages 60–62) 3. 1

4. 2

5. 1

@f

6. Answers will vary.

9. 1

@d 12. Answers will vary. 13. 4 14. 4 15. %f 16. 3 !f = 2 %f 17. 3 !f – 1 #f = 2 %f – 1 #f = 1 @f 18. 3 19. 3 20. $d 21. 4 !d = 3 $d 22. 4 !d – 2 @d = 3 $d – 2 @d = 1 @d 10. 2

11. 1

Let’s EXERCISE! (page 63)

(k 6. 4 $j 11. !s 16. 4 Qq Qw 1. 5

AoA 7. 3 @g 12. 6 @g 17. 4 sUf

&h 8. 7 !d 13. 10 &k 18. 4 !s

2. 6

3. 1

Au: 9. 1 $l 14. 7 &l 19. 4 @s)a 4. 8

#f 10. 1 !s 15. 1 aTs 20. !a!s 5. 1

Show Your EXPERTISE! (page 64) 1. 2

&k – !s = 2 #k cups

2. 1

@d – !s = 1 !h cups

3. 2

&k – %h = 2 sQf cups

4. Answers will vary but should include the fact that he forgot to find a common denominator. Unit 5: Multiplying and Dividing Mixed Numbers and Fractions Multiplying Fractions Let’s Explore! (pages 66–67) 1. A. 2. A.

%l x @g ^j x &k

B. B.

%l x @g = Qr Pt ^j x &k = Rt Wy

C. C.

Qr Pt = @l Rt Wy = #f



3. The product of two fractions can be another fraction, a mixed number, or a whole number.

© Mark Twain Media, Inc., Publishers

89

Exploring Fractions

Answer Keys

Let’s Exercise! (page 68)

$h x aIs = Eu Ww = $l 5. Qe Tw 9. lI; = fRg 1.

!f x aI; = fI; = !g 6. Wr To 10. fUk 2.

q Yt r = jEj 7. aQkY; = fRg 11. fUgP; = fUg 3.

dYl = aWd 8. sEaP; = !j 12. Qt Yq 4.

13. 1 Show Your Expertise! (page 69)

@l x &l = Qi Rq yard 3. #f x aUs = Wr Qi = aUh yard 1.

%j x %k = Wt Ty yard 4. *l x $g = Er Wt yard 2.

Dividing Fractions Let’s Explore! (pages 70–71)

$g !h

$g ^a StF

StF $g

^j !l

^j (a GuF

GuF %j

1. A. ÷ B. x = C. =4 D. See student’s work. Four towels should be cut.

2. A. ÷ B. x = C. =7 D. See student’s work. Seven ribbons should be cut. 3. Yes; Answers may vary.

Let’s Exercise! (page 72)

^k ÷ !f = ^k x $a = WkR = 3 3. aE; x %a = Qq Tp = 1 !s 5. $l x Qq Ew = aT;Wk = Qw Eu 7. Qq Et x %s = Ye Tp = 2 !h 9. %h x $d = Wq Pi = 1 !l 1.

$g ÷ Qq Qy = $g x Qq Yq = Yt Rt = 1 gOg 4. @j x AtF = We It = $g 6. aIa x #s = Ww Rw = 1 aQa 8. &k x *g = Tr Yp = 1 @g 10. aY; x @a = Qq Wp = 1 !g 2.

11. 1 Show Your Expertise! (page 73)

#f ÷ #k = #f x *d = Wq Rw = 2 3. *l ÷ !l = *l x (a = JoS = 8 1.

© Mark Twain Media, Inc., Publishers

@d ÷ !h = @d x ^a = AeS = 4 4. ^j ÷ @j = ^j x &s = Rq Wr = 3 2.

90

Exploring Fractions

Answer Keys

Multiplying Mixed Numbers Let’s Explore! (pages 74–75) 1. A. 15

!s x 2 #f

B. 2 x 15 + 1 = 31;

D. 341 ÷ 8 = 42 r 5

E. 42

DwA; 4 x 2 + 3 = 11; ArA C. DwA x ArA = E Ri Q

%k

2. Answers may vary. Let’s Exercise! (page 76)

AyJ = 2 %h 4. Du: x AwL = GqJr:; 40 %j 7. LiG x AoG = Q Ru Ww T = 19 !s(f 1.

ArD = 3 !f 5. SeG x ArL = FqJwG; 39 aUs 8. FyJ x SiL = Q Er Yi E = 28 Qr Oi 2.

*g x AeA = Iq It ; 5 Qq Et 6. AuK x AuL = DrFoS; 6 Rr Io 3.

9. The whole number needs to be made into a fraction by making it have a denominator of 1. Show Your Expertise! (page 77)

@g x 15 !s = AtJ x DwA = GqSpJ = 52 aU; feet 2. 1 !s x 15 !s = #s x DwA = LrD = 23 !f feet 3. 2 @d x 15 !s = *d x DwA = W Ry I = 41 @h = 41 !d feet 4. 117 Qy Up feet 1. 3

Dividing Mixed Numbers Let’s Explore! (pages 78–79)

!s ÷ 3 !f B. FwG x aRd = AwKy:

1. A. 22

C.

