Objectives
To provide practice using a rate table to record rate
information; and to provide practice solving rate problems.
1
materials
Teaching the Lesson
Key Activities
Students use a version of the “What’s My Rule?” table, called a rate table, to solve
rate problems.
Key Concepts and Skills
•
•
•
•
Find multiples. [Number and Numeration Goal 3]
Solve multiplication and division facts. [Operations and Computation Goal 3]
Multiply and divide decimals by whole numbers. [Operations and Computation Goal 4]
Use repeated addition and scaling to model multiplication problems.
ⵧ Math Journal 2, pp. 312 and 313
ⵧ Study Link 12 1
䉬
ⵧ Teaching Aid Master (Math Masters, p. 389)
ⵧ Transparency (Math Masters, p. 454; optional)
ⵧ slate
See Advance Preparation
[Operations and Computation Goal 7]
• Use patterns and rules to solve rate problems. [Patterns, Functions, and Algebra Goal 1]
Key Vocabulary rate table • unit rate
Ongoing Assessment: Informing Instruction See page 917.
Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip.
[Patterns, Functions, and Algebra Goal 1]
2
materials
Ongoing Learning & Practice
Students play the Credits/Debits Game (Advanced Version) to practice adding and
subtracting integers.
Students practice and maintain skills through Math Boxes and Study Link activities.
ⵧ Math Journal 2, p. 314
ⵧ Student Reference Book, p. 239
ⵧ Study Link Master (Math Masters, p. 341)
ⵧ Game Master (Math Masters, p. 469)
ⵧ penny; deck of number cards
3
materials
Differentiation Options
READINESS
Students describe and
illustrate situations
involving rates.
ENRICHMENT
Students plot points
corresponding to a linear
relationship on graph
paper and connect the
points using a straight
line.
EXTRA PRACTICE
Students use rate tables
to solve rate problems.
Additional Information
Advance Preparation For Part 1, make an overhead transparency of Math Masters,
page 454 or draw several blank rate tables on the board. For the optional ELL Support
activity in Part 3, you need a copy of Each Orange Had Eight Slices: A Counting Book by
Paul Giganti, Jr. (Greenwillow Books, 1992).
914
Unit 12 Rates
ⵧ Math Journal 2, pp. 312 and 313
ⵧ Teaching Aid Masters (Math Masters, pp. 388,
443, and 454)
ⵧ Each Orange Had Eight Slices: A Counting Book
ⵧ colored pencils or crayons; straightedge
See Advance Preparation
Technology
Assessment Management System
Exit Slip
See the iTLG.
Getting Started
Mental Math and Reflexes
Math Message
Pose rate problems. Suggestions:
Flashlight batteries are on
sale at 2 packages for $3.00. What is
the cost of 6 packages of batteries?
Each bunch of balloons costs $6. What is the cost of
2 bunches? $12
3 bunches? $18
5 bunches? $30
Kesia types 60 words per minute. At that rate, how many words can she type in
7 minutes? 420 words
80 minutes? 4,800 words
400 minutes? 24,000 words
Hei walks 2.2 miles per hour. At that rate, how many hours will it take her to travel
13.2 miles? 6 hours
15.4 miles? 7 hours
52.8 miles? 24 hours
Study Link 12 1
Follow-Up
䉬
Ask students to share the examples
of rates they have brought from home
and to add them to the Rates All Around
Museum, if you have one. Encourage
students to continue to look for
examples of rates.
1 Teaching the Lesson
䉴 Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Ask students to share solution strategies. Draw pictures on the
board to illustrate their strategies.
Some possible strategies might include the following:
䉯 6 packages is 3 times as much as 2 packages.
$3.00
$3.00
$3.00
$9.00
䉯 If 2 packages cost $3.00, then 1 package costs half as much:
1
of $3.00 $1.50. Six packages cost 6 times as much as
2
1 package: 6 ⴱ $1.50 $9.00.
$1.50 $1.50 $1.50 $1.50 $1.50 $1.50 $9.00
Tell students that in this lesson they will use rate tables to help
them organize information in rate problems.
Lesson 12 2
䉬
915
Batteries: 2 packages/$3.00
Packages
Cost
䉴 Introducing Rate Tables
WHOLE-CLASS
ACTIVITY
The information provided by a rate can be extended to show
other equivalent rates. Make a rate table and fill in the cost of
2 packages and 6 packages of batteries. (See margin.) Students
should recognize this as a “What’s My Rule?” table: The in
numbers (packages) are known; most of the out numbers (cost)
are to be found.
