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Core State Standards
637A Unit 7 Fractions and Their Uses; Chance and Probability
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 143, 144
Multiplying Fractions by Whole Numbers
Objective
To apply and extend previous understandings of multiplication to multiply a fraction by a whole number.o
Key Concepts and Skills
• Use a number line to represent a fraction.
[Number and Numeration Goal 2]
• Understand a fraction _ ab as a multiple of _1b . [Number and Numeration Goal 3]
• Determine between which two whole numbers a fraction lies.
[Number and Numeration Goal 6]
• Solve number stories involving multiplication of a fraction by a whole number.
[Operations and Computation Goal 7]
• Write equations to model number stories.
[Patterns, Functions, and Algebra Goal 2]
Key Activities
Students use a number line as a visual fraction model to represent a fraction _ab multiplied by a whole number n as the product n ∗ ( _ab ) or _(n ∗ a)b . They solve number stories involving multiplication of a fraction by a whole number by using visual fraction models and equations to represent the problems.
Ongoing Assessment:
Informing Instruction See page 637D.
Key Vocabulary multiple equation Materials
Math Journal 2, pp. 217A–217E Study Link 7 12
half-sheets of paper calculator (optional)
Math Boxes 712a Math Journal 2, p. 217F
Students practice and maintain skills through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement Use Math Boxes, Problem 3.
[Operations and Computation Goal 5]
Study Link 712a Math Masters, p. 242A
Students practice and maintain skills through Study Link activities.
READINESS
Skip Counting to Show Multiples of Unit Fractions
Math Masters, p. 242B calculator
Students use calculators to skip count by unit fractions.
ENRICHMENT
Visual Models for Multiplying a Fraction by a Whole Number
Student Reference Book, p. 58 Math Masters, pp. 242C and 242D Students explore alternative visual fraction models for multiplying a fraction by a whole number.
EXTRA PRACTICE 5-Minute Math
5-Minute Math™, pp. 22 and 23
Students practice multiplying fractions by whole numbers.
Teaching the Lesson Ongoing Learning & Practice Differentiation Options
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Lesson 712a 637B
Adjusting the Activity
Provide students with calculators to assist with skip counting. See the Part 3 Readiness activity for additional information.
A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L
NOTE In Third Grade Everyday Mathematics children participated in skip-counting activities to help them memorize the multiplication facts.
While completing these activities, they were finding multiples. A multiple of a number is the product of a counting number and the number itself.
Date Time
Multiples of Unit Fractions
LESSON
712a
For Problems 1–3, fill in the blanks to complete an equation describing the number line.
1.
1
8 2
8 3
8 4
8 5
8 6
8 7
0 8 1
Equation: 5 ∗ 1_8 = _58
2.
1 6
2 6
3 6
4 6
5
0 6 1
Equation: 3 ∗ 1_ 6 = _36 , or 1_2
3.
1 3
2 3
3 3
4 3
5 3
6 0 3
Equation: 4 ∗ _13 =
For Problems 4–6, use the number line to help you multiply the fraction by the whole number.
4.
1 4
2 4
3
0 4 1
Equation: 2 ∗ _ 14 = 2_4 , or _12
5. 1
10 2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
10 0 10
Equation: 6 ∗ _ 10 1 =
6.
1 5
2 5
3 5
4 5
5 5
6 5
7 5
8 5
9 5
10 0 5
Equation: 7 ∗ 1_ 5 =
58
4 _3 , or 1 _13
__10 6 , or 3_5
7
_5 , or 1 2_5
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Math Journal 2, p. 217A
Student Page
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS DISCUSSIONAsk students how they determined the next three multiples in each sequence. Possible strategies:
Think of the problem as skip counting by _ 101 s. To get the next multiple, add _ 101 to the previous fraction. For example, 1
_ 10 + _ 101 = _ 102 ; _ 102 + _ 101 = _ 103 ; _ 103 + _ 101 = _ 104 ; and so on.
Think in terms of equal groups. For example, 1 group of _ 14 is _ 14 ; 2 groups of _ 14 is __ 2
4 ; 3 groups of _ 14 is _ 34 ; 4 groups of _ 14 is _ 44 ; and so on.
