1. Trail Mix Investigation

2. Technology You will need access to Desmos for this lesson.
Go to

3. Ask students what they think makes trail mix good. Bad. Expensive
Ask students what they think makes trail mix good? Bad? Expensive? Cheap?
Healthy? Unhealthy?
Focus the discussion around healthy/unhealthy and frame the guiding questions.

4. Questions in Focus
Which nutritional components of trail mix are the best predictor of the calorie content? Does the amount of fat in the trail mix reliably predict the calorie content? Does the amount of carbs? How could we figure this out? Why would someone care about this relationship?
Discussion time … then explain that we chose to focus on fat vs. calories

5. Narrowing Our Investigation
We will focus on this question today:
Can the amount of fat in a trail mix reliably predict the calorie content?

6. Displaying Bivariate Data
Can we begin to answer our question by looking at the relationship between the grams of fat and calories in a ...
dot plot?
scatter plot?
frequency table?
What is most appropriate for this investigation and why?

7. Displaying Bivariate Quantitative Data
Use technology to construct a scatter plot for the fat and calories from the nutrition labels from your trail mix.
Consider and discuss the following questions:
How will you set up and scale your axes?
What do you notice about the data?
What is the “center” of this data?
How do we measure the variability of this data?

8. Scatter Plot
Here is a table and graph of the fat and calorie content of several different trail mixes.
[Link to Handout]

9. Focus Question:
Can the amount of fat in a trail mix reliably predict the calorie content?
How does the scatter plot help us to answer this question?
What else do we need to do to answer this question?
Guiding Questions: Are fat and calories associated? How can we describe the association between fat and calories? Can we estimate the linear fit (with a line of best fit?)? How would we model this without guessing?

10. Line of Best Fit
Estimate the line of best fit for the Trail Mix data.

11. Testing Your Line of Best Fit
Variability in bivariate quantitative data can be quantified by calculating the residuals and creating a residual plot.
Residuals: the vertical distances from the actual y-value of each data point to the corresponding y-value generated by a linear model; actual - predicted
Residual Plot: a scatter plot created by plotting the x-values on the horizontal axis, and the corresponding residuals on the vertical axis

12. Variability
On your scatter plot or in a spreadsheet, find the residuals for your line of fit.
Create a residual plot by hand.
Pay attention to patterns if any emerge … this may indicate that a linear model is not appropriate for this data set.
Things to note:
Patterns in residual plots indicate something
Squaring and summing the residuals may lead to a conceptual understanding of the LSRL

13. Variability
Go back to your scatter plot on which you drew the residuals.
Sketch and shade in squares generated from the residuals. Compare your squares to some of your classmates. Who has the smallest total area of squares? What do you think that means?

14. Line of Best Fit → LSRL
The Least Squares Regression Line (LSRL) will give us the true line of best fit.
It does this by minimizing the sum of the squares created by the residuals to generate the equation for the best line of fit.
You can play with this idea here:

15. Line of Best Fit → LSRL
Use desmos.com to find the LSRL of the trail mixes’ fat vs. calories data by typing in y1 ~ mx1+b
Link:

16. How Good Is Our Model? Correlation Coefficient, r
The correlation coefficient, r, is a measure of how much or how little data is scattered around the LSRL. If you have already plotted the residuals and decided that a linear model is a good fit, the correlation coefficient, r, is a measure of the strength of a linear association, and is a value from +1 (positive, perfectly linear association) to -1 (negative, perfectly linear association).

17. How Good Is Our Model?
r-Squared: The correlation coefficient squared (r2) gives us a percentage that helps us understand the variability in the data. r2 % of the variability in the dependent variable can be explained by a linear relationship with the independent variable.
Lower r-squared % → higher chance that other variables are affecting the dependent variable (in this case, perhaps carbs, protein, etc. are affecting the calorie content in the trail mixes).

18. Answering Our Guiding Question
Can the amount of fat in the trail mix reliably predict the calorie content? Why or why not?