## How Should I Move?

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.

This lesson is intended to be an introduction to various function
types, including linear, exponential, quadratic, and piecewise.
Therefore, use this activity to pique students’ interest in functions
and to allow them to deduce algebraic equations. However, as you work
through the lesson, do not expect exact answers. Rather, focus on
allowing students to develop an understanding of the multiple
representations of functions and concepts such as constant vs. variable
movement. Lesson 2 in this unit, *How Did I Move?*, works well when you
are introducing the slope-intercept form of linear equations.

Note that you may wish to repeat the corresponding parts of this lesson at that time.

### Preparation

The lesson is designed to allow students to investigate six graph pairs over two days. Use Day 3 to reflect on the content of Days 1 and 2.

How Should I Move? Graphs Activity Sheet

Before Day 1, choose 6 graph pairs from the How Should I Move? Graphs Activity Sheet. Graph Pairs 1–6 deal with linear graphs, while Graph Pairs 7–9 include exponential and quadratic graphs. The linear graphs are generally easier for students to explore algebraically. If you wish to include a discussion of constant vs. variable movement, then include at least one of Graph Pairs 7–9.

If only one motion detector is available, prepare the How Should I Move? Overhead by writing the 6 graph pair numbers in the 6 empty boxes. For example, if you choose to use Graph Pairs 1–4, 7, and 8, the completed table make look like this:

Groups | Graph Pairs | |

Day 1 | Day 2 | |

Groups 1 & 2 | 1 | 4 |

Groups 3 & 4 | 2 | 7 |

Groups 5 & 6 | 3 | 8 |

Before class begins on Day 1, set up the motion detector to collect data and project the results to the front of the classroom. Set the motion detector to collect data in feet, to work best with the activity sheets provided.

### Introducing the Activity

Ask a student who arrives early to walk in front of the motion detector so the resulting graph can be displayed. Students’ interest will be piqued as they enter the classroom if they see a student’s movements creating a graph. Allow a few minutes for students to play with the motion detector and graphs before beginning the formal lesson. Note that graphs begin at 2 feet from the motion detector.

Explain to the class that over the next 2 days, they will be trying to re-create graphs by moving in front of the motion detector. They will work in groups, and groups will be paired so that they compete with another group to create the best graph. Each group will be given 2 chances to make the best graph. For example, group 1 tries to create Graph A twice. Then, group 2 tries to create the same graph. Group 2 will then begin first for Graph B. Refer to one of the graphs they created when they first came in, or show the sample graph below:

*If only one motion detector is available:*

Divide the class into 6 groups. Each group should have at least 3 students so that one student runs the motion detector, the second student writes down the strategy, and the third student moves to create the graph. If it is not practical in your class to have six groups, separate the class into equal-sized groups so that all students will have a chance to use the motion detector. Other students in the group could be assigned the task of suggesting the movement if group size exceeds 3.

Display the How Should I Move? Overhead with the appropriate graph pair numbers filled in. Ensure students know their group assignments and the graph pairs they are to create on the motion detector. All students in the class should complete the How Should I Move? Questions Activity Sheet for each graph pair, using the graphs created by the groups assigned to that graph pair. Once all students have an opportunity to complete the activity sheet for a graph pair, the group responsible for creating the graphs on the motion detector should share their answers.

How Should I Move? Questions Activity Sheet

*If multiple motion detectors are available:*

Separate the class into groups of 3. Let students know that during each trial, one student runs the motion detector, the second student writes down the strategy, and the third student moves to create the graph. Pair groups together so they can compete to create the best graph. Then, let the class know which graph pairs they will be responsible for and distribute the motion detectors and graphing calculators or computers to each pair of groups.

### Main Activity (Days 1 and 2)

How Should I Move? Graphs Activity Sheet

How Should I Move? Questions Activity Sheet

Distribute the How Should I Move? Graphs Activity Sheet to each group. Place a pile of the How Should I Move? Questions Activity Sheets in a convenient location; each student will need a copy for each of the graph pairs being explored.

Have students follow the How Should I Move? Questions Activity Sheet in order. If only one motion detector is available, each student should make 2 conjectures about the type of movement that may be needed to create the graphs (Question 2) before the assigned groups try to create the graphs on the motion detector. If multiple motion detectors are available, groups may work at their own pace.

Now the competition will begin. The assigned groups should take turns competing to create the best graphs. All students will sketch the best graphs from the motion detector and record the group number for the best walker. If the two groups are not successful, allow other groups to try to create the graphs. Do not provide any hints.

After the groups have created the first pair of graphs, ask all the groups to read their conjectures. Allow for a discussion, asking the groups that participated in the trials to provide answers to Questions 5–8. When answering Question 7, encourage students to discuss their starting position, speed, and the times and locations where they had to change their movements. For example, Graph Pair 1 has the same starting location for both graphs, but a faster speed is required to create Graph B.

How Should I Move? Questions Answer Key

When 3 graph pairs have been completed, have students attempt Question 9. Refer to the How Should I Move? Questions Answer Key as you circulate through the room and assist students. They may use either the original graph or their sketch to complete the table. Students will very likely struggle with the equation(s). It may help to first write an explanation of the movements using numbers, and then go back to try the equation(s). You may give students some guidance using one of the graphs as an example, and have them attempt the others on their own. Move around the room and ask students about the patterns they see in their data.

