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Overview:
In this lesson, students will learn to model situations mathematically. Specifically, students will perform investigations on the growth of populations and use exponential functions to represent that growth.
Time Allotment:
Two 50-minute class periods
Subject Matter:
Exponential Growth and Mathematical Modeling
Learning Objectives:
Students will be able to:
Standards:
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
http://standards.nctm.org/document/chapter6/alg.htm
NCTM Algebra Standard for Grades 9-12
http://standards.nctm.org/document/chapter7/alg.htm
Supplies:
Teachers will need the following:
Students will need the following:
Steps
Introductory Activity:
1. If students are not already in groups, divide them into teams of four.
2. As an introduction to various types of growth, show video clips. The first video shows a simple model of cellular growth, increasing by one each time; the second video shows a doubling pattern. You can also demonstrate these growth patterns using chips and an overhead projector. For the first model, place one chip on the transparency, and continue adding one chip at a time. For the second model, start with one chip and then double the number each time.
3. Discuss the various types of growth that may occur in nature. Ask students what mathematical operations they could use to represent growth. (Answer: addition and multiplication.) Explain to students that they will be using Skeeters to model population growth during this lesson.
Learning Activities:
1.Distribute the handout for the lesson.
2. Ask one group, “Based on the information in the table, how many populations do we have?” Select one person in the group to explain the five populations that are represented in the table.
3. Ask another group, “What is the initial population for each different color?” Select one student to explain that green has an initial population of one; yellow, one; orange, 40; red, five; and purple, five.
4. Choose a student to read aloud the growth characteristic of the green population.
5. Demonstrate a “shake” of the green population by placing one green Skeeter in the box to represent the initial population and then shaking the box. At the end of the shake, add two green Skeeters for each Skeeter in the box as the growth characteristic. Ask students, “What is the population now, at the end of Shake 1?” Students should respond that the population is now three green Skeeters.
6. Demonstrate a second shake, and ask, “How many green Skeeters should I now add?” Students should say that you should add six Skeeters, because you are to add two Skeeters for each Skeeter in the box.
7. Ask, “What is the total population at the end of Shake 2?” Students should respond that the population is now nine Skeeters.
8. Have each group conduct an exploration for one of the Skeeter populations. (An effective way to do this is to form “exploration stations” by filling ten containers with colored Skeeters – two containers for each color. Distribute one container to each group.) As they conduct the exploration, have students create a scatterplot for the population, with the shake number along the horizontal axis and the Skeeter population along the vertical axis. (Students should graph the points using paper and pencil, and enter the data for the populations using the STAT feature of their graphing calculators.)
9. Instruct students to complete the “Pattern” column for Tables 2-6 using the information for the population with which they worked. Say, “By looking at the Skeeter population, find the pattern that can be used to predict the population for each consecutive shake.” Tell students to consider whether the pattern is growing by addition or multiplication.
10. Have a student read aloud the description of how to complete the “Process” column. Then present an example of how to complete this column. First, ask one group: “What pattern did you notice in your population?” If that group worked with green, they should respond that the population multiplied by three, or tripled, with each shake. With student input, represent this on the board or overhead projector in the following way:
By replacing the variable “Initial Population” with its value, 1, rewrite these equations as:
When students are ready for the next step, you may wish to demonstrate or elicit from them that they can represent the last line as:
Pop(2) = 1 � 3^{2}
11. Give the groups several minutes to complete Tables 2-6 for the particular population with which they worked.
12. Have students rotate the exploration stations so that each group receives a new color. Allow students time to complete the exploration for each of the five colors, creating scatterplots for each exploration. By the end, they should have completed all of Tables 2-6.
13. Randomly select a group to present their findings for each of the populations. A student from the group should give each color’s population for n shakes. Record the equations on the board or overhead projector as follows:
(For the purple population, be sure to request the equations generated by several groups, as the answers will likely differ.)
14. Ask students, “Why do you think the equations for the purple populations vary?” Students should respond that the population is less predictable because the number of Skeeters added is based on the number that land with a mark showing, and that number may be different for each group after each shake.
15. Ask students, “What percent of the Skeeters [in the purple population] would you expect to show a mark after each shake?” Students should respond that about 50 percent of them will show a mark each time.
16. Ask students, “How is that 50 percent represented in the equation? What is the value that your groups got for the number that is raised to a power in the equation for purple?” Students should respond that the value is around 1.5, which is the decimal number used to represent a 50-percent increase.
17. With the class, convert the green population equation to a function in x and y. (It may be helpful to refer to the scatterplot, which plotted the number of shakes along the x-intercept and Skeeter population along the y-axis. The class should come up with the following equation:
18. Give the groups one minute to come up with equations in x and y for the other four colors. They should report the following:
19. Ask students, “What did the shape of the graph for each of these populations look like?” Students should notice that the graphs for orange and red were lines (i.e., linear functions), while the graphs for green, yellow, and purple were curves (i.e., exponential functions).
20. Conclude by telling students: “The purpose of the lesson was to use patterns to form mathematical models. In each equation, the first number represents the initial population. The operation [multiplication or addition] explains how the population grows. And the number attached to x – either the coefficient or the base – explains how quickly the population grows. For these five equations, we have lines and curves. The curves result from multiplication, and the lines result from addition.”
Culminating Activity/Assessment:
Refer to the videos (or overhead presentation) shown at the beginning of the lesson. Show each video three times to make sure that students are able to identify the pattern. Then, ask students to generate an equation that models the population growth shown in each video. Have them come up with these equations individually, not as a group.
The questions below dealing with exponential functions and 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.
Amelia recorded the time it took for candles of different lengths to burn out. Her results are shown in the table below.
- On a grid, create a scatterplot of her data and draw a line of best fit for the data.
- Write an equation for your line of best fit.
- Explain what the slope represents within the context of this problem.
Solution: The graph below shows the scatterplot with a possible line of best fit.
The line of best fit in the graph above has a y-intercept at approximately (0, 3). In addition, the line appears to pass through (1, 10), so the slope of the line is . Therefore,the equation of the line is y = 7x + 3.
Slope is the ratio of the change along the vertical axis to the change along the horizontal axis. The vertical axis represents time, and the horizontal axis represents length of the candle. Consequently, the slope represents the time required for a candle to burn a certain amount. Above, the slope was found to be 7, which means that it takes 7 minutes for a candle to burn 1 inch.
Mitch wants to use 40 feet of fencing to enclose a flower garden. Which of these shapes would use all the fencing and enclose the largest area?
A. A rectangle with a length of 8 feet and a width of 12 feet
B. An isosceles right triangle with a side length of about 12 feet
C. A circle with a radius of about 5.6 feet
D. A square with a side length of 10 feet (correct answer)
Mr. Brady asked his algebra class to make a scale drawing of the classroom, which is shaped like a rectangle with a width of 20 feet and a length of 24 feet. Travis made the width of the classroom in his scale drawing 5 inches. What should be the length in Travis’s scale drawing?
A. 6 inches (correct answer)
B. 8 inches
C. 9 inches
D. 10 inches
Teacher Commentary:
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.