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Insights Into Algebra 1 - Teaching For Learning
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Workshop 8 Mathematical Modeling Lesson Plans
Lesson Plans:

Introduction

Lesson Plan 1: Mathematical Modeling, Circular Movement and Transmission Ratios

Lesson Plan 2: Skeeter Populations and Exponential Growth
Download the Workshop 8 Guide


Tool Box
Journal
Graphing Calculator
Channel-Talk
NCTM Standards


Lesson Plan 1: Mathematical Modeling, Circular Movement and Transmission Ratios

Overview Procedures For Teachers Related Standardized Test Questions Materials

Supplies:

Teachers will need the following:
  • Overhead transparency with a diagram of the fan, the crankshaft, and the alternator pulleys in an engine
Students will need the following:
  • Graphing calculator
  • Peg board
  • Drivers and followers (at a minimum, circles with radii of 1 cm, 2 cm, 4 cm) that can be inserted into the peg board
  • Rubber bands
Note: Sarah Wallick obtained the pulleys she used in the video lesson for this workshop from Glencoe McGraw Hill, publisher of the Core Plus curriculum materials.

Steps

Introductory Activity:

1. Display the transparency diagram on the overhead projector. Briefly discuss how the fan, crankshaft, and alternator pulleys in an engine work together.

2. With the class, discuss the following questions:
  • How does the speed of the crankshaft affect the speed of the fan? How does it affect the speed of the alternator?

    If the idling speed of the crankshaft of a four-cylinder sports car is about 850 rpm, how far, in centimeters, would a point on the edge of the fan pulley travel in one minute? Do you think that a point on the alternator pulley would travel the same distance in one minute? Why or why not?

  • Describe another situation in which the speed of one rotating object affects the speed of another object.
Learning Activities:

1. Introduce the setup of a follower and driver, and explain that students will rotate the driver pulley to determine the affect on the follower pulley. Specifically, students will determine the distance traveled by the follower when the driver pulley makes one complete revolution.

Ask students to experiment with different sizes of followers and record their observations in a two row table. They should record the number of driver rotations in the first row and the corresponding number of follower rotations in the second row.

2. Have students experiment with a driver of radius 2 cm and a follower of radius 1 cm. Check to be sure their data is close to the following:

Rotations of driver 1 2 3 4   5
Rotations of follower 2 4 6 8 10

Students should graph the data in a scatterplot using a graphing calculator. Have them describe the patterns in the data, and, as a class, find algebraic models that might fit the data. Discuss various equations or representations they could use; these might include the following:
    y = 2x
    Next = Now + 2, Start at 0 (the value for 0 rotations of the driver)
3. Have students repeat the experiment with a driver of radius 1 cm and a follower of radius 2 cm. Their data should be close to the following:

Rotations of driver 1 2 3 4 5
Rotations of follower .5 1 1.5 2 2.5

Students should look at the patterns in this second set of driver follower data. Again, they should graph the data in a scatterplot using a graphing calculator. Have them describe the patterns in the data, and, as a class, find algebraic models that might fit the data. Discuss various equations or representations they could use; these might include the following:
    y = 0.5 x
    y = 1/2 x
    y = x/2
    Next = Now + 0.5, Start at 0 (the value for 0 rotations of the driver)
4. Ask students to work in groups to generate a formula to determine the transmission factor for any ratio of driver-to-follower size. Reconvene the class for a discussion and select students to share and justify their answers

5. Assign student groups a set of problems related to the diagram they investigated in the introductory activity and have them prepare their solutions on poster paper. Choose several students to present solutions to the whole class. As part of this discussion, students should be able to correct, or justify, their intuitive answers to Question #2 from the introductory activity.

Culminating Activity/Assessment:

At the end of class, have students recount what they learned in their math journals. Then ask students to share their thoughts with the whole class. To summarize, create a list of things students have learned on the overhead projector.

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Next: Related Standardized Test Questions
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