# Lesson 5

Goodness of Fit

These materials, when encountered before Algebra 1, Unit 3, Lesson 5 support success in that lesson.

## 5.1: What’s the Rate? (5 minutes)

### Warm-up

The goal of this activity is for students to determine the rate of change for situations. This activity prepares students to interpret the slope of linear models in the associated Algebra 1 lesson.

### Student Facing

Each situation can be modeled using a linear equation. Describe the rate of change for each situation.

1. Andre started his no-interest savings account with $1,000. He makes the same deposit each week, and there is$1,600 in the account after 6 weeks.
2. Kiran starts with $748 in his checking account. After 4 weeks of spending the same amount each week, he has$716 left.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The goal of this activity is for students to practice finding the rate of change in a given scenario. Discuss how students can find the rate of change. Here are sample questions to promote class discussion:

• “How do you determine the rate of change for Andre’s savings account?” (First, I think about the difference between how much money he had in the beginning and the end of the situation. Once I determine the difference, then I use the 6 weeks to divide to get the rate of change.)
• “How do you determine the rate of change for Kiran’s checking account?” (First, I think about the difference between how much money he had in the beginning and the end of the situation. Once I determine the difference, I set up a proportion to describe what he spent over 4 weeks and to calculate how much that meant he spent each week.)
• “How does rate of change relate to a linear model?” (The rate of change relates to a linear model because linear models show a constant increase, decrease, or no change in the variables. The rate of change tells me how much a variable increases, decreases, or if it does not change at all. Similarly, in a linear model, the slope tells us how much the line increases, decreases, or if does not change at all. The rate of change and the slope give us the information on how a variable or line changes, respectively.)

## 5.2: Goodness-of-Fit (20 minutes)

### Activity

The purpose of this activity is for students to think about how to determine if a linear model is a better or worse fit for data displayed in a scatter plot. This activity prepares students to create lines of best fit in the associated Algebra 1 lesson.

### Launch

Display the scatter plot for all to see.

Ask students which line appears to fit the data best. Ask them to explain their reasoning. Listen for students who describe the trend formed by the data or uses the term line of best fit. Line 1 is a better fit to the data than line 2 because it follows the same trend as the one formed by the points. For students who have forgotten about fitting lines, this image can help them recall what makes a line a good fit for a data set.

### Student Facing

Here are 3 copies of the same scatter plot. Each student tries to draw a line that models the data well.

Noah says his line fits the data well because the line connects the leftmost point to the rightmost point.

Andre says his line fits the data well because it passes directly through as many points as possible.

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Lin says her line fits the data well because the points are somewhat evenly arranged around the line with about half the points above the line and half the points below the line.

Do you agree with any of these students? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The goal of this discussion is for students to think about what makes a linear model a good fit for a data set. Discuss what would make a linear model a good fit. Here are sample questions for discussion:

• “What do you think makes a line a good fit for a data set?” (A good fit line goes through the middle of the data and follows the trend formed by the points. A bad fit would not show the same trend as the one formed by the points or not go through the middle of the data.)
• “What do you think is an advantage of using a linear model to represent data?” (Using a linear model to represent data allows you to get an overall view of the trend of the data. You can tell if the data are increasing or decreasing overall, even if they do not do so with each new data point. You can also predict or estimate additional information because the linear model helps to establish a quantifiable pattern in the data.)

## 5.3: What Fits? (15 minutes)

### Activity

The purpose of this activity is for students to learn how to create a linear model that is a good fit to data. Students first determine which scatter plots are best represented by a linear model. Then, students create their own linear model that is fit to the data, and compare their lines with a partner’s.

### Student Facing

1. Look at the scatter plots, and determine which one is best modeled by a linear model.
2. Draw a linear model that fits the data well on the appropriate scatter plot. Compare your line with a partner’s. If your lines are different, determine which line is the better fit line.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Discuss how students arrived to their answers. Here are sample questions for discussion:

• “How do you determine if a scatter plot is best modeled by a linear model?” (The scatter plot with points that are closest to forming the shape of a straight line is the scatter plot that is best modeled by a linear model.)
• “For groups that had different lines, how did you determine whose line was more accurate?” (We looked closely at each of the lines and decided that the better fit line was the one that had the points more evenly arranged around it.)