PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
This is Google's cache of http://www.cehd.umn.edu/rationalnumberproject/89_4.html. It is a snapshot of the page as
it appeared on Jul 23, 2011 19:19:04 GMT. The current page could have changed in the meantime. Learn more
These terms only appear in links pointing to this page:
http www cehd umn edu rationalnumberproject images 89_4 sheet_2 pdf
Text-only version
Rational Number Project Home Page
Cramer, K., Behr, M., & Bezuk, N. (1989, October). Proportional
Relationships and Unit Rates. Mathematics Teacher, 82 (7), 537-544.
PROPORTIONAL RELATIONSHIPS AND UNIT
RATES
KATHLEEN CRAMER
University of Wisconsin-River Falls, River Falls, WI 54022
NADINE BEZUK
San Diego State University, San Diego, CA 92115
MERLYN BEHR
Northern Illinois University, DeKalb, IL 60115
Teacher's Guide
Introduction: This set of lessons extends ideas developed in the
September "Activities" (Cramer, Post, and Behr 1989). In those lessons
situations involving proportional and nonproportional relationships were
modeled through physical experiments and interpreted using tables and
graphs. Students learned to discriminate between proportional and
nonproportional relationships in two ways: (1) the graph of a
proportional relationship forms a straight line through the origin and (2)
the formula or number sentence used to describe the set of data points
in a proportional relationship involves only multiplication or division.
The activities that follow interpret proportional relationships using a unitrate approach found to be more meaningful than the traditional crossmultiplication algorithm presented in most textbooks (Post, Behr, and
Lesh 1988). Students have an intuitive understanding of unit rate from
their shopping experiences. One-step multiplication and division story
problems introduced in third grade can be thought of as unit-rate
problems. In multiplication problems the unit rate is given and becomes
one of the factors. For example,
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 1 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
Mary bought 3 boxes of pencils. Each box contained 5 pencils.
How many pencils did Mary buy?
The unit rate in this problem is 5 pencils per box. The solution is 3 times
this unit rate. In division problems the unit rate is asked for and
becomes the answer. For example,
Mary bought 3 boxes of pencils and ended up with 15 pencils.
How many pencils were in each box?
The solution, 15 pencils/3 boxes = 5 pencils/ 1 box, is a unit rate.
The missing-value problems introduced in later grades can be seen as
extensions of these one-step multiplication and division problems. The
foregoing problems can be combined as follows:
Mary bought 3 boxes of pencils and ended up with 15 pencils.
How many pencils would she have purchased if she had bought
7 boxes of pencils?
Here it is useful to find out how many pencils are in one box (unit rate
= 5 pencils/1 box) and then to multiply this unit rate by the seven
boxes of pencils.
One should note that the foregoing problem actually involves two unit
rates: 5 pencils/l box and (1/5 box)/1 pencil. In solving missing-value
problems only one unit rate is appropriate. Students need experience
interpreting each possible unit rate so that they can determine which
rate is appropriate to the problem. The unit rate expressing how many
boxes for one pencil can be used in the following situation:
Mary bought 3 boxes of pencils and ended up with 15 pencils.
How many boxes did she buy if she bought 30 pencils?
The problem can be solved by multiplying the unit rate by the number of
pencils; (1/5 box)/1 pencil X 30 pencils.
Changing the context and numbers involved will change the difficulty
level of these missing-value problems. The following activities introduce
the concept of unit rate in the familiar buying context but also use the
less familiar money-exchange context. These activities, combined with
the activities from "Interpreting Proportional Situations" referenced
earlier, give students an expanded view of proportional relationships and
meaningful alternatives to solving traditional missing-value problems.
Grade levels: 6-8
Materials: Activity sheets 1-4 for each student, transparencies of these
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 2 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
sheets for class discussion, and calculators (optional)
Objectives: Students will develop an understanding of the unit rates
associated with a proportional relationship. Students will also develop
the ability to determine the appropriate rate to use in solving a problem
and to use the corresponding unit rate to solve missing-value problems.
Directions: Sheet 1 displays a teacher-led lesson, The pictures help
students see that division is used to calculate a unit rate, as well as help
students determine which of the two parts of the rate relationship
should be the divisor in the division problem.
The teacher should model how to
use the picture to find the unit
rate. In problem 1, the teacher
points out that to find the cost of
one apple, the money has to be
divided into three equal parts. A
simple mapping, such as that in
figure 1, can show the cost of one
apple. The teacher should record
the information depicted by the
mapping (30 cents/1 apple), then
write a sentence interpreting this
rate.
Students should find the unit rate for problem 2 in a similar manner,
then write the interpretive statement. In this instance, the cost of one
orange does not equal an integral amount.
The less familiar contexts of problems 3 and 4 can be addressed in a
similar manner. Two possible rates are generated for the same
relationship (2 British pounds = 3 U.S. dollars). Each results in its own
diagram.
The following questions can be used to help students solve the problems
on sheet 1:
1. What operation was modeled in each solution?
2. For each problem, what quantity was divided and into how many
equal groups? Can you explain why?
3. What computation can be set up to solve each problem?
4. How can each division problem be written using fractional notation?
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 3 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
On Sheet 2, students are asked to generate two possible rates for each
relationship, compute the equivalent unit rates, and write a short
sentence interpreting each of the unit rates. The rates should be written
in the form "a/b." Be sure that students do not "drop" the labels. They
are necessary to interpret the rate. Students must understand that a
unit rate always describes "how many for one." Some unit rates have
strange or nonfunctional meanings, For example, 3 tennis balls/1 can is
easier to envision than (1/3 can)/1 tennis ball.
