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 rectangles are 3 units long by 1 unit wide, 6 units long by 2 units wide and 9 units long by 3 units wide.

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 \((^\circ\text{F})\)
    20
    4
    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)
    10
    8
    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?
    side
    length
    total
    edge length
    3
    5
    \(9\frac12\)
    \(s\)
  2. What is the surface area of each cube?
    side
    length
    surface
    area
    3
    5
    \(9\frac12\)
    \(s\)
  3. What is the volume of each cube?
    side
    length
    volume
    3
    5
    \(9\frac12\)
    \(s\)
  4. Which of these relationships is proportional? Explain how you know.
  5. Write equations for the total edge length \(E\), total surface area \(A\), and volume \(V\) of a cube with side length \(s\).



  1. 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?
  2. A different rectangular solid has length \(\ell\), width 10, height 5, and volume \(V\). Is the relationship between \(\ell\) and \(V\) a proportional relationship?
  3. 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.

\(y = 4 + x\)

\(y = \frac{x}{4}\)

\(y = 4x\)

\(y = 4^{x}\)

\(y = \frac{4}{x}\)

\(y = x^{4}\)

Six identical tables, each with 3 columns and 4 rows of data. 
  1. Predict which of these equations represent a proportional relationship.
  2. Complete each table using the equation that represents the relationship.
  3. Do these results change your answer to the first question? Explain your reasoning.
  4. 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.