Week 11 — Assignment (Adaptive Learning) · "Fractions All the Way Down"

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 7 (rational expressions & equations) · 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 11 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 15.

Integrity note. Do your own thinking; the coach is there to help and to grade. Submitting a report you didn't actually earn 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 11 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) — Simplify rational expressions ────────────
SHOW ME: "Simplify each rational expression completely. State any excluded values.
(a) (x² − 16) / (x² − x − 12)
(b) (2x² + x − 6) / (x² − 4)"

VETTED ANSWER (from w11_verify.py, PASS):
(a) Factor numerator: x² − 16 = (x + 4)(x − 4). Factor denominator: x² − x − 12 = (x − 4)(x + 3). Cancel (x − 4): (x + 4) / (x + 3), x ≠ 4, x ≠ −3.
(b) Factor numerator: 2x² + x − 6 = (2x − 3)(x + 2). Factor denominator: x² − 4 = (x − 2)(x + 2). Cancel (x + 2): (2x − 3) / (x − 2), x ≠ 2, x ≠ −2.

RUBRIC: 12 points each.
Full 12 = factored correctly, canceled the right factor, stated correct excluded values.
Partial (7–10): right factors but missing one exclusion, or one sign error in a factor.
Minimal (3–5): attempted to factor but incorrect; correct answer without any work shown.
Zero if tried to cancel terms without factoring.

FRESH VARIANT:
"(a) (x² − 25) / (x² − 2x − 15) (b) (3x² − x − 2) / (x² − 1)"
Answers (verified): (a) Factor: (x−5)(x+5) / [(x−5)(x+3)] = (x+5)/(x+3), x≠5, x≠−3. (b) Factor: (3x+2)(x−1) / [(x−1)(x+1)] = (3x+2)/(x+1), x≠1, x≠−1. Same rubric.

──────────── PROBLEM 2 (26 points) — Multiply & divide rational expressions ────────────
SHOW ME: "Simplify completely. State any excluded values.
(a) [(x² − 9) / (x + 2)] · [(x + 2) / (x − 3)]
(b) [(x² − 1) / (x + 3)] ÷ [(x − 1) / (x + 3)]"

VETTED ANSWER (verified):
(a) Factor numerator of first: (x+3)(x−3). Write product: (x+3)(x−3)/(x+2) · (x+2)/(x−3). Cancel (x+2) and (x−3): (x + 3), x ≠ −2, x ≠ 3.
(b) Flip second fraction: [(x²−1)/(x+3)] · [(x+3)/(x−1)]. Factor: (x−1)(x+1)/(x+3) · (x+3)/(x−1). Cancel: (x + 1), x ≠ ±1, x ≠ −3.

RUBRIC: (a) 13 points, (b) 13 points.
Full = factored, correctly canceled across the product (for mult) or after flipping (for div), stated exclusions.
Partial (7–10): correct process, one sign or exclusion error.
Minimal (3–5): flipped correctly for div but couldn't complete factoring.
Zero for (a) if multiplied without factoring or canceling.

FRESH VARIANT:
"(a) [(x² − 4) / (x + 5)] · [(x + 5) / (x − 2)] (b) [(x² − 9) / (x − 4)] ÷ [(x + 3) / (x − 4)]"
Answers (verified): (a) (x−2)(x+2)/(x+5) · (x+5)/(x−2) → (x+2), x≠2, x≠−5. (b) Flip: (x−3)(x+3)/(x−4) · (x−4)/(x+3) → (x−3), x≠3, x≠−3, x≠4. Same rubric.

──────────── PROBLEM 3 (24 points) — Add & subtract with an LCD ────────────
SHOW ME: "Perform the operation and simplify.
(a) 3/(x + 2) + 5/(x − 1)
(b) x/(x² − 4) − 1/(x − 2)"

VETTED ANSWER (verified):
(a) Denominators: (x+2) and (x−1) — no common factor. LCD = (x+2)(x−1). Build fractions: 3(x−1)/[(x+2)(x−1)] + 5(x+2)/[(x+2)(x−1)]. Add: [3(x−1)+5(x+2)]/[(x+2)(x−1)] = [3x−3+5x+10]/[(x+2)(x−1)] = (8x+7)/[(x+2)(x−1)], x≠−2, x≠1.
(b) Factor: x²−4 = (x−2)(x+2). LCD = (x−2)(x+2). Second fraction: 1/(x−2) = (x+2)/[(x−2)(x+2)]. Subtract: [x − (x+2)]/[(x−2)(x+2)] = (x − x − 2)/[(x−2)(x+2)] = −2/[(x−2)(x+2)], x≠2, x≠−2.

RUBRIC: 12 points each.
Full = correct LCD, correct equivalent fractions, correct numerator (watch sign on subtraction), simplified result.
Partial (7–10): right LCD, one sign error in numerator.
Minimal (3–5): used product of denominators (not factored LCD) but otherwise correct method.
Zero for adding numerators over sum of denominators (wrong fraction rule).

