Lesson 5

Connections between Representations

These materials, when encountered before Algebra 1, Unit 5, Lesson 5 support success in that lesson.

5.1: Math Talk: Evaluating Expressions (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating expressions at a given value of the variable. These understandings help students develop fluency and will be helpful later in this lesson when students will need to evaluate similarly-structured expressions.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Evaluate mentally:

\(6,\!400 - 400x\)  when \(x\) is 0

\(6,\!400 - 400x\)  when \(x\) is 2

\(6,\!400 \boldcdot \left(\frac{1}{10}\right)^x\)  when \(x\) is 0

\(6,\!400 \boldcdot \left(\frac{1}{10}\right)^x\)  when \(x\) is 2

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. Be sure to draw attention to evaluating operations in the conventional order, and the fact that \(\left(\frac{1}{10}\right)^2\) means \(\frac{1}{10} \boldcdot \frac{1}{10}\). To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’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 \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

5.2: A Good Night’s Sleep (20 minutes)

Activity

In this activity, students are given an equation, and generate a corresponding table and graph. Then, they respond to some questions where 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.

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.

Graphing technology could be a helpful tool if students choose to use it (MP5), unless the focus becomes which buttons to press 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. So, use your judgment about whether it would be productive to allow use of graphing technology.

Student Facing

Is more sleep associated with better brain performance? A researcher collected data to determine 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 \(n = 40 - 4t\) models the relationship between \(t\), the time in hours a student slept the night before, and \(n\), the number of errors the student made in the problem-solving task.

  1. Use the equation to find the coordinates of 5 data points on a graph representing the model. Organize the coordinates in the table.
  2. Create a graph that represents the model.
    hours of sleep, \(t\)  number of errors, \(n\)
       
       
       
       
       
    Blank coordinate grid. Horizontal axis labeled t from 0 to 15. Vertical axis labeled n from 0 to 50.
  3. In the equation \(n = 40 - 4t\), what does the 40 mean in this situation? Where can you see it on the graph?
  4. In the equation \(n = 40 - 4t\), what does the -4 mean in this situation? Where can you see it on the graph?
  5. How many errors would you expect a person to make who had slept 3.5 hours the night before?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share their responses. If not already mentioned in students’ explanations, highlight the connections between the equation, the graph, and the quantities in the situation. In particular, show the point \((0,40)\) alongside the equation \(40=40-4(0)\), 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.

Possible discussion questions:

  • “How can we use an equation to express the number of errors after 3.5 hours without sleep?” (\(n=40-4(3.5)\))
  • “Where on the graph is the number of errors made 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.)

5.3: What’s My Equation? (15 minutes)

Activity

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 make connections between graphs and equations.

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.

Student Facing

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:

A

Decreasing line with 3 points plotted. 

B

0 comma 81, 1 comma 27, 2 comma 9 plotted on a parabola 

  1. Write an equation for Model A. Be prepared to explain how you know. Explain what the numbers mean in your equation.
  2. 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 from 0 hours of sleep is that?
  3. Complete the table for Model B for 3, 4, and 5 hours of sleep.

    \(t\) 0 1 2 3 4 5
    \(n\) 81 27 9      
  4. Which is an equation for Model B? If you get stuck, test some points!

\(n=81-3t\)

\(n=81-\frac13t\)

\(n=81 \boldcdot \left(3 \right)^t\)

\(n=81 \boldcdot \left(\frac13 \right)^t\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share their responses and their reasoning. Record their responses to show connections between each graph and the corresponding equation.