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 MapMyRun. 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 MapMyRun, or a different route tracker of your choice (there are many app choices available).
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 route tracking site of your choice. The goals of this lesson can be achieved effectively with students working alone or in small groups.
- 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 MapMyRun.
Agoura Great Race Excel File
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.
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
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.
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
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.
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
[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?
answers: stock market data analysis, heart rate data from heart rate
monitors, roller coasters (which could lead into a discussion of
instantaneous slopes and derivatives) and sales figures for various
- 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?
- 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.
NCTM Standards and Expectations
- Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.
- Use graphs to analyze the nature of changes in quantities in linear relationships.
- Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.
Common Core State Standards – Mathematics
Grade 6, Expression/Equation
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
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
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
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
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
Make sense of problems and persevere in solving them.
Model with mathematics.
Use appropriate tools strategically.
Look for and make use of structure.