Week 13 — Doubling, Halving, and Interest on Interest: A Short Exponential Report

Assignment 12 of the term — evaluate an exponential at a negative input, track a halving sample, price a phone that loses value, compound a savings balance, and tell someone new to money math why interest on interest matters.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 1 you learned that a negative exponent means a reciprocal, and in Week 3 you learned to evaluate a function and read its range; this week both return as exponential functions. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

You will use four pieces of this week's work: the exponential function f(x) = 3 · 2ˣ, the 100-gram sample A(t) = 100 · (1 ⁄ 2)ᵗ that halves every period, the 1,200-dollar phone V(t) = 1200 · (0.75)ᵗ that loses 25% of its value each year, and 1,000 dollars at 6% interest compounded annually, A = 1000 · (1.06)ᵗ.

Work these four problems, showing your arithmetic:

1. Evaluate f(−2) for the exponential function f(x) = 3 · 2ˣ. What is its value?
2. A 100-gram sample halves every period, so A(t) = 100 · (1 ⁄ 2)ᵗ with t in half-lives. How many grams remain after 3 half-lives?
3. A 1,200-dollar phone loses 25% of its value each year, so V(t) = 1200 · (0.75)ᵗ. How many dollars is it worth after 2 years?
4. 1,000 dollars earns 6% interest compounded annually, so A = 1000 · (1.06)ᵗ. How many dollars are in the account after 2 years?

Beside each answer, add one brief interpretation: what that value shows and one use or limit of it. For problem 1, explain why a negative exponent gives a small positive output and why the function still shows growth. For problem 2, name the decay factor and say why the mass never reaches 0. For problem 3, explain why the factor is 0.75 and not 0.25, working one year at a time. For problem 4, compare your balance with the simple-interest balance after two years and explain where the difference comes from.

Explain it to a non-expert

A younger cousin has opened a first savings account and asks why the bank keeps talking about "compound" interest. Write a 3–5 sentence message: why compound interest grows faster than simple interest, what multiplying by the same growth factor each year does, and one honest limit or caveat, such as a rate that can change or a gap that is small in the first years. Explain any technical term in everyday words. More than one good caveat can be defended.

For a different, ungraded example, imagine a sourdough starter that doubles in size every 8 hours. Model answer: "The starter does not add the same amount each time; it doubles whatever is already there. So after three doublings it is 2 · 2 · 2 = 8 times as big, not 3 times as big. That is why a baker watches it more closely as the day goes on. One caveat: the doubling only lasts while the starter has fresh flour, so it cannot go on forever."

Use that pattern for the savings account, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for your cousin before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about exponential functions: the initial value and the growth factor, evaluating with the exponent first, negative exponents as reciprocals, growth and decay by the base, the horizontal asymptote, percent rates as factors, and compound versus simple interest. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the exponential function f(x) = 3 * 2^x, the 100-gram sample A(t) = 100 * (1/2)^t that halves every period, the 1,200-dollar phone V(t) = 1200 * (0.75)^t that loses 25% of its value each year, and 1,000 dollars at 6% interest compounded annually, A = 1000 * (1.06)^t.
PROBLEM P1: Evaluate f(−2) for the exponential function f(x) = 3 · 2ˣ. What is its value?
PROBLEM P2: A 100-gram sample halves every period, so A(t) = 100 · (1 ⁄ 2)ᵗ with t in half-lives. How many grams remain after 3 half-lives?
PROBLEM P3: A 1,200-dollar phone loses 25% of its value each year, so V(t) = 1200 · (0.75)ᵗ. How many dollars is it worth after 2 years?
PROBLEM P4: 1,000 dollars earns 6% interest compounded annually, so A = 1000 · (1.06)ᵗ. How many dollars are in the account after 2 years?
After the four answers, I must briefly interpret each one: what that value shows and one use or limit of it; for P1, why a negative exponent gives a small positive output and why the function still shows growth; for P2, the decay factor and why the mass never reaches 0; for P3, why the factor is 0.75 and not 0.25, working one year at a time; for P4, a comparison with the simple-interest balance after two years and where the difference comes from. I must then write a 3-5 sentence message for a younger cousin with a first savings account, explaining why compound interest grows faster than simple interest, what multiplying by the same growth factor each year does, and one honest limit or caveat. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct exponential and interest computations (40 points), reasoning about exponential models (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct value of f(-2), mass after 3 half-lives, value after 2 years, or balance after 2 years = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what that value shows plus a use or limit = 2; exactly one of the two = 1; neither = 0. For P1, a value without the reciprocal reading and the growth-by-the-base reason does not count as the account; for P3, a value without the 1 - r factor reasoning does not count as the account. For P4, BOTH an accurate comparison with the simple-interest balance with the reason for the gap AND a final message that explains why compound interest grows faster, what the growth factor does, and an honest caveat = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message for my cousin AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised exponential report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised exponential report with all four answers, their brief interpretations, and your message for your cousin — written, or recorded as audio or video with a transcript, or as a labelled sketch with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 13 skills — evaluating an exponential function at a negative input, telling growth from decay by the base, building a model from a percent rate, and computing a compound-interest balance and comparing it with simple interest — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message for your cousin. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised exponential report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

Built with The Course Maker · samples.thecoursemaker.com