Insights Into Algebra 1: Teaching for Learning
Mathematical Modeling Lesson Plan 1: Mathematical Modeling, Circular Movement and Transmission Ratios
In this lesson, students will learn to use mathematical models to represent real life situations. In particular, they will use tables and equations to represent the relationship between the number of revolutions made by a “driver” and a “follower” (two connected gears in a system), and they will explain the significance of the radii of the gears in regard to this relationship.
Two 50-minute class periods
Transmission factors, explicit and recursive equations, and mathematical modeling
Students will be able to:
- Understand the significance of the transmission factor in the design of rotating objects (gears) that are connected.
- Determine the transmission factor using the radii of connected pulleys.
- Describe the differences between negative and positive transmission factors.
- Determine the distance a point on the circumference of a pulley travels, given the speed at which the pulley is turning.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
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.
Teachers Activities and Assignment
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.
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:
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:
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
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.
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.
Related Standardized Test Questions
The questions below dealing with mathematical modeling have been selected from various state and national assessments. Although the lesson above may not fully equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
- Taken from the New York Regents Exam, Mathematics (August 2002):
In the accompanying diagram, x represents the length of a ladder that is leaning against a wall of a building, and y represents the distance from the foot of the ladder to the base of the wall. The ladder makes a 60° angle with the ground and reaches a point on the wall 17 feet above the ground. Find the number of feet in x and y.
Solution: The triangle formed by the ladder, the wall, and the ground is a 30-60-90 triangle. Consequently, feet, and feet. Another solution method would be for students to use right-triangle trigonometry.
- The graph below shows the height of Cindy’s model rocket during the course of its flight.
- Taken from the New York Regents Exam, Mathematics (August 2002):
Which of these equations can be used to find the height of the rocket at any time during its flight?
A. y = 9x
B. y = x2 – 81
C. y = x2 + 9x
D. y = 9x – x2 (correct answer)
- Taken from the Virginia End of Course Exam, Algebra I (Spring 2001):
Mary published her first book. She was given $10,000.00 and an additional $0.10 for each copy of the book that sold. Her earnings, d, in dollars, from the publication of her book are given by:
d = 10,000 + 0.1n
where n is the number of copies sold. During the first year Mary earned $35,000.00 from the publication and sale of her book. How many copies of her book sold in the first year?
C. 250,000 (correct answer)
“The Power of the Circle” (PDF)
From Contemporary Mathematics in Context: A Unified Approach, Course 2, Part B, copyright (c) 2003 by Glencoe/McGraw-Hill. Used with permission.
Student Work: Transmission Factors Assignment
I’m very pleased with the work. The student has demonstrated her ability to analyze the problem and relate the data she collected in the table to a mathematical model using both an explicit equation and a recursive equation; from these she was able to develop a graph. Finally she developed a general model of transmission factor.
My only concern with the work is just below the line “Divide the driver by the follower”. She represents the relationship as a fraction, then equates it to slope. This is incorrect, since slope = delta y/delta x. The transmission factor = x/y.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.