1
$1.50
2
$3.00
3
$4.50
4
$6.00
5
$7.50
6
$9.00
7
$10.50
Fill in the rest of the table with the help of the class. As you do
so, ask questions such as the following:
8
$12.00
●
9
$13.50
If 2 packages cost $3.00, how can you find the cost of
1 package? Divide $3.00 by 2.
●
How can you find the cost of 3 packages? Multiply the cost of
1 package by 3: 3 ⴱ $1.50 $4.50. Or add the cost of 1 package
to the cost of 2 packages: $3.00 $1.50 $4.50.
●
How can you find the cost of 8 packages? Multiply the cost of
2 packages by 4: 4 ⴱ $3.00 $12.00.
Rate Table
Links to the Future
Representing rates with tables of values, formulas, and graphs is a
Grade 5 Goal.
䉴 Solving Rate Problems
WHOLE-CLASS
ACTIVITY
(Math Masters, p. 454)
NOTE The unit rate is a version of the rate
that tells how many of some thing for each 1
of a second thing. For example, 27 minutes
for 9 buttons is not a unit rate; 3 minutes for
1 button is a unit rate. Similarly, $3.00 for
2 packages is not a unit rate; $1.50 for
1 package is a unit rate. It is usually easier to
fill in the other cells of a rate table if the unit
rate is filled in first.
Use an overhead transparency of Math Masters, page 454 or the
blank rate tables you have drawn on the board.
1. Enter the given rate in a rate table.
2. Fill in the rest of the cells in the rate table.
3. Answer the question posed in the problem.
Example:
Mr. Rankin sews buttons on shirts. He sewed 9 buttons
in 27 minutes. At that rate, how many buttons did he sew
after 15 minutes?
1. Make a rate table and enter the given rate: 9 buttons
in 27 minutes.
Buttons
Minutes
916
Unit 12 Rates
1
2
3
4
5
6
7
8
9
27
2. Fill in the rest of the cells in the rate table.
Adjusting the Activity
a. Fill in the cell for 1 button first; this is called a unit rate.
Buttons
1
Minutes
3
2
3
4
5
6
7
8
9
27
b. Then fill in the other cells.
Ask students to state a rule for
finding the number of buttons sewn in a given
number of minutes. Divide the number of
minutes by 3. Now ask students to use
variables to state the rule as a formula.
Sample answer: b m / 3, where b is the
number of buttons sewn and m is the number
of minutes
䉬
AUDITORY
Buttons
1
2
3
4
5
6
7
8
9
Minutes
3
6
9
12
15
18
21
24
27
KINESTHETIC
䉬
TACTILE
䉬
VISUAL
3. Answer the question posed in the problem: At that rate,
Mr. Rankin sewed 5 buttons after 15 minutes.
Work with the class to solve these problems:
●
A building has a height of 60 feet. If each story is 12 feet high,
how many stories does the building have? 5 stories. The unit
rate 12 ft/1 story is given. To support English language
learners, discuss the meaning of story in this context.
●
DeAndre receives $4 per week as an allowance. If he saves all
his money, how much will he have after one year? $208. The
unit rate $4/1 week is given.
●
In one week, Jill mowed 12 lawns and made $72. If she
charged each customer the same amount, how much did she
charge per lawn? $6. The given rate $72 per 12 lawns is not a
unit rate. Division is needed to find the unit rate: 72 12 $6 per 1 lawn. How much had she earned after she mowed
4 lawns? $24. Since the unit rate is $6 per lawn, multiplication
is used to find an equivalent rate: 4 ⴱ 6 $24 per 4 lawns or
repeated addition 6 6 6 6 $24 per 4 lawns.
Student Page
Date
Time
LESSON
12 2
䉬
Rate Tables
47
For each problem, fill in the rate table. Then answer the question below the table.
Ongoing Assessment: Informing Instruction
1. Bill’s new car can travel about 35 miles on 1 gallon of gasoline.
Gasoline mileage: 35 miles per gallon
Watch for students who note that a rate table is like a “What’s My Rule?” table
turned on its side. Encourage students to consider the strategies they used to
complete “What’s My Rule?” tables and to find the rules when they are solving
rate problems.
Miles
Gallons
35
1
70 105 140 175 210 245 280
2
3
4
5
At this rate, about how far can the car travel on 7 gallons of gas?
