Learning Math: Patterns, Functions, and Algebra
Nonlinear Functions Part B: Exponential Growth (40 minutes)
Session 7, Part B
In This Part
- A Salary Situation
- Population Growth
A Salary Situation
Suppose you are given two different options for salary at a temporary job.
|Plan A:||You can earn $2,000 each week, or|
|Plan B:||You can earn 1 penny the first week, 2 pennies the next week, 4 pennies the next week, and so on, doubling your salary each week.|
The job lasts 25 weeks, and your goal is to earn the largest total salary for the 25 weeks.
Which salary plan would you choose and why? Write down your initial reaction.
Use a spreadsheet or some other method to figure out which salary plan is better.
Tip: When using a spreadsheet, create three columns, one for the week number, one for plan A, and one for plan B. When creating the rule for plan B, remember that the salary is doubling each week, and that the first week’s salary is one penny, not one dollar.
Are you surprised that the second salary grew so rapidly? Use the spreadsheet to determine how much money you would make from plan B in 40 weeks.
Tip: To make a column wider, click and hold down on the right edge of the column heading letter, then drag the mouse to the right.
In this video segment, on screen participants discuss the differences between salary plans A and B in Problems B1 and B2 above. In particular, they explore the differences between linear and exponential functions. Watch this segment after you have worked on Problems B1 and B2. If you get stuck, you can watch the video segment to help you.
Describe the difference in the ways salary plans A and B grow. Which one is a proportional change, and which is an absolute change? For more on this, see Session 4, Part A.
You can find this segment on the session video, approximately 7 minutes and 51 seconds after the Annenberg Media logo.
Many populations — human, plant, and bacteria — grow exponentially, at least at first. In time, these populations start to lose their resources (space, food, and so on).
Here’s an example:
“Whale Numbers up 12% a Year” was a headline in a 1993 Australian newspaper. A 13-year study had found that the humpback whale population off the coast of Australia was increasing significantly. The actual data suggested the increase was closer to 14 percent!
When the study began in 1981, the humpback whale population was 350. Suppose the population has been increasing by about 14 percent each year since then. To find an increase of 14 percent, you could do either of the following (P stands for population):
Pthis year = Plast year + (0.14) * Plast year
Pthis year = (1.14) * Plast year
Each of these is a recursive rule. The first rule says that to know this year’s population, start with last year’s population (which is Plast year), then add the population growth. Since the growth is 14 percent of last year’s population,
add (0.14) * Plast year.
The second rule is used more frequently because it’s easier to calculate — it incorporates the adding in one calculation. This equation shows that the second computation is equivalent to the first:
x + (0.14) * x = (1.14) * x
The second computation fits the format of an exponential function, because successive outputs have the same ratio (in this situation, the ratio is 1.14).
Make a table that shows the estimated whale population for the 5 years after 1981.
|Years after 1981||Estimated Population|
If the whale population continues to grow by 14 percent per year, predict how many whales there would be in 2001 (20 years after 1981).
Tip: See if you can do this problem without extending your table from Problem B4. You would need to use a closed-form rule to “jump” directly to the 2001 answer.
Take It Further
How many years does it take the whale population to double if it grows at this rate? Does your answer depend on the starting value of the whale population?
In this video segment, taken from the “real world” example at the end of the Session 7 video, Mary Bachman of the Harvard School of Public Health discusses the exponential model of population growth, the factors that affect the model, and the uses of population modeling.
Do you think the whale population discussed in Problems B3-B6 could increase at 14 percent per year forever? Why or why not?
You can find this segment on the session video, approximately 22 minutes and 40 seconds after the Annenberg Media logo.
Take It Further
Look at the following toothpick pattern. The number of toothpicks needed to build each stage of the pattern is a linear function.
Create a toothpick pattern in which the number of toothpicks you need for each stage is an exponential function.
Tip: Think of a pattern that would require you to double the number of toothpicks you use at each step. That would form an exponential function, because successive outputs have a common ratio (2).
The “Population Growth” problem taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p.186. www.glencoe.com/sec/math
In this section, we’ll explore situations that give rise to increasing exponential functions.
Groups: Begin by putting up an overhead of the description of two different salary options for Problems B1 and B2. Discuss your gut reactions: Which is the better choice? No need to justify your answers; go instead with your first instincts.
Work with a spreadsheet or calculator to figure out the final value of each salary scenario. Remember that the most important factor in this comparison is the total amount earned over the 25 weeks.
