##### 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 = AP^{rt} d) A = Pe^{rt}

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)^{4x}into 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.

**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)^{4x}and write it in function form using y = f(x).

- f(x)=100(1+.05/4)
^{4x}This 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)
^{4t}We 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?