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

Three rectangles on a coordinate grid. The dimensions are as follows: Top rectangle, length 3 units; width 1 unit. Middle rectangle, length 6 units; width 2 units. Bottom rectangle, length 9 units, width 3 units.
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.

  1. 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 $(^\circ\text{F})$
    row 1 20  
    row 2 4  
    row 3 175  
  2. 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)
    row 1 10  
    row 2 8  
    row 3 3$\frac12$  
  3. 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.

Three cubes of different sizes: first cube has side length 3, second cube side length 5, and thrid cube has side length 9 and 1/2
  1. How long is the total edge length of each cube?
row 1 side
length		 total
edge length
row 2 3  
row 3 5  
row 4 $9\frac12$  
row 4 $s$  
  1. What is the surface area of each cube?
row 1 side
length		 surface
area
row 2 3  
row 3 5  
row 4 $9\frac12$  
row 5 $s$  
  1. What is the volume of each cube?
row 1 side
length		 volume
row 2 3  
row 3 5  
row 4 $9\frac12$  
row 5 $s$  
  1. Which of these relationships is proportional? Explain how you know.

  2. Write equations for the total edge length $E$, total surface area $A$, and volume $V$ of a cube with side length $s$.

8.4: All Kinds of Equations

Here are six different equations.

$y = 4 + x$

$y = \frac{x}{4}$

$y = 4x$

$y = 4^{x}$

$y = \frac{4}{x}$

$y = x^{4}$

  1. Predict which of these equations represent a proportional relationship.
  1. Complete each table using the equation that represents the relationship.
Six identical three column tables with 4 rows of data: The first column is labeled "x", the second column is labeled "y", and the third column is labeled "the fraction y over x". Row 1: x, 2. Row 2: x, 3. Row 3: x, 4. Row 4: x, 5. Each table has an equation above it, as follows: Table 1, Equation 1: y equals 4 + x; Table 2, Equation 2: y equals 4x; Table 3, Equation 3: y equals the fraction 4 over x; Table 4, Equation 4: y equals x the fraction x over 4; Table 5, Equation 5: y equals 4 to power x; Table 6, Equation 6: y equals x to the power 4;
  1. Do these results change your answer to the first question? Explain your reasoning.
  2. 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.

row 1 $a$ $b$ $\frac{b}{a}$
row 2 20 100 5
row 3 3 15 5
row 4 11 55 5
row 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. 

Practice Problems ▶