FwG; 4 x 3 + 1 = 13; ArD 180 ÷ 26 = 6 r 24 E. 6 Ww Ry = 6 Qq We

2 x 22 + 1 = 45; D.

2. Answers may vary.

© Mark Twain Media, Inc., Publishers

91

Exploring Fractions

Answer Keys

Let’s Exercise! (page 80)

DuJ; dUj 2. Eq Op ; Qe Po 5. AeJ ÷ &h; AeJ x ^j = Aw:qS = 4 ^j 7. DtH ÷ ArD; DtH x aRd = AyFtF = 2 Qy Rt 1.

LoG; lOg 4. JiA; jIa 6. GyD ÷ AoL; GyD x aOl = Rq Uq Ur = 4 dUk 8. AiG ÷ #s; AiG x @d = Ew Pr = 1 !f 3.



9. The mixed number should be changed into an improper fraction. Then invert the fraction and multiply. 10. No Show Your Expertise! (page 81)

#f ÷ 4 #f = Q Er O ÷ ArL; Q Er O x aRl = GuGyH = 7 aYl ; 7 boards 2. 12 !s ÷ 3 @d = SwG ÷ AeA; SwG x aEa = Uw Tw = 3 sOs ; 3 boards 3. 47 !f ÷ 3 @d = Q Ir O ÷ AeA; Q Ir O x aEa = GrHrJ = 12 Er Or ; 12 boards 1. 34

© Mark Twain Media, Inc., Publishers

92

Look for these Mark Twain Media books for grades 4–8+ at your local teacher bookstore.

* * * * * *

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SCIENCE

Rocks, Minerals, & Fossils Meteorology Earthquakes & Volcanoes Science Experiments: Chem./Physics Science Experiments: Earth Science Science Experiments: Biology/Ecology The Atom Physical Science Resourceful Rain Forest Microorganisms Science Fair Projects Your Body and How It Works Elements and the Periodic Table Learning About Invertebrates Simple Machines Chemistry Atmosphere and Weather Rocks and Minerals Electricity and Magnetism Learning About DNA Light and Color Sound Learning About Atoms The Solar System Learning About Vertebrates Learning About Our Solar System Jumpstarters for Science Science Tutor: Chemistry Science Tutor: Life Science Science Tutor: Physical Science Science Tutor: Earth & Space Science Easy Science Experiments: Weather Easy Science Experiments: The Earth's Surface Easy Science Experiments: Water, Airplanes, … Learning About Cells Amazing Facts About Mammals Amazing Facts in Science Discovering Ecology Life Science Quest for Middle Grades Jumpstarters for Life Science Jumpstarters for Meteorology Strengthening Physical Science Skills

SOCIAL STUDIES

Civil War: The War Between the States Greek and Roman Mythology Medieval Times: 325-1453 Understanding the U.S. Constitution Explorers of the New World World War II The Industrial Revolution Egypt and the Middle East Democracy, Law, and Justice Seven Wonders of the World and More Economics and You Mayan, Incan, and Aztec Civilizations The American Revolution Greek and Roman Civilizations Holocaust South America Renaissance Elections Africa Basic Economics Mexico Personal Finance U.S. History Maps U.S. Constitution: Preparing for the Test 50 U.S. States and Territories World Civilizations and Cultures Amazing Facts in U.S. History U.S. Presidents: Past and Present Constitutional Puzzlers Discovering and Exploring the Americas Life in the Colonies The American Revolution The California Gold Rush The Lewis and Clark Expedition The Oregon and Santa Fe Trails The Westward Movement

CD-1529 CD-1530 CD-1531 CD-1532 CD-1533 CD-1550 CD-1556 CD-1560 CD-1561 CD-1562 CD-1563 CD-1572 CD-1584 CD-404026 CD-404031 CD-404032 CD-404033 CD-404036 CD-404037 CD-404038 CD-404039 CD-404040 * CD-404080

Abraham Lincoln and His Times Industrialization in America Slavery in the United States The American Civil War The Reconstruction Era We the People: Government in America The RoaringTwenties/Great Depression Coming to America: Immigration America in the 1960s and 1970s America in the 1980s and 1990s World War II and the Post-War Years Understanding Investment/Stock Market Amazing Facts in World History Jumpstarters for U.S. History Jumpstarters for the U.S. Constitution Wonders: A Journey Around the World Mysteries: A Journey Around the World U.S. History: People Who Helped Make the Republic Great: 1620–Present U.S. History: Inventors, Scientists, Artists, & Authors U.S. History: People and Events in African-American History U.S. History: People and Events: 1607–1865 U.S. History: People and Events: 1865–Present Jumpstarters for World History

