Exponential Functions – How to make money the easy way!
AUTHOR: Ashley Welch
TEAM MEMBERS: Ashley Welch, Julie McPhail, Katie May
DATE LESSON TO BE TAUGHT: During "Will You live to be 110?" Unit
GRADE LEVEL: Algebra II
SOURCE OF THE LESSON: Various websites:
http://www.math.com/tables/general/interest.htm - compound interest
http://www.answers.com/topic/logistic-function
TEKS:
3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.
2A.11) Exponential and logarithmic functions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
The student is expected to:
(B) use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior;
(D) determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods;
(F) analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem.
CONCEPT(S): Students will get a sense for what an exponential function is, as well as what types of real-life situations are modeled by exponential functions.
OBJECTIVES (LEARNER OUTCOMES):
Students will be able to:
- Identify the shape of an exponential function
- Predict the future values for compound-interest accounts
- Be able to compare and contrast a graph of an exponential function vs. a linear function
MATERIALS LIST and ADVANCED PREPARATIONS:
SAFETY: Students will be working with calculators and pencil and paper. Nothing seems inherently dangerous.
SUPPLEMENTARY MATERIALS: none
ENGAGEMENT:
12 minutes
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What the Teacher Will Do |
Eliciting Questions |
Student Responses |
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Give brief intro to the topic and the task |
What kind of functions are we used to seeing so far? |
Linear, quadratic |
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What can you tell me about a linear function? |
The graph is a straight line |
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Can anyone give me any examples of relationships in our world that are linear? |
Sediment accumulation vs. time, etc. |
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For linear equations, what do you notice about when you add a certain amount to the independent variable, say time. What happens to the dependent variable? |
You add some quantity to the dependent variable. |
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Ask student to draw the linear relationship between number of meters vs. centimeters on the board. |
Who can draw the graph of the number of centimeters vs. meters on the board? |
Student volunteers, and draws a linear graph |
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What do you notice about this graph? |
It's straight. |
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"Let's pull out some data points and put them in a table, and then have a look at it." Stand at the board and ask students to read off various points, and tabulate. Explain function patterns |
When you add two meters to the independent variable, what do you do to the dependent variable? |
You add 200 centimeters. |
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Does this always work with this linear function? Pick another starting point. |
Yes! |
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"Okay, so it seems that we've noticed a pattern here." |
What is the pattern? |
When you ADD something to the independent variable, you ADD something to the dependent variable. |
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Today we're going to learn about a different type of equation, which some of you may have seen before. But first, I've got a question for you. |
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Suppose that you have obtained a one-month summer job. Your employer has given you three salary options. You can either earn a penny the first day, and then each day that goes by, your money will double (so on the first day, you have a penny. On the second, you have two pennies, the third day, you've made 4 pennies). Or, you could make 30,000 each day. |
Which would you choose? The $30,000 per day, or the doubling-money, starting at one penny? |
The million bucks! OR Wait a minute, she wouldn't be asking if it weren't the penny option... |
EXPLORATION: 20 minutes
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What the Teacher Will Do |
Eliciting Questions |
Student Responses |
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I want you to now make a table with the days of that month as your independent variable and your month-to-date earnings as the dependent variable. Make the table neat and legible, because we're going to be focusing on it later. |
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Calculating how much the students are earning each day, making a table, and then a graph. |
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When students are about done, begin to ask questions about what they notice. |
Who wants to change their mind after making these graphs? |
Whoever chose the $30,000 a day probably will! |
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What do you notice about the two graphs in front of you? What is their shape? |
One is a straight line, and the other one is a curve that gets huge!
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How much money did the $30,000 per day worker make total? How about the person who started at a penny but whose earnings doubled each day? |
The person with the $30K salary brought in $900,000. The person who started at a penny made over five million! $5,368,709.12, to be exact! |
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In mathematical terms, how would you describe the function you made of the employee who gets $30,000 a day, and why? |
Linear, because it is a straight line. |
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Does anyone know what the graph of the person who started at a penny is called? |
Exponential. |
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It's called an "exponential function". |
Does anyone have a guess why it's called exponential? Who can tell me what an exponent is? |
An exponent something that you are raising something to, so with five squared, 2 is the exponent. |
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Direct the focus back to the shape of the graphs. |
What else can you say about the graph of our exponential function? How fast or slow does it rise? |
It rises really fast! |
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Does it increase rapidly the whole time? What about here in the beginning? |
It's gradual at first.
