Lesson 8Comparing Relationships with Equations
Learning Goal
Let’s develop methods for deciding if a relationship is proportional.
Learning Targets
I can decide if a relationship represented by an equation is proportional or not.
Warm Up: Notice and Wonder: Patterns with Rectangles
Problem 1
Do you see a pattern? What predictions can you make about future rectangles in the set if your pattern continues?
Activity 1: More Conversions
Problem 1
The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
Use the equation
, where represents degrees Fahrenheit and represents degrees Celsius, to complete the table. temperature
temperature
Use the equation
, where represents the length in centimeters and represents the length in inches, to complete the table. length (in)
length (cm)
Are these proportional relationships? Explain why or why not.
Activity 2: Total Edge Length, Surface Area, and Volume
Problem 1
Here are some cubes with different side lengths. Complete each table. Be prepared to explain your reasoning.
How long is the total edge length of each cube?
side
lengthtotal
edge lengthWhat is the surface area of each cube?
side
lengthsurface
areaWhat is the volume of each cube?
side
lengthvolume
Problem 2
Which of these relationships is proportional? Explain how you know.
Problem 3
Write equations for the total edge length
Are you ready for more?
Problem 1
A rectangular solid has a square base with side length
Problem 2
A different rectangular solid has length
Problem 3
Why is the relationship between the side length and the volume proportional in one situation and not the other?
Activity 3: All Kinds of Equations
Problem 1
Here are six different equations.
Predict which of these equations represent a proportional relationship.
Complete each table using the equation that represents the relationship.
Do these results change your answer to the first question? Explain your reasoning.
What do the equations of the proportional relationships have in common?
Lesson Summary
If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of
Notice that the quotient of
If quantity
Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.