# 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, $$n$$

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:

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.

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

$$n=81-3t$$

$$n=81-\frac13t$$

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

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