Fraction Division
Objective To introduce division of fractions and relate the operation of division to multiplication. o
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Teaching the Lesson
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Common Core State Standards
Ongoing Learning & Practice
Key Concepts and Skills
Math Boxes 8 12
• Find common denominators for pairs of fractions. [Number and Numeration Goal 5]
Math Journal 2, p. 290 Students practice and maintain skills through Math Box problems.
• Use diagrams and visual models for division of fractions problems. [Operations and Computation Goal 5]
• Solve number stories involving division of a fraction by a whole number, division of a whole number by a fraction, and division of a fraction by a fraction.
Study Link 8 12
Math Masters, p. 248 Students practice and maintain skills through Study Link activities.
[Operations and Computation Goal 5]
• Write equations to model number stories. [Patterns, Functions, and Algebra Goal 2]
Key Activities Students use diagrams and visual models to divide fractions. They solve number stories involving division of a fraction by a whole number, division of a whole number by a fraction, and division of a fraction by a fraction. Students use visual fraction models and equations to represent the problem.
Curriculum Focal Points
Interactive Teacher’s Lesson Guide
Differentiation Options READINESS
Playing Build-It Student Reference Book, p. 300 Math Masters, pp. 446 and 447 per partnership: 1 six-sided die Students compare and order fractions and rename mixed numbers as fractions. ENRICHMENT
Exploring the Meaning of the Reciprocal Math Masters, p. 249 calculator Students explore the meaning of the reciprocal. EXTRA PRACTICE
Dividing with Unit Fractions Math Masters, p. 253B Students practice using visual models to divide fractions.
Ongoing Assessment: Informing Instruction See page 683.
Ongoing Assessment: Recognizing Student Achievement Use journal page 289. [Operations and Computation Goal 5]
Materials Math Journal 2, pp. 288–289B transparency of Math Masters, p. 440B Student Reference Book, pp. 79–80B Study Link 8 11 slate or half-sheets of paper
Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 144–147
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Getting Started Mental Math and Reflexes
Math Message
Pose questions about unit fractions. Suggestions: 1 1 1 How many _2 s are in 1 whole? 2 How many _2 s are in _2 ? 1 1 1 1 How many _2 s are in 4? 8 How many _4 s are in _2 ? 2 1 1 1 How many _2 s are in 2_2 ? 5 How many _ s are in 1? 3 3 1 _ How many s are in 2? 8
Solve Problems 1 and 2 on journal page 288.
4
1 1 How many _4 s are in 1_ ?6 2 3 1 _ _ How many s are in ? 3 4
Study Link 8 11 Follow-Up
Have partners share answers and resolve differences. Ask volunteers to explain their solution strategies for Problem 9.
4
1 Teaching the Lesson WHOLE-CLASS DISCUSSION
▶ Math Message Follow-Up (Math Journal 2, p. 288: Math Masters, p. 248)
Discuss students’ solutions. Use a transparency of Math Masters, page 440B to illustrate Problems 1a–1c. Problem 1a
0 inches
1
2
3
4
5
6
2
3
4
5
6
Problem 1b
Student Page 0 inches
1
Date
Time
LESSON
8 12
Fraction Division
Math Message
Problem 1c
1.
Use the ruler to solve Problems 1a –1c.
0 inches
0 inches
1
2
3
4
5
6
1
2
3
1. a. How many 2s are in 6? 1 s are in 6? b. How many _ 2
3? 1 s are in _ c. How many _ 8 4
2. a. How many 2s are in 10? 1 s are in 10? b. How many _
5
6
3 segments 12 segments _ _ of an inch? 6 segments c. How many -inch segments are in a.
How many 2-inch segments are in 6 inches?
b.
1 How many _ 2 -inch segments are in 6 inches? 3 4
1 8
Point out that each problem on the journal page asks: How many x’s are in y? Ask students to translate each problem into a question of this form. Record the questions on the board.
4
2. a.
How many 2-pound boxes of nuts can be made from 10 pounds of nuts?
5 boxes
Use the visual below to help you solve the problem.
Sample answer: 2 3 4
1 1 lb b.
1 lb
1 lb
1 lb
1 lb
1 lb
1 lb
1 lb
5 1 lb
1 lb
1 How many _ 2 - pound boxes can be made from 10 pounds of nuts?
20 boxes
Draw a picture to support your answer.
