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The Line Runner

  • Lesson
6-8
1
AlgebraData Analysis and Probability
Michael Weingarden
Camarillo, CA

Students studying linear equations inevitably explore the concept of slope. Slope is a recurring topic found in algebra when studying linear equations, in trigonometry when studying the tangent ratio, and in calculus when studying the derivative. Lead students to a deeper understanding of slope and linear equations using elevation and distance data from a popular mapping website.

The goal of this lesson is to have students learn how slope and linear equations are related to elevation. There are websites that return elevation data when users enter a route. This lesson is based on the data provided by Map My Run. The examples in this lesson use data from the Agoura Great Race.

This lesson is most effective in a computer lab. However, if one is not available, use your own computer and projector or project data gathered ahead of time. Modifications are suggested if a computer lab is not available.

Before the lesson, make yourself comfortable with Map My Run. Make sure that you can find the desired destination, plot the route, and download the elevation and distance data. Work through the lesson by yourself once and consider presenting the lesson to students in the classroom before allowing them to work individually in the lab.

In the computer lab, direct students to http://www.mapmyrun.com. The goals of this lesson can be achieved effectively with students working alone or in small groups.

Students should:

  • Map a new run by plotting points on a map and ensuring that distance and elevation data can be seen.
  • Map a route that includes at least one hill with both an uphill and a downhill.
  • Download distance and elevation data to their spreadsheet application. (Make sure the units are the same.)
  • Plot the points on a scatterplot, which will closely resemble the elevation profile of the route.

As an alternative to having students plot their own course, students can use data from the Agoura Great Race by using the data included in the Agoura Great Race Excel File, which was downloaded from Map My Run.

Note that the scatterplot of data will very much look like a line graph. So many points are used that it appears to be a solid path. Use this opportunity to reinforce that a line consists of an infinite number of points.

Students should look at the graph of their data and choose three sections to investigate:

  • A long uphill portion of the route.
  • A shorter but very steep section of the uphill route.
  • A long downhill portion of the route.

Students will analyze these three sections of the route using a spreadsheet. You may wish to circulate and help students select good sections of their route to investigate.

Students will highlight all of the cells in the spreadsheet between (and including) the bottom and top points for the slope of a portion of the run with the greatest uphill rise. Students should then create a scatterplot chart of the data using the spreadsheet. When finished, they should see a graph of just the uphill slope.

Now have students add a trend line and display the linear equation.

Students can now answer Questions 1-6 on the Line Runner Activity Sheet and the first part of Question 7.

 

pdficon Line Runner Activity Sheet 

Students then repeat the above steps for the steepest slope on the uphill run and the slope on a downhill run. Students should now completely answer Question 7 and go on to Questions 8 and 9 on the activity sheet.

The following is a pictorial summary of what students will find. The image below shows the XY‑scatterplot and trend lines using the Agoura Great Race Excel File.

3176 lp1

The red line represents the average slope of one major uphill portion of a run. The green line represents the steeper slope of one small subsection of the uphill run. The light blue line represents the downhill (or negative) slope of a different section. Students might get a green line that is almost identical to the red line, but it is desirable for some students to get a green line that is significantly steeper than the red line. The very steep slope of the green line would be much harder to run than a course with a constant slope equal to the red line.

Students should recognize that positive slopes represent going uphill and negative slopes represent going downhill. They also should be able to look at a large group of data points and recognize varying steepness of slope.

Assessments 

  1. Have students who have made unexpected or innovative conclusions present their findings to the rest of the class. Allow other students to volunteer if they feel that have something unique to contribute.
  2. Have students summarize the activity on a poster or in a PowerPoint presentation by showing how their trendlines and equations represent their path. They should include a scatterplot and an analysis of three areas, one with average steepness, one with extreme steepness up, and another with extreme steepness down.
  3. Have students explain how a runner would need to set a treadmill incline to train for a race over the route that you analyzed. (Incline on a treadmill is measured in percent.)

Extensions 

  • Have students use techniques presented in this lesson to measure the grade (also known as slope) of a nearby road. Steep roads usually have a sign to express the steepness. Grade is expressed in percent. For example, a 5% grade means that the road rises 5 feet for every 100 feet of run. You also can use the data from Conejo Grade. Students should download the data, analyze with the spreadsheet application and determine the slope of the Conejo freeway. The sign on the freeway says 7%, so students should find the slope to be close to 0.07.
  • Have students pretend to be surveryors or civil engineers. They will measure a slope of a hill from top to bottom. Then have them gather several intermediate slopes. Have them calculate the average of the intermediate slopes and compare it to the overall slope from top to bottom. This could be done using a handheld GPS device.
  • Tie this activity to a physical education program. Students can analyze their local runs and possibly suggest new runs depending on whether they want to make the runs more or less difficult.
  • This activity could also be tied in with other types of web-based data and spreadsheet slope analysis including stock market data from web sites such as Yahoo! Finance. Have students predict what the graphs will look like by drawing lines representing growing sales for a company, stronger growth, or a decline. Then have the produce the graphs to see how close they were.
  • The graphing calculator could be used to analyze the data instead of or in addition to the spreadsheet application.

Questions for Students

1. When one slope is steeper than another, is the number that represents the slope bigger or smaller?

[The steeper the slope, the bigger the number.]

2. In most lessons on slope, slope is represented as a fraction with rise over run. Is slope always a fraction? What other ways can slope be represented?

[From this lesson, we can see that slope can be represented by a decimal or percent as well.]

3. If a line was drawn on the graph and the slope was an integer, would the line be more steep or less steep than the slopes you worked on in this lesson?

[All of these slopes have rises that are smaller than the runs. So, an integer slope should be a lot steeper.]

4. How did this lesson show that knowledge about slope and linear equations can be applied in the real world?

[To calculate the steepness of roads or paths.]

5. Are there other applications for the work we did with the spreadsheet today?

[Sample answers: stock market data analysis, heart rate data from heart rate monitors, rollercoasters (which could lead into a discussion of instantaneous slopes and derivatives) and sales figures for various companies.]

Teacher Reflection 

  • What could be done to improve student success in the computer lab?
  • How well did students work as teams?
  • Could this activity work better as a multiple day or after-school project?
  • Did any students find a local route that could be used the next time you use this activity?
  • Which students produced the most interesting results and why were the results so interesting?
  • Is it possible for students to self-assess or use peer assessment with this assignment?

Learning Objectives

Students will:

  • Relate slope and linear equations using real world data
  • Compare and contrast the steepness of multiple lines visually and numerically
  • Compare and contrast positive and negative slopes

Common Core State Standards – Mathematics

Grade 6, Expression/Equation

  • CCSS.Math.Content.6.EE.A.3
    Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Grade 7, Expression/Equation

  • CCSS.Math.Content.7.EE.A.2
    Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

Grade 8, Expression/Equation

  • CCSS.Math.Content.8.EE.B.5
    Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Grade 8, Expression/Equation

  • CCSS.Math.Content.8.EE.B.6
    Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Grade 8, Functions

  • CCSS.Math.Content.8.F.B.4
    Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.