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Overview:
This lesson provides students with an introduction to exponential functions. The class first explores the world population since 1650. Students then conduct a simulation in which a population grows at a random yet predictable rate. Both situations are examples of exponential growth.
Time Allotment:
Two 50-minute class periods
Subject Matter:
Exponential functions
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.Divide students into groups of four. In each group, assign the roles of captain, recorder, reporter, and timekeeper.
2.Explain that the class will examine population growth.
3.Distribute the handout.
4.Have students read aloud the introductory paragraphs for the lesson.
5.Ask students to consider what things they can do mathematically to make predictions about the future. Students should suggest that collecting data, making graphs, and looking for patterns would be useful in making predictions.
6.Show PowerPoint data regarding the world population from 1650 to 1850. (The PowerPoint slide show uses animation to proceed through a series of questions for students, which correspond to steps 7-14 below. If PowerPoint is not available, these images can be used to present similar material on an overhead projector.)
7.In groups of four, have students describe any patterns they notice in the changes in world population from 1650 to 1850.
8.Have student groups predict the world population in 1950.
9.Have reporters state their team’s prediction for 1950. Record the various predictions on the chalkboard or overhead projector. Be sure that students explain how they made the prediction, and have students discuss the various predictions. (Once students agree on which predictions are reasonable, you may wish to have them take the average of these predictions to come up with a whole class prediction.)
10.Have student teams plot their 1950 point on the graph containing the points for 1650, 1750, and 1850. Because the three points for 1650, 1750, and 1850 lie somewhat along a straight line, have students check the reasonableness of their prediction by noticing if it lies along the same line.
11.Reveal the actual population in 1950. (The student predictions will likely have been much lower.) Then, ask them to use this new information to predict the world population in 2000. Again, have students discuss this problem in their groups.
Learning Activities:
1.Have a student read the directions for the exploration:
To help make predictions in real-world situations, researchers often use experiments known as simulations. The results of the simulations are gathered and analyzed. This data is then compared with known information about the actual population. If the result seems questionable, the simulation may be revised.
2.Have students explain what a simulation is in their own words. Elicit from students that a simulation is a model of a real-world situation.
3.Have several students give examples of simulations.
4. Have a student continue reading under the Exploration section:
This modeling process can be summarized by the following five steps:
- creating a model
- translating the model into mathematics
- using the mathematics
- relating the results to the real-world situation
- revising the model
In the following exploration, you investigate this modeling process using a population of Skeeters.
5.Have students read the directions for the exploration. Then, give them 30 minutes to run the simulation and complete the portion of the handout under the heading Discussion 1.
6. Record the teams’ predictions for “Shake 20.” (Students make this prediction in Discussion 1, Part c.1 of the handout.)
7.Lead a class discussion about the predictions. Have students explain the patterns they noticed as they ran the simulation, and how they used those patterns to make their prediction.
8.Give students 10 minutes to complete Discussion 2. (In Discussion 2, student teams decide the best way to describe the shape of the graph.)
9.Ask the reporter from each team to share the team’s description of the shape of the graph. Record their descriptions on the chalkboard or overhead projector. Discuss the descriptions and elicit from students that the graph is a curve.
10.Give students the remainder of the class period to record what they learned in their journals.
You may wish to follow this lesson with an activity that allows students to compare linear growth with exponential growth. A lesson plan for such an activity is provided in Workshop 8 Part II.
12.Record the groups’ predictions for 2000 on the chalkboard or overhead projector. Be sure to have students state how they arrived at their predictions, and allow them to discuss the reasonableness of these predictions.
13.Reveal the actual population in 2000.
14.Explain that things can grow in different ways, following different patterns and in ways we might not expect. Consequently, we adjust our predictions based on new information. Explain that students will conduct an exploration.
The questions below dealing with exponential functions have been selected from various state and national assessments. Although the lesson above may not fully equip students 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.
The number of bacteria in a culture doubles each hour. Which graph below best represents this situation?
Answer: Choice D
This table shows the 1997 and estimated 2002 populations of the four most populous countries in the world. Jessica and Katy are using the data for a presentation of their world geography report.
- Jessica says that between 1997 and 2002 the population of China increased faster than that of the United States. Katy insists that the United States population increased more quickly. Explain how each of them can be correct. Justify your answer mathematically.
- Suppose the table was extended for additional five year periods (2007, 2012, 2017, etc.). If the population of the United States continues to grow at the percentage rate that is predicted in this table, after how many five year periods will the population reach 300 million? Show or explain how you found your answer.
Solution:
- The population of China increased by 50.7 million, but the population of the United States only increased by 11.5 million; hence, Jessica stated that the population of China was increasing faster. On the other hand, the population of China increased by only 4.1 percent, while the population of the United States increased by 4.3 percent; consequently, Katy said that the population of the United States was increasing faster.
- The population of the U.S. in 2007 will be 279.5 � 1.043 291.5 million. In 2012, the population will be 291.5 � 1.043 304.1 million. In 10 years, or after two more five year periods, the population will be over 300 million.
Dave’s Electronics is having a sale on radios. Each day the price of every radio will be reduced 5% from the previous day’s price. If the price of a radio before the sale was $50, which expression can be used to find its price on the n^{th} day of the sale?
A. 50 – 0.05n
B. 50 – 50n(0.95n)
C. 50(0.05)^{n}
D. 50(1-0.05)^{n} (correct answer)
Teacher Commentary:
I am very pleased with this sample of student work as far as reaching the traced goals.
Looking at the work, I have a couple of comments. Under the second discussion, answer C, I will have to ask the student to clarify the meaning of “that it is curved but eventually it will become a straight line”. Some questions I might ask are: What would be the implications of that straight line? And what happens to the pattern of the population growth? Is that possible?
I think that probably it is still not clear that while the growth rate increases, we still have some limitations.
On the second page, on the answer to d (2), it is still not clear to me that the student really understands how to get the dimensions of the box. I would ask questions like: what is the area that one Skeeter occupies in the box? What is the area of the region (the pizza box) where the Skeeters grow? Are there any limitations of space? If yes, how many Skeeters will fit in there?