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 problemsolving 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, \(t\) number of errors, \(n\)  In the equation \(n = 40  4t\), what does the 40 mean in this situation? Where can you see it on the graph?
 In the equation \(n = 40  4t\), what does the 4 mean in this situation? Where can you see it on the graph?
 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:
 Write an equation for Model A. Be prepared to explain how you know. Explain what the numbers mean in your equation.
 Model B is exponential.
 How many errors did participants make with 0 hours of sleep?
 How many errors with 1 hour of sleep?
 What fraction of the errors from 0 hours of sleep is that?

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 
Which is an equation for Model B? If you get stuck, test some points!
\(n=813t\)
\(n=81\frac13t\)
\(n=81 \boldcdot \left(3 \right)^t\)
\(n=81 \boldcdot \left(\frac13 \right)^t\)