The purpose of this activity is for students to apply what they know about functions and their representations in order to investigate the effect of a change in one dimension on the volume of a rectangular prism. Students use graphs and equations to represent the volume of a rectangular prism with one unknown edge length. They then use these representations to express what happens to the volume when one of the edge lengths is doubled. Groups make connections between the different representations by pointing out how the graph and the equation reflect an edge length of a prism that is doubled and by explaining the relationships between that length and the volume of the prism (MP2).

If students are not sure how to start labeling their graph, consider asking:

• “Can you explain how you wrote your equation between and ?
• “How could making a table of values help you make a graph?”

Use this discussion to help students make connections between the context, equation, and graph to support the idea that the volume doubles when is doubled.

Invite previously selected groups to share what happens to the volume when is doubled. Display their graph and equation for all to see, and ask students to point out where they see the effect of doubling in the graph (when looking at any two edge lengths that are double each other, their volume will be double also). Ask students:

• “Which of your variables is the independent? The dependent?” (The side length, , is the independent variable, and volume, , is the dependent variable.)
• “Which variable is a function of which?” (Volume is a function of side length.)

If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the edge length . Display volume equation for all to see. Ask, “How can we write this equation to show that the volume doubled when doubled?” (Using algebra, we can rewrite as . Since the volume for was , this shows that the volume for is twice the volume for ).

Building on their thinking about an edge length and volume of a prism, in this activity students consider a similar problem with a cylinder. Using different representations of the function relating the volume of a cylinder to its height given a fixed radius, students continue to identify the effect of the changing dimension on the graph and the equation of this function.

In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). 

The goal of this discussion is for students to connect that the relationship discussed in this activity about cylinders and in a previous activity about prisms are both examples of linear relationships.

Begin by asking previously selected groups to share their graphs and equations alongside some of the graphs from the earlier activity with the prism. Ask:

• “Compare the graph in this activity to the graph in the last activity. How are they alike? How are they different?” (They both look like a line going through the origin. The lines do not have the same slope.)
• “Compare the equation in this activity to the equation in the last activity. How are they alike? How are they different?” (Both equations have the same form of volume equaling a number times a letter. The numbers, 15 and 78.5, are not the same.)
• “How can you tell that this is a linear relationship?” (I can tell it is a linear relationship because it is written as , only for both, since the volume of a shape with a side or height of 0 is 0.)

Conclude the discussion by using Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “What happens to the volume if you halve the height, ?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say . . . ?”
• “Can you describe that another way?”
• “Can you say more about . . . ?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

The purpose of this activity is for students to use representations of functions to explore more about the volume of a cone. This activity differs from the previous ones because students are given a graph that shows the relationship between the volume and height of cones with a fixed radius. They use the graph to reason about the coordinates and their meaning and, along with the formula for the volume of a cone, they calculate the unknown radius (MP2).

Students may round at different points in their work, leading to slightly different answers.

Students can solve the equation  with different series of steps. If they choose to first multiply the numbers on the side with , they may round the expression to . This would give , making slightly less than 15, though it rounds to 15. Solving by multiplying each side by yields , making equal 15 exactly (if we accept 3.14 as the value of ). This is an opportunity to discuss how rounding along the way while working toward a solution can introduce imprecision.

The purpose of this discussion is for students to share the strategies they used to write the equation represented by the graph. Consider asking some of the following questions:

• “How did you write your equation relating and ?” (Since the formula for the volume of a cone is , I used , , and to get , which means . So, .)
• “How do you know that the relationship between volume and height for these cones is a function? How is this shown in the graph? In the equation?” (The equation is a linear equation, so it must be a function. Each input value has one and only one output value .)
• “Identify the independent and dependent variables in this relationship. If they were switched, would we still have a function? Explain how you know.” (The independent variable is the height of the cone and the dependent variable is the volume of the cone . If they were switched, it would still be a function since each value of results in one and only one possible value for .)