Scatter Plots, Spaghetti, and

Predicting the Future

Image used with permission from commons.wikimedia.com

Warming up for the lesson1. What is the slope of the line that contains the

points (-3, 7) and (6, 1) ?

2. What is the equation of the line referred to in #1? Write the equation in slope-intercept form.

3. Sketch a graph of this line. Explain what the slope means and how the

slope of the line is related to how the graph looks.

Solutions and discussion1. What is the slope of the line that contains the

points (6, 1) and (-3, 7) ?

2. What is the equation of the line in slope-intercept form?

€

m = y2 − y1x2 − x1

= 7 −1−3−6

= 6−9

= − 23

Use y = mx + b, solve for b: 1 = (-2/3)(6) + b 1 = (-4) + b b = 5 y = (-2/3)(x) + 5Use point-slope form:y – 1 = (-2/3)(x – 6)y – 1 = (-2/3)x + 4y = (-2/3)x + 5

3. Sketch the graph:

What have we learned in the past about

scatter plots?

Scatter plots show a graph of a set of points.

Scatter plots help us to see if there is a relationship between the two variables.

Positive CorrelationAs x increases, y increases.

Negative CorrelationAs x increases, y decreases.

No Correlationx and y are not related

How is correlation related to slope of a line?

Reviewing the vocabulary of Scatter plots

• x-axis• y-axis• independent variable• dependent variable• correlation: positive, negative, or

none

LostJeremy decided to go on a diet and start exercising in order to lose weight. His goal was to lose 20 pounds. At the end of every week he weighed himself and recorded his weight in a chart.

week 1 2 3 4 5 6 7 8 9 10 11 12

weight 273 268 266 263 262 264 261 260 259 257 256 253

We can make a scatter plot and use it to organize and analyze this information, and maybe even predict the future!

Lost week 1 2 3 4 5 6 7 8 9 10 11 12

weight 270 268 266 263 262 264 261 260 259 257 256 253

Do the points appear to have a positive or negative correlation, or neither?Explain your answer.

0 2 4 6 8 10 12 14240

245

250

255

260

265

270

275

Jeremy's weight loss

Week

Wei

ght i

n Po

unds

“Trend line” When the points in a scatter plot seem to be in a pattern of a line, we can sketch a line that is near as many of the points as possible. This is called a “Trend Line”. There should be about the same number of points above as below the line.

0 2 4 6 8 10 12 14240

245

250

255

260

265

270

275

Jeremy's weight loss

Week

Wei

ght i

n Po

unds

Spaghetti without the sauceLet’s all decide on a good Trend Line together.

Using a piece of spaghetti, locate a line that you think is as close as possible to as many of the points as possible.

Your line should pass through at least two of the points in the scatter plot.

Let’s all use the points (7, 261) and (10, 257).

Use a straight edge and draw the line, being sure it contains the two points we selected.

Complete question 4 on your papersLet’s review our work. slope:

Using the points (7, 261) and (10, 257)

So the slope equals about -1.3

€

m = y2 − y1x2 − x1

= 257 −26110 − 7

= −43

Slopes and Trend lines

Slope is a very important value. It explains the rate of change that is occurring with the variables. Slope includes information about how the variables are increasing or decreasing in relation to each other.

In this example about Jeremy, how would you describe or explain the slope as a rate of change?

A closer look at SlopeWhat does slope mean in this example?????

Assign units to the slope. Since slope compares the change in y with the

change in x , we will use the units of y and x along with the value for slope.

If m = -1.3, what does that mean?€

m = y2 − y1x2 − x1

= change in weightchange in weeks

The equation of the trend line

We can use our slope (-1.3) and either one of the two points on our trend line ((7, 261) or (10, 257)) to find the equation of the line.

y = mx + b261 = -1.3(7) + b261 = -9.1 + b270 ≈ by = -1.3x + 270

y = mx + b257 = -1.3(10) + b257 = -13 + b270 = by = -1.3x + 270

The equation of the trend line

Our trend line has the equation y = -1.3x + 270What is the y-intercept of this line?

What does the y-intercept tell us about Jeremy?(hint: what value of x corresponds to the y-intercept?)

Does this match the data that we were given? Explain why or why not.

Predicting the future using the trend line

Using the equation for our trend liney = -1.3x + 270,

we can predict the future!Questions:What if Jeremy continued his diet for another

week? How could you use the equation of the trend line

to predict his weight?

Do you think you can use the equation of the trend line to predict Jeremy’s weight a year from now?

Why or why not?

How can we use the equation of the trend line to predict Jeremy’s weight after 14 weeks?

How can we use the equation of the trend line to predict Jeremy’s weight after 14 weeks?Substitute 14 in place of x: y = (-1.3)(14) + 270y = 251.8 pounds

Do you think you can use the equation of the trend line to predict Jeremy’s weight a year from now? Why or why not?

It depends on if Jeremy continues his diet and exercising for that length of time.