FRESH VARIANT:
"(a) 2/(x + 1) + 4/(x − 3) (b) x/(x² − 9) − 1/(x − 3)"
Answers (verified): (a) LCD = (x+1)(x−3). Numerator: 2(x−3)+4(x+1) = 2x−6+4x+4 = 6x−2. Result: (6x−2)/[(x+1)(x−3)], x≠−1, x≠3. (b) Factor: (x−3)(x+3). LCD = (x−3)(x+3). Subtract: [x−(x+3)]/[(x−3)(x+3)] = −3/[(x−3)(x+3)] → −3/[(x−3)(x+3)], x≠3, x≠−3. Same rubric.

──────────── PROBLEM 4 (26 points) — Solve rational equations (incl. extraneous check & work/rate) ────────────
SHOW ME: "Solve each equation. Show all steps, including the excluded values you identify before solving and the check you perform after. Note any extraneous solutions.
(a) 2/(x − 3) + 1/2 = 5/[2(x − 3)]
(b) x/(x − 1) = 1/(x − 1) + 2
(c) WORK/RATE: Pipe A can fill a tank in 3 hours; Pipe B can fill the same tank in 6 hours. How long does it take if both pipes work together?"

VETTED ANSWERS (verified, w11_verify.py, PASS):
(a) Excluded: x = 3. LCD = 2(x−3). Multiply: 4 + (x−3) = 5 → x−3 = 1 → x = 4. Check: x = 4 ≠ 3. ✓ Solution: x = 4.
(b) Excluded: x = 1. LCD = (x−1). Multiply: x = 1 + 2(x−1) = 2x−1 → −x = −1 → x = 1. Check: x = 1 is the excluded value → extraneous. No solution.
(c) Rate A = 1/3 tank/hr; Rate B = 1/6 tank/hr. Together: 1/3 + 1/6 = 1/t. LCD = 6t: 2t + t = 6 → 3t = 6 → t = 2 hours. (Check: 1/3 + 1/6 = 3/6 = 1/2 → fills in 2 hr. ✓)

RUBRIC:
(a) 8 points: 3 for identifying excluded value, 3 for correct LCD-clear and solving, 2 for check.
(b) 8 points: 3 for identifying excluded value, 3 for finding x=1 algebraically, 2 for recognizing it as extraneous and writing "no solution."
(c) 10 points: 4 for setting up 1/3 + 1/6 = 1/t correctly, 4 for solving correctly, 2 for verification.
Partial for each: right setup, arithmetic slip = half the points for that part.
Zero for (b) if student writes x = 1 as the answer without checking.

FRESH VARIANT:
"(a) 3/(x + 1) + 1/2 = 5/[2(x + 1)] (b) x/(x − 2) = 4/(x − 2) + 3 (c) Worker A finishes a job in 4 hr; Worker B in 12 hr. How long together?"
Answers (verified): (a) Excluded x=−1. LCD = 2(x+1): 6+(x+1)=5 → x+1=−1 → x=−2. Check: x=−2 ≠ −1. ✓ x = −2. (b) Excluded x=2. Multiply: x=4+3(x−2)=3x−2 → −2x=−2 → x=1. Check: x=1 ≠ 2. ✓ x = 1. (c) 1/4+1/12=1/t → 3t+t=12... LCD=12t: 3t+t=12 → 4t=12 → t=3 hr. 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").
• Say specifically what I got right, then TEACH the gap — show the correct step so I actually learn.
• OFFER A RE-ATTEMPT: "Want to raise your score? I'll give you a similar problem." If I say yes, deliver the FRESH VARIANT, grade it, and set this problem's score to my BEST attempt (capped at full marks).
• Move on when I'm satisfied.
- If I ask about the material, answer briefly, then return to the current problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY against the rubric. Grade only against the vetted key above. For Problem 4(b), if I write "x = 1" without noting it's extraneous, do NOT give full credit — the check is part of the rubric.

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 11 ASSIGNMENT — Fractions All the Way Down
Student: [name] | Date: ___
Problem 1 (Simplify rational expressions): a/24 — [one line]
Problem 2 (Multiply & divide): b/26 — [one line]
Problem 3 (Add & subtract with LCD): c/24 — [one line]
Problem 4 (Solve rational equations + work/rate): 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; Problem 4(b) is the key check — the report should mention "extraneous" and "no solution" for that part. A student who wrote x = 1 without that language probably didn't work through the check.
• The answer key + rubric live inside the student prompt, and every answer is pre-computed and independently re-verified (w11_verify.py, PASS, 34 checks, 0 failures). Known weak point (H5/H7): AI-self-scored grades submitted by share link are gameable; acceptable here as one assignment among many.

Canvas placement block

canvas_object    = Assignment
title            = "Week 11 Assignment — Fractions All the Way Down (adaptive)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = adaptive
submission_types = [online_text_entry, online_url]   # paste the report + the chat share link
due_offset_days  = 6   # Sun Nov 15
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-11 assignment is the AI-coached, self-scored version in I-assignment-and-rubric-week-11.md. This file shows the same Week-11 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 7 (rational expressions & equations) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade

The Assignment

This week is about the algebra of rational expressions — fractions with polynomial numerators and denominators. In four parts, you'll show you can simplify by factoring and canceling, operate by multiplying, dividing, and combining with an LCD, and solve rational equations while catching any extraneous solutions. Show all your steps. Submit your work as a document upload or text entry in Canvas. You'll be graded on the rubric below — read it before you start.