1 lb 1 2
1 2
Sample answer: 1 lb 1 lb 1 lb 1 2
1 2
1 2
1 2
1 2
1 2
1 lb 1 2
1 2
1 lb
1 lb
1 lb
1 lb
1 lb
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
2
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Refer students to the illustrations and questions for Problems 1 and 2, and ask what division open number sentence fits the first question. 6 ÷ 2 = s Continue for the other problems, writing the division open number sentence next to each question on the board. Ask students to refer to the visual models, if needed. 1. a. How many 2s are in 6? 6 ÷ 2 = s 1 s are in 6? 6 ÷ _ 1 =s b. How many _ 2
2
3? _ 3 ÷_ 1 s are in _ 1 =s c. How many _ 8 4 4 8
2. a. How many 2s are in 10? 10 ÷ 2 = b 1 s are in 10? 10 ÷ _ 1 =b b. How many _ 2
2
NOTE The division number models use b (for the number of boxes) and s (for the number of segments) to represent the unknowns. Students may prefer to use other letters or symbols. To avoid confusion in this introduction to division of fractions, the number models use ÷ rather than / to show division.
▶ Dividing with Unit Fractions
WHOLE-CLASS DISCUSSION
(Math Journal 2, p. 289; Student Reference Book, pp. 79 and 80A)
Read and discuss the first example on page 79 of the Student Reference Book on dividing a whole number by a unit fraction. A unit fraction is a fraction with a numerator of 1. Briefly discuss the solution. Draw 3 rectangles on the board, and ask students to copy the rectangles on a sheet of paper or slate.
Ask students to use the rectangles to illustrate the following problem: Jane has 3 loaves of banana bread to share with her 1 s, how many quarter loaves friends. If she cuts each loaf into _ 4 will she have to share with her friends?
Student Page Date
Time
LESSON
Dividing with Unit Fractions
8 12
1.
Four pizzas will each be sliced into thirds. Use the circles to show how the pizzas will be cut. Find how many slices there will be in all. The drawings show that there will be 1 = 4÷_ 3
2.
slices in all.
Edith has 2 inches of ribbon. She wants to cut the ribbon 1 1 _ into _ 4 -inch pieces. How many 4 pieces can she cut?
8
Edith can cut
3.
12
Students should conclude that one way to illustrate the solution is to divide each of the rectangles into 4 equal parts.
12
1 = pieces. So, 2 ÷ _ 4
8
0
inches
1
2
.
1 1 1 1 4 4 4 4
1 Two students equally share _ 3 of a granola bar. Divide the rectangle at the right to show how much of the bar each will get.
1 1 1 1 4 4 4 4
1 1 1 1 4 4 4 4
1 _
Each student will get 1 ÷ = _ 2 3
4.
1 _
6
of a granola bar.
1 3÷_ 4 = 12
6
Ask:
1 Three students equally share _ 4 of a granola bar. Divide the rectangle at the right to show how much of the bar each will get.
1 _
Each student will get 1 ÷ = _ 3 4
5.
1 _
12
●
of a granola bar.
12
What number model represents this problem? 3 ÷ _14 = 12 1 = 12 below their rectangles. Ask students to write 3 ÷ _ ●
When you divide a whole number by a unit fraction (less than 1), is the quotient larger or smaller than the whole number? Explain.
Sample answer: The quotient is larger than the whole number because you are finding how many small parts fit into something that is larger. 6.
1 s are in 3? 12 How many _ 4
4
When you divide a unit fraction (less than 1) by a whole number, is the quotient larger or smaller than the fraction? Explain.
Sample answer: The quotient is smaller than the fraction because you are dividing up the fraction into an equal number of smaller parts.
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A unit fraction can also be divided by a whole number. Read and discuss the examples on page 80A of the Student Reference Book. Draw a rectangle on the board, and divide it into 5 equal parts with _15 shaded. Ask students to draw the same diagram on a piece of paper or slate.
Tell students that you can represent a unit fraction (such as _15 ) being divided by a whole number (such as 3) by drawing a model for the fraction and then cutting it up into smaller equal parts. 1 Pose the following problem: Three family members equally share _ 5 of a loaf of corn bread. How much of the loaf of corn bread will each person get? Have students divide their rectangles to show how the corn bread can be divided to find the solution to the problem.
