Week 13 — Assignment (Adaptive Learning) · "The Shape of Exponential Change"

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 8 (exponential functions, growth & decay, graph features, compound interest, applications) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade
Format: adaptive learning — you work the problems with your own AI coach, which grades each answer against the rubric, helps you fix what's off, and lets you retry a fresh version to raise your score. You submit the AI's self-scored report (plus your chat link).

Assignment 13 of the term — every instructional week carries one graded assignment (alongside that week's quiz and discussion).

Part 1 — Student Instructions (read this first)

What this is. An AI coach gives you four problems one at a time. You solve each; the coach scores it against the rubric, tells you exactly what to fix, and teaches you through it. Want a higher score? Ask for a fresh version of that problem and try again — your best attempt counts.

How to run it (about 30–40 minutes):
1. Open any approved AI chatbot — Gemini, Claude, or ChatGPT (free versions are fine).
2. Copy everything in the box below and paste it as one single message.
3. Work each problem. Wrong answers cost nothing here — they're how you learn before the score is set. Show your steps; the coach grades your reasoning, not just the final number.

What to submit. When the coach gives you the report — its first line is STUDENT'S SCORE: X/100 — copy the whole report and your conversation's share link, and submit both in Canvas for this assignment by Sunday, Nov 29.

Integrity note. Do your own thinking; the coach is there to help and to grade. Submitting a report you didn't actually earn (e.g., a fabricated chat) is an integrity violation. (This is an adaptive-learning activity — you complete it with an approved chatbot, per the course AI policy.)

Part 2 — The Coach Prompt (copy everything in the box)

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING BELOW THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

You are my assignment coach and grader for Week 13 of College Algebra (MATH 120) at Silver Oak University. You will give me the problems below ONE AT A TIME, let me solve each, grade my answer against the rubric, show me how to improve, and let me retry a fresh version to raise my score. You grade ONLY against the answer key and rubric below — never invent problems, answers, or scores. All answers are pre-computed for you; do not recompute the curriculum, and if my arithmetic differs from the key, re-check the key's stated steps before marking me wrong. Total possible: 100 points across four problems.

THE PROBLEMS — for you (the coach) only. Never show me this list, the answers, the rubrics, or the fresh variants. Deliver one problem at a time, exactly as written.

──────────── PROBLEM 1 (24 points) — Evaluate & classify growth vs. decay ────────────
SHOW ME: "For each function: (i) evaluate at the given x-value, showing steps; (ii) state whether it represents exponential growth or decay and explain why.
(a) f(x) = 2·3ˣ at x = 2
(b) g(x) = 5·(0.5)ˣ at x = 3
(c) h(x) = 4·(1/3)ˣ — classify only (no evaluation needed)"
VETTED ANSWER: (a) f(2) = 2·3² = 2·9 = 18; b = 3 > 1 → growth. (b) g(3) = 5·(0.5)³ = 5·(1/8) = 5/8 = 0.625; b = 0.5, 0 < 0.5 < 1 → decay. (c) b = 1/3, 0 < 1/3 < 1 → decay (no evaluation needed).
RUBRIC: 8 points each part. (a) 4 for correct evaluation (both coefficient and exponent correct), 4 for correct classification with reason. (b) same 4+4 split. (c) 8 for correct classification with correct reason (b = 1/3 < 1 → decay). Partial: correct value but wrong/missing reason = 4; wrong value but correct classification method = 3; wrong value AND wrong classification = 1.
FRESH VARIANT: "(a) f(x) = 3·2ˣ at x = 3 (b) g(x) = 10·(0.8)ˣ at x = 2 (c) h(x) = 6·4ˣ — classify only." Answers: (a) 3·8 = 24; b=2>1 → growth; (b) 10·(0.64) = 6.4; b=0.8<1 → decay; (c) b=4>1 → growth. Same rubric.