On Sheet 3, students learn how to select an appropriate unit rate and
use the unit rate to solve missing-value problems. Additional practice in
selecting the appropriate rate can be furnished by posing missing value
questions for each of the situations presented on sheet 2. This additional
experience may be necessary for many students before they attempt the
applications on Sheet 4.
Students who have studied slope should note that the unit rate in a
proportional relationship is the same as the slope of the corresponding
line. To summarize these two related activities, students should
generate a list of characteristics of proportional relationships on the
basis of the ideas developed in both sets of activities.
The links will open a new window with the activity sheets in .pdf format
Sheet
Sheet
Sheet
Sheet
1
2
3
4
Answers:
Sheet 1
Sheet 2
1. 6 bags/30 pounds
30 pounds/6 bags
(1/5) bag/1 pound
One pound of flour fills 1/5 of a bag,
5 pounds/1 bag
One bag holds 5 pounds of flour.
2. 9 balls/3 cans
3 cans/9 balls
3 balls/1 can
One can contains 3 balls.
(1/3) can/1 ball
One ball fills 1/3 of a can.
3. 5 gallons/$6.50
$6.50/5 gallons
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 4 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
0.769 gallons/$1
One dollar will purchase 0.769 gallons.
$1.30/1 gallon
One gallon costs $1.30.
4. 25 minutes/10 laps
10 laps/25 minutes
2.5 minutes/1 lap
One lap is run in 2.5 minutes.
0.4 laps/1 minute
In one minute 0.4 laps are run.
Sheet 3
Problem A
1. 3 apples/90 cents
90 cents/3 apples
2. (1/30) apple/1 cent
30 cents/1 apple
3. The amount of apple for one cent; the cost of one apple
4. 30 cents/1 apple
5.
Apples
Cost
1
2
3
4
5
.30
.60
.90
1.20
1.50
6. The price increases by thirty cents for each apple; the cost of apples
can be found by multiplying the number of apples by thirty cents.
7.
30 cents/1 apple x 7 apples = $2.10-1 (unit price) x (number of
apples) = cost .
Problem B
1. 2 pounds/3 dollars;
3 dollars/2 pounds
2. 2/3 pound/1 dollar;
1.5 dollars/1 pound
3.
The number of British pounds that equal one U.S. dollar; the
number of U.S. dollars that equal one British pound
4. 2/3 pound/1 dollar
5.
Dollars
Pounds
1
2
3
2/3
4/3
6/3=2
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 5 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
4
5
6.
7/28/11 1:59 PM
8/3
10/3
The number of pounds can be found by multiplying the number of
dollars by 2/3.
7. 2/3 pounds/1 dollar X 20 dollars = 40/3 pounds ~= 13.33
pounds;(unit price) x (number of U.S. dollars) = number of pounds.
Sheet 4
1.
3 cans white/2 cans
blue
1.5 cans white/1 can
blue
Anne should mix 1.5 cans of white paint with each can of blue. (1.5
cans white/1 can blue) x 6 cans blue = 9 cans white.
2.
5 cups water/2 scoops
mix
2.5 cups water/1 scoop
mix
Ryan should use 2.5 cups of water for each scoop of mix. (2.5 cups
water/1 scoop mix) X (9 scoops mix) = 22.5 cups water.
3. 10 minutes/6 laps
5/3 minutes/1 lap
Donna runs 1 lap in 5/3 minutes. (5/3 minutes/1 lap) x (5 laps) = 8
1/3 minutes.
4. 12 laps/4 minutes
3 laps/1 minute
Mark's train travels 3 laps each minute. (3 laps/1 minute) X (9
minutes) = 27 laps.
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 6 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
REFERENCES
Cramer, Kathleen, Thomas R. Post, and Merlyn Behr. "Activities:
Interpreting Proportional Situations." Mathematics Teacher 82
(September 1989):445-452.
Post, Thomas R., Merlyn J. Behr, and Richard Lesh. "Proportionality and
the Development of Prealgebra Understandings." In The Ideas of
Algebra, K-12. 1988 Yearbook of the National Council of Teachers of
Mathematics, edited by Arthur F. Coxford and Albert P. Shulte. Reston,
Va.: The Council, 1998.
This paper is based in part on research supported by the National
Science Foundation under grants DPE840077 and TE1-8652431 (the
Rational Number Project). Any opinions, findings, and conclusions
expressed are those of the authors and do not necessarily reflect the
views of the National Science Foundation.
Edited by Robert A. Laing and Dwayne E. Channell, Western Michigan
University, Kalamazoo, MI 49008
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 7 of 8
PROPORTIONAL RELATIONSHIPS AND UNIT RATES
7/28/11 1:59 PM
This section is designed to provide mathematical activities in
reproducible formats appropriate for students in grades 7-12. This
material may be reproduced by classroom teachers for use in their own
classes. Readers who have developed successful classroom activities are
encouraged to submit manuscripts, in a format similar to the 'Activities"
already published, to the editorial coordinator for review. Of particular
interest am activities focusing on the Council's curriculum standards, its
expanded concept of basic skills, problem solving and applications, and
the uses of calculators and computers.
file:///Users/millsv/Library/Caches/TemporaryItems/Outlook%20Temp/PROPORTIONAL%20RELATIONSHIPS%20AND%20UNIT%20RATES.webarchive
Page 8 of 8