In this activity, students analyze the relationship between two quantities that form a ratio, and think about how tables, graphs, and equations represent the relationship. The work here activates students’ prior knowledge of ratios, ways to reason about them, and the language used to describe ratios.

The familiar context of mixing paint is used to elicit language related to ratios and rates, but students may also use language that is less formal to describe one quantity in terms of the other. This will serve as the basis for thinking about dependent and independent variables and for writing equations with two variables.

Students begin by completing a table of equivalent ratios and plotting the values on two coordinate grids, creating two graphs that represent the same situation. Then students describe the relationship between the two quantities in as many ways as they can. Thinking about how one quantity relates to the other in turns allows students to interpret and write equations with variables. Along the way, students reason quantitatively and abstractly (MP2). 

In the Activity Synthesis, the terms “dependent variable” and “independent variable” are introduced to describe the two variables.

In the digital version of the activity, students can choose to use an applet to plot points on a graph to show the relationship between cups of red paint and cups of yellow paint in the mixture. The applet allows students to place a point directly onto the graph, using the axis labels to identify the correct location. Then they can double-click the point to check or change the point’s coordinates. 

If students don't have individual access, consider displaying the applet for all to see after students create their own graphs on paper.

Invite students to share whether they agree that the equation Lin wrote relates the amount of yellow paint to red paint and explain their reasoning. Then discuss the equation that students wrote for the last question. 

If no students mentioned using the values in the table to check Lin’s equation or to help write a second equation, ask them to discuss this idea. 

Explain that in a situation that has two quantities that change in relation to each other, we can use the words “dependent” and “independent” to describe the two variables:

• If we know the amount of red paint, we can use it to calculate the amount of yellow paint, as Lin has done by writing . 

  In this case, we say that the amount of yellow paint depends on the amount of red paint, so is the dependent variable and is the independent variable. 

• If we know the amount of yellow paint, we can use it to calculate the amount of red paint, as students have done by writing . 

  In this case, we say that the amount of red paint depends on the amount of yellow paint so, is the dependent variable and is the independent variable

Draw students’ attention to the two graphs they created. Ask: 

• “Which equation is represented by the first graph?” () “How can we tell?” (Multiplying the horizontal or first value of the ordered pairs by gives the second or vertical value.)