Lesson 8
Comparing Relationships with Equations
Let’s develop methods for deciding if a relationship is proportional.
8.1: Notice and Wonder: Patterns with Rectangles

Do you see a pattern? What predictions can you make about future rectangles in the set if your pattern continues?
8.2: More Conversions
The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
- Use the equation F =\frac95 C + 32, where F represents degrees Fahrenheit and C represents degrees Celsius, to complete the table.
temperature (^\circ\text{C}) temperature (^\circ\text{F}) 20 4 175 - Use the equation c = 2.54n, where c represents the length in centimeters and n represents the length in inches, to complete the table.
length (in) length (cm) 10 8 3\frac12 - Are these proportional relationships? Explain why or why not.
8.3: Total Edge Length, Surface Area, and Volume
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 length3 5 9\frac12 s - What is the surface area of each cube?
side
lengthsurface
area3 5 9\frac12 s - What is the volume of each cube?
side
lengthvolume 3 5 9\frac12 s - Which of these relationships is proportional? Explain how you know.
-
Write equations for the total edge length E, total surface area A, and volume V of a cube with side length s.
- A rectangular solid has a square base with side length \ell, height 8, and volume V. Is the relationship between \ell and V a proportional relationship?
- A different rectangular solid has length \ell, width 10, height 5, and volume V. Is the relationship between \ell and V a proportional relationship?
- Why is the relationship between the side length and the volume proportional in one situation and not the other?
8.4: All Kinds of Equations
Here are six different equations.
y = 4 + x
y = \frac{x}{4}
y = 4x
y = 4^{x}
y = \frac{4}{x}
y = x^{4}

- 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?
Summary
If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of a and b, two quantities that are in a proportional relationship.
a | b | \frac{b}{a} |
---|---|---|
20 | 100 | 5 |
3 | 15 | 5 |
11 | 55 | 5 |
1 | 5 | 5 |
Notice that the quotient of b and a is always 5. To write this as an equation, we could say \frac{b}{a}=5. If this is true, then b=5a. (This doesn’t work if a=0, but it works otherwise.)
If quantity y is proportional to quantity x, we will always see this pattern: \frac{y}{x} will always have the same value. This value is the constant of proportionality, which we often refer to as k. We can represent this relationship with the equation \frac{y}{x} = k (as long as x is not 0) or y=kx.
Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.