Lesson 5Comparing 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

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?

Activity 1: More Conversions

Problem 1

The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.

  1. Use the equation , where represents degrees Fahrenheit and represents degrees Celsius, to complete the table.

    temperature

    temperature

  2. Use the equation , where represents the length in centimeters and represents the length in inches, to complete the table.

    length (in)

    length (cm)

  3. 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.

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

  2. What is the surface area of each cube?

    side
    length

    surface
    area

  3. What is the volume of each cube?

    side
    length

    volume

Problem 2

Which of these relationships is proportional? Explain how you know.

Problem 3

Write equations for the total edge length , total surface area , and volume of a cube with side length .

Are you ready for more?

Problem 1

A rectangular solid has a square base with side length , height 8, and volume . Is the relationship between and a proportional relationship?

Problem 2

A different rectangular solid has length , width 10, height 5, and volume . Is the relationship between and a proportional relationship?

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.

  1. Predict which of these equations represent a proportional relationship.

  2. 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;
  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?

Lesson Summary

If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of and , two quantities that are in a proportional relationship.

Notice that the quotient of and is always 5. To write this as an equation, we could say . If this is true, then . (This doesn’t work if , but it works otherwise.)

If quantity is proportional to quantity , we will always see this pattern: will always have the same value. This value is the constant of proportionality, which we often refer to as . We can represent this relationship with the equation (as long as is not 0) or .

Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.