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Unit 6, Lesson 4

Connections between Representations

Extra Support Materials for Algebra 1

4.1

Warm-up

Standards Alignment

Building On

Addressing

Instructional Routines

Math Talk MLR8: Discussion Supports

Materials

None

Activity Narrative

This Math Talk focuses on evaluating expressions at a given value of the variable. It encourages students to think about number relationships and to rely on what they know about operations to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students evaluate similarly structured expressions.

To evaluate expressions at a given value of the variable, students need to look for and make use of structure (MP7).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Activity

None

Evaluate mentally.

when is 0
when is 2
when is 0
when is 2

Activity Synthesis

The goal of this discussion is to review students’ strategies for evaluating expressions for a given value of the variable. To involve more students in the conversation, consider asking:

“Who can restate ’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

Be sure to draw attention to evaluating operations in the conventional order, and to the fact that means .

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:

“First, I because .”
“I noticed , so I .”

Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

4.2

Activity

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

None

Activity Narrative

In this activity, students are given an equation, and then they generate a corresponding table and graph. Then they respond to some questions in which they interpret these representations in terms of the situation. The purpose of this exercise is to reinforce meaningful connections between different representations of the same situation.

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). However, we don’t recommend technology if the focus turns to pressing buttons to get answers, without understanding. Students also may be better equipped to attempt the analysis questions if they create the table and graph by hand. Consider these points before deciding whether it would be productive to allow use of graphing technology.

This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.

4.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

This activity is a preview of the work in the associated Algebra 1 lesson. Students have a scaffolded opportunity to determine a decay factor from a given graph, and to make connections between graphs and equations. Making connections between representations is part of reasoning abstractly and quantitatively (MP2).

Launch

The context and types of questions continue from the previous activity, so students can continue to work individually or with a partner without much interruption.

Activity

None

Launch

Ask students to think about this question: “How is your day different when you’ve had plenty of sleep the night before, compared to when you didn’t get enough sleep the night before?” Then, invite them to share their thoughts with a partner. Monitor for students who mention that getting enough sleep has to do with performing better the next day. Also listen for conversations about how we’d define “enough sleep.” Select a few students to share their thoughts with the whole class.

Tell students that in this activity, a researcher has tried to create a model for performance on a problem-solving task based on hours of sleep the previous night. Allow students to work individually or in pairs.

Activity

None

Is more sleep associated with better brain performance? A researcher collected data to figure out if there was an association between hours of sleep and ability to solve problems. She administered a specially designed problem-solving task to a group of volunteers, and, for each volunteer, recorded the number of hours slept the night before and the number of errors made on the task.

The equation models the relationship between , the time in hours a student slept the night before, and , the number of errors the student made in the problem-solving task.

Use the equation to find the coordinates of 5 data points on a graph representing the model. Organize the coordinates in the table.
Create a graph that represents the model.

hours of sleep, number of errors,
In the equation , what does the 40 mean in this situation? Where can we see it on the graph?
In the equation , what does the -4 mean in this situation? Where can we see it on the graph?
How many errors would you expect a person to make who had slept 3.5 hours the night before?

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

The purpose of this discussion is to make connections between the verbal description, equation, table, and graph.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to what the 40 and -4 mean in the situation and where the numbers can be seen in the graph. 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 clarify and strengthen their partner’s 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?”

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.

After Stronger and Clearer Each Time, invite students to share their responses. Highlight the connections between the equation, the graph, and the quantities in the situation. In particular, show the point alongside the equation , and demonstrate on the graph how the graph shows that the rate of change in this situation is -4 errors per additional hour of sleep.

Here are some questions for discussion:

“How can we use an equation to express the number of errors after 3.5 hours of sleep?” ()
“Where on the graph is the number of errors made by someone who got no sleep?” (The vertical intercept)
“Will the graph continue to decrease indefinitely?” (No. The number of errors is limited to the number of questions on the task. Also, humans can’t sleep indefinitely.)

The sleep researcher repeated the study on two more groups of volunteers, collecting different data. Here are graphs representing the equations that model the different sets of data:

<p>Decreasing line with 3 points plotted. </p> — A

<p>0 comma 81, 1 comma 27, 2 comma 9 plotted on a parabola </p> — B

Write an equation for Model A. Explain what the numbers in your equation mean.
Model B is exponential.
1. How many errors did participants make with 0 hours of sleep?
2. How many errors with 1 hour of sleep?
3. What fraction of the errors with 0 hours of sleep is your answer to the last question?
Complete the table for Model B for 3, 4, and 5 hours of sleep.

0 1 2 3 4 5

81 27 9
Which is an equation for Model B? If you get stuck, test some points!

hours of sleep,	number of errors,

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Lesson 4

Connections between Representations

Warm-up

Standards Alignment

Building On

Addressing

6.EE.A.2.c

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Building Toward

Standards Alignment

Building On

Addressing

8.F.B

Building Toward

HSF-LE.A.2

Standards Alignment

Building On

Addressing

Building Toward

HSF-LE.A.2