Part 1 — Simplify rational expressions (24 pts). Simplify each completely and state all excluded values:
(a) (x² − 16) / (x² − x − 12)
(b) (2x² + x − 6) / (x² − 4)
(c) (x² + 6x + 9) / (x² − 9)
(d) (3x² − 12) / (x² + x − 6)

Part 2 — Multiply & divide (26 pts). Simplify completely; state excluded values:
(a) [(x² − 9) / (x + 2)] · [(x + 2) / (x − 3)]
(b) [(x² − 1) / (x + 3)] ÷ [(x − 1) / (x + 3)]
(c) [(x² − 4) / (x + 5)] · [(x + 5) / (x − 2)]
(d) [(2x² + 3x − 2) / (x + 4)] ÷ [(2x − 1) / (x + 4)]

Part 3 — Add & subtract with an LCD (24 pts). Perform the operation and simplify:
(a) 3/(x + 2) + 5/(x − 1)
(b) x/(x² − 4) − 1/(x − 2)
(c) 2/x + 3/x
(d) 1/x + 1/(x + 1)

Part 4 — Solve rational equations: show excluded values, solution, and check (26 pts).
(a) 2/(x − 3) + 1/2 = 5/[2(x − 3)]
(b) x/(x − 1) = 1/(x − 1) + 2 ← identify any extraneous solution
(c) 1/x = 4
(d) WORK/RATE: Pipe A fills a tank in 3 hours; Pipe B fills the same tank in 6 hours. How long does it take if both pipes run together? Write and solve the rational equation; show the check.

Integrity & AI note. This is your own work, submitted for grading. You may use an approved chatbot (Gemini, Claude, or ChatGPT) to help you think — check a factoring step, verify an LCD — but submitting AI-generated solutions 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-11.md.)

Rubric — 100 points

Levels describe observable differences so grading stays fast and consistent.

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

(All values pre-computed and independently re-verified — w11_verify.py, PASS, 34 checks, 0 failures.)

Part 1:
(a) (x+4)(x−4)/[(x−4)(x+3)] = (x+4)/(x+3), x≠4, x≠−3.
(b) (2x−3)(x+2)/[(x−2)(x+2)] = (2x−3)/(x−2), x≠2, x≠−2.
(c) Numerator: (x+3)². Denominator: (x−3)(x+3). Cancel: (x+3)/(x−3), x≠3, x≠−3.
(d) Factor: 3(x²−4)/(x²+x−6) = 3(x−2)(x+2)/[(x+2)(x−3)]. Cancel: 3(x−2)/(x−3) = (3x−6)/(x−3), x≠−2, x≠3.

Part 2:
(a) (x+3)(x−3)/(x+2) · (x+2)/(x−3) → cancel → (x+3), x≠−2, x≠3.
(b) (x−1)(x+1)/(x+3) · (x+3)/(x−1) → cancel → (x+1), x≠±1, x≠−3.
(c) (x−2)(x+2)/(x+5) · (x+5)/(x−2) → cancel → (x+2), x≠2, x≠−5.
(d) Factor numerator: 2x²+3x−2 = (2x−1)(x+2). Flip: (2x−1)(x+2)/(x+4) · (x+4)/(2x−1) → cancel → (x+2), x≠1/2, x≠−4.

Part 3:
(a) LCD=(x+2)(x−1). 3(x−1)/LCD + 5(x+2)/LCD = (3x−3+5x+10)/LCD = (8x+7)/[(x+2)(x−1)], x≠−2, x≠1.
(b) LCD=(x−2)(x+2). x/[(x−2)(x+2)] − (x+2)/[(x−2)(x+2)] = (x−x−2)/LCD = −2/[(x−2)(x+2)], x≠±2.
(c) Same denominator: (2+3)/x = 5/x, x≠0.
(d) LCD=x(x+1). (x+1+x)/[x(x+1)] = (2x+1)/[x(x+1)], x≠0, x≠−1.

Part 4:
(a) Excluded: x=3. LCD=2(x−3): 4+(x−3)=5 → x=4. Check 4≠3. ✓ x=4.
(b) Excluded: x=1. Clear: x=1+2(x−1)=2x−1 → x=1. x=1 is excluded → no solution (x=1 is extraneous).
(c) Excluded: x=0. Multiply by x: 1=4x → x=1/4. Check 1/4≠0. ✓
(d) 1/3+1/6=1/t. LCD=6t: 2t+t=6 → t=2 hours. Check: 1/3+1/6=3/6=1/2; two hours → fills tank. ✓

Canvas placement block

canvas_object    = Assignment
title            = "Week 11 Assignment — Fractions All the Way Down (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 15
published        = true
rubric_ref       = "week-11-assignment-rubric"
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"