Lesson 5

Connections between Representations

  • Let’s look at the relationship of verbal descriptions, equations, tables, and graphs.

5.1: Math Talk: Evaluating Expressions

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

5.2: A Good Night’s Sleep

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?

5.3: What’s My Equation?

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!

\(n=81-3t\)

\(n=81-\frac13t\)

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

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

Summary