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When is the rate of increase the fastest? i.e., where does the graph have the steepest slope? |
At the far right |
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Thinking about this in terms of values for our independent variables, in this case days that have passed in the month, what can you say about the growth of this function? |
As the independent variable is larger, the growth is larger. If the independent variable is very small, the growth is smaller. |
EXPLANATION:
8 minutes
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What the Teacher Will Do |
Eliciting Questions |
Student Responses |
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Can someone come up with a creative way of writing out how much the person who started at a penny makes? |
I'm hoping someone will do (.01)*2*2*2*2*2.... |
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If nobody came up with it, lead the class to see that it is (.01)*2*2*2*2*2.... |
Now, what's another, more condensed, way to write this? |
You can raise 2 to the number of times you multiply it. |
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So if this job only lasted 5 days, how would you write the equation? How about 7? 10? |
(.01) * 2*2*2*2, or (.01)2^4, etc.
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What changes in this equation? |
The exponent! |
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Revisit function patterns. Please take out your tables of data. Sometimes we don't have the luxury of having a graph to look at, but we have tables of data, like an accounting book. It can be useful to recognize what kind of equation you're dealing with so that you can make informed decisions. |
What did we say about the pattern of a linear function? |
When you add a unit to the independent variable, you add a different unit (that stays constant,) to the dependent variable. |
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So on your tables, what are we adding to the independent variable each day? |
One day. |
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And what are we adding to the amount? |
$30,000
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Does this change between day 3 and 4, say, and between 27 and 28? |
No, both jump $30,000. |
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Let's take a look at the penny salary. |
Do you notice the same pattern? |
Yes, when you add something to the days, you add something to the money. |
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That's true, you are adding quantities as each day goes by, but is the quantity the same? |
No. |
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How else might we think of this? Does anyone see a different kind of pattern? |
Each day you double it! |
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What's another way to say doubling? |
You multiply by two. |
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You MULTIPLY! That's right. It turns out that whenever you have a table of data, and you see that as you add the same unit to the independent variable, you actually MULTIPLY the dependent variable by something, that's a good indication that you're looking at an exponential function! |
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ELABORATION: 15 Minutes
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What the Teacher Will Do |
Eliciting Questions |
Student Responses |
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Introduce the idea of compound interest. |
Where in life might you come across and exponential situation? |
There are tons, but I'm hoping someone will say a bank account with interest. |
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Describe what accumulating interest is, in the context of a savings account. Go to http://www.math.com/tables/general/interest.htm to show the equations. P = C (1 + r) t |
Let's figure out a hypothetical situation. Suppose that you have $1000 today, and you want to put it in a savings account with 7% interest. The equation you would use would be P = 1000(1.07)^x, where x is the number of years you let this thing accumulate interest. You can use your graphing calculators to see what happens. How much money do you have after 10 years? 20? 30? 50? |
$1967, $3870, $7612, $29,457. |
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Wow, that's pretty good. With only a thousand dollars now, you'd have nearly $30,000 in fifty years, when you're thinking about retiring. Let's go beyond just 50 years, though. Say that you're starting an account that you want your grandkids or their kids to have. |
Where are you at at 100 years? 120 years? 200 years? |
$867,716, $3,357,788 $752,931,621
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Holy smokes, after 200 years, your descendents would have nearly a billion dollars from just $1000 started today! |
Why is this happening? Why were you seeing modest gains at first, but much more rapid increases later? |
The more money you've got in the bank, the greater the increase! |
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There's a proven link between wealth and longevity. It may be that people with more money have better access to doctors, or more resources to stay healthy throughout their lives. But one thing stays constant, money tends to increase one's life span. |
So if you want to live as long as you can, what's a smart thing to do now? |
Start saving money |
EVALUATION:
5 minutes today, 20 minutes during future class period
As an end of day assessment, I will be asking students to write a one-sentence summary, with the subject "Exponential Functions". The students will need to address who? (exponential functions), does what? wo what/whom? when? where? how? And why?, all in one summarizing sentence. We will be giving a quiz or other type of assessment the week after this lesson, which will include exponential functions. Format TBD, but will probably involve a function pattern recognition exercise, a qualitative description of linear vs. exponential vs. logistic function graphs, matching equations with one of the three categories, and real-life examples of each type of function.