Ongoing Assessment: Informing Instruction 1 Watch for students who record the answer as _3 . Have students draw the line for thirds to extend across the rectangle in order to visualize the total number of parts out of 15.
Have a volunteer come to the board to show the solution. The student should divide the shaded fifth into three equal parts using horizontal lines. If necessary, model the lines extended all the way across the larger rectangle, with one small part double shaded. Explain that because the family only has _15 of a loaf to begin with, when it is divided into three equal parts, each part of the corn 1 bread that is cut up is _ of the entire loaf. So each person will 15 1 _ get 15 of the loaf of corn bread.
1 5
÷3=
1 15
1 ÷3=_ 1 Ask: What number model represents this problem? _ 5 15 1 ÷3=_ 1 ” below their rectangles. Ask students to write “_ 5
15
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Student Page Date
Relationship Between Multiplication and Division
8 12
1.
Each division number sentence on the left can be solved by using a related multiplication sentence on the right. Draw a line to connect each number sentence on the left with its related number sentence on the right.
a.
Division
Multiplication
1 = 5÷_ n 4
1 = _ n ∗ 4_ 4 51 5
1 _ b. 5
÷4=n
1 n∗4=_ 5
1 _ c. 4 ÷ 5 = n
1 = n∗_ 5 4
9 _ d. 10
9 3 =_ n∗_ 5 10
3 _ e. 5
f. 2.
3 ÷_ =n 5
9 ÷_ =n 10
Have students solve Problems 1–6. Circulate and assist. Briefly discuss solutions.
1 = n∗_ 4 5
1 ÷ _ 4_ 4 15 = n 5
3 9 =_ n∗_ 5 10
Solve the following division number sentences (from above). Use the related multiplication sentences to help you find each quotient. a.
1 = 5÷_ 4
1 _ c. 4 ÷ 5 =
20 20
1 _ b. 5
Ongoing Assessment: Recognizing Student Achievement
1 _
÷4=
1 1 _ _ d. 4 5 ÷ 4 5 =
20
1
4.
Sample answers: Dividing 1 1 _ How is dividing 5 by _ 4 different from dividing 5 by 4? _1 1 5 by _ 4 results in a quotient that is greater than 5. Dividing 5 by _1 , you are 1 ÷ 4 results in a quotient that is less than _ . With 5 4 _1 5 1 finding how many _ 4 s are in 5. With 5 ÷ 4, you are finding how _ 1 many 4s are in 5 , which is a very tiny number. 1 Answers vary. Write a number story for 5 ÷ _ 4.
5.
1 ÷ Write a number story for _ 4. 5
3.
Have students read through Problems 1–4 on journal page 289. Ask them to describe how Problems 1 and 2 are different from Problems 3 and 4. Sample answer: In Problems 1 and 2, you are dividing a whole number by a unit fraction. In Problems 3 and 4, you are dividing a unit fraction by a whole number.
Time
LESSON
Journal Page 289
Use journal page 289, Problems 1 and 2 to assess students’ ability to divide a whole number by a unit fraction using a visual model. Students are making adequate progress if they are able to solve Problems 1 and 2. Some students may be able to solve Problems 3 and 4, which involve dividing a unit fraction by a whole number. [Operations and Computation Goal 5]
Answers vary.
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▶ Relationship between
WHOLE-CLASS DISCUSSION
Multiplication and Division (Math Journal 2, p. 289A)
Another way to solve a fraction division problem is to think about it as a related fraction multiplication problem. Remind students of the relationship between multiplication and division. For example, to solve 63 ÷ 7, you can think: What number times 7 is 63? or n ∗ 7 = 63. 9 Write the following problems on the board to show how the relationship helps when dividing with fractions. 1 ÷ 5, think: What number times 5 is _ 1 ? Or n ∗ 5 = _ 1. ● To solve _ 10 10 10 1 _ 50
● ● ●
1 , think: What number times _ 1 is 6? Or n ∗ _ 1 = 6. 30 To solve 6 ÷ _ 5 5 5 2 2 2 = 6. 9 _ _ To solve 6 ÷ , think: What number times is 6? Or n ∗ _ 3
3
3
3 , think: What number times _ 1 ÷_ 3 is _ 1? To solve _ 10 10 10 10 1 3 =_ 1._ Or n ∗ _ 10
10 3
Ask students to solve the problems on journal page 289A. Circulate and assist.