GEOGRAPHY

CD-1314 Map Reading, Latitude, Longitude, … CD-1551 World Geography CD-1555 Exploring Asia CD-1556 Exploring Africa CD-1566 Exploring Europe CD-1567 Exploring South America CD-1569 Exploring North America CD-1570 Exploring Antarctica CD-1571 Exploring Australia CD-1573–CD-1576 Discovering the World of Geography: Grades 4–8 CD-404060 Jumpstarters for Geography * CD-404095 Daily Skill Builders: World Geography

LANGUAGE ARTS

CD-1850 How to Prepare and Give a Speech CD-1857 Grammar and Composition CD-1862 Writing to Inform and Persuade CD-1300 Phonics for Middle-Grade Students CD-1366 Debate Skills CD-1375 Reading Comprehension: Grade 7 CD-1381 Confusing Words CD-1382 Synonyms and Antonyms CD-1383 Using a Dictionary CD-1386 Challenges Galore:Vocabulary Building CD-1393 Proofreading CD-1394 Story Writing CD-1399 Poetry Writing CD-1543–CD-1546, CD-1553 Writing Engagement: Grades 4–8 CD-1547 Essential Words CD-1549 Report and Term Paper Writing CD-1554 English Warm-ups CD-1590–CD-1595 Student Booster Writing series CD-1620–CD-1624 Reading Tutor series CD-404008 Diagraming Sentences CD-404011 Jumpstarters for Grammar CD-404012 L.A. Tutor: Capitalization/Punctuation CD-404013 Language Arts Tutor: Grammar CD-404015–CD-404019 Reading Engagement: Grades 3–8 CD-404027 Jumpstarters for Writing CD-404035 Lessons in Writing CD-404051 Writing a Persuasive Essay CD-404053 Jumpstarters for Language Arts CD-404054 Jumpstarters for Vocabulary Building CD-404055 Adventures in Writing CD-404061–CD-404063 Daily Skill Builders: Grammar: Grades 3–6 CD-404064–CD-404066 Daily Skill Builders: Spelling & Phonics: Grades 3–6 CD-404067–CD-404069 Daily Skill Builders: Vocabulary: Grades 3–6

CD-404070–CD-404072 Daily Skill Builders: Reading: Grades 3–6 * CD-404073 Jumpstarters for Figurative Language * CD-404078 Jumpstarters for Capitalization & Punctuation * CD-404081 Jumpstarters for Root Words, Prefixes, & Suffixes

STUDY SKILLS

CD-1859 Improving Study & Test-Taking Skills CD-1898 Listening Skills CD-1321 Library Skills CD-1597 Note Taking: Lessons to Improve Research Skills & Test Scores CD-1625–CD-1630 Preparing Students for Standardized Testing: Grades 3–8

MATH

CD-1874 Algebra CD-1876 Pre-Algebra CD-1879 Statistics and Probability CD-1310 Understanding Graphs & Charts CD-1325 Pre-Calculus CD-1329 Word Problems CD-1331 Applying Pre-Algebra CD-1332 Basic Geometry CD-1333 Fractions, Decimals, and Percentages CD-1388 Math-O Games CD-1540–CD-1542 Math Twisters: Gr. 5–7 CD-1578–CD-1582 Math Engagement: Gr. 4–8 CD-1589 Math Projects CD-1615–CD-1619 Math Tutor series CD-404000–404004 Math Games: Gr. 4–8 CD-404009 Math Challenges CD-404020 Helping Students Understand Algebra CD-404021 Helping Sts. Understand Pre-Algebra CD-404022 Jumpstarters for Algebra CD-404023 Jumpstarters for Math CD-404028 Helping Students Understand Algebra II CD-404029 Helping Students Understand Geometry CD-404030 Jumpstarters for Pre-Algebra CD-404041 Pre-Algebra Practice CD-404042 Algebra Practice CD-404043 Algebra II Practice CD-404044 Geometry Practice CD-404057 Jumpstarters for Fractions & Decimals CD-404058 Jumpstarters for Geometry CD-404059 Jumpstarters for Math Word Problems * CD-404074 Math Logic * CD-404083 Daily Skill Builders: Algebra * CD-404084 Daily Skill Builders: Division * CD-404085 Daily Skill Builders: Fractions & Decimals * CD-404086 Daily Skill Builders: Pre-Algebra * CD-404087 Daily Skill Builders: Word Problems * CD-404088 Exploring Fractions * CD-404089 Math Reference for Middle Grades

FINE ARTS

Music: a.d. 450–1995 Great Artists and Musicians American Popular Music Theater Through the Ages Music of Many Cultures Musical Instruments of the World Everyday Art for the Classroom

CD-1890 CD-1891 CD-1892 CD-1893 CD-1894 CD-1596 CD-1632

HEALTH & WELL-BEING

CD-1895 CD-1896 CD-1897 CD-1819 CD-1339 * CD-404079 * CD-404090

Life Skills Self-Management: Promoting Success Promoting Positive Values Health, Wellness, and Physical Fitness Developing Life Skills Jumpstarters for the Human Body Healthy Eating and Exercise

CROSS-CURRICULUM

CD-1306 CD-404010 CD-404056 * CD-404075

Career Search Professional Teacher Calendar Using Primary Sources Research *Denotes New Release