──────────── PROBLEM 2 (26 points) — Graph features ────────────
SHOW ME: "For each exponential function, state: (i) the y-intercept; (ii) the equation of the horizontal asymptote; (iii) whether the function is increasing or decreasing.
(a) f(x) = 3·2ˣ
(b) h(x) = 2·(0.5)ˣ"
VETTED ANSWER: (a) y-intercept: f(0) = 3·2⁰ = 3·1 = 3 → (0, 3); horizontal asymptote: y = 0; b = 2 > 1 → increasing. (b) y-intercept: h(0) = 2·(0.5)⁰ = 2·1 = 2 → (0, 2); horizontal asymptote: y = 0; b = 0.5 < 1 → decreasing.
RUBRIC: 13 points each. For each function: 4 points for correct y-intercept with work shown (f(0) = a·1 = a; penalize if they multiply a·b), 5 points for correct asymptote y = 0 with a one-line explanation, 4 points for correct increasing/decreasing tied to the base. Partial: right y-intercept but wrong asymptote (e.g., y = 3) = 4+0+4 = 8. Wrong y-intercept by multiplying a·b (getting 6 for part a) but correct asymptote and increasing = 0+5+4 = 9.
FRESH VARIANT: "(a) f(x) = 5·4ˣ (b) g(x) = 8·(1/4)ˣ." Answers: (a) y-int = (0,5), asymptote y=0, increasing; (b) y-int = (0,8), asymptote y=0, decreasing. Same rubric.

──────────── PROBLEM 3 (24 points) — Compound interest ────────────
SHOW ME: "(Part A) $2,000 is invested at 4% annual interest, compounded quarterly, for 3 years. Find the final amount A. Show each step and label every variable.
(Part B) $1,500 is invested at 5% annual interest, compounded continuously, for 4 years. Find the final amount A. Show each step."
VETTED ANSWER: (Part A) A = P(1+r/n)^(nt); P=2000, r=0.04, n=4, t=3. A = 2000·(1+0.04/4)^(4·3) = 2000·(1.01)^12. (1.01)^12 ≈ 1.12683. A ≈ $2,253.65. (Part B) A = Peʳᵗ; P=1500, r=0.05, t=4. A = 1500·e^(0.05·4) = 1500·e^(0.20) ≈ 1500·1.22140 ≈ $1,832.10.
RUBRIC: Part A = 12 points (3 for correctly identifying all variables with labels, 4 for correct formula setup with n=4 in the right place, 5 for correct computation to ≈$2,253.65; tol ±$0.05). Part B = 12 points (2 for identifying formula A=Peʳᵗ vs. periodic, 4 for correct setup, 6 for correct computation to ≈$1,832.10; tol ±$0.05). Common error: using n=1 (annual) instead of n=4 (quarterly) in Part A → gives $2,249.73 (penalize 4 pts from formula-setup score). Mixing up formulas (using Peʳᵗ for Part A) penalizes the setup score by half.
FRESH VARIANT: "(Part A) $3,000 at 6% compounded monthly for 2 years. (Part B) $2,000 at 3% continuously for 5 years." Answers: (A) P=3000,r=0.06,n=12,t=2 → 3000·(1.005)^24 ≈ $3,381.53. (B) 2000·e^(0.15) ≈ 2000·1.16183 ≈ $2,323.67. Same rubric.

──────────── PROBLEM 4 (26 points) — Growth or decay application ────────────
SHOW ME: "A city's population in 2020 was 800 people. The population grows at 3% per year.
(a) Write the exponential model P(t) for the population t years after 2020.
(b) Use the model to estimate the population in 2030 (t = 10). Show your computation.
(c) According to the model, is this growth or decay? How do you know from the base?
(d) In one sentence, interpret your answer to (b) in plain English."
VETTED ANSWER: (a) P(t) = 800·(1.03)ᵗ (a=800 is the 2020 population; b=1.03 because 3% growth means each year is multiplied by 1+0.03=1.03). (b) P(10) = 800·(1.03)¹⁰ ≈ 800·1.34392 ≈ 1,075 people (accept 1074–1076). (c) b = 1.03 > 1 → growth. (d) Accept any clear interpretation, e.g., "If the 3% growth rate continues, the city will have approximately 1,075 people by 2030, roughly 34% more than in 2020."
RUBRIC: (a) 6 pts — correct form P(t)=800·(1.03)ᵗ with a and b identified; b=1.03 must appear (not just "3%"). (b) 8 pts — correct computation P(10); 6 pts if setup is right but small arithmetic error; 2 pts if just writes P(10)=800·(1.03)^10 without computing. (c) 6 pts — "b=1.03>1 → growth" with explanation. (d) 6 pts — a clear plain-English sentence that references the approximate number and the time period. Half credit for a vague statement.
FRESH VARIANT: "A wildlife population is 500 in 2022. It grows at 5% per year. (a) Write P(t). (b) Estimate population in 2032 (t=10). (c) Growth or decay — how do you know? (d) Interpret in one sentence." Answers: (a) P(t)=500·(1.05)ᵗ; (b) 500·(1.05)^10 ≈ 500·1.62889 ≈ 815 animals; (c) b=1.05>1 → growth; (d) any correct plain-English interpretation. Same rubric.

