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Unit 7, Lesson 7

Accurate Representations

Extra Support Materials for Algebra 1

7.1

Warm-up

10 mins

Possible Rectangles

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

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 discuss values that do and don’t make sense to use for the length of the rectangle. When students decide what values do and do not make sense they reason abstractly and quantitatively (MP2).

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 length is 1 centimeter? 5 centimeters?” It’s not necessary to develop the function 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.” They should be able to explain a reason. Is it possible that the given length is the length of this rectangle?

Activity Synthesis

The goal of this discussion is to explore students’ reasoning about what lengths are possible. Invite students to explain their reasoning for their answers. Here are some questions for discussion:

“What types of numbers are not possible lengths of the rectangle?” (Negative numbers, 0, numbers greater than or equal to 20)
“What are some other numbers that are possible lengths of the rectangle?” (15, 2.7, )
“How could you describe all possible lengths of the rectangle?” (, where is the length)

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.

7.2

Activity

15 mins

Which Graph Represents It the Best?

Instructional Routines

MLR3: Critique, Correct, Clarify

Materials

None

Activity Narrative

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 by 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 they have to attend to precision in the words they choose when they explain why one graph is a better representation than the others (MP6).

This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.

Launch

Read the directions and first situation aloud. Display the graphs for the first situation. Give students a minute of quiet think time, then ask them to share their reasoning with a partner.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the first situation by correcting errors, clarifying meaning, and adding details.

Display this first draft:
Graph B represents the situation the best because the -axis represents time in days and negative days do not make sense in this situation.
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

After Critique, Correct, Clarify for the first problem, tell students to complete the remaining problems.

7.3

Activity

15 mins

Refining the Representation

Instructional Routines

None

Materials

None

Activity Narrative

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. When students analyze graphs to make sure they best represent a situation, they attend to precision (MP6).

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.

Engagement: Provide Access by Recruiting Interest. Invite students to share their experience at concession stands and the numbers of items they can buy.
Supports accessibility for: Conceptual Processing, Memory

Activity

None

Student Task Statement

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

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

A

B

C

Graph of plotted points on coordinate plane. Horizontal axis by fours. Vertical axis by ones. Plotted points start at origin and increase in a straight line to 24 comma 6.

D

Graph of plotted points on coordinate plane. Horizontal axis by fours. Vertical axis by ones. Plotted points start at left edge of graph, go through origin and end at right edge of graph.
A tank that starts with 25 gallons of water drains at a rate of 2 gallons per minute. represents time in minutes, and represents volume of water in the tank in gallons.

A

Graph of line on coordinate plane. Horizontal axis by twos from negative 4 to 16. Vertical axis by fours. Line decreases and goes through y axis at point 0 comma 25. Goes through x axis at point 12 point 5 comma 0.

B

Graph of line on coordinate plane. Horizontal axis by twos from negative 4 to 16. Vertical axis by fours. Line decreases and goes through y axis at point 0 comma 24 point 5. Ends at point 12 point 5 comma 0.

C

Graph of line on coordinate plane. Horizontal axis by twos from negative 4 to 16. Vertical axis by fours. Line decreases and starts at point 0 comma 24 point 5. Ends at point 12 point 5 comma 0.

D

Graph of plotted points in a straight line on coordinate plane. Horizontal axis by twos from negative 4 to 16. Vertical axis by fours. Plotted points decrease in a line. Points at 0 comma 24 point 5, 1 comma 23, 2 comma 21, 3 comma 19, 4 comma 17, 5 comma 15, 6 comma 13, 7 comma 11, 8 comma 9, 9 comma 7, 10 comma 5, 11 comma 3, 12 comma 1.
Someone folds a paper in half, then in half again repeatedly. After each fold, the thickness of the folded paper increases. represents the number of folds, and represents the thickness in layers.

A

Graph of a function. X axis from negative 6 to 6, by 2s. Y axis from 0 to 35, by 5s. Function begins around negative 6 comma 0, intersects the y axis around 1, and goes through approximately 4 comma 15 and 5 comma 30.

B

C

Plotted points graphed on coordinate plane. Horizontal axis by twos, from negative 6 to 6. Vertical axis by fives. Plotted points at negative 6 comma 0, negative 5 comma 0, negative 4 comma 0, negative 3 comma 0, negative 2 comma point 5, negative 1 comma point 75, 0 comma 1, 1 comma 2, 2 comma 4, 3 comma 8, 4 comma 16, 5 comma 32.

D

Plotted points in a graph on coordinate plane. Horizontal axis by two from negative 6 to 6. Vertical axis by fives. Plotted points at 0 comma 1, 1 comma 2, 2 comma 4, 3 comma 8, 4 comma 16, 5 comma 32.
A t-shirt company offers deals on bulk purchases. Shirts cost $5 each if you buy less than 10, and they cost $4 each if you buy 10 or more. represents the number of shirts bought, and represents the cost of shirts in dollars.

A

B

Graphs of two lines on coordinate plane. Horizontal axis by twos. Vertical axis by tens. Black line goes through the origin and increases to end at 10 comma 50. Blue line starts at 10 comma 40 and increases to end at 22 comma 88.

C

D

Graphs of two sets of plotted points in straight lines on coordinate plane. Horizontal axis by twos. Vertical axis by tens. Black starts at origin and increases to end at 9 comma 45. Blue starts at 10 comma 40 and increases to end at 22 comma 88.