## How Did I Move?

6-8,9-12
2

A common problem when students learn about the slope-intercept equation y = mx + b is that they mechanically substitute for m and b without understanding their meaning. This lesson is intended to provide students with a method for understanding that m is a rate of change and b is the value when x = 0. This kinesthetic activity allows students to form a physical interpretation of slope and y-intercept by running across a football field. Students will be able to verbalize the meaning of the equation to reinforce understanding and discover that slope (or rate of movement) is the same for all sets of points given a set of data with a linear relationship.

### Preparation

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

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 classroom.

### 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

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.

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.

Assessment Options

1. Ask groups to present their results from the football activity to the class. They could compare graphs and discuss how they are the same and how they are different. Finally, the class can determine which student had the fastest speed overall for this activity.
2. Ask each group to present the answer to one of the Questions for Students.

Extensions

1. You may use remote-controlled cars instead of individual students to compare speeds.
2. Have students create a problem to The Winning Goal activity and pass it to another group to find the faster player.
3. Ask students to create other rate of change scenarios, such as how much they earn at their jobs vs. the amount of time they work. They should represent their data using tables, graphs, equations, and written explanations.
4. Move on to the next lesson, Road Rage.

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 opposite happens.]

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.]

Teacher Reflection

• 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?

### How Should I Move?

6-8, 9-12
This investigation uses a motion detector to help students understand graphs and equations. Students experience constant and variable rates of change and are challenged to consider graphs where no movements are possible to create them. Multiple representations are used throughout the lesson to allow students to build their fluency with in how graphs, tables, equations, and physical modeling are connected. The lesson also allows students to investigate multiple function types, including linear, exponential, quadratic, and piecewise.

9-12
In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds to the cars crashing. Multiple representations are woven together throughout the lesson, using graphs, scatter plots, equations, tables, and technological tools. Students calculate the time and place of the crash mathematically, and then test the results by crashing the cars into each other.

### Learning Objectives

Students will:

• 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

• 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.