6
7
245
8
miles
2. Jennifer earns $8 for every 4 hours she helps out around the house.
Earnings: $8 in 4 hours
䉴 Practicing with Rate Problems
PARTNER
ACTIVITY
(Math Journal 2, pp. 312 and 313)
Assign journal pages 312 and 313. The unit rate is given for
Problem 1. For the remaining problems, students should find the
unit rate first and then fill in the remainder of the rate table. For
Problem 7, have students discuss their solution strategies.
Dollars
2
4
6
8
10
12
14
16
Hours
1
2
3
4
5
6
7
8
At this rate, how much money does Jennifer earn per hour? $ 2.00
3. A gray whale’s heart beats 24 times in 3 minutes.
Gray whale’s heart rate: 24 beats in 3 minutes
Heartbeats
8
16
24
32
40
48
56
64
Minutes
1
2
3
4
5
6
7
8
16
At this rate, how many times does a gray whale’s heart beat in 2 minutes?
times
4. Ms. Romero paid $1.80 for 3 pounds of grapes.
Cost of grapes: 3 pounds for $1.80
Pounds
1
2
0.60 1.20
3
4
5
6
7
8
2.40 3.00 3.60 4.20 4.80
At this rate, how much do 5 pounds of grapes cost? $ 3.00
Dollars
1.80
Math Journal 2, p. 312
Lesson 12 2
䉬
917
Student Page
Date
Time
LESSON
Rate Tables
12 2
䉬
continued
Ongoing Assessment:
Recognizing Student Achievement
5. Malia bought 6 yards of fabric for $15.00.
Cost of fabric: 6 yards for $15.00
Dollars
2.50
5
7.50
1
2
3
Yards
10 12.50
4
1
At this rate, how much will 72 yards of fabric cost? $
17.50 20
15.00
5
6
7
8
Use an Exit Slip (Math Masters, page 389) to assess students’ ability to
describe a rule for a pattern and use the rule to solve problems. Have
students describe the pattern in the rate table in Problem 3 on journal page 312.
Students are making adequate progress if their explanation demonstrates an
understanding of the pattern; that is, the numbers in the top row of the rate
table are all multiples of 8, and each number in the top row can be found by
multiplying the number in the bottom row by 8. Some students may note that
they can use the pattern to find values for numbers not given in the table. For
example, a gray whale’s heart beats 120 times in 15 minutes, or 36 times in
1
42 minutes.
18.75
3
6. Alden bought 4 of a pound of cheese for $6.
3
Cost of cheese: 4 of a pound for $6
Pounds
1
4
1
2
3
4
1
14
12
14
2
Dollars
2
4
6
8
10
12
14
16
1
At this rate, how much will 14 pounds cost? $
1
3
1
10.00
Try This
7. The Jefferson family plans to sit down to Thanksgiving dinner at 6:00 P.M. They have
an 18-pound turkey. The turkey needs to cook about 20 minutes per pound.
a. At what time should the turkey go in the oven?
Exit Slip
12 noon
[Patterns, Functions, and Algebra Goal 1]
b. Explain what you did to solve the problem.
Sample answer: The turkey will take a total of
18 ⴱ 20 360 minutes to cook. 360 / 60 6 hours.
I counted back 6 hours from 6:00. The turkey
should go in at noon.
2 Ongoing Learning & Practice
Math Journal 2, p. 313
䉴 Playing the Credits/Debits
PARTNER
ACTIVITY
Game (Advanced Version)
(Student Reference Book, p. 239; Math Masters, p. 469)
Students play the Credits/Debits Game (Advanced Version) to
practice adding and subtracting integers. See Lesson 11-6 for
additional information.
䉴 Math Boxes 12 2
䉬
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 314)
Mixed Practice Math Boxes in this lesson are linked with
Math Boxes in Lessons 12-4 and 12-6.
Student Page
Date
Time
LESSON
12 2
䉬
Math Boxes
1. a. Complete the table.
Number
of Pizzas
1
Number
of Servings
3
2. Complete.
2
6
a. 3 gal 3
4
9
12
5
b. 4 qt 15
d.
b. How many pizzas are
needed for 279 servings?
