Teaching Phase: How will the teacher present the concept or skill to students?
As a bell ringer activity, display the following formulas on the board (LCD)
a) A = P(1+r/n)t b) A = P(1+r/n)nt c) e = APrt d) A = Pert
Ask the students to identify the correct compound interest and continuous interest formulas.
Circulate the room and check the students' work.
Once students have correctly identified the formulas b and d, have them label the variables using appropriate units.
A- total amount of money including interest in dollars r- interest rate
P- principal amount in dollars(amount you begin with) t- time in years
n- number of times compounded per year
*note e is not a variable, e is approximately 2.71828
Check with students and find out how many of them have graphed using a graphing calculator. Pair any novice students with an experienced student for the next activity.
Have the students input the equation y = 100(1+.05/4)4xinto the graphing calculator. Using the window screen [0,20] x scale 1 and [0, 500] y scale 50, graph the equation and find the value of y when x is 13. (190.78)
Pause here and ask students if anybody needs assistance. When all students are able to come up with the correct graph and y value it is time to start the lesson.
Tell students the goal of the lesson today is to integrate function notation into our compound interest formulas. Begin with the compound interest formula A = P(1+r/n)nt
Step 1. Choose t as the independent variable and A as the dependent variable.
- Make P = 100 dollars,
- r = 5% (.05)
- n = 4 (compounded quarterly)
Step 2. Plug in the given values and notice the similarity to the graphing activity.
Step 3. Take the equation y = 100(1+.05/4)4xand write it in function form using y = f(x).
- f(x)=100(1+.05/4)4xThis is function, f, in terms of x.
- recall the values of x are considered the domain of the function f.
- Ask students questions about the function
- What would the domain be in step 2? (time in years)
- Is there a domain value that cannot be used? (values less than zero)
- How is the output(range) related to the input(domain)? (the greater the input the greater the output)
Step 4. Now we will write our compound interest formula in function form where t is the independent variable.
- A(t)=100(1+.05/4)4tWe now have function A in terms of t years.
Step 5. Let's find the monetary values for different years (inputs in the function domain)
- t = 15 years
- t = 32.8 years
- A(32.8)=$510.30)*students may need help here--have them change the x window screen to encompass 32.8 years
- t = 9 months (.75 years)
Step 6. Sketch the graph of this function making sure to plot the values above.
- Does this agree with our conclusions from step 3?
Step 7. Have students write a word problem using the initial parameters from above.
- Bill purchased a one hundred dollar CD that will earn five percent interest compounded quarterly. How much money will Bill have in the CD at the end of fifteen years?