Algebra 1/Integrated I Grade Level Lesson PlanContext: Trail Mix

The focus of this context is to have students explore a linear association in bivariate quantitative data. Several important areas within the high school Idaho Content and Practice Standards can be addressed through this context.

ESSENTIAL QUESTION: Can the amount of fat in a trail mix reliably predict the calorie content? Why or why not?

Guiding Questions:

1. 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?

An investigation into this question could involve students collecting, representing, and analyzing the nutritional content of a variety of trail mixes. Students can sample several different types of trail mix, weigh the individual ingredients, and approximate the nutritional contents of their sample. This connects to their understanding of proportional relationships from previous grades. Students can also use the manufacturer’s nutrition labels to generate a data set that suits their investigation. High school students can be pressed to connect their informal understanding of approximating the center of bivariate quantitative data with a line of best fit with the least squares regression line, and begin to describe and quantify the variability with residual plots and the correlation coefficient. This work fosters connections to other domains in the standards, such as Number and Quantity, Geometry, and Functions.

2. What patterns or relationships do you see when the data is displayed in a scatter plot? This question presses students to consider the independent variable (the amount of fat) and the dependent variable (total calories) and informally determine if a relationship between them exists. Student may use arguments such as “the higher the fat content, the higher the calories” to defend their claim of a linear association between the two variables. This is an excellent place to start the formal investigation into this relationship.

3. What does the equation for the line of best fit (or the least squares regression line) mean in this context? What does the slope suggest about trail mix? What does the y-intercept suggest about trail mix?

These questions will press students to consider what the slope of the LSRL means in terms of the number of calories that are expected to be in the trail mix per gram of fat. In this particular lesson plan, the slope of the LSRL is 7.67, which is different than the expected 9 calories per gram of fat many students learn about in health or nutrition classes. This could generate an interesting discussion about variability and/or the appropriateness or strength of the linear model we are using to describe this relationship. The y-intercept could also generate some interesting discussion (“If there are 0 grams of fat in the trail mix, there would still be 178 calories in it?”), and allows an opportunity to discuss interpolation and extrapolation of a data set.

4. How can you determine the “goodness” of our linear model?

An investigation into this question will require students to consider the residuals, use technology to generate the least squares regression line (LSRL), and interpret the correlation coefficient for the LSRL, all for the first time in their statistical experience. When investigating the residuals, conceptual connections can be made to the mean absolute deviation (MAD) that students will have worked with in Grades 6 and 7. The LSRL minimizes the total area of the squares generated by the residuals, which connects well with the squares generated by the formula for the standard deviation.

SAMPLE LESSON PLAN:

Though a full range of High School standards can be targeted through use of this context, this sample lesson plan and instructional sequence targets the following:

HS.S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a) Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

b) Informally assess the fit of a function by plotting and analyzing residuals.c) Fit a linear function for a scatter plot that suggests a linear association.

HS.S-ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

HS.S-ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

HS.S-ID.9. Distinguish between correlation and causation.

When considering all of the work students will be required to do to answer the Essential Question, it is likely that they will meet all eight of the Standards for Mathematical Practice. However, the bolded practices below are specifically targeted in this sample lesson:

1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

Materials:● Access to desmos.com/calculator or NCTM Core Math Tools● Handout: https://idoteach.boisestate.edu/modal/files/2016/06/TrailMixHandoutAlg1IntI.pdf ● Slides: https://idoteach.boisestate.edu/modal/files/2016/06/Trail-Mix-Alg-1_Int-I-Slides.pptx ● Data in Desmos: https://www.desmos.com/calculator/zgh10m1xzm

● LSRL Student e-Tool in Desmos: https://www.desmos.com/calculator/lywhybetzt Instructional Sequence:

Activity Details Teacher Notes

Use the Slides to introduce the trail mix investigation and frame the Essential Question.

Slide 3

Guiding Questions: 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?

On Slide 5, pose the Essential Question: Can the amount of fat in a trail mix reliably predict the calorie content? Why or why not?

On Slide 3, 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.

