Corequisite Support Activity for 6.A: Exploring Lines of Best Fit

what you’ll need to know:

In this support activity you’ll become familiar with the following:

Use a given trend to create an appropriately labeled graph from a data table.
Interpret an ordered pair (x, y) that represents a data point on a graph.
Use a given graph to make a linear prediction.
Write a linear equation to describe a given scenario.
Identify the explanatory and response variables of a scenario.
Interpret the slope and y-intercept in the context of statistics given the equation of linearly related variables.

In the next preview assignment and in the next class, you will need to know how to write a linear equation that describes a situation where there is a constant rate of change. In this activity, you’ll prepare for that by following steps to model data from two variables using a linear equation. You will create a table of variable values then use that data to plot points on a graph. Then you’ll use the graph and data to write a linear equation that describes the scenario. You may have seen linear equations before. See the recall box below for a refresher on the slope-intercept form of a linear equation and how to represent linearly related variable values as ordered pairs in order to create a graph that models the data.

Recall

When modeling data from bivariate data (two variables) with a linear relationship, we need to understand how to write and graph a linear equation.

Core skill:
Identify the parts of a linear equation in slope-intercept form
- Given two variables, let’s name them [latex]x[/latex] and [latex]y[/latex], that have a relationship in which the value of [latex]y[/latex] is dependent upon the value of [latex]x[/latex], we can describe the relationship using a linear equation.
- Ex. If a certain tree grows two inches in a year, we can say that a 7 inch tall sapling will grow 2 inches each year thereafter by letting the number of years be represented by [latex]x[/latex] and the height of the tree in inches be represented by [latex]y[/latex].
  - “The height [latex]y[/latex] is equivalent to the starting height plus 2 inches per year.”
  - Mathematically, we can write [latex]y = 7 + 2x[/latex].
  - Equivalently, we can write [latex]y = 2x + 7[/latex] which is in a form we call “slope-intercept form.”
- The familiar slope-intercept form of a linear equation you probably learned about in algebra is [latex]y=mx+b[/latex], where
  - [latex]x[/latex] and [latex]y[/latex] are, respectively, the explanatory and response variables,
  - [latex]m[/latex] represents the constant rate of change (the slope), and
  - [latex]b[/latex] represents the initial value in a given scenario (which can be zero or negative).

Core skill:

Graph a linear equation

[Coordinate Plane], [Slope] [<– link to these from DC Student Resources please, in the places indicated above]

In this activity, we’ll follow several steps to see how to model bivariate data that is linearly related. Let’s begin by watching the story of a special dog’s journey to a healthier life. The video below follows the dog Kai on a one year journey to lose 100 pounds. As you watch the video, note how Kai’s small, constant changes eventually add up to a big change.

Life Lessons from Kai

A small and constant change can create a straight path to a happier life!

The first two questions below may help you identify and process any emotions related to watching the video. Watching Kail’s journey may stir thoughts about something in your life that you would like to change. It could be anything at all, not just a physical characteristic like Kai’s. Maybe you would like to work on punctuality or study skills, getting more sleep, or hitting the gym more regularly. Whatever it is, reframe your thoughts and answer Question 1 below individually. You don’t have to write down or share your answer with anyone.

question 1

1) Reflect on how the video made you feel. Is there something in your life that you would like to change? [You are not required to write this answer down if you would prefer to keep it to yourself.]

Hint

Now, answer Question 2 on your own then compare your answers with a partner.

question 2

2) List some phrases or ideas from this video that might help you meet a goal or life challenge. [This is required.]

Hint

Modeling and Interpreting a Linear Relationship

When Kai met Pam, he was almost 100 pounds overweight, and for this reason alone, his owner wanted him euthanized. Kai’s life was in danger and he was miserable, but Pam rescued him. Here are the details of Kai’s journey:

Initial Weight: Kai weighed 173 pounds before Pam agreed to foster him. She modified his diet and began walking him three times daily.

Constant Rate of Change: A reasonable goal was for Kai to lose about two pounds each week.

Question 3

3) Assuming Kai meets his goal of losing two pounds per week, let’s track his weight from week to week for the first six weeks. Complete the following chart.

Week	Kai’s weight
0	173
1	171
2
3
4
5
6

Hint

Question 4

4) Create a graph to keep track of Kai’s weight loss for the first six weeks, assuming he manages to consistently lose two pounds each week. Label your axes accordingly.

A grid

Hint

question 5

5) What do your horizontal and vertical axes represent?

Hint

question 6

6) Explain the ordered pair (5 , 163) in the context of Kai’s situation.

Hint

question 7

7) What pattern is made when you have a constant rate of change (two pounds lost per week)?

Hint

Making a Prediction From a Graph

question 8

8) At this constant rate of weight loss, how much will Kai weigh after 10 weeks? After 25 weeks?

Hint

question 9

9) Describe how you arrived at your answers to Questions 8. Use mathematical operations in your description.

Hint

Writing an Equation to Describe a Model

question 10

10) Use your description in Question 9 to write a linear equation that describes Kai’s situation. Make sure to explain what your variables mean.

Hint

Linear Equations in the Context of Statistics

You may recall the slope intercept form of a linear equation from algebra as [latex]y=mx+b[/latex], where [latex]m[/latex] represents slope and [latex]b[/latex] represents the y-intercept. While the components of the equation are the same in statistics, the letters representing slope and the y-intercept are different. Tip: don’t let the different use of the letter [latex]b[/latex] in the statistics formula trip you up! See the explanation below for the form of the equation used in statistics: [latex]y = b + ax[/latex].

example

Let’s practice converting an algebraic linear equation from the form [latex]y=mx+b[/latex] into a form commonly used in statistics, [latex]y = b + ax[/latex].

Ex. A linear equation representing the relationship between two variables is given as

[latex]y = 1.5x + 12[/latex].

What is the slope of the equation? What is the y-intercept?
Rewrite the equation in the form [latex]y = b + ax[/latex].
In the context of statistics, what does the letter [latex]b[/latex] represent? How about [latex]a[/latex]?

Show Answer

In statistics, we use specific letters to represent variables in linear equations. The following are the common naming conventions:

x represents the explanatory variable represented on the horizontal axis.

y represents the response variable represented on the vertical axis.
a represents the y-intercept, which is the y-coordinate of the point where a line intersects the y-axis.
b represents the constant rate of change or slope of the line.

question 11

11) For Kai’s situation, identify the following:

Part A: Explanatory variable

Hint

Part B: Response variable

Hint

Part C: Slope

Hint

Part D: y-intercept

Hint

Part E: Rewrite your equation from Question 10 using this new naming convention.

Hint

Hopefully, you feel comfortable modeling linearly related bivariate data by creating a table and graph and writing an equation to describe the relationship. It’s time to move on to the course section and activity now.

Alpha Module 4: Modeling Linear Relationships