Regression Lines (Lesson 3.5)
Chapter 3  Day 6
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Learning Targets

Make predictions using regression lines, keeping in mind the dangers of extrapolation.

Calculate and interpret a residual.

Interpret the slope and y intercept of a regression line.
Experience First
This activity will be using the Barbie bungee context from Lessons 3.2 and 3.4. Students will use the 2 Quantitative Variables applet to find a regression line that they will then use to make some predictions.
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The Barbie data is a great context for slope and yintercept because they both have a very tangible physical meaning. The slope of the regression line tells us that for every additional rubber band added, we predict the distance that Barbie’s head reaches to increase by 7.64 cm. They yintercept tells us the predicted distance travelled is 25.33 cm when there are 0 rubber bands. Of course, this is really just a predicted value of Barbie’s height.
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Formalize Later
When we have students write out the equation for the regression line, we have them (1) use context instead of x and y and (2) put a “hat” over the y variable to indicate that we are predicting y from x.
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predicted lowest height=25.33+7.46(# rubber bands)
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These two small changes will make predictions and residuals come much easier for students. The notation change also helps with the interpretation of slope and yintercept.
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To help students remember the correct interpretation of slope, we took them back to Algebra 1, where they learned slope as rise/run or (change in y) / (change in x). Then we took our slope of 7.46 and wrote it as 7.46/1. When students think of slope as (change in y) / (change in x), the interpretation becomes much easier to come up with (rather than memorize!) We also made sure that students were saying “predicted distance” rather than just “distance” when interpreting slope and yintercept.
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You can preview tomorrow’s lesson by asking students how they think the Applet is figuring out which line is the “best”. You can also preview Lesson 3.7 by making students aware of the s (standard deviation of the residuals) and r^2 (coefficient of determination) in the applet output, both of which will be interpreted later.