93
c. 6 pt 11
e. 64 gal pizzas
12
8
12
qt
pt
c
1 c 23 c
512 pt
pt
47
3. Find the solution of each open sentence.
a. m 40 60
m
b. 55 q 40
q
c. (23) s 0
s
d. p (36) 80
p
100
15
23
44
137
4. Complete the name-collection box.
Sample answers:
8.01
8 0.01
10 1.99
6 2.01
2.67 ⴱ 3
32.04 4
148
5. Fill in the blank with one of
the words below.
impossible
unlikely
䉴 Study Link 12 2
䉬
149
INDEPENDENT
ACTIVITY
(Math Masters, p. 341)
6. Calculate.
A
D
C
A
D
B
A
A
B
A
A
B
a. 10% of 460 b. 5% of 120 c. 40% of
likely
unlikely
Without looking, it is
that
a B block will be pulled from the bag.
d.
e.
80
Math Journal 2, p. 314
918
Writing/Reasoning Have students write a response to the
following: Look carefully at the bag of blocks in Problem 5.
Describe two events that are equally likely. Express the
probability of each event as a fraction. Sample answers: Pulling a
B block and pulling a C or D block are equally likely events. The
3
1
probability of each event is 12, or 4. Pulling an A block and pulling
a B, C, or D block are equally likely events. The probability of each
6
1
event is 12, or 2.
Unit 12 Rates
f.
30
50
38
10
46
6
Home Connection Students solve problems involving
rates. In some problems, students need to calculate the
unit rate when it is not given.
4
% of 20 6
% of 92 46
% of 150 57
38
39
Study Link Master
Name
3 Differentiation Options
READINESS
䉴 Illustrating Rate Problems
Rates
12 2
䉬
PARTNER
ACTIVITY
175 176
1.
Hotels R Us charges $45 per night for a single room.
At that rate, how much does a single room cost per week?
$
315
2.
The Morales family spends about $84 each week for
food. On average, how much do they spend per day?
$
12
Sharon practices playing the piano the same amount
of time each day. She practiced a total of 4 hours
on Monday and Tuesday combined. At that rate,
how many hours would she practice in a week?
14
3.
15–30 Min
Literature Link To explore rate situations using a visual
model, have students read Each Orange Had Eight
Slices: A Counting Book by Paul Giganti, Jr. (Greenwillow
Books, 1992). Ask students to use rate language to describe each
situation. For example: “Two juicy oranges. Eight slices per orange.
Two seeds per slice.”
In a Math Log, have students illustrate their own rate situation
and use rate language to describe it.
INDEPENDENT
ACTIVITY
Hours
2
4
6
8
Days
1
2
3
4
hours
10 12 14
5
6
7
Try This
4.
People in the United States spend an average of 6 hours and 4 minutes
each week reading newspapers.
364
a.
That’s how many minutes per week?
b.
At that rate, how much time does an average
person spend reading newspapers in a 3-day period?
Minutes
Days
minutes per week
156
minutes
52 104 156 208 260 312 364
1
2
3
4
5
6
7
Practice
5.
7.
䉴 Representing Rates with
Time
Solve the problems.
(Math Masters, p. 388)
ENRICHMENT
Date
STUDY LINK
9,096 24 º 379
81 R4
652 8 6.
870 º 63 8.
546 42 54,810
13
Math Masters, p. 341
5–15 Min
Line Graphs
(Math Journal 2, pp. 312 and 313; Math Masters, p. 443)
To further explore representing rates with line graphs,
have students choose a problem on journal page 312 or
313 and display the values found in the rate table on a
line graph. Remind students to choose a reasonable title and
labels for the graph.
NOTE Continuous quantities can be divided into smaller and smaller amounts.
Measures, such as the height and weight of a student in your class, are
continuous quantities. Discrete quantities are countable things that cannot be
broken up into smaller amounts, such as the number of students in your class.
The rates in this lesson can be represented by line graphs, since most of the
variables represent continuous, not discrete, quantities.
EXTRA PRACTICE
䉴 Solving Rate Problems
Jennifer’s Earnings
16
14
Earnings (dollars)
Students should notice that when the dots on the graph are
connected, they form a straight line. Discuss how to obtain values
from the graph. In the example to the right, to find out how much
1
money Jennifer earns in 22 hours, move along the horizontal axis
1
to 22, which is halfway between 2 and 3 hours, and mark the point
1
on the line that is directly above 22. Then move left from there to
1
5 on the vertical axis. Thus, Jennifer earns $5.00 in 22 hours.
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
Time (hours)
INDEPENDENT
ACTIVITY
5–15 Min
(Math Masters, p. 454)
To provide practice solving rate problems, have students use rate
tables to record given rate information and generate equivalent
rates. Use Math Masters, page 454 to create problems to meet the
needs of individual students, or have students create and solve
their own problems.
Lesson 12 2
䉬
919