Tell students that in this lesson they will use their understanding of multiples to multiply fractions by whole numbers.
Using a Visual Fraction Model to
PARTNER ACTIVITYMultiply a Unit Fraction by a Whole Number
(Math Journal 2, pp. 217A)
Draw the number line below on the board or overhead.
1 2
2 2
3 2
4 2
5 2
6
0
2Have volunteers explain how they could use the number line and their understanding of multiples to help them solve the problem 3 ∗ __ 1
2 .
Getting Started
Mental Math and Reflexes
Have students name the next three multiples in a sequence. Suggestions:
8, 16, 24, ... 32, 40, 48 50, 60, 70, ... 80, 90, 100 25, 50, 75, ... 100, 125, 150 82, 84, 86, ... 88, 90, 92 56, 60, 64, ... 68, 72, 76 18, 27, 36, ... 45, 54, 63 70, 140, 210, ... 280, 350, 420
600; 1,200; 1,800; ... 2,400; 3,000; 3,600 125, 250, 375, ... 500, 625, 750
Math Message
Name the next three multiples in each sequence.
1 __
10 , __ 10 2 , __ 10 3 , … 10 __4 , 10 __5 , __ 10 6 1
__
4 , __2 4 , __ 4 3 , … _44 , _54 , 6_4
Study Link 7
12 Follow-Up
Have small groups compare the results of the penny toss experiment. Ask volunteers to share their answers for Problem 5. Have students indicate thumbs-up if they agree.
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637C Unit 7 Fractions and Their Uses; Chance and Probability
217C
Date Time
Multiplying Fractions by Whole Numbers
LESSON
712a 58
Use number lines to help you solve the problems.
1. 5 ∗ 1_ 6 = 5_6
6 6
1 6
0
1 6
1 6
1 6
1 6
2. 6 ∗ 1_ 3 = _ 63 , or 2
9 3 6
3 3
3
1 3
0
1 3
1 3
1 3
1 3
1 3
3. 3_8 = 3 ∗ 1_ 8
8 8 4
8 1
8
0
Sample answer:
1 8
1 8
4. 2 ∗ 4_ 3 = _83 , or 2 2_3
9 3 6
3 3
3
4 3
0
4 3
5. 12__8 , or 1 4_8 , or 1 1_2 = 4 ∗ _ 38
8 8
16 8 4
8
12 0 8
3 8
3 8
3 8
3 8
6. __10 6 , or 3_5 = 3 ∗ _ 10 2
10 10 5
10 2
10 2 10
2 10
0
Sample answer:
185-218_EMCS_S_MJ2_G4_U07_576426.indd 217C 3/3/11 12:39 PM
Math Journal 2, p. 217C
Student Page
217B
Date Time
An Algorithm for Multiplying a Fraction by a Whole Number LESSON
712a
Example 1: Equation: 6 ∗ 1_ 5 = 6_ 5
5 5 1
5
10 0 5
1 5
1 5
1 5
1 5
1 5
Example 2: Equation: 3 ∗ 2_ 5 = 6_ 5
5 5 2
5
10 0 5
2 5
2 5
Write an equation to describe each number line.
1. a.
4
4 8
0 4 1 4
1 4
1 4
1 4
1 4
1 4
6 ∗ 1_4 = 6_4
b .
4 4
8 0 4
3
4 3
4
2 ∗ 3_4 = 6_4
2. a.
9 3 6
3 3
0 3 1 3
1 3
1 3
1 3
1 3
1 3
1 3
1 3
8 ∗ 1_3 = 8_3
b .
9 3 6
3 3
0 3 2
3 2
3 2
3 2
3
4 ∗ 2_3 = 8_3
3. Study the pairs of number lines above. Use the patterns you see to describe a way to multiply a fraction by a whole number.
Sample answer: If I take the whole number, multiply it by
the numerator of the fraction and then write the product over the denominator, that is my answer.
58
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Student Page
One way is to visualize jumps or hops on the number line, starting at 0. The fraction tells the size of the jump; the whole number tells the number of jumps. Thus, 3 ∗ _ 12 is 3 jumps, each _ 12 unit long. You end up at _ 32 . So, 3 ∗ _ 12 = _ 32 , or 1 _ 12 .
