Financial Math Foundations

Overview

In this third lesson of the unit, students build on prior work with exponents to model financial growth. They will use exponent rules to rewrite annual-to-period interest expressions, then apply these ideas to present value and annuity-type calculations used in actuarial contexts.

Learning intentions

Students will be able to:

• use exponent properties to rewrite expressions involving compound interest rates
• interpret and compute present value using an exponent form
• apply the idea of repeated payments to set up and evaluate a simple annuity expression
• explain whether results make sense in context (e.g., higher rates lower present value)

Success criteria

Students can:

• correctly transform an expression like (1.15^t) into an equivalent monthly/period form using exponent rules
• compute present value of a single future value by using the reciprocal of the growth factor
• set up a repeated-payment present value calculation and get a correct numeric answer
• justify their approach using clear mathematical language and correct notation

Curriculum links

• CCSS.MATH.CONTENT.HSA-SSE.B.3c: Use exponent properties to transform expressions for exponential functions (annual to monthly/period equivalents, showing approximate equivalence)
• CCSS.MATH.PRACTICE.MP1: Explain problem meaning, choose an approach, and monitor reasonableness
• CCSS.MATH.PRACTICE.MP2: Decontextualize (symbolize the situation) and contextualize (interpret symbols as rates, time, and payment timing)
• CCSS.MATH.PRACTICE.MP4: Apply mathematics to real-world financial modeling and interpret results in context
• CCSS.MATH.PRACTICE.MP6: Communicate precisely with correct notation, units (per year/per month), and labeled answers

Lesson structure (30 minutes)

1. 0–4 min · Quick opener (context + entry). Teacher displays two statements: “Annual interest 15%” and “Monthly interest.” Students do a quick think-write: What information must change when converting annual to monthly?

2. 4–12 min · Direct teach: exponent rewrite for rates. Teacher models converting (1.15^t) to a monthly-period equivalent using exponent rules (e.g., rewrite as ((1.15^{1/12})^{12t}) and note it shows the monthly growth factor raised to the number of months); then connects to “time” in the exponent. Students follow the algebra steps in pairs and circle what each part represents (rate factor vs. time exponent).

3. 12–19 min · Guided practice: present value from a growth factor. Teacher explains: for a single future amount (FV) due after (n) periods, the present value is (PV = \frac{FV}{(1+i)^n}). Teacher works one example: “An amount of $10,000 is expected in 3 years at 6% compounded annually” (or a class-chosen simple variation), showing exponent form and units. Students complete a second example independently, then compare with a partner using a “does this make sense?” check (if rate increases, PV decreases).

4. 19–26 min · Mini-activity: simple annuity setup and computation. Teacher presents a concrete scenario: “You will receive $500 at the end of each month for 12 months at a monthly interest rate (i). Find the present value.” Teacher emphasizes setting up the repeated-payment sum using exponential growth/discount reasoning. If students have seen the geometric-series form previously in the unit, teacher guides them to the standard result:

• (PV = PMT\cdot \frac{1-(1+i)^{-n}}{i}) Students compute the present value with a provided (i) (already converted earlier in class) and (n), then write one sentence explaining what the exponent (-n) represents (discounting back n periods).

5. 26–30 min · Exit ticket (alignment + reasoning). Students answer two prompts on one page:

• Transform: Rewrite (1.08^{5}) into an equivalent form with monthly periods using exponent rules (assume 12 months per year).
• Compute/interpret: If (PV = \frac{FV}{(1+i)^n}), state what happens to PV when (i) increases and give a numeric reason using a specific (FV, i, n) from a short template.

Resources

• Lesson 3 worksheet (exponent rewrite, PV, annuity)
• Timer/slide with the monthly-vs-annual conversion example
• Student calculators (optional if allowed)
• Graph paper or lined paper for clear equation writing
• Reference card: (PV=\frac{FV}{(1+i)^n}) and annuity present value formula
• Exit ticket slips

Assessment

• During guided practice, teacher checks exponent transformations for correct placement of (1/12) and correct exponent multiplication
• Listen for PV reasoning: students can describe discounting (negative exponent or division by growth factor)
• Exit ticket review: confirm each student can (1) rewrite an exponential expression correctly and (2) justify a reasonableness claim about present value vs. interest rate

Differentiation

• Support: Provide sentence frames for explanations (“The exponent represents ___ periods,” “When the interest rate increases, the discount factor ___, so PV ___”).
• Support: Offer one partially completed example for the annuity setup (with the correct formula skeleton filled in).
• Extension: For students who finish early, ask them to compare two interest rates (e.g., 5% vs. 8%) and compute two PVs, then describe the percent difference.
• EAL/SEN: Allow use of a calculator for arithmetic while requiring full algebra steps and clear notation (rate per period, time in periods, and correct exponent sign).