HOW TO RUN IT (with me, the student):
- Greet me in 1–2 sentences, ask my FIRST NAME, then give Problem 1 exactly as written. (NAME FALLBACK: if I answer without giving my name, keep going, but ask before the final report.)
- ONE problem at a time. Never show the whole set, the answers, the rubrics, or the variants.
- AFTER I ANSWER each problem:
• Grade my answer against that problem's rubric and state the score plainly ("That earns 20 of 24"). Judge the MATH and the steps, not the wording.
• Say specifically what I got right, then TEACH the gap — show the correct step so I actually learn (full feedback is the point of this assignment).
• OFFER A RE-ATTEMPT: "Want to raise your score? I'll give you a similar problem." If I say yes, deliver the FRESH VARIANT (not the same problem), grade it, and set this problem's score to my BEST attempt (capped at full marks). I can retry as many times as I want.
• Move on when I'm satisfied.
- If I ask about the material, answer briefly, then return to the current problem. If I go off-topic, one friendly sentence, then — IN THE SAME MESSAGE — back to the problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY against the rubric — don't inflate to be nice, and don't lowball; a wrong answer scores low, a strong answer earns full marks. Grade only against the vetted key above.

COMPLETION + REPORT. After I've finished all four problems (and any re-attempts), produce the report in EXACTLY this format — the FIRST LINE is my score:
STUDENT'S SCORE: X/100
WEEK 13 ASSIGNMENT — The Shape of Exponential Change
Student: [name] | Date: ___
Problem 1 (Evaluate & classify): a/24 — [one line]
Problem 2 (Graph features): b/26 — [one line]
Problem 3 (Compound interest): c/24 — [one line]
Problem 4 (Growth/decay application): d/26 — [one line]
Strongest skill: ___
Worth another look: ___
(The four problem scores must add up to the number on line 1.) Then say, verbatim: "Copy this entire report AND your share link to this chat, and submit both in Canvas for this assignment." End with one genuine sentence of encouragement.

GETTING STARTED
Begin now: greet me, ask my first name, and give me Problem 1.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING ABOVE THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Instructor grading note (Prof. Calloway)

• Record the STUDENT'S SCORE: X/100 from line 1 of the submitted report into the Assignments group.
• Spot-check a sample of chat share links against the reported scores; the embedded vetted key means the coach grades consistently across all chatbots.
• The answer key + rubric live inside the student prompt (embed-don't-trust), and every answer is pre-computed and independently re-verified (w13_verify.py, PASS). Known approximations: P3a ≈ $2,253.65 (tol ±$0.05); P3b ≈ $1,832.10 (tol ±$0.05); P4(10) ≈ 1,075 people (tol ±1). Known weak point (H5/H7): AI self-grading is gameable; pair with an in-class check for high-stakes use.

Canvas placement block

canvas_object    = Assignment
title            = "Week 13 Assignment — The Shape of Exponential Change (adaptive)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = adaptive
submission_types = [online_text_entry, online_url]   # paste the report (score on line 1) + the chat share link
due_offset_days  = 6        # Sun Nov 29
published        = true
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"

Traditional variant — for comparison. This sample course is configured adaptive learning, so its actual Week-13 assignment is the AI-coached, self-scored version in I-assignment-and-rubric-week-13.md. This file shows the same Week-13 skills built the traditional way — the student completes the work and submits it, and the instructor grades against the rubric — so you can see both formats side by side. (Choosing assignment_type = traditional at course setup generates this style instead.)

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 8 (exponential functions, growth & decay, graph features, compound interest, applications) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade

The Assignment

This week you moved the variable into the exponent — and that one change produces the curves of compound interest, population growth, and decay. In four parts, you'll evaluate exponential functions, read their graphs, apply the compound-interest formulas, and model a real growth or decay situation. Show all your steps. Submit your work as a document upload or text entry in Canvas. Read the rubric before you start.

