Number Models for
Ratio Number Stories
Objective To introduce writing number models for ratio
number stories.
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Ongoing Learning & Practice
Key Concepts and Skills
Dividing with Unit Fractions
• Use equivalent fractions and ratio
models to solve ratio number stories. Math Journal 2, p. 414; Student
Reference Book, pp. 80A and 80B
Students practice dividing with unit
fractions and whole numbers.
[Number and Numeration Goal 5]
• Model and solve ratio problems. [Operations and Computation Goal 7]
Key Activities
Students use tiles to model and solve number
stories involving ratios of part of a set to the
whole set. They write and solve number
models that represent ratio number stories.
Math Boxes 12 5
Math Journal 2, p. 415
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Ongoing Assessment:
Informing Instruction See page 938.
Use Math Boxes, Problem 3. Ongoing Assessment:
Informing Instruction See page 939.
Study Link 12 5
Materials
Math Journal 2, pp. 412 and 413
Study Link 124
Square Tiles slate
[Operations and Computation Goal 6]
Math Masters, p. 361
Students practice and maintain skills
through Study Link activities.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 64–68
936
Unit 12
Probability, Ratios, and Rates
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Introducing Cross Multiplication
for Solving Ratio Problems
Math Masters, p. 362
Students use cross multiplication
to solve ratio number stories.
EXTRA PRACTICE
Solving Ratio Number Stories
Math Masters, p. 363
Students write and solve ratio
number stories.
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP6
Getting Started
Content Standards
5.NF.7a, 5.NF.7b, 5.NF.7c
Mental Math and Reflexes
Math Message
Have students find equivalent fractions. Ask volunteers
to explain how they used the multiplication rule or the
division rule in their solutions. Suggestions:
Model this problem using your tiles. Then write
the solution as a faction to express the ratio. Josie tosses a
penny 32 times. It lands heads up 5 out of 8 times. How many
20
times does the penny land heads up? _
Find equivalent fractions with a denominator of 4.
3 _
6
_
16
4 _
_
6
12 _
_
2 4
1 4
8 4
32
Study Link 12 4 Follow-Up
Find equivalent fractions with a denominator of 12.
6 _
48
36
2
24 _
_
_
3_
36 12
6 12
12
Have partners share their answers and resolve
differences.
Find equivalent fractions with a numerator of 8.
8
2 _
_
16 _
8
_
88 _
8
_
7 28
20 10
121 11
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Ask volunteers to draw their tile models on the board and explain
their solution strategies. Encourage students to use ratio language
when sharing their solutions. For example:
If the penny landed heads up 5 out of 8 tosses, it must have
landed heads up on 20 out of 32 tosses.
For every 8 tosses, the penny landed heads up 5 times. So it
landed heads up 20 times in 32 tosses.
▶ Introducing Number Models
for Ratio Number Stories
HEADS: 20
TAILS: 12
Square Tiles from Math Masters, page 431
WHOLE-CLASS
ACTIVITY
ELL
Algebraic Thinking Show students how to represent the Math
Message problem with a number model. Josie tossed a penny
32 times. It landed heads up 5 out of 8 times. How many times
did the penny land heads up?
5
Ask a volunteer to express the ratio 5 out of 8 times as a fraction. _
8
Write the fraction on the board or a transparency. This ratio is
equivalent to an unknown number of times the penny landed
heads up out of 32 tosses. To support English language learners,
discuss the meaning of unknown in this context, and explain that
it can be represented by using a symbol as shown in the display.
Add this information to the display on the board or transparency.
Explain that when writing a number model for a ratio number
story, it is important to keep in mind what the numerator and
denominator of the fraction represent. One approach is to write key
words next to the numerator and denominator. Add the words
heads and tosses to the display. (See margin.)
(heads) 5
□
=
(tosses) 8
32
Lesson 12 5
937
Student Page
Date
Time
LESSON
Tell students that to find the missing number in the number
model, they need to find a fraction that has 32 as its denominator
5 . Ask: Would you use the multiplication rule
and is equivalent to _
8
or the division rule to find the equivalent fraction? The
multiplication rule Why? 8 is less than 32, so multiply. Think: 8
times what number equals 32? Since 8 ∗ 4 = 32, multiply the
denominator, 8, and the numerator, 5, by 4. Modify the display on
the board or transparency:
More Ratio Number Stories
12 5
You can solve ratio number stories by first writing a number model for the story.
Example: Sidney missed 2 out of 9 problems on the math test. There were
36 problems on the test. How many problems did he miss?
(missed) 2
Write a number model: (total) 36
9
36
Find the missing number.
Think: 9 times what number equals 36?
9 º 4 36
Multiply the numerator, 2, by this number:
2º48
(missed) 2 º 4
(total) 9 º 4
8
36
Answer: Sidney missed 8 out of 36 problems.