1 2
2 2
3 2
4 2
5 2
6
0
2 1 21 2
1 2
Partners complete journal page 217A. Tell students that an equation is a number sentence with an equals sign, such as 3 ∗ _ 12 = _ 32 . As you circulate and assist, pose questions such as the following:
● Which number in the equation tells you the size of the jump?
The first fraction
● Which number in the equation tells you the number of jumps?
The whole number
● Can you name the products in Problems 3 and 6 as mixed numbers? _ 43 = 1 _ 13 ; _ 75 = 1 _ 25
Using a Visual Fraction Model
WHOLE-CLASS ACTIVITYto Multiply Any Fraction by a Whole Number
(Math Journal 2, pp. 217B and 217C)
Have partners study the examples at the top of journal page 217B.
On a half-sheet of paper, students should record any similarities and differences they see between the equations modeled on the number lines.
Expect students to share observations such as the following:
Both equations involve multiplication of a fraction by a whole number.
Both equations have the same product.
The factors in the equations are different, but _ 25 is a multiple of _ 15 and 6 is a multiple of 3.
It takes more jumps of _ 15 to get to _ 65 than it does jumps of _ 25 because the jumps of _ 15 are smaller than the jumps of _ 25 .
The whole number factor in 6 ∗ _ 15 = _ 65 is twice as much as the whole number factor in 3 ∗ _ 25 = _ 65 . The fraction factor in 6 ∗ _ 15 = _ 65 is half as much as the fraction factor in 3 ∗ _ 25 = _ 65 .
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Lesson 712a 637D
Date Time
Solving Number Stories
LESSON
712a
Suma and her sister Puja are making 12 blueberry-wheat muffins for breakfast. The recipe lists the following ingredients:
1 cup flour 1 egg
1_
2 cup whole-wheat flour 1_ 2 cup skim milk 2 teaspoons baking powder 2_ 3 cup honey 3_
4 cup blueberries 1_ 4 cup cooking oil 1_
4 teaspoon salt 3_ 8 teaspoon cinnamon Use the list of recipe ingredients to help you solve the number stories below. For each problem, write an equation to show what you did.
1. The sisters decided to double the recipe.
a. How many cups of whole-wheat flour do they need now?
2_2 , or 1 cup(s) Equation: 2 ∗ _ 12 = _22
b. How many cups of blueberries do they need now?
_64 , or 1 2_4 , or 1 1_2 cup(s) Equation: 2 ∗ _ 34 = _64
c. How many cups of honey do they need now?
_43 , or 1 1_3 cup(s) Equation: 2 ∗ _ 23 = _43
2. Suma and Puja decide to make 48 muffins instead of 12.
a. How many teaspoons of salt do they need now?
4_4 , or 1 teaspoon(s) Equation: 4 ∗ 1_ 4 = 4_4
b. How many teaspoons of cinnamon do they need now?
12__8 , or 1 4_8 , or 1 1_2 teaspoon(s) Equation: 4 ∗ 3_ 8 = 12__8
c. How many cups of skim milk do they need now?
4_2 , or 2 cup(s) Equation: 4 ∗ _ 12 = _42
58
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Math Journal 2, p. 217D
Student Page
NOTE In Lesson 3-8, students used number models to model number stories. A number model is a number sentence or part of a number sentence. A number model can include an equal sign, but it is not required. An equation is a number sentence with an equal sign. See Section 10.2 in the Teacher’s Reference Manual for more information.
Have partners complete Problems 1 and 2 on journal page 217B by writing a multiplication equation to describe each number line.
When students have completed Problem 3, bring the class together to discuss the algorithm for multiplication of a fraction by a whole number. The pattern can be expressed as: n ∗ a__
b = (n ∗ a)_ b .
Have students complete journal page 217C for additional practice multiplying fractions by whole numbers. Encourage students to use the pattern they discovered on journal page 217B to check their answers.
Solving Number Stories
PARTNER ACTIVITY(Math Journal 2, pp. 217D and 217E)
Pose the following number story:
When Carlos goes to the gym, he exercises for _ 34 of an hour and burns about 200 calories. Last week he went to the gym 5 times.
How many hours did Carlos spend at the gym last week?