▶ Introducing Common
WHOLE-CLASS ACTIVITY
Denominator Division (Math Journal 2, p. 289B)
Draw four circles on the board, and ask students to copy these circles on a sheet of paper. Ask them to solve the problem 4 ÷ _23 and to illustrate their solution using the four circles.
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After a few minutes, bring the class together to discuss their solutions. Use the students’ responses to emphasize that one method for obtaining the solution is to divide each of the circles into 3 equal parts. As you illustrate this method on the board, point out that dividing each circle into 3 equal parts is equivalent 3 . This shows that 4 wholes is to renaming each whole as _ 3 12 . equivalent to _ 3 2 – 3
2 – 3
2 – 3
2 – 3
2 – 3
2 – 3
12 2 _ ÷_=6 3
3
12 ÷ _ 2 = 6 under the circles on the board. Students will Write _ 3 3 2 . Emphasize that the readily see that there are 6 groups of _ 3 answer is the result of dividing the numerators 12 ÷ 2 = 6.
Guide the discussion toward the following algorithm for division with fractions: Step 1 Rename the numbers using a common denominator. Step 2 Divide the numerators, and divide the denominators. Discuss the examples at the top of journal page 289B. Point out that this method works for fractions divided by fractions or for mixed numbers or whole numbers divided by fractions. Use the following example to show that this method also works for fractions divided by whole numbers. 48 1 ÷6=_ 1 ÷_ _ 8
= = =
8 8 1 ÷ 48 _ 8÷8 1 ÷ 48 _ 1 1 _ 48
Solve Problems 1−3 on journal page 289B as a class. Ask students to come up to the board to record their steps.
Student Page Date
Time
LESSON
Common Denominator Division
8 12
One way to divide fractions is to use common denominators. This method can be used for whole or mixed numbers divided by fractions. Step 1 Rename the fractions using a common denominator. Step 2 Divide the numerators, and divide the denominators.
Problem
Solution
4 4÷_ 5 =?
20 4 4 _ _ 4÷_ 5 = 5 ÷ 5 = 20 ÷ 4 = 5
5 1 _ _ 6 ÷ 18 = ?
5 15 1 1 _ _ _ _ 6 ÷ 18 = 18 ÷ 18 = 15 ÷ 1 = 15
5 1 _ 3_ 3 ÷ 6 =?
5 10 5 20 5 1 _ _ _ _ _ 3_ 3 ÷ 6 = 3 ÷ 6 = 6 ÷ 6 = 20 ÷ 5 = 4
Examples: 3 3 12 ÷ _ =_ 3÷_ 4 4 4 12 ÷ 3 =_
=
1.
4 = 4÷_ 5
1 _ 4. 2
1 = ÷_ 8
3 3 18 3 ÷_ =_ ÷_ 3_ 5 5 5 5 18 ÷ 3 =_ 5÷5
6 =_ 1 , or 6
2
8.
1 _ 5. 6
1 = ÷_ 18
15
3.
5 1 ÷_ = 3_ 3 6
6.
1 = 6÷_ 4
1 _ ÷4=
24
4 24
bags
necklaces
Eric is planning a pizza party. He has 5 large pizzas. He wants to cut 3 each pizza so that each serving is _ 5 of a pizza. How many people can get a full serving of pizza?
8
10.
4
5 _ 2. 6
Regina is cutting string to make necklaces. She has 15 feet of string 1 and needs 1_ 2 feet for each necklace. How many necklaces can she make?
10
9.
5
1 Chase is packing flour in _ 2 -pound bags. He has 10 pounds of flour. How many bags can he pack?
20
The algorithm introduced in this lesson focuses students on the meaning of division with fractions. The standard algorithm that involves multiplying by the reciprocal will be introduced in Sixth Grade Everyday Mathematics.
=
6÷6 2 _ 1 , or
Solve.
7.
Links to the Future
1 ÷ _ 1 =_ 2 ÷ _ 1 _ 3 6 6 6 2÷1 =_
4÷4 4 _ 1 , or 4
people
3 3 _ 2 A rectangle has an area of 3 _ 10 m . Its width is 10 m. What is its length?