A calculator can be used to create the same table as in a spreadsheet. Even if you are filling in the table by hand, there are shortcuts to using the calculator. For example, type in “.01” then “Enter.” Then type “x 2” and “Enter.” Repeatedly hitting the Enter key will produce the list of outputs for the second salary option. Keep track of each output, and then sum the outputs at the end. For the first salary option, because the salary never changes, simply multiply the salary ($2,000) by the number of weeks (25) to get the total.
Groups: When finished the comparison, share your results and reactions. Many will probably be surprised that starting with such a small amount — a penny compared with $2,000 — you could end up with more than 10 times as much total money, and in only half a year.
The salary activity segues nicely into the population model. If a population reproduced by doubling, it would quickly run out of resources, even starting with a small number. Though exponential models are used for some populations, the bases (the constant multiple between outputs) is usually much closer to 1.
Groups: Read through the whale problem. Spend a moment discussing why increasing by 14 percent is mathematically equivalent to multiplying by 1.14. Percents are not a focus of this session, but it’s worth spending a little time on this idea in order to understand why it is true. You can also relate it to other things the students probably know:
- If you have a 10 percent decrease, you can calculate 0.9 x (original number). This is the same as (1 – 0.1) x (original number), or (original number) – 0.1 x (original number).
- To compute the final price of an item when you have to pay 10 percent sales tax, you can use 1.1 x price, which is the same as (1 + 0.1) x price, or (price) + 0.1 x (price).
- 3 x (number) + 5 x (number) = (3 + 5) x number
- or in symbols: 3n + 5n = (3 + 5)n = 8n
- or in words: If you have 3 of something and add 5 of that thing, you end up with 8 of the thing.
These are specific cases of an important algebraic idea: the distributive property of multiplication over addition.
Here, we have
1 x (population) + 0.14 x (population) = (1 + 0.14) x (population) = 1.14 x (population)
If working on a computer, open a new worksheet to model the situation.
Groups: Work in pairs on Problems B4-B6.
Think about a strategy for calculating how long the population takes to double. The Fill Down command on the spreadsheet can be used until a population of 700 is reached. To answer the question of whether it depends on the initial population, change that starting number in the spreadsheet and see if it doubles in the same place. The doubling time does not depend on the starting value; thus an exponent n can be found so that 1.14n = 2.
Here’s one way to see that the time to double doesn’t depend on the starting value, and it also highlights some important algebraic thinking.
You’re looking for a year where 1.14 x 1.14 x 1.14x … x 1.14 x n = 2 x n. There is an n multiplied on each side, so the only thing that could possibly make the multiple of 2 is all those 1.14s multiplied together. You just have to find the right number of them, and the number of 1.14s only depends on the year.
This also tells you that, for example, if you get a 5 percent raise at your job every year, the number of years it takes you to double your salary is fixed, and it doesn’t depend on how much you start out earning.
Groups: Problem B7 relates back to earlier work with toothpick patterns, but requires a bit of creativity. You may want to have actual toothpicks available. If you have difficulty coming up with patterns that work, think of a particular case. For example, try to come up with a pattern that uses twice as many toothpicks at each successive stage. To maintain a pattern visually, it helps to think about making copies of the shape at any stage, and arranging them in some regular way.
Here are a couple of possible solutions:
Groups: To wrap up this part, talk about how “growing exponentially” is used as slang in the press to mean “growing very fast,” but in fact “exponentially” has a specific mathematical definition. Think of possible definitions for “growing exponentially” in your own words, and add exponential functions to the list of nonlinear functions started at the beginning of the session.
Most initial reactions are to choose plan A.
The total for 25 weeks of plan A, at $2,000 per week, is $50,000. Plan B, which starts at 1 penny and doubles each week totals $335,544.31, including $167,772.16 in the final week.
After 40 weeks, the grand total at this job would be $10,995,116,277.75, including just under $5.5 billion in the last week. Before the end of the year, you would exhaust the total money supply of the United States.
Here is the completed table.
You can extend the spreadsheet, but an easier way to obtain an answer would be to calculate 350 * (1.14)20 = 4,810 whales. One way to estimate this is to observe that the population seems to be nearly doubling in 5 years (doubling would be 700 whales). This means that we could expect the population to nearly double three more times in a total of 20 years. This doubling would be 350 -> 700 -> 1,400 -> 2,800 -> 5,600, so an estimate is roughly 5,000 whales.
It takes a little more than 5 years. And no, it doesn’t matter what the initial population value was, because the ratio of the population after 5 years from now relative to starting population is created only by multiplying 1.14 five times.
One possible pattern is to build, at each stage, a triangle of toothpicks with sides twice as long as the previous triangle.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.