## Running Races

3-5
1

Students generate and compare paths which model given problem situations on graphing grids.

To begin, request that the students refer to their fact mastery records. Check to verify that they have memorized the two facts that they chose the day before. Have them cover the facts that they just learned, then list the facts that they have left to memorize. Ask them to make a plan for memorizing the remaining facts.

Show the students a table of values (with values filled in), and ask volunteers to graph the two lines on the chalkboard or overhead projector. Invite students to describe the race that was graphed. You might wish to use the table of values shown below.

On the overhead projector, place a blank graphing grid and table of values, and display the Distance, Speed, and Time Simulation.

Call on volunteers to position the runner at a starting point other than 0, and call on other volunteers to set the stride lengths. Ask another volunteer to enter the starting points in the table of values. Now advance each runner one stride at a time, and record each distance in the table of values.

Call students’ attention to the graph in the right hand corner of the simulation after each stride. Ask volunteers to graph each path on the graphing grid and to describe what they graphed.

Distribute three copies of the Graphing Grid Activity Sheet and one copy of the Table of Values (Seconds) Activity Sheet to each student. Tell the students to assume that each stride takes one second, and invite students to set the parameters of a new simulation, including starting points which are not 0.

As one student runs the simulation, encourage other students to enter the distances in their table of values and graph the paths on their grids. Run the simulation once or twice more, setting different stride lengths, starting points, and number of strides. Call on different volunteers to graph the races on the overhead as the other students graph them on their own grids. For each race, discuss the entries in the table of values, the placement of each point, and the resulting line that models each runner’s path. When all students have finished, ask them to describe the race orally and then to write a description under the graph.

Now tell the students that the race will go all the way to the tree, and that they should watch the graph in the right hand corner of the simulation very carefully. Identify volunteers to position the runners at the starting point and to set the stride lengths. Run the simulation, this time choosing the PLAY button to start the race. (Clicking this button causes the entire race to happen at once, instead of step-by-step.) Invite discussion of the paths that show what the runners did. Repeat using other settings.

Play Button

Now pose the following challenges to the students, one at a time, and discussion student solutions for each one:

• Make a graph that shows a race in which the boy and the girl started from the same position, but the girl got to the tree first. [There are many solutions, but the stride length of the girl must be greater than the stride length of the boy.]
• Make a graph that shows a race in which the boy starts behind the girl, but the boy gets to the tree first. [There are many solutions, but the stride length of the boy must be significantly greater than the stride length of the girl.]
• Make a graph that shows a race in which the boy starts at the tree and the girl starts at the house, but the girl reaches the tree before the boy reaches the house. [There are many solutions, but the stride length of the girl must be greater than the stride length of the boy.]
• Make a graph that shows the girl running for 2 seconds with a stride length of 3, and then she runs for 5 seconds with a stride length of 3. [The graph will change directions after 2 seconds. It will become steeper, because the stride length increased.]

Assessment Option

By the end of this unit, students should be able to do the following:

• Graph sets of ordered pairs on a graphing grid
• Compare graphs of linear equations

The Questions For Students will assist you in assessing your students’ level of understanding of the concepts in this lesson and unit. Because this information will be useful during parent conferences, when preparing individual educational plans, and when planning future instruction, you should make notes on the Class Notes Recording Sheet, and you should save all of the sheets that you completed during this unit for future reference.

Questions for Students

1. Suppose one the runners started at 8 and took six strides of 2 while the other runner started at 1 and took six strides of 4. Who do you predict would be closer to the tree after six strides? Why?

[The first runner would get to 8 + 6 × 2 = 20, while the second runner would get to 1 + 6 × 4 = 25. Therefore, the second runner would be closer to the tree.]

2. What is similar about these graphs? What is different?

[Both graphs are straight lines, but the graph for the runner who took strides of length 4 is steeper. The graph for the runner who started at 8 begins higher than the graph of the runner who started at 1. The lines cross somewhere between 3 and 4 seconds.]

3. If a runner with a stride length of 4 started at 2 and ran 5 seconds, where would that runner be? How about if the stride length were 6?

[In the first case, the runner would end at 2 + 5 × 4 = 22. In the second case, the runner would end at 2 + 5 × 6 = 32.]

4. A boy with a stride of 2 started at 14 and raced a girl with a stride of 3 who started at 0. Who would be ahead after 5 seconds? Who would be ahead after 10 seconds? After 15 seconds? Can you show this with a table of values? With a graph?

[Both a table of values and a graph would show that the boy would be ahead for the first 14 seconds; after that, the girl would be in the lead. So, the boy would be leading after 5 and 10 seconds, but the girl would be leading at 15 seconds.]

5. Which of the multiplication facts did you learn this week? How many do you have left to learn? What is your plan for doing so?

[Answers will vary, but it is important to have students answer this question and generate a plan for learning the remaining facts.]

Teacher Reflection

At the completion of this unit, it will be helpful to look back on what was successful and what could be improved. The following set of questions may help you determine the focus of future instructional activities, and documenting the level of each student’s understanding makes accurate information available for planning appropriate subsequent activities.

• Were the majority of the students comfortable using the applet? Did all of them have equal access to the computer?
• Which students met all the objectives of this unit? What extension activities are appropriate for those students?
• Which students did not meet all the objectives of this unit? What additional instructional experiences do they need?
• Did all students display understanding of the relationship between the table of values and the graph?
• Can students explain orally how to graph using a table of values? Can they write an explanation?
• Do all the students recognize the facts they know by heart and those they have yet to learn?
• What were the greatest challenges for the students in this unit?

The next set of questions will be especially useful as you move forward and develop a long-range plan for the rest of the year.

• How can you help students continue to focus on the important ideas from this unit?
• What other investigations would extend their experiences with graphing?
• How can you connect the key ideas of this unit with lessons about similar mathematical content?
• What learning experiences would be of assistance to students not yet comfortable with these concepts?
• Which activities will help the students retain their command of the multiplication facts?

### Number Line Journeys

3-5
In this lesson, students generate products using a number line model. Students are encouraged to predict the products and to answer puzzles involving multiplication.

### Setting the Pace

3-5
Students continue their investigation of modeling multiplication on the number line using the Distance-Speed-Time Simulation from the NCTM E-Examples.

### The First Race

3-5
Again using the E-Example simulation, students will model multiplication facts on the number line and compare various representations.

### Telling Racing Stories

3-5
In this lesson, students model races in which runners start from various positions. They enter numbers in a table of values, model races on a coordinate grid, and compare the results. Students begin to develop an understanding of linear relationships.

### Learning Objectives

Students will:

• Graph sets of ordered pairs on a graphing grid.
• Compare graphs of linear equations.

### NCTM Standards and Expectations

• Describe, extend, and make generalizations about geometric and numeric patterns.
• Represent and analyze patterns and functions, using words, tables, and graphs.
• Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.

### Common Core State Standards – Mathematics

Grade 5, Algebraic Thinking

• CCSS.Math.Content.5.OA.B.3
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule ''Add 3'' and the starting number 0, and given the rule ''Add 6'' and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.