11
meters
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Student Page Date
Time
LESSON
1.
▶ Practicing Common
Math Boxes
8 12
2.
Find the whole set. a.
1 10 is _ 5 of the set.
b.
3 12 is _ 4 of the set.
50 16
d.
5 15 is _ 8 of the set.
e.
3 9 is _ 5 of the set.
Decimal
Percent
0.8
80% 12.5% 55%
5
_1
28
2 _ c. 8 is 7 of the set.
Denominator Division
Complete the table. Fraction 4 _
0.125
8
11 _
0.55 0.6
20
24
2 _
3 857 _ 1,000
15
Add.
5 1_
7 6_ 8 4 10 _
2 Ongoing Learning & Practice
1 14 _ = 5_ 8 + 8 3 1 _ = 4_ 10 + 6 2
5
e.
580 boxes
13 9_ 15
1 2 _ _ c. 6 5 + 3 3 =
d.
A worker can fill 145 boxes of crackers in 15 minutes. At that rate, how many can she fill in 1 hour?
3 5 _ =_ 8 + 6
24
b.
19 20 108 109
70
5.
Write a fraction or a mixed number for each of the following:
15 1 _ or _ 4 40 2 _ , or _
a.
15 minutes = 60 ,
b.
40 minutes = 60
45 _ 60 , or 25 _ , or
c.
45 minutes =
d.
25 minutes = 60
e.
Assign Problems 1–10 on the journal page. Have students work with a partner. Circulate and assist. Briefly share solutions as needed.
2 66 _ 3%
89 90
4.
1 4_ 4
3 1 _ 2_ 4 + 12 =
a.
(Math Journal 2, p. 289B)
85.7%
0.857
74 75
3.
12 _ 60 , or
12 minutes =
PARTNER ACTIVITY
6.
▶ Math Boxes 8 12
Measure the line segment below to the 1 nearest _ inch. 4
(Math Journal 2, p. 290)
hour
2
3 hour 3 _ 4 hour 5 _ 12 hour 1 _ 5 hour
INDEPENDENT ACTIVITY
in.
62 63
Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-10. The skill in Problem 6 previews Unit 9 content.
183
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▶ Study Link 8 12
INDEPENDENT ACTIVITY
(Math Masters, p. 248)
Home Connection Students practice operations with fractions and mixed numbers.
3 Differentiation Options Name
Date
STUDY LINK
8 12 1. a.
b.
2. a.
Time
Mixed-Number Review
▶ Playing Build-It
32
1
, so there will be
To practice comparing and ordering fractions and renaming mixed numbers as fractions, have students play this variation of Build-It. If students did not keep their Fraction Cards, they will need to cut the cards from Math Masters, page 446.
1 Two families equally share _ of a garden. Show how they can 3
divide their portion of the garden. b.
_1
The drawing shows that _ ÷2= 3 _1 6 gets of the total garden. 1
6
, so each family
Common Denominator Division Step 1 Rename the numbers using a common denominator.
Students play the game as introduced in Lesson 8-1 except that at the end of each round, they toss a six-sided die to determine a whole-number part for each of their 5 fractions. Students then rename the mixed numbers as fractions. For example, after 7 , and _ 5 would become 3_ 1, _ 1, _ 1, _ 1 , 3_ 1, tossing a 3, the fractions _ 5 4 3 12 6 5 4 7 , and 3_ 5 . Renamed as fractions, the list would be _ 16 , _ 13 , _ 10 , 1 , 3_ 3_
Step 2 Divide the numerators, and divide the denominators. Solve. Show your work. 3.
2 5÷_ =
5.
4_ ÷_ = 8 4
3
15 _ _1 2 , or 7 2 11 _ _1 2 , or 5 2
3
1
4.
3 4 ÷ _ _ = 5
6.
6_ ÷_ = 3 9
7
2
7
20 _ 21
60 _ _4 7 , or 8 7
Practice
5
7.
1 4_ = 3_
9.
3 1 1_ + 2_ =
11.
8 4 7_ - 5_ = 9 9
13.
3 2 5_ + 2_ =
15.
3 3 ∗ 3_ =
4
4
5
5
3
4
4
8.
3_45 1_5
10.
9
12.