(heads) 5 ∗ 4 20
=
(tosses) 8 ∗ 4 32
Write a number model for each problem. Then solve the problem.
1.
Of the 42 animals in the Children’s Zoo, 3 out of 7 are mammals. How
many mammals are in the Children’s Zoo?
Number model:
(mammals) 3
(animals) 7
or x
42
Answer:
18 mammals
(unit)
2.
Five out of 8 students at Kenwood School play an instrument. There are
224 students at the school. How many students play an instrument?
(play instrument) 5
8
(students)
Number model:
Give another example:
or x
224 Answer: 140 students
(unit)
3.
Mr. Lopez sells subscriptions to a magazine. Each subscription costs $18.
For each subscription he sells, he earns $8. One week, he sold $198
worth of subscriptions. How much did he earn?
Number model:
(earnings) 8
(subscriptions) 18
198
or x
Answer:
The penny landed heads up 20 times.
$88
Math Journal 2, p. 412
●
Marcus received 3 votes for every 5 votes cast. If he received
18 votes, how many votes were cast?
Ask students to write a number model for the problem. When
most students have finished, ask a volunteer to express the ratio
3 Write the fraction on
3 votes for every 5 votes cast as a fraction. _
5
the board or a transparency. Ask volunteers to explain how to
complete writing the number model for the problem.
(Marcus’s votes) __
18
3 ___
=
□
(total votes) 5
Ask: How would you find the missing number? Think: 3 times
what number equals 18? Since 3 ∗ 6 = 18, multiply the
numerator, 3, and the denominator, 5, by 6. Modify the display on
the board or transparency:
(Marcus’s votes) 3
∗ 6 ___
18
_____
=
(total votes) 5 ∗ 6 30
In all, 30 votes were cast.
Student Page
Date
More Ratio Number Stories
12 5
4.
Ongoing Assessment: Informing Instruction
Time
LESSON
Watch for students who write 18 in the denominator instead of the numerator.
Have them read the parts of the problem as they work. For example, Marcus
received 3 of every 5 votes. Marcus received 18 votes, so 18 is the numerator.
The denominator is the unknown.
continued
Make up a ratio number story. Ask your partner to solve it.
Number model:
Answer:
Find the missing number.
5.
1
3
x
39
x
7.
7
8
f
56
f
9.
5
6
m
11.
m
42
6.
13
8.
49
21
y
1
5
13
n
n
10.
35
3
4
y
9
25
s
s
100
▶ Solving Ratio Number Stories
28
65
(Math Journal 2, pp. 412 and 413)
There are 48 students in the fifth grade at Robert’s school. Three out of
8 fifth graders read two books last month. One out of 3 students read
just one book. The rest of the students read no books at all.
52 books
(unit)
Explain what you did to find the answer.
3
36
18
. So 18 students read two books each.
8
86
48
48
1 16
1
16
. So 16 students
That is 36 books in all. 3 48
3 16
48
∑
∑
read 1 book each. The total number of books read was
36 16 52.
Math Journal 2, p. 413
Unit 12
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IIN
NG
36
How many books in all did the fifth graders read last month?
938
WHOLE-CLASS
ACTIVITY
Probability, Ratios, and Rates
Algebraic Thinking Pose ratio number stories for students to solve.
Have them write number models for the number stories on their
slates and then solve the number models. Discuss solutions after
each problem.
Student Page
Suggestions:
Date
Maria saved $2 out of every $5 she earned delivering
newspapers. How much did she save from earnings of $45? $18
Dividing with Unit Fractions
12 5
1.
Two out of 3 pieces of candy in a bag are lemon flavored. The
bag contains 24 lemon-flavored pieces of candy. How many
pieces of candy are in the bag? 36 pieces
Five pizzas will each be sliced into fourths. Use the circles to show how
the pizzas will be cut. Find how many slices there will be in all.
1
The blueberries in _
5 of a farmer’s field need to be
picked. Four workers will share the work equally.
Divide the rectangle to show how much of the
field each worker will pick.
1
_
Each worker will pick
1
_
5 ÷4=
3.
20
of the field.
1
_
20
Each division number sentence on the left can be solved by using a related
multiplication number sentence on the right. Write the letter for the multiplication
sentence next to its related division sentence.
Division
Multiplication
d
a
c
b
1
8÷_
3 =n
1
_
8 ÷ 10 = n
1
8÷_
10 = n
1
_
3 ÷8=n
4.
Watch for students who have difficulty writing the number models. Have them
work with a partner to model the number stories with their square tiles.
slices.
20
1
5÷_
4 =
A choir has 50 members. Twenty members are sopranos. How
many sopranos are there for every 5 members of the choir?
2 sopranos
Ongoing Assessment: Informing Instruction
20
The pizzas will be cut into
2.
