Create a set of index cards for each group of three students,
with a different position at time 0 on each card. Each group should
have unique set of index cards. Each group member will be assigned a
different role on the football field for each of the three tasks. If a
football field is not readily available, a hallway or other space can
be used. Students should have a simple method for measuring their
distance (e.g., number of blocks on the wall, tiles on the floor, etc.)
so that they focus on the concept of movement as a rate of change
rather than spending time measuring distance. Use the following
criteria to create the three index cards for each group:
- Task 1: Specify a starting location and a time.
- Task 2: Specify a different starting location and a distance.
- Task 3: Specify the same starting and ending location, along with a time.
An example set of index cards is shown below.
Football Field Activity
How Did I Move? Activity Sheet
Distribute page 1 of the How Did I Move? Activity Sheet, which includes a drawing of the field. Read the
instructions on the page with students. Stress that students can only
run from left to right, and they must begin at the field position
specified on their index card. Inform students that they must move in a
forward direction; they cannot run forward and then back again. Also,
let them know that the remaining pages of the activity sheet will have
them analyze the data they collect on the football field. Allow
students to ask questions to clarify the tasks. When all student
questions have been answered, take them to the football field.
At the field, provide each group with a set of 3 index cards, a
stopwatch, and a pencil. Ask students to rotate through the roles of
football player, recorder, and timer. Each player should run or act out
the situation that corresponds to the data provided on his or her index
card. When groups complete their tasks, distribute the remaining pages
of the activity sheet. Students can either work on this at the field
or, when all students have finished gathering their data, back in the
Completing the How Did I Move? Activity Sheet
Students should work together in their groups to complete
pages 2 and 3 of the activity sheet. Circulate among groups and help
them with any difficulties they may have.
- Question 2: When students plot their data, make sure each
student plots 3 lines – one for each member of their group. Remind them
to label each line with the runner's name.
- Question 3:
- a) Students should see that the steepest line gains yardage
faster. If students have difficulty determining which runner was
fastest, they could use 2 pencils or 2 rulers to compare the steepness
of the lines.
- b) This is an interesting question that depends on
students’ interpretation. They could say that the student who didn’t
move was slowest, or they could say it was the student with the flatter
line. Strictly speaking, both answers are correct, so listen to the
discussion and encourage students to justify their answer.
- c) The graph for the group member who did not move is a horizontal line.
- Question 4: Based on the data and graphs in the
previous questions, students should be able to calculate the speeds and
determine the linear equations, but be prepared to provide guidance if
necessary. Students should use the slope formula to determine the slope
of each line, and then use the slope to write the linear equation. Some
students may also choose to use , but the scale of the graph may make this difficult.
- Question 6: Students should be able to determine that the greater the value of m, the faster the student ran.
- Question 7: Remind students that y = 100 when a
touchdown is scored, because a football field is 100 yards long.
Students may be confused by the equation for the third member of their
group (the one who didn't move), but they should be able to determine
that this player would never score a touchdown.
By the end of the activity, students should be able to relate yards per second to their movement (m) and their beginning position (b) to their position at time 0. Clarify for students that m does not stand for movement and b does not stand for beginning. This is just a mnemonic for remembering the role slope and y-intercept play in the equation y = mx + b.
Coleman's Touchdown and The Winning Goal
Coleman's Touchdown Activity Sheet & Answer Key
In the Coleman's Touchdown Activity Sheet,
students are presented with 7 questions that help to reinforce the
concepts from the previous activity. They predict when Coleman will
score a touchdown, and discover — either visually or computationally —
that his speed (or rate of change) is the same between any two points
on the graph. This may be confusing or unexpected for some students.
The Winning Goal Activity Sheet & Answer Key
If time allows, allow students to work on The Winning Goal Activity Sheet. The activity allows students to compare two different
forms of data for two players on a field hockey team. Kaitlin’s data
are presented in a table, and Brea’s data are presented in a graph.
Students analyze these data, create the slope-intercept equations, and
make a recommendation to the field hockey coach regarding which player
to substitute based on a mathematical analysis of each player's speed.
This helps reinforce the concepts learned earlier in the lesson.
Questions for Students
1. How can you tell by looking at a graph which student is fastest?
[The steepest line corresponds to the fastest student.
Students with steeper slopes traveled a greater distance during the
time interval. This is an ideal time to bring up the visual
representation of rise/run, where rise is the distance and run is the time.]
2. What happens to a line if run is changed in the formula m = rise/run?
[If run is increased, the fraction becomes smaller. This
would correspond to a traveling the same distance in a great time
resulting in a slower speed in this scenario. If rise decreases, the
3. Why is y = mx + b called the slope-intercept form of a linear equation?
[The value of m represents slope and b represents the y-intercept. In other words, without doing any calculations, you can see the slope and y-intercept of a line just by looking at the equation.]
4. What is a real-world example that demonstrates the meaning of slope?
[In this lesson, slope represents speed in yards per
second. Other examples of slope include miles per gallon, dollars per
hour, or cost per minute. In general, slope refers to the rate of
change in a linear equation.]
5. What is a real-world example that demonstrates the meaning of y-intercept?
[In this lesson, the y-intercept is position 0 on
the football field, i.e., the goal line of the opposing team. Another
common example is a cell phone plan—often, the monthly charge includes
a fixed cost plus some cost per minute. The y-intercept is the fixed cost.]
6. What did you notice about the slope between any 2 points on the line representing Coleman’s position? Why did this happen?
[The slopes are all the same in the activity. This is
because the slope between any two points on any given line is the same.
This relates to the constant motion result in Lesson 1.]
7. If a player were at position 0 and position 100 simultaneously at time 0, what would the slope of that player's line be?
[There would be no slope. On a graph, this would be
represented by a vertical line. The situation is impossible because a
person cannot physically at 2 places at the same time. You may wish to
ask students to compare this scenario with the one experienced by the
student who stayed in one place in the How Did I Move? activity.]
- How did your lesson address auditory, tactile and visual learning styles?
- How did students demonstrate understanding of the materials presented?
- Did students make the connection between slope, speed, and rate of change?
- How did students communicate that they understand the meaning of the slope-intercept equation?
- What were some of the ways in which students illustrated that they were actively engaged in the learning process?
- What, if any, issues arose with classroom management? How did
you correct them? If you use this lesson in the future, what could you
do to prevent these problems?
- Collect data in an activity on a football field.
- Compare movement and starting positions based on the data.
- Create the slope-intercept equations relating m to their movement and speed and b to their beginning running location.
- Use the equation to predict the distance at any given time.
NCTM Standards and Expectations
- Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.
- Use graphs to analyze the nature of changes in quantities in linear relationships.
- Use symbolic algebra to represent and explain mathematical relationships.
- Approximate and interpret rates of change from graphical and numerical data.
Common Core State Standards – Mathematics
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.
Grade 8, Stats & Probability
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Grade 8, Stats & Probability
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.