5 17 _ 7_ 12 , or 8 12 9 1 9_, or 11_ 4
4
14. 16.
22 1_68 , or 1_34 8 8 38 3 4_ , or 5_ 2 4 35 + 1 _ = 35 3_ 7 5 1 _ 2 3 7 _ = 3_ 5
15–30 Min
(Student Reference Book, p. 300; Math Masters, pp. 446 and 447)
Four pizzas will each be cut into eighths. Show how they can be cut to find how many slices there will be in all. The drawing shows that 4 ÷ _ = 8 32 slices in all.
PARTNER ACTIVITY
READINESS
Study Link Master
3
5
4 - 1_ = 4 6 2 _ ∗ = 4_ 7 3
12
43 , and _ 23 . _ 12 6
3 5 3_ - 1_ =
6
5
4
3
4
28 _ 7 , or 4
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Teaching Master ENRICHMENT
▶ Exploring the Meaning
PARTNER ACTIVITY 15–30 Min
of the Reciprocal
Name
Time
Exploring the Meaning of the Reciprocal
8 12
Lamont and Maribel have to divide fractions. Lamont doesn’t want to use common denominators. He thinks using the reciprocal is faster, but he’s not sure what a reciprocal is. Maribel looks it up on the Internet and finds this: One number is the reciprocal of another number if their product is 1.
(Math Masters, p. 249)
To explore the relationship between a number and its reciprocal, have students use what they know about fractions, fraction multiplication, and their calculators to find the reciprocals of numbers.
Date
LESSON
1.
Example 1:
Example 2:
3º?=1
_1 º ? = 1 2
3 1 3 º _3 = _3 = 1
_1 º 2 = _2 = 1 2 2
_1 is the reciprocal of 3 3
1 2 is the reciprocal of _2
1 3 is the reciprocal of _3
_1 is the reciprocal of 2 2
Find the reciprocals. _1 _1 6 a. 6 b.
2.
1 _
7
7
c.
5 What do you think would be the reciprocal of _6 ?
9
_1 d. 9
20 _6 5
20
Reciprocals on a Calculator
When students have finished the Math Masters page, ask them to describe the pattern for finding the reciprocal of a number. Guide students to see that the reciprocal of a fraction is the fraction with the numerator and denominator interchanged, or inverted. 9 , and _ 9 =_ 36 = 1. The 4 is _ 4 ∗_ For example, the reciprocal of _ 9 4 9 4 36 reciprocal of a whole number is a unit fraction that has the whole 1, number as its denominator. For example, the reciprocal of 8 is _ 8 8 = 1. 1 =_ so 8 ∗ _ 8 8
On all scientific calculators, you can find a reciprocal of a number by raising the number to the -1 power. 3.
Write each number in standard notation as a decimal and a fraction. 1 _1 _ 0.04 , 25 8-1 0.125 , 8 b. 5-2 c. 2-3
a. 4.
Write the key sequence you could use to find the reciprocal of 36.
5.
3 Write the key sequence you could use to find the reciprocal of _7 .
6.
What pattern do you see for the reciprocal of a fraction?
3
6 n
3
1
(–)
7
F D , or 3 (–)
d
1
0.125 ,
_1 8
1
6
, or 3
F D
7
Once the original number is written as a fraction, the reciprocal is the original fraction written with the numerator as the denominator and the denominator as the numerator.
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EXTRA PRACTICE
▶ Dividing with Unit Fractions
3/25/11 2:00 PM
PARTNER ACTIVITY 5–15 Min
(Math Masters, p. 253B)
Students practice using visual models to divide fractions.
Teaching Master Name
Date
LESSON
8 12
1.
2.
Number Stories: Division with Fractions
Five pies will each be sliced into fourths. Ira would like to find out how many slices there will be in all. a.
Show how the pies will be cut.
b.
1 The drawings show that 5 ÷ _ = 4 slices in all.
Jake has a 3-inch strip of metal. He would like to find out how many 1 _ -inch strips he can cut. 2
6
Jake can cut 3.
Time
20
0
, so there will be
20
1
2
inches
6
1 strips. So, 3 ÷ _ = 2
3
.
Two students equally share _ of a granola bar. They would 4 like to know how much of the bar each will get. 1
a.
Show how the piece of granola bar will be cut.
b.