A parking garage reserves 1 out of 11 parking spaces for cars
with handicapped permits. If the garage has 99 parking spaces
in all, how many are reserved for handicapped spaces?
9 parking spaces
Time
LESSON
a.
1
n ∗ 10 = _
8
b.
1
n∗8=_
3
c.
1
n∗_
10 = 8
d.
1
n∗_
3 =8
Solve the following division number sentences (from above). Use the related
multiplication number sentences to help you find each quotient.
1
_
24
a.
1
8÷_
3 =
c.
1
8÷_
10 =
80
1
_
b. 8
÷ 10 =
1
_
d. 3
÷8=
80
1
_
24
Math Journal 2, p. 414
393-427_EMCS_S_MJ2_G5_U12_576434.indd 414
4/12/11 4:17 PM
Have students complete journal pages 412 and 413. Remind
students to label their number models for the ratio problems with
key words next to the numerators and denominators.
2 Ongoing Learning & Practice
▶ Dividing with Unit Fractions
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 414; Student Reference Book,
pp. 80A and 80B)
Student Page
Students practice dividing a whole number by a unit fraction, and
a unit fraction by a whole number. Students use the relationship of
multiplication and division to solve problems. Remind students to
review Student Reference Book, pages 80A and 80B as needed.
Date
Time
LESSON
12 5
1.
Rewrite each fraction pair with common
denominators.
2
3
and 1
1
6
6
a. and 3
2
15
8
3
2 b. and 20 and 20
4
5
18
6
9 2
24 and 24
and
c. 8
▶ Math Boxes 12 5
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 415)
Math Boxes
2.
List the factors of 142.
1, 2, 71, 142
12
65
3.
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 12-7.
10
Estimate the answer for each problem.
Then solve the problem.
4.
Estimate
Solution
15,000
17,214
a.
302 57
b.
599 9
5,400
5,391
c.
701 97
70,000
67,997
d.
498 501
250,000
249,498
There are 270 students in the soccer
league. Two out of three students are
boys. How many students are boys?
180 students
108 109
247
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 3
Use Math Boxes, Problem 3 to assess students’ ability to estimate and solve
multidigit multiplication problems. Students are making adequate progress if their
estimates demonstrate the use of magnitude and their solutions are correct.
[Operations and Computation Goal 6]
Complete the table.
Graph the data and
connect the points
with line segments.
Maryanne’s Earnings
Hours
Earnings
2
$24.00
$48.00
100
4
Maryanne earns
$12 per hour.
Rule:
Earnings 12 number of hours
5
7
60
9
$108.00
84
Earnings ($)
5.
80
60
40
20
0
0 1 2 3 4 5 6 7 8 9
Number of Hours
124 231
232
Math Journal 2, p. 415
Lesson 12 5
939
Study Link Master
Name
Date
▶ Study Link 12 5
Time
Ratio Problems
STUDY LINK
12 5
(Math Masters, p. 361)
Find the missing number.
x
40
x
m
54
m
f
32
f
1
1. 5
5
3. 6
5
5. 8
8
45
20
2
2. 3
16
y
y
1
4. 4
15
n
n
13
6. 50
g
100
g
106–109
243–245
24
60
26
INDEPENDENT
ACTIVITY
Home Connection Students write and solve number
models for ratio problems.
Write a number model for each problem. Then solve the problem.
Of the 115 students in the sixth grade, 2 out of 5 belong to the Drama Club.
How many students are members of the Drama Club?
7.
115
2
115
5
Number model:
Answer:
46 students
(unit)
Three out of 4 students at Highland School ordered a hot lunch today. There
are 156 students at the school. How many students ordered a hot lunch?
8.
3
115
4
156
Number model:
Answer:
3 Differentiation Options
117 students
(unit)
Gina and the other members of her troop sell cookies for $3 a box. For each
box they sell, the troop earns $1.50. One week, Gina’s troop sold $90 worth
of cookies. How much did the troop earn?
9.
1.50
3
90
Number model:
ENRICHMENT
$45.00
Answer:
▶ Introducing Cross Multiplication
30% of the tickets sold by a movie theater for the Friday night show were
children’s tickets at $4 each. The rest of the tickets were sold at the full price
of $8.50. The movie theater collected $360 just for the children’s tickets. How many
tickets did they sell in all?
10.
Number model:
15–30 Min
for Solving Ratio Problems
360 4 90; Answer: 300 tickets
90
30
x
100
SMALL-GROUP
ACTIVITY
(unit)
(Math Masters, p. 362)
Practice
216
63 11.
12.
36 729
13.
21,600
63 102 Math Masters, p. 361
Algebraic Thinking To explore using cross multiplication to
solve ratio problems, have students solve ratio number stories.