Each group should present their answer to Question 9. Have students list the equation(s) they came up with on the board, and discuss how they are the same and different, how they relate to the movements, and how they relate to the graphs of time vs. distance.

Day 2 should largely mimic Day 1 with students competing to create the best graphs for 3 graph pairs. As students struggle with more difficult graphs or graphs that are impossible to create using the motion detector, encourage them to explore the activity without providing hints. Again, do not expect exact equations. Tables, equations, and explanations that approximate the given graphs are sufficient to understand the relationships between the representations.

**Analysis** (Day 3)

Have students share the equations they've deduced over the past
two days. If students were unable to come up with any equations, begin
sharing with students algebraic representations of graphs. For example,
in Graph Pair 7, discuss the doubling and halving patterns. Ask how
doubling and halving might be represented symbolically, working up to
the representation 2^{x}. Ask students to volunteer
their equations, or provide correct equations and have students verify
that they represent the graphs accurately. Do this until all 6 graph
pairs have been discussed.

Comparing Graph Pairs Activity Sheet

Have students complete the Comparing Graph Pairs Activity Sheet as a summary activity. Depending on which graph pairs you chose to use for this activity, some questions may be left blank. You may choose to review the answer key (provided with the activity sheet) with the class, and share with students which questions cannot be answered. Alternatively, you may let students consider all the questions and decide for themselves which do not have answers. For example, if you did not choose Graph Pairs 4 or 9, there will be no answer to Question 3, but you may leave it to the students to discover this. When everyone has completed this activity sheet, have students share their answers with the class. Lead a discussion of the answers as a closure activity for the lesson.

- CBR2 or other motion detector
- TI Interactive!, TI-84 graphing calculator, TI Nspire handheld, or other compatible graphing calculator or computer with compatible software
- How Should I Move? Overhead
- How Should I Move? Graphs Activity Sheet
- How Should I Move? Questions Activity Sheet (several copies per student)
- How Should I Move? Questions Answer Key
- Comparing Graph Pairs Activity Sheet and Answer Key
- Graphing Equations Activity Sheet (optional)

**Assessment Options**

- Ask students to write journal entries about their experience in this activity. When did they struggle? How did they change their approach to create a better graph? How did struggling help them understand the content better?
- Ask each group to present the answer to one of the questions on the Comparing Graph Pairs Activity Sheet.
- Have students complete the Graphing Equations Activity Sheet in class or assign it as homework.

Graphing Equations Activity Sheet - Have students write explanations about how they may have created the graphs from the 3 graph pairs that were not explored in class.

**Extensions**

- More complex graphs, suggested by students or by you, can be attempted. These could be additional examples of the function types included in this lesson (linear, exponential, quadratic, and piecewise), or different function types that fit with your curriculum. Later in the year, when you introduce a new function type, consider having a shorter motion detector activity.
- Ask students to suggest what each graph might represent in real life and to tell a story that fits the graph. For example, for the piecewise functions included in Graph Pairs 5 and 6, students might explain why a person would be walking, stop, and then start walking again.
- Students could use computer-based motion detector activities and compare the result to their movements.
- Move on to the next lesson,
*How Did I Move?*

**Questions for Students**

1. How could you move to create the graph?

[Answers will vary, but students should discuss position, speed, and direction.]

2. How did you move differently to create this graph than you did to create the previous graph?

[Answers will vary, but students should again discuss changes in position, speed, and direction.]

3. How can you tell by looking at a graph whether you should move at a constant rate or a variable rate?

[Encourage students to make the connection from the graph to the movement and from the movement back to the graph. After a few graphs, students should begin to recognize that a line or line segment is created using constant movement, and a curve is created using variable movement.]

4. How do different function types lead to different graphs, tables, and equations?

[Answers will vary. This is a question students should revisit several times as they work through the activities.]

5. Which graphs were unique?

[Answers will vary, but encourage students to discuss distinctive characteristics of graphs such as changes in movement (e.g., moving for part of the graph and standing still for another part).]

6. What other kinds of graphs do you think could be created using a motion detector?

[Only functions can be created on a motion detector. This is a great time to discuss how the

vertical line testrelates to this activity. A vertical line on the graph represents all the distances at a given time. A graph fails the vertical line test if more than one point of the graph goes through a vertical line. This means a person would have to be at more than one distance from the motion detector at a point in time.]

**Teacher Reflection**

- How did you ensure that students understood the relationships among the movement, graph, table, and algebraic equation?
- Were students actively engaged in this lesson?
- Did students meet the objectives of the lesson? If not, what should be done in subsequent lessons?
- Was students’ level of enthusiasm/involvement high or low? Explain why.
- How did students demonstrate understanding of the materials presented?
- 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 Did I Move?

*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.

### Road Rage

### Learning Objectives

Students will:

- Provide a conjecture on the type of motion that creates a provided graph.
- Use a motion detector to re-create provided graphs.
- Compare pairs of graphs by describing the similarities and differences between them in physical modeling and symbolic representations.
- Create tables and equations for the graphs.
- Compare the results of several graph investigations to one another.

### 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

- 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.

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.2

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

- CCSS.Math.Content.8.SP.A.3

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.