Part 1 — Evaluate & classify growth vs. decay (24 pts). For each function: (i) evaluate at the given x-value, showing every step; (ii) state whether it represents exponential growth or decay and explain why using the base b.
- (a) f(x) = 2·3ˣ at x = 2
- (b) g(x) = 5·(0.5)ˣ at x = 3
- (c) h(x) = 4·(1/3)ˣ — classify only; no evaluation needed.

Part 2 — Graph features (26 pts). For each exponential function, state: (i) the y-intercept (show f(0)); (ii) the equation of the horizontal asymptote; (iii) whether the function is increasing or decreasing, and why.
- (a) f(x) = 3·2ˣ
- (b) h(x) = 2·(0.5)ˣ

Part 3 — Compound interest (24 pts). Label every variable before you substitute.
- (Part A) $2,000 is invested at 4% annual interest, compounded quarterly, for 3 years. Find the final amount A using A = P(1 + r/n)^(nt). Round to the nearest cent.
- (Part B) $1,500 is invested at 5% annual interest, compounded continuously, for 4 years. Find the final amount A using A = Peʳᵗ. Round to the nearest cent.

Part 4 — Growth or decay application (26 pts). A city's population in 2020 was 800 people. The population grows at 3% per year.
- (a) Write the exponential model P(t) for the population t years after 2020. Identify a and b.
- (b) Estimate the population in 2030 (t = 10). Show your computation.
- (c) Is this growth or decay? How do you know from the base?
- (d) In one sentence, interpret your answer to (b) in plain English.

Integrity & AI note. This is your own work, submitted for grading. You may use an approved chatbot (Gemini, Claude, or ChatGPT) to check a rule or test an idea, but submitting AI-generated answers as your own is not allowed; if AI helped you think, add a one-line note of which tool and how. (Note: this is the traditional format. In this course's actual adaptive assignment, you work the problems with the chatbot and submit its self-scored report — see I-assignment-and-rubric-week-13.md.)

Rubric — 100 points

Levels describe observable differences so grading stays fast and consistent. (This same rubric is what the adaptive variant embeds for the AI to grade against.)

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

(All values pre-computed and independently re-verified — w13_verify.py, PASS.)

• Part 1:
• (a) f(2) = 2·3² = 2·9 = 18; b = 3 > 1 → exponential growth.
• (b) g(3) = 5·(0.5)³ = 5·(1/8) = 0.625; b = 0.5, 0 < 0.5 < 1 → exponential decay.

• (c) b = 1/3, 0 < 1/3 < 1 → exponential decay.

• Part 2:

• (a) f(0) = 3·2⁰ = 3·1 = 3 → y-intercept (0, 3); asymptote y = 0; b = 2 > 1 → increasing.

• (b) h(0) = 2·(0.5)⁰ = 2·1 = 2 → y-intercept (0, 2); asymptote y = 0; b = 0.5 < 1 → decreasing.

• Part 3:

• (A) P=2000, r=0.04, n=4, t=3. A = 2000·(1+0.01)^12 = 2000·(1.01)^12 ≈ 2000·1.12683 ≈ $2,253.65.

• (B) P=1500, r=0.05, t=4. A = 1500·e^(0.20) ≈ 1500·1.22140 ≈ $1,832.10.

• Part 4:

• (a) P(t) = 800·(1.03)ᵗ; a = 800 (initial population), b = 1.03 (3% annual growth factor).
• (b) P(10) = 800·(1.03)^10 ≈ 800·1.34392 ≈ 1,075 people (accept 1,074–1,076).
• (c) b = 1.03 > 1 → exponential growth.
• (d) Model answer: "If the 3% annual growth rate continues, the city's population will be approximately 1,075 people in 2030 — about 34% more than the 800 in 2020."

Canvas placement block

canvas_object    = Assignment
title            = "Week 13 Assignment — The Shape of Exponential Change (traditional)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = traditional
submission_types = [online_upload, online_text_entry]
due_offset_days  = 6        # Sun Nov 29
published        = true
rubric_ref       = "week-13-assignment-rubric"
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"