Illuminations: Running Races

Running Races


The First Race

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

Learning Objectives

 
Students will:
  • Compare representations of multiplication on the number line
  • Find the product of two factors 0-9

Materials

 

Instructional Plan

Before class, interview students individually to verify that they have memorized the two facts that they chose the day before. If they have, have them blacken those two facts on their Fact Mastery Record. Then, request that they choose two facts to learn next. As in the last lesson, have them draw a number line model of each fact on a file card, write each fact on the back on the appropriate file card, and encourage them to review these facts several times during the day and to take them home and practice them with their family. (The next day, again test each student privately on the two new facts. If the answers are given rapidly and correctly, direct the student to indicate these facts on their Fact Mastery Record.)

Pair students and distribute two copies of the Table of Values (Strides) activity sheet to each pair.

Table of Values (Strides) Table of Values (Strides)

Open the Distance, Speed, and Time Simulation.

Set the simulation so that each runner is facing to the right, and move each runner to 0. Next, set the length of each runner’s stride, which will be the second factor in the equation. (The first factor is the number of strides.) Ask a volunteer to suggest how many strides each runner should take, and have students circle the boxes corresponding to that number of strides on the Table of Values (Strides) activity sheet.

Ask students to predict which runner will be farther along after each takes the suggested number of strides. [The runner with the greater stride length will be farther along.] Then, run the simulation. Have students enter the position of each runner in the appropriate box as each stride is taken. For instance, if the number of suggested strides is 4, the stride length of the boy is set at 2, and the stride length of the girl is set at 3, then students’ tables should be completed as follows:

When the suggested number of strides has been taken and the positions recorded, ask the students to verify their prediction, and explain why it was or was not correct. Ask them to represent the race on the first two number lines on the Number Lines activity sheet. Then, have them write their conclusions and share them with a friend.

Number Lines Activity Sheet Number Lines Activity Sheet

Run the simulation again, using different stride lengths and number of strides. Call students’ attention to the graph after each run of the simulation. Ask them to record the values from each race on a different pair of number lines.

After the simulation has been run four times, ask the class to compare the positions of the runners in each race. Ask them if they notice a pattern in the rows of the table. [If recorded correctly, each row will represent skip counting by the stride length. For instance, a stride length of 3 will represent skip counting by 3.]

Questions for Students

 
How would you model a race where each runner takes five strides, with the boy taking strides of 4 and the girl taking strides of 2?
[The boy would get to 5 × 4 = 20, but the girl would only get to 5 × 2 = 10.]

How would you model a race where each runner takes seven strides of 3? What equation will represent this situation? Who would be the winner?

[Each runner would get to 7 × 3 = 21, so the race would be a tie.]

What numbers would you find in a table of values if the runner took eight strides of 5?

[5, 10, 15, 20, 25, 30, 35, 40]

If the boy takes strides of 3 and the girl takes strides of 4, what would their positions be after 10 strides? ...after 20 strides?

[After 10 strides, the boy would be at 30, and the girl would be at 40. After 20 strides, the boy would be at 60, and the girl would be at 80. It is interesting to note that 20 is double 10, and the distance after 20 strides is double the distance after 10 strides.]

Which multiplication facts are you now sure of? Which two facts will you learn next? What is your plan for learning them?

[Answers will vary, but it is important to have students answer this question and commit to learning two new facts each day.]

Assessment Options

 
At this point in the unit, students should be able to do the following:
  • Compare representations of multiplication on the number line
  • Find the product of two factors
Your entries on the Class Notes
sheet begun earlier in the unit may be helpful as you plan remediation or extension activities for individual students. In addition to your observations, you may wish to collect work samples from the students.

Teacher Reflection

 
  • Did all students have a chance to use the applet? If not, how can you assure that the others have a chance to use it?
  • Which students could not model multiplication on the number line and record the product? What instructional experiences do they need next?
  • Which students have learned two more multiplication facts by heart since yesterday? How can you motivate the students who did not?
  • Which students have memorized all the multiplication facts? What retention activities can be done to help them hold on to these facts?
  • What adjustments should be made the next time this lesson is taught?

NCTM Standards and Expectations

 
Algebra 3-5
  1. Describe, extend, and make generalizations about geometric and numeric patterns.
  2. Represent and analyze patterns and functions, using words, tables, and graphs.
  3. Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
This lesson prepared by Grace M. Burton.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics


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