Lesson 7

Accurate Representations

These materials, when encountered before Algebra 1, Unit 6, Lesson 7 support success in that lesson.

7.1: Possible Rectangles (10 minutes)

Warm-up

In this lesson, students analyze graphs that represent situations, and explain why the coordinates of certain points on the graph either do or don’t make sense in the situation. In this warm-up, students encounter a straightforward mathematical context of the perimeter of a rectangle, and simply discuss values that do and don’t make sense to use for the length of the rectangle.

Launch

Before analyzing whether the lengths given in the activity are valid, help students refamiliarize themselves with the meaning of the perimeter of a rectangle by asking, “A rectangle has a perimeter of 40 centimeters. What is the width of the rectangle if its width is 1 centimeter? 5 centimeters?” It’s not necessary to develop the function $$20-x$$ to come up with these widths—it’s enough that students understand why the corresponding widths in centimeters are 19 and 15.

Give students a minute to read the task statement, and ensure that they understand that the answer to each part is just “yes” or “no,” and they should be able to explain a reason. Is it possible that the given length is the length of this rectangle?

Student Facing

A rectangle has a perimeter of 40 centimeters.

Here are some lengths in centimeters. Determine if each one could be a length of this rectangle. Be prepared to explain your reasoning.

1. 10
2. 4.5
3. 20
4. 32
5. -10
6. 0
7. $$\frac{21}{4}$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

First, invite one or more students to explain why the first two could be the length of the rectangle, displaying any diagrams or calculations they use to support their reasoning. Then, focus on the next two lengths. Ensure students understand that these are not possible, reasoning either that there is not enough left over for the other sides, or that if we calculated the width using $$20-x$$, the width would be a negative number. In this situation where the numbers represent the length of the side of a rectangle, negative values don’t make sense. Finally, the -10 and the 0 don’t make sense as inputs, because the sides of a rectangle must be a positive number. Time permitting, the last length of $$\frac{21}{4}$$ is not only an opportunity for some arithmetic on fractions, but also an opportunity to discuss that there is no reason why the lengths must be a whole number.

To summarize, the numbers that are valid depend on the context, where the numbers are measurements of the sides of a rectangle. Before moving on to the next activity, remind students that when we are talking about allowable inputs to a function, we are talking about its domain. How can we best describe the allowable lengths of this rectangle? (Any length between 0 cm and 20 cm is allowable.) An inequality can also be used to describe the lengths: $$0 < \ell <20$$. Consider displaying a graph of $$y=20-x$$ and discuss how the portion of the graph in Quadrant 1 shows the possible values of $$x$$.

7.2: Which Graph Represents It the Best? (15 minutes)

Activity

For each of four given situations, students decide which version of a graph represents it the best. The graphs differ by whether they are discrete or continuous, and the domain that is graphed. In order to successfully complete this activity, students need to relate the features of the graphs to the situation they represent (MP2), and have to attend to precision in the words they choose when they explain why one graph is a better representation than the others (MP6).

Launch

Before students start working, ensure they understand that they are choosing one of four possible graphs that represent it the best. Give them a few minutes of quiet work time, then an opportunity to share their reasoning with a partner. Follow with a whole-class discussion.

Student Facing

For each situation, several graphs are given. Which graph represents the situation the best? Be prepared to explain your reasoning.

1. The fine for an overdue book at the library is $0.25 per day, up to a maximum of$6. $$x$$ represents time in days, $$y$$ represents the fine in dollars.

2. A tank that starts with 25 gallons of water drains at a rate of 2 gallons per minute. $$x$$ represents time in minutes, and $$y$$ represents volume of water in tank in gallons.

3. Someone folds a paper in half, then in half again repeatedly. After each fold, the thickness of the folded paper increases. $$x$$ represents number of folds, and $$y$$ represents thickness in layers.

4. A t-shirt company offers deals on bulk purchases. Shirts cost $5 each if you purchase less than 10, and they cost$4 each if you purchase 10 or more. $$x$$ represents number of shirts purchased, and $$y$$ represents cost of shirts in dollars.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The focus of this discussion should be explaining how students decided which graph best represents each situation. Encourage students to attend to precision in their explanations, using words like domain, positive, negative, reasonable, and whole numbers. An explanation for each situation might sound like:

1. I chose C because only positive values are graphed for number of days and the amount of the fine. Also, the description says that $6 is the maximum fine, and this is the largest value for $$y$$ in graph C. Finally, I chose the graph made of disconnected points, because the fine increases by$0.25 each day without any values in between.
2. I chose C because it doesn’t include any negative values for time, and it doesn’t include any volumes greater than 25, which is the maximum that the tank holds. I chose the graph that is a continuous line segment rather than points, because the tank is draining at a constant rate, which means that all of the numbers between 0 and 12.5 make sense in the domain, not just the whole numbers.
3. I chose D because it consists of points rather than a continuous curve, and it makes sense that the number of folds can only be a whole number. The largest value for number of folds plotted is 5, which makes sense, because it’s really hard to fold a piece of paper 5 times, never mind more than 5 times.
4. I chose D because it shows a different function when $$x$$ is less than 10 and greater than 10, it only shows whole-number values for number of t-shirts, and it doesn’t include any negative values for the number of t-shirts or the price, which wouldn’t make sense in this situation.

Note that many students may not realize there’s a limit on how many times you can fold a piece of paper in half. It may require some discussion or a demonstration to convince them that 5 is a reasonable maximum for the domain.

7.3: Refining the Representation (15 minutes)

Activity

In this activity, students practice both explaining why certain points on a graph don’t make sense in a situation, and sketching a new graph that better represents the situation. In the second question, they encounter a quadratic function to prepare them for the functions they are studying in the associated Algebra 1 lesson.

Launch

Arrange students in groups of 2. Display the graph for the first question for all to see. Explain that we need to decide which points on the graph make sense in a situation.

Provide access to calculators for students who may need support finding the coordinates of points on the graph of the quadratic function in the second question.

Student Facing

1. At the concession stand, popcorn costs \$2 and bananas cost \$1. Clare spent \\$16 on popcorn and bananas for her family.

1. Explain why each of these points on the graph do not make sense in the situation:
1. $$(\text-2, 20)$$
2. $$(1.5, 13)$$
3. $$(10, \text-4)$$
2. Sketch a graph that better represents the situation. Explain your reasoning.

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2. The height in feet of a baseball over time, in seconds, is modeled by the function $$h$$ given by the equation $$h(t)=3+60t-16t^2$$. A graph of the function is shown.
1. Choose two points that are on the graph but do not make sense with the context. Explain your reasoning.
2. Sketch a graph that better represents the situation.

3. What point represents the baseball landing on the ground?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Invite students to share the sketches of their graphs, and explain why they made choices about how to sketch them. Building on the previous activity, encourage students to attend to precision in their explanations.

• “Why is one graph continuous and the other discrete?” (It doesn’t make sense for someone to order part of a banana or a partial order of popcorn, so the inputs and outputs must be whole numbers. The height graph’s input is time, so we are not limited to whole number inputs.)
• “What do both graphs have in common?” (Both graphs only make sense for points in the first quadrant or on the positive half of each axis.)
• “How would you describe the domain that makes sense for the function that gives the baseball’s height?” (between 0 and about 3.8, or $$0 \le t \le 3.8$$)