Lesson 21

The purpose of this warm-up is for students to jump back into recognizing functions and determining if two quantities are a function of each other. Students are asked questions similar to these throughout this lesson, and the discussion of this warm-up is meant to get students using the language of functions, which continues throughout the rest of the activities.

Give students 1–2 minutes of quiet work time, followed by a whole-class discussion.

If students struggle with the second and third questions, ask them if this is an example of a proportional relationship. Remind them what they learned previously about proportional relationships and how that can be used to answer these questions.

Invite students to share their answers and their reasoning for why gas burned is a function of distance traveled. Questions to further the discussion around functions:

• “Looking at the graph, what information do you need in order to determine how much gas was used?” (We need to know the number of miles traveled.)
• “What is the independent variable? Dependent variable? How can you tell from the graph?” (The independent value is the distance traveled. The dependent value is the gas burned. By convention, the independent is on the \(x\)-axis and the dependent is on the \(y\)-axis.)
• “What are some ways that we can tell from the graph that the relationship between gas burned and distance traveled is proportional?” (The graph is a line that goes through the origin, we can see a constant ratio between \(y\) and \(x\) in some points like \((80,10), (160, 20)\), and \((240, 30)\).)

This activity is optional.  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 investigate 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 that is doubled. Identify groups who make these connections and ask them to share during the discussion.

If students struggle to see how the change in volume is reflected in the equation, have them start with values from the graph and put those into the equation to see how the volume changes.

The goal of the discussion is to ensure that students use function representations to support the idea that the volume doubles when \(s\) is doubled.

Select previously identified groups to share what happens to the volume when you double \(s\). Display the graph and equation for all to see and have students point out where they see the effect of doubling \(s\) in the graph (by 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, \(s\), is the independent variable, and volume, \(V\), is 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 \(s\). Display volume equation \(V=15(2s)\) for all to see. Ask, “How can we write this equation to show that the volume doubled when \(s\) doubled?” (Using algebra, we can rewrite \(V=15(2s)\) as \(V=2(15s)\). Since the volume for \(s\) was \(15s\), this shows that the volume for \(2s\) is twice the volume for \(s\)).

This activity is optional.  This activity is similar to the previous (optional) one from a function point of view, but now students investigate the volume of a cylinder instead of a rectangular prism. Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height. The exploration and representations resemble what was done in the previous activity, and students continue to identify the effect of the changing dimension on the graph and the equation of this function. As groups work on the task, identify those who make the connection between the graph and equation representations, and encourage groups to look for similarities and differences between what they see in this activity and what they saw in the previous activity. Invite these groups to share during the whole-class discussion.

Ask previously identified groups to share their graphs and equations. 

If students completed the optional activity, display student graphs from both activities and ask:

• “Compare the graph in this activity to the graph in the last activity. How are they alike? How are they different?”
• “Compare the equation in this activity to the equation in the last activity. How are they alike? How are they different?”
• “How can you tell that this is a linear relationship?”

For the last question, make sure students understand that “looks like a line” is insufficient evidence for saying a relationship is linear since the scale of the axes can make non-linear graphs look linear. It is important students connect that the equation is linear (of the form y=mx+b) or relate back to the relationship between the height and volume: that the volume is \(25\pi\), or about 78.5, times the height.

If students did not complete the optional activity, ask:

• “Which of your variables is the independent? The dependent?” (The height, \(h\), is the independent variable, and volume, V, is dependent variable.)
• “Which variable is a function of which?” (Volume is a function of height.)
• “How can you tell that this is a linear relationship?”

See notes above on the last question in this list. If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the height \(h\). Display volume equation \(V=78.5(2h)\) for all to see. Ask how we can write this equation to show that the volume doubled when \(h\) doubled (using algebra, we can rewrite \(V=78.5(2h)\) as \(V=2(78.5h)\). Since the volume for \(h\) was \(78.5h\), this shows that the volume for \(2h\) is twice the volume for \(h\)).

This activity is optional.  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 of a cone and the height of a cone. They use the graph to reason about the coordinates and their meaning. Students use the equation for the volume of a cone and coordinates from the graph to determine the radius of this cone and discuss the answer to this last question with their partners. Identify students to share their strategies during the discussion.

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by time to discuss the answer and their strategies for the last question with their partner. Follow with a whole-class discussion.

Students may round \(\pi\) at different points during their work, leading to slightly different answers.

Students can solve the equation \(2,355=\frac13 \pi r^2 10\) with different series of steps. If they choose to first multiply the numbers on the side with \(r\), they may round the expression \(\frac13 \pi r^2 10\) to \(10.47 r^2\). This would give \(224.9=r^2\), making \(r\) slightly less than 15, though it rounds to 15. Solving by multiplying each side by \(3\boldcdot \frac{1}{\pi}\boldcdot \frac{1}{10}\) yields \(r^2=225\), making \(r\) equal 15 exactly (if we accept 3.14 as the value of \(\pi\)). This is an opportunity to discuss how rounding along the way in a solution can introduce imprecision.

Ask previously identified students to share their answers and their strategies for finding the radius in the last question.

To further the discussion, consider asking some of the following questions:

• “How did you deal with the \(\frac13\) in the equation?”
• “How did you deal with \(\pi\) when you were trying to solve for the radius?”
• “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?”
• “Identify the independent and dependent variables in this relationship. If they were switched, would we still have a function? Explain how you know.”