Lesson 12Using Graphs to Compare Relationships

Let’s graph more than one relationship on the same grid.

Learning Targets:

  • I can compare two related proportional relationships based on their graphs.
  • I know that the steeper graph of two proportional relationships has a larger constant of proportionality.

12.1 Number Talk: Fraction Multiplication and Division

Find each product or quotient mentally.

\frac23 \boldcdot \frac12

\frac43 \boldcdot \frac14

4 \div \frac15

\frac96 \div \frac12

12.2 Race to the Bumper Cars

Diego, Lin, and Mai went from the ticket booth to the bumper cars. Descriptions and tables representing their journeys are below.

1. Read each description and complete each table.

  • Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
  • Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
  • Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
Diego's time (seconds) Diego's distance (meters)
0
15
30 50
1
Lin's time (seconds) Lin's distance (meters)
0
25
20 50
1
Mai's time (seconds) Mai's distance (meters)
0
25
40 50
1

2. Draw a graph for each person’s journey, including Tyler's (from before).

  • Drag the names to the correct lines to label them.
  • If you choose to, you can use the Paint Brush tool to change the color of each line. Select the tool, click on a color in the palette below the graph, and then click on a line. Click on the Move tool (the arrow) before changing to a new paint brush color.
  • You can hide any points you create with the checkbox below the graph.

3. For Tyler, Diego, and Lin, which person is moving the most quickly? How is that reflected in the graph?

Are you ready for more?

Write equations to represent each person’s relationship between time and distance.

12.3 Space Rocks and the Price of Rope

  1. Meteoroid Perseid 245 and an unknown asteroid were traveling through the solar system.

    Explore the applet to learn about the distance they had each traveled after a given time.

    Is Asteroid x traveling faster or slower than Perseid 245? Explain how you know.

  2. The graph shows the price, p , of different lengths, L , of two types of rope.

    Two lines graphed in the coordinate plane labeled "Cotton" and "Nylon". There is a horizontal L axis and a vertical p axis. The line labeled "Nylon" begins at the origin and moves steadily upward and to the right. The line labeled "Cotton" also begins at the origin moves steeply upward and to the right.

    If you buy $1.00 of each kind of rope, which one will be longer? Explain how you know.

Lesson 12 Summary

Here is a graph that shows the price of blueberries at two different stores. Which store has a better price?

We can compare points that have the same x value or the same y value. For example, the points (2, 12) and (3, 12) tell us that at store B you can get more pounds of blueberries for the same price.

The points (3, 12) and (3, 18) tell us that at store A you have to pay more for the same quantity of blueberries. This means store B has the better price.

We can also use the graphs to compare the constants of proportionality. The line representing store B goes through the point (1, 4) , so the constant of proportionality is 4. This tells us that at store B the blueberries cost $4 per pound. This is cheaper than the $6 per pound unit price at store A.

Lesson 12 Practice Problems

  1. Match each equation to its graph.

    1. y = 2x
    2. y = \frac45 x
    3. y = \frac14 x
    4. y = \frac23 x
    5. y = \frac43 x
    6. y = \frac32 x

    1

    A line is graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 5 are indicated on the horizontal axis. The numbers 0 through 5, are indicated on the vertical axis. The line begins at the origin. It moves steadily upwards and to the right passing through the points with coordinates 1 comma four fifths, and 5 comma 4.

    2

    Graph 2 is a line graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 5 are indicated on the horizontal axis. The numbers 0 through 5 on the vertical axis. There are two evenly spaced horizontal gridlines between each integer. The line begins at the origin. It moves steadily upward and to the right passing through the points with coordinates 1 comma two-thirds, and 3 comma 2.

    3

    Graph 3 is a line graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 5 are indicated on the horizontal axis. The numbers 0 through 5 are indicated on the vertical axis. There are 2 evenly spaced horizontal gridlines between each integer. The line begins at the origin. It moves steadily upward and to the right passing through the points with coordinates 1 comma 1 and one-third, and 3 comma 4.

    4

    Graph 4 is a line graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 5 are indicated on the horizontal axis. The numbers 0 through 5 are indicated on the vertical axis. There are three evenly spaced horizonatal gridlines between each interger. The line begins at the origin. It moves gradually upwards and to the right passing through the points with coordinates 1 comma one-fourth, and 4 comma 1.

    5

    Graph 5 is a line graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 5 are indicated on the horizontal axis. The numbers 0 through 5 are indicated on the vertical axis. There are also horizontal gridlines midway between each integer. The line begins at the origin. It moves steeply upward and to the right, passing through the points with coordinates 1 comma 2, and 2 comma 4.

    6

    Graph 6 is a line graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 5 are indicated on the horizontal axis. The numbers 0 through 5 are indicated on the vertical axis. There are horizontal gridlines midway between each integer. The line begins at the origin. It moves steadily upward and to the right, passing through the points with coordinates 1 comma 1 point 5 and 2 comma 3.
  2. The graphs below show some data from a coffee shop menu. One of the graphs shows cost (in dollars) vs. drink volume (in ounces), and one of the graphs shows calories vs. drink volume (in ounces).

    __________________ vs volume

    A coordinate plane with the origin labeled “O”. The horizontal axis is labeled “volume in ounces” and the numbers 0 through 26, in increments of 2, are indicated. The vertical axis has the numbers 0 through 6 indicated. The points with coordinates 10 comma 3 point 7 5, 12 comma 4, 16 comma 4 point 5, and 24 comma 4 point 9 5 are indicated.

    _____________________ vs volume

    A coordinate plane with the origin labeled “O”. The horizontal axis is labeled “volume in ounces” and the numbers 0 through 26, in increments of 2, are indicated. The vertical axis has the numbers 0 through 350, in increments of 50 indicated. The points 10 comma 150, 12 comma 180, 15 point 9 comma 2 hundred 38 point 5, and 24 comma 360 are indicated.
    1. Which graph is which? Give them the correct titles.
    2. Which quantities appear to be in a proportional relationship? Explain how you know.
    3. For the proportional relationship, find the constant of proportionality. What does that number mean?
  3. Lin and Andre biked home from school at a steady pace. Lin biked 1.5 km and it took her 5 minutes. Andre biked 2 km and it took him 8 minutes. 

    1. Draw a graph with two lines that represent the bike rides of Lin and Andre. 
    2. For each line, highlight the point with coordinates (1,k) and find k .
    3. Who was biking faster?