1 The drawing shows that _ ÷2= 4
_1 8
student will get 4. a.
_1 8
, so each
of a granola bar.
Drawing A can be used to find _ ÷ 5. 3 1 1 of _ , Drawing B can be used to find _ 5 3 1
1 _ 1 ∗ . Use the drawings to show or _ 5 3 1 1 _ 1 ∗ . ÷5=_ that _ 5 3 3
b.
Complete. 1 _ 1 _ ∗ = 5 3 1 _ ÷5=
A
B
1 _ 15 1 _
3
1 1 _ ∗ ÷5=_ 3 3
15
_1 5
1 _ =
15
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Lesson 8 12
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Name
Date
STUDY LINK
Mixed-Number Review
8 12
1. a.
b.
2. a.
b.
Time
Four pizzas will each be cut into eighths. Show how they can be cut to find how many slices there will be in all. The drawing shows that 4 ÷ _ = 8 slices in all. 1
, so there will be
Two families equally share _ of a garden. Show how they can 3 divide their portion of the garden. 1
The drawing shows that _ ÷2= 3 gets of the total garden. 1
, so each family
Common Denominator Division Step 1 Rename the numbers using a common denominator. Step 2 Divide the numerators, and divide the denominators. Solve. Show your work. 3.
2 5÷_ =
4.
3 4 ÷ _ _ = 5
5.
4_ ÷_ = 8 4
6.
6_ ÷_ = 3 9
8.
7 _ = 3_
3
1
3
7
2
7
7.
1 4_ = 3_
9.
3 1 + 2_ = 1_ 5 5
10.
3 5 3_ - 1_ = 8 8
11.
8 4 7_ - 5_ = 9 9
12.
2 4 + 1_ = 3_ 7 5
13.
3 2 + 2_ = 5_ 3 4
14.
3 4 - 1_ = 4
15.
3 3 ∗ 3_ = 4
16.
∗_= 4_ 7 3
4
4
5
2
5
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Practice
6
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Name
Date
LESSON
Time
Solving Mixed-Number Addition Problems
82
Add. Write each sum as a mixed number in simplest form. Show your work. 1.
+ 2_ = 5_ 5 5
2.
+ 5_ = 3_ 5 10
3.
3 1 + 2_ = 4_ 4 12
4.
3 2 + 2_ = 4_ 3 4
5.
6.
1
4
2
3
Josiah was painting his garage. Before lunch, he painted 1 _ walls. 3 2 _ After lunch, he painted another 1 3 walls. How many walls did he paint during the day? 2
Julie’s mom made muffins for Julie and her friends to share. 3 1 _ Julie ate 1 _ muffins. Her friends ate 3 muffins. How many 4 2 muffins did Julie and her friends eat altogether?
Without adding the mixed numbers, insert , or =. Explain how you got your answer. 3 2 + 6_ 1_ 8 3
8.
5
8
1 7 2_ + 2_ 5 8
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7.
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Name
Date
LESSON
8 12
1.
2.
Number Stories: Division with Fractions
Five pies will each be sliced into fourths. Ira would like to find out how many slices there will be in all. a.
Show how the pies will be cut.
b.
The drawings show that 5 ÷ _ = 4 slices in all. 1
0
1
inches
strips. So, 3 ÷ _ = 2 1
2
3
.
Two students equally share _ of a granola bar. They would 4 like to know how much of the bar each will get. 1
a.
Show how the piece of granola bar will be cut.
b.
The drawing shows that _ ÷2= 4 1
student will get 4. a.
Copyright © Wright Group/McGraw-Hill
, so there will be
Jake has a 3-inch strip of metal. He would like to find out how many 1 _ -inch strips he can cut. 2 Jake can cut
3.
Time
of a granola bar.
Drawing A can be used to find _ ÷ 5. 3 1 1 _ of , Drawing B can be used to find _ 5 3 1
1 _ 1 ∗ . Use the drawings to show or _ 5 3 1 1 _ 1 _ ∗ . that 3 ÷ 5 = _ 5 3 b.
, so each
A
B
Complete. 1 _ 1 _ ∗ = 5 3 1 _ ÷5= 3
1 1 _ ∗ ÷5=_ 3 3
=
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Copyright © Wright Group/McGraw-Hill
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