Ongoing Assessment: Informing Instruction
Watch for students who are distracted by the “extra” 200 in the number story.
Encourage them to eliminate irrelevant information by determining exactly what they want to find out, what information they already know, and what they might need to know in order to solve the problem.
On a half-sheet of paper, have students draw a visual fraction model to represent the number story. Expect drawings such as the following:
0
44 84 124 164 2043 4
3 4
3 4
3 4
3 4
Then have students write a multiplication equation to represent the problem. 5 ∗ _ 34 = __ 15
4
Ask students to determine between which two whole numbers of hours the product lies. 3 and 4 hours Have them explain their strategy for finding the answer. Possible strategies:
Use the number line drawn to represent the number story. Note that the product lies between 12_ 4 , or 3, and _ 164 , or 4.
The fraction _ 154 can be renamed as the mixed number 3 _ 34 by dividing the numerator, 15, by the denominator, 4: 15
4 → 3 R3.The quotient, 3, is the whole number part of the mixed number.
The remainder, 3, is the numerator of the fraction part of the mixed number. It tells how many fourths are left over after making as many wholes as possible.
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637E Unit 7 Fractions and Their Uses; Chance and Probability
Date Time
Solving Number Stories continued
LESSON
712a
The Hillside Elementary School walking club meets every Monday after school.
The table below shows how far some students walked at their last meeting.
Student
Miles _ 1 3 _ 9
10 _ 5
4 5 _
2 _ 4
3 _ 5
6
Katie Mahpara Nikhil Cole Maria Jack
Use the information in the table to solve the number stories.
3.a. If Katie walks the same distance at every
meeting, how far will she walk after 2 meetings? miles
b. After 7 meetings? _73 , or 2 1_3 miles
c. After 7 meetings, Katie will have walked between . Circle the best answer.
1 and 2 miles 2 and 3 miles 3 and 4 miles
4.a. If Jack walks the same distance at every
meeting, how far will he walk after 3 meetings? miles
b. After 3 meetings, Jack will have walked between . Circle the best answer.
1 and 2 miles 2 and 3 miles 3 and 4 miles
5.a. If Mahpara walks the same distance at every
meeting, how far will she walk after 4 meetings? miles
b. After 4 meetings, Mahpara will have walked between . Circle the best answer.
1 and 2 miles 2 and 3 miles 3 and 4 miles
6. If Cole walks the same distance at every meeting and wants to
walk a total of _ 152 miles, how many meetings will he need to attend? 3 meetings 7. Make up your own multiplication number story about Nikhil or Maria.
Answers vary.
Try This
2
_
3
15
__6 , or 2 3_6 , or 2 1_2
36
__10 , or 3 __10 6 , or 3 3_5
185-218_EMCS_S_MJ2_G4_U07_576426.indd 217E 3/24/11 9:28 AM
Math Journal 2, p. 217E
Student Page
Have partners complete journal pages 217D and 217E. Encourage students to use visual fraction models, such as number lines, to help them solve the problems. When reviewing answers, pose questions such as the following:
● Which of the products on journal page 217D can you rename as whole numbers? Problem 1a: _ 22 = 1; Problem 2a: _ 44 = 1;
Problem 2c: _ 42 = 2
● Between which two whole numbers does the product in Problem 2b lie? 1 and 2
● In Problem 2, how did you decide which whole number you would multiply the recipe ingredients by? Sample answer: The recipe makes 12 muffins. If the sisters want 48 muffins they will need to quadruple the recipe because 12 ∗ 4 = 48.
● How did you solve Problem 6? Sample answer: Let the letter a stand for the number of meetings Cole would need to attend and write the equation a ∗ _ 52 = _ 152 . Use the algorithm for multiplying a fraction by a whole number and think: What number times 5 will give me 15? 3 ∗ 5 = 15, so 3 ∗ _ 52 = _ 152 . Cole will need to attend 3 meetings.
● In Problem 6, between which two whole-number distances does the distance _ 152 miles lie? Between 7 and 8 miles
Allow time for students to share and solve the number stories they wrote for Problem 7. For each problem, pose questions such as the following:
● Between which two whole numbers does the answer lie?
● Can you use a visual fraction model or an equation to represent the problem?
