Lesson 9Two Graphs for Each Relationship

Learning Goal

Let’s use tables, equations, and graphs to answer questions about proportional relationships.

Learning Targets

  • I can interpret a graph of a proportional relationship using the situation.

  • I can write an equation representing a proportional relationship from a graph.

Lesson Terms

  • coordinate plane
  • origin

Warm Up: True or False: Fractions and Decimals

Problem 1

Decide whether each equation is true or false. Be prepared to explain your reasoning.

Activity 1: Tables, Graphs, and Equations

Problem 1

Explore the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.

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  1. What stays the same and what changes in the table? in the equation? on the graph?

    • Choose one row in the table and write it here. To what does this row correspond on the graph?

  3. Grab and drag the point until you see the equation .

    • Do not move the point. Choose three rows from the table, other than the origin. Record and , and compute .

  4. What do you notice? What does this have to do with the equation of the line?

  5. Do not move the point. Check the box to view the coordinates . What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?

  6. Drag the point to a different location. Record the equation of the line, the coordinates of three points, and the value of .

    Equation of the line:

  7. Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.

Print Version

Your teacher will assign you one of these three points:

, , .

A blank xy-plane where the numbers 0 through 10 appear on both axes.

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  1. On the graph, plot and label only your assigned point.

  2. Use a ruler to line up your point with the origin, . Draw a line that starts at the origin, goes through your point, and continues to the edge of the graph.

  3. Complete the table with the coordinates of points on your graph. Use a fraction to represent any value that is not a whole number.

  4. Write an equation that represents the relationship between and defined by your point.

  5. Compare your graph and table with the rest of your group. What is the same and what is different about:

    • your tables?

    • your equations?

    • your graphs?

  6. What is the -coordinate of your graph when the -coordinate is 1? Plot and label this point on your graph. Where do you see this value in the table? Where do you see this value in your equation?

  7. Describe any connections you see between the table, characteristics of the graph, and the equation.

Are you ready for more?

Problem 1

The graph of an equation of the form , where is a positive number, is a line through and the point .

  1. Name at least one line through that cannot be represented by an equation like this.

  2. If you could draw the graphs of all of the equations of this form in the same coordinate plane, what would it look like?

Activity 2: Hot Dog Eating Contest

Problem 1

Andre and Jada were in a hot dog eating contest. Andre ate 10 hot dogs in 3 minutes. Jada ate 12 hot dogs in 5 minutes.

  1. The points shown on the first set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.

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  2. Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point. Write an equation for each line. Use to represent time in minutes, and to represent number of hot dogs.

  3. What does the constant of proportionality tell you in each case?

  4. The points shown on the second set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.

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  5. Draw lines from the origin through each of the two points. Write an equation for each line. What does the constant of proportionality tell you in each case?

Print Version

Andre and Jada were in a hot dog eating contest. Andre ate 10 hot dogs in 3 minutes. Jada ate 12 hot dogs in 5 minutes.

Here are two different graphs that both represent this situation.

Two graphs in the coordinate plane: The graph on the left has two points. The horizontal axis is labeled "time in minutes" and the vertical axis is labeled "number of hotdogs." The first point is to the right of the vertical axis and high above the horizontal axis. The second point is to the right and above the first point. The graph on the right has two points. The horizontal axis is labeled "number of hotdogs" and the vertical axis is labeled "time in minutes." The first point is far to the right of the vertical axis and above the horizontal axis. The second point is to the right and above the first point.

  1. On the first graph, which point shows Andre’s consumption and which shows Jada’s consumption? Label them.

  2. Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point.

  3. Write an equation for each line. Use to represent time in minutes and to represent number of hot dogs.

    1. Andre:

    2. Jada:

  4. For each equation, what does the constant of proportionality tell you?

  5. Repeat the previous steps for the second graph.

    • Andre:

    • Jada:

Lesson Summary

Imagine that a faucet is leaking at a constant rate and that every 2 minutes, 10 milliliters of water leaks from the faucet. There is a proportional relationship between the volume of water and elapsed time.

  • We could say that the elapsed time is proportional to the volume of water. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of of a minute per milliliter.

  • We could say that the volume of water is proportional to the elapsed time. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of 5 milliliters per minute.

Let’s use to represent volume in milliliters and to represent time in minutes. Here are graphs and equations that represent both ways of thinking about this relationship:

Two line graphs: The first line graph has a horizontal v-axis and a vertical t-axis. The line begins at the origin and moves upwards and to the right passing through the points with coordinates (1, 1/5) and (10, 2). The graph is labeled with the equation t=1/5v. The second line graph has a horizontal t-axis and vertical v-axis. The line begins at the origin and moves upwards and to the right passing through the points with coordinates (1,5) and (2,10). The graph is labeled with the equation v =5t.

Even though the relationship between time and volume is the same, we are making a different choice in each case about which variable to view as the independent variable. The graph on the left has as the independent variable, and the graph on the right has as the independent variable.