0.2.7 Comparing Proportional and
Nonproportional Relationships (Part 2)
Lesson Objectives
Understand that a relationship between two
quantities represented by and is proportional
if and only if there is some
such that
.
Required Materials
snap cubes
four-function calculators
Recognize when a relationship represented by a
table is not proportional and explain why.
Recognize when a relationship represented by a
table could be proportional and explain why.
Recognize when a relationship represented by an
equation is or is not proportional and explain why.
Setup: Notice and Wonder: Patterns (5 minutes)
Students in groups of 2. 1 minute of quiet think time, followed by partner and whole-group discussions.
Statement
Anticipated Responses
Do you see a pattern? What predictions can you
make about future rectangles in the set if your
pattern continues?
Setup: More Conversions (10 minutes)
Statement
Anticipated Responses
The other day you worked with converting meters,
centimeters, and millimeters. Here are some more
1. Temperature ( F): 68; 39.2, 347.
unit conversions.
2. Length (cm): 25.4; 20.32; 8.89.
1. Use the equation
, where
represents degrees Fahrenheit and represents
degrees Celsius, to complete the table.
3. The temperature conversion does not determine
a proportional relationship because the number
of degrees Fahrenheit per degree Celsius is not
the same. The length conversion does determine
a proportional relationship because the number
temperature ( F)
of centimeters per inch is the same.
temperature ( C)
20
4
175
2. Use the equation
, where
represents
the length in centimeters and represents the
length in inches, to complete the table.
length (in)
length (cm)
10
8
3
3. Are these proportional relationships? Explain why
or why not.
Setup: Total Edge Length, Surface Area, and Volume (10 minutes)
Statement
Anticipated Responses
Here are some cubes with different side lengths.
1. A cube has 12 edges. Total edge lengths: 36, 60,
Complete each table. Be prepared to explain your
reasoning.
114,
.
2. A cube has 6 faces each with an area of
units. Surface areas: 54, 150,
,
.
square
3. The bottom layer of a cube fits
cubic units and s
of these layers make up the cube. Volumes: 27,
125,
, .
4. The relationship between side length and total
edge length is proportional but the relationships
between side length and surface area, and
between side length and volume are not.
1. How many edges does a cube have? How long is
the total edge length of each cube?
5.
,
,
side length
total edge length
3
5
2. How many faces does a cube have? What is the
area of one face? What is the surface area of each
cube?
side length
surface area
3
5
3. How many cubic units fit on the bottom layer of
each cube? How many of these layers make up the
cube? What is the volume of each cube?
side length
3
5
volume
4. Which of these relationships is proportional?
Explain how you know.
5. Write equations for total edge length , total
surface area , and volume of a cube with side
length .
Setup: All Kinds of Equations (10 minutes)
Statement
Here are six different equations.
1. Complete each table using the equation that
represents the relationship.
Anticipated Responses
2. Which of the equations represent proportional
relationships? Which equations do not?
3. What do the equations of the proportional
relationships have in common?
4. What do the equations of the nonproportional
relationships have in common?
Setup: Tables and Chairs (5 minutes)
Cool-down (5 minutes)
Anticipated Responses
Andre is setting up rectangular tables for a party. He
1.
(or
). This relationship is proportional
can fit 6 chairs around a single table. Andre lines up
10 tables end-to-end and tries to fit 60 chairs around
because it can be represented with an equation of
the form
(or
). There are 6 chairs per
them, but he is surprised when he cannot fit them all.
table no matter how many tables.
1. Write an equation for the relationship between
the number of chairs and the number of tables
if the tables are apart from each other
2.
(or
). This relationship is not
proportional because the number of chairs per
table changes depending on how many tables
there are. The quotient of chairs and tables is not
constant. The relationship can not be expressed
with an equation of the form
.
if the tables are placed end-to-end
2. Is the first relationship proportional? Explain how
you know.
3. Is the second relationship proportional? Explain
how you know.
Lesson Summary (5 minutes)
We saw many equations that represent proportional relationships. What do they all have in common?