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Lesson 712a 637F
Date Time
Math Boxes
LESSON
712a
4. Write an equivalent fraction, decimal, or whole number.
Decimal Fraction
a. 0.60 b. 0.65 _100 65 c. 1.0 _50 50 d. 0.9
1. Karen used 60 square feet of her back yard for a garden. Vegetables fill _ 35 of her garden space. Tomato plants fill 1_ 6 of the space taken up by vegetables. How many square feet are used for tomatoes?
6 square feet
2. Multiply. Use a paper-and-pencil algorithm.
3,741= 87 ∗ 43
3. a. Lukasz drew a line segment that was 2 _ 28 inches long. Then he extended it another 2 _ 38 inches. How long is the line segment now?
4 5_8 inches
b. Sybil drew a line segment 3 1_ 8 inches long. Then she extended it another 2 _ 34 inches. How long is the line segment now?
5 7_8 inches
18 19 59
55–57
162–166 129
61 62
60 ___
100
6. Complete.
a. 42 in.= 3 ft 6 in.
b. 16 ft = 192 in.
c. 67 in. = 5 ft 7 in.
d. 22 ft = 7 yd 1 ft e. 1 1_2 yd = 4 ft 6 in.
5. Complete the table and write the rule.
Rule: -3.49
in out
104.16 100.67 87.35 83.86
45.72 42.23
55.41 51.92
77.69 74.20
9 __
10
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Math Journal 2, p. 217F
Student Page
242A
Name Date Time
58
Multiplying Fractions by Whole Numbers
LESSON
712a
Use the number lines to help you solve the problems.
1. 5 ∗ _ 15 = 5_ 5 , or 1
10 5 1
5 2 5
3 5
4 5
5 5
6 5
7 5
8 5
9 0 5
1 5
1 5
1 5
1 5
1 5
2. 3 ∗ _ 49 =
18 9 9
0 9
4 9
4 9
4 9
3. 6 ∗ _ 36 = _ 186 , or 3
18 6 12
6 6
0 6
3 6
3 6
3 6
3 6
3 6
3 6
Write a multiplication equation to represent the problem and then solve.
4. Rahsaan needs to make 5 batches of granola bars. A batch calls for _ 12 cup of honey.
How much honey does he need? Equation: 5 ∗ _ 12 = _ 52 , or 2 1_ 2 cups
5. Joe swims _ 106 of a mile 5 days a week. How far does he swim every week?
Equation: 5 ∗ _ 10 6 = _ 3010 , or 3 miles
How far would he swim if he swam every day of the week?
Equation: 7 ∗ _ 10 6 = _ 4210 , or 4 _ 10 2 , or 4 _ 15 miles
Practice
6. a. List the factor pairs of 5. 1 and 5 b. Is 5 a prime number? yes
7. a. List the factor pairs of 21. 1 and 21; 3 and 7
b. Is 21 a prime number? no 12
_ 9 , or 1 _ 39 , or 1 _ 13
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Math Masters, p. 242A
Study Link Master
2 Ongoing Learning & Practice
Math Boxes 7
12a
INDEPENDENT ACTIVITY(Math Journal 2, p. 217F)
Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 7-9 and 7-11. The skill in Problem 6 previews Unit 8 content.
Ongoing Assessment:
Math Boxes Problem 3Recognizing Student Achievement
Use Math Boxes, Problem 3 to assess students’ ability to solve mixed-number addition problems. Students are making adequate progress if they are able to solve Problem 3a, which involves mixed numbers with like denominators. Some students may be able to solve Problem 3b, which involves mixed numbers with unlike denominators, by using equivalent mixed numbers with like denominators, using manipulatives, or drawing pictures.
[Operations and Computation Goal 5]
Study Link 7
12a
INDEPENDENT ACTIVITY(Math Masters, p. 242A)
Home Connection Students use number lines to multiply fractions by whole numbers.
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637G Unit 7 Fractions and Their Uses; Chance and Probability
Name Date Time
Skip Counting by a Unit Fraction
LESSON
712a
1.Use your calculator to count by 1_ 2 s. Complete the table below.