In a variation of this lesson, students actually took samples of different trail mixes, weighed all of the different ingredients individually, and calculated the total fat and calories for each trail mix. Their data was used for the remainder of the lesson instead of the sample data provided in this sample lesson plan.

After a discussion surrounding the questions posed on Slides 6 & 7, give students the handout that displays the data in a table and a scatter plot.

A modification to the lesson could include students creating the scatter plots themselves with just the data table given to them. This is entirely up to the teacher’s discretion and time available. Creating a scatter plot by hand may take a significant amount of time, as students may run into issues setting up the scales for the axes, struggle with accuracy in plotting the points, etc. However, if students have limited experience creating scatter plots by hand, this could prove to be valuable. Discussion about how the scales and intervals of the graphs can affect their appearances would be interesting.

Slide 6

Handout

“When making the scatter plots by hand, some students ‘rounded’ their data to more friendly numbers (intervals of 2 or 5, for example) and this greatly affected the graph.Other students were more precise in the plotting of the data, and their graphs shows more scatter (a.k.a. more variation in the data and a better representation of the actual data).”

Using the questions on Slide 9, frame Guiding Question 2 and discuss: What patterns or relationships do you see when the data is displayed in a scatter plot?

The hope is that students will suggest a linear association between the fat content and total calories of the trail mixes in the sample. This should lead into the approximation of the line of best fit, and a discussion about these Guiding Questions:What does the equation for the line of best fit mean in this context? What does the slope suggest about trail mix? What does the y-intercept suggest about trail mix?

Having students share their approximations and their equations allows the students to see how the variability in the data set has an affect on the approximated “center.” Time should be spent exploring their work at this point in the lesson. These differences will also allow for a discussion on the interpretations of the slopes and y-intercepts of the estimated equations. This also may organically generate another important guiding question: How do we know which line is the BEST fit for the data? This will be “answered” with the investigation into the least squares regression line later in the lesson.

Prior to presenting Slide 11, Guiding Question 4 could be posed: How can you determine the “goodness” of our linear model?This may stir some debate and generate the need for an investigation into the residuals.

“Students in my class struggled with understanding residuals. We had to slow down and go point by point under the document camera until some aha’s happened around the room.”

When calculating residuals for the first time, teachers may need to pay more attention to the conceptual connections to the mean absolute deviation (MAD) that students will have worked with in Grades 6 and 7 to help students understand what they are doing (i.e. finding the vertical distance from each point to the line of best fit).

This may or may not be the best time to discuss patterns in a residual plot, and the implications of such patterns on the appropriateness of a linear model for a data set. This can be saved for a follow-up lesson.

At this point in the lesson, pose the question “How do we know which line is the BEST fit for the data?” again. Work through Slides 13-15 and the animated LSRL Student e-Tool at https://www.desmos.com/calculator/lywhybetzt

LSRL Student e-Tool

Once students have the equation for the LSRL, teachers can ask students to compare the LSRL to their approximated line of best fit. What are the differences in the slopes? What are the differences in the y-intercepts?

“This student created rectangles instead of squares on their scatter plot. The rectangles were generated by the residuals and the corresponding distance from the y-axis. I might have students play with the e-Tool before trying to draw their own squares to avoid this misconception.”

What does that mean in terms of the trail mixes?

Notes: The PARAMETERS give the slope and the y-intercept for the LSRL. Under RESIDUALS, students can click the “plot” button and see the residual plot for the LSRL.

Slides 16 and 17 introduce the correlation coefficient and r2 values. These are the final pieces of information students need to answer the essential question, which is posed again on Slide 18.

Students can answer the essential question in a variety of ways (discussion, written response, video response, etc.). They should be pressed to use the LSRL, the residuals, and the correlation coefficient to make their claim.

POTENTIAL FOLLOW-UP ASSIGNMENT/ASSESSMENT:Students can determine which is a better predictor of calorie content in trail mix, fat or total carbohydrates. The data and graph for the carbohydrates vs. calories investigation can be found here:https://www.desmos.com/calculator/gml16yzg7r

Students should compare the scatter plot, LSRL, residuals, and the correlation coefficient to make and support their claim.