Ask students to write the number model for the problem from
this lesson’s Math Message and use the variable x for the
missing number.
(HEADS) __
x
5 ___
=
(tosses) 8 32
5 was changed
When this problem was solved earlier in the lesson, _
8
5 ∗ 4.
to the equivalent fraction _
8∗4
Another strategy is based on the idea of the quick common
denominator. Remind students that a quick common denominator
of two fractions is the product of their denominators.
(HEADS) 5
∗ 32
x∗8
______
= ______
32 ∗ 8
(tosses) 8 ∗ 32
Teaching Master
Name
Date
LESSON
Since the two fractions are equivalent, the renamed numerators
must be equal.
Time
Solving Ratio Problems with Cross Multiplication
12 5
Cross multiplication is a strategy for
solving ratio problems that is based
on the quick common denominator.
Solution:
32
x
5
=
32
8
Cross multiply:
Example:
Josie tosses a penny 32 times.
It lands heads up 5 out of 8 times.
How many times does the penny land
heads up?
5 ∗ 32 = x ∗ 8
So 160 = x ∗ 8
x
5
8
Number Model:
8x
=
8x
x
160
20
5 32
The missing number is 20. The penny lands on HEADS 20 times.
20 times
Answer:
Ask: What number times 8 equals 160? 20
(unit)
Use cross multiplication to solve the following problems. Let the variable x
represent the missing number in each problem.
1.
Jeremy received 3 votes for every 5 votes cast. If he received 18 votes, how
many votes were cast?
3
5
Number model:
Cross multiply:
1x8
3 x 5 18
Solution:
3 x 90; x 30
Answer:
30 votes
(unit)
2.
The restaurant at the mall sold 324 lunches. For every 9 lunches served,
3 were fish plates. How many fish plates were served?
3
9
Number model:
Cross multiply:
x
324
9 x 3 324
Solution:
9 x 972; x 108
Answer:
108 fish lunches
(unit)
3.
The Nature Center has a total of 87 amphibians on display. For every 6
amphibians, 2 are types of salamanders. How many salamanders are there?
2
6
Number model:
Cross multiply:
8x7
6 x 2 87
Solution:
6 x 174; x 29
Answer:
29 salamanders
(unit)
4.
Write a ratio number story for your partner to solve using cross multiplication.
Math Masters, p. 362
940
Unit 12
Probability, Ratios, and Rates
Compare the number model to the equation obtained from the
quick common denominator. (See following page.) Point out that
the equation can be obtained by multiplying the numerator of the
second fraction by the denominator of the first fraction, and
multiplying the denominator of the second fraction by the
numerator of the first fraction. This process, called cross
multiplication, is a convenient way to solve equations in which
two fractions are equal.
Teaching Master
Have students complete the master. Remind students that the
strategy of cross multiplication is not a trick. It is based on the use
of multiplication to find a common denominator for equivalent
fractions. Circulate and assist.
quick common
denominator
equation:
Name
LESSON
12 5
1.
Of the 90 fifth-grade girls at Lincoln School, 1 out of 6 reported that they
jumped rope 3 times last week. How many girls jumped rope 3 times
last week?
Number model:
(jumped rope) 1
(didn’t jump) 6
Answer:
15 girls
x
90
(unit)
2.
cross-multiplication
number model:
Time
Write a number model for each problem. Include key words for the numerators
and denominators. Then solve the problem.
x ∗8
5 ∗ 32
=
8 ∗ 32
32 ∗ 8
(HEADS )
(tosses)
Date
Solving Ratio Number Stories
The 175 seniors at Kennedy High School voted for the color of caps and
gowns they would wear at the graduation ceremony. Six out of 7 voted for
silver. How many students voted for silver?
Number model:
5
x
=
8
32
Answer:
6
(silver)
(other color) 7
x
175
150 students
(unit)
3.
8 ∗ x = 5 ∗ 32
Melissa’s brother, Sidney, was explaining his college food budget to her. He
told Melissa that he budgeted $160 a month for restaurants, but he spent 3
out of every 4 dollars at the campus pizza parlor. How much did he spend at
the pizza parlor?
Number model:
(pizza) 3
(other) 4
Answer:
$120.00
x
160
(unit)
EXTRA PRACTICE
▶ Solving Ratio Number Stories
PARTNER
ACTIVITY
15–30 Min
(Math Masters, p. 363)
4.
A survey was conducted at Sidney’s college to find out how the 640 students
budgeted their food money. Five out of 8 students reported that they spent less
than $160 a month on food. How many students spent less than $160 on food?
Number model:
Answer:
(less than $160) 5
(at least $160) 8
x
640
400 students
(unit)
Math Masters, p. 363
Algebraic Thinking Students solve ratio number stories. Then
they write a ratio number story for partners to solve.
Lesson 12 5
941