One 1 _ 2 Two
1 _ 2 s
Three 1 _ 2 s
Four 1 _ 2 s
Five 1 _ 2 s
Six 1 _ 2 s
Seven 1 _ 2 s
Eight 1 _ 2 s
Nine 1 _ 2 s
Ten 1 _ 2 s 1
_ 2 2_ 2 3_ 2 _ 42 _5 2 _6
2 7_ 2 _8
2 _9 2 10_
2 2. Use your calculator to count by 1_ 3 s. Complete the table below.
One 1 _ 3 Two
1 _ 3 s
Three 1 _ 3 s
Four 1 _ 3 s
Five 1 _ 3 s
Six 1 _ 3 s
Seven 1 _ 3 s
Eight 1 _ 3 s
Nine 1 _ 3 s
Ten 1 _ 3 s 1
_ 3 2_ 3 3_ 3 _ 43 _5 3 _6
3 7_ 3 _8
3 _9 3 10_
3 3. Use your calculator to count by 1_ 5 s. Complete the table below.
One 1 _ 5 Two
1 _ 5 s
Three 1 _ 5 s
Four 1 _ 5 s
Five 1 _ 5 s
Six 1 _ 5 s
Seven 1 _ 5 s
Eight 1 _ 5 s
Nine 1 _ 5 s
Ten 1 _ 5 s 1
_ 5 2_ 5 3_ 5 4_
5 _5 5 _6
5 7_ 5 _8
5 _9 5 10_
5
4. Use your calculator to count by 1_ 8 s. Complete the table below.
One 1 _ 8 Two
1 _ 8 s
Three 1 _ 8 s
Four 1 _ 8 s
Five 1 _ 8 s
Six 1 _ 8 s
Seven 1 _ 8 s
Eight 1 _ 8 s
Nine 1 _ 8 s
Ten 1 _ 8 s 1
_ 8 _2 8 3_
8 4_ 8 _5
8 _6 8 7_
8 _8 8 _9
8 10_ 8
5. Use your calculator to count by _ 101 s. Complete the table below.
One 1 _ 10
Two 1 _ 10 s
Three 1 _ 10 s
Four 1 _ 10 s
Five 1 _ 10 s
Six 1 _ 10 s
Seven 1 _ 10 s
Eight 1 _ 10 s
Nine 1 _ 10 s
Ten 1 _ 10 s 1
_ 10 _2
10 _3 10 _4
10 _5 10 _6
10 _7 10 _8
10 _9 10 10_
10
6. How is skip counting by 1_ 3 s on your calculator from 0 to nine 1_ 3 s the same as finding the product 9 ∗ _ 13 ?
Sample answer: When you skip count by _1
3 from 0 nine times, you are finding nine groups of
1_
3 . This is the same as 9 ∗
1_
3 .
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Math Masters, p. 242B
Teaching Master
3 Differentiation Options
READINESS SMALL-GROUP
ACTIVITY
Skip Counting to Show
15–30 MinMultiples of Unit Fractions
(Math Masters, p. 242B)
To explore multiples of unit fractions, have students skip count on the calculator. Remind students that when you skip count by a number, your counts are the multiples of that number.
Review the steps for counting by 5s on the calculator. Students can program their calculator using the following steps:
TI-15:
1. Press On/Off and Clear simultaneously. This clears your calculator display and memory.
2. Press Op1 + 5 Op1. This tells the calculator to count up by 5s.
3. Press 0. This is the starting number.
Casio fx-55:
1. Press . This clears your calculator display and memory.
2. Press 5. This tells the calculator to count by 5s.
3. Press . This tells the calculator to count up.
4. Press 0. This is the starting number.
Now the calculator is ready to count by 5s. Without clearing their calculators, have students press the Op1 key or the key. Press the Op1 key or the key repeatedly as the students count together by 5s.
Next have students skip count by the unit fraction _ 1
4 . You may first need to remind students of the steps to enter a fraction on their calculators.
To enter _ 1 4 :
On a TI-15: 1 n 4 d .
On a Casio fx-55: 1 4.
Have students skip count by unit fractions to complete the tables on Math Masters, page 242B. Afterward, discuss how Problem 6 highlights the concept that a fraction such as _ 9
3 means the same thing as 9 ∗ ( _ 1
3 ). In general, _ a
b = a ∗ ( _ 1 b ).
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