Week 7 — Assignment (Adaptive Learning) · "Four Methods, One Toolkit"

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 6 (factoring, square root property, completing the square, quadratic formula, discriminant) · 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 7 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, Oct 18.

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 7 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. 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) — Factoring & zero-product property ────────────
SHOW ME: "Solve each equation by factoring. Show every step and verify at least one solution by substituting back. (a) x² − 7x + 10 = 0 (b) (2x − 1)(x + 3) = 0 (c) 3x² − 12x = 0"
VETTED ANSWER: (a) find factors of +10 adding to −7: (−2)(−5) → (x−2)(x−5)=0 → x=2 or x=5. Check: 4−14+10=0 ✓. (b) zero-product directly: 2x−1=0 → x=1/2; x+3=0 → x=−3. (c) factor GCF: 3x(x−4)=0 → x=0 or x=4 (never divide by x).
RUBRIC: (a) 8 pts — full credit for both solutions with correct factoring; 4–6 for correct solution but no factor shown or one sign error; 0–2 for wrong factors. (b) 8 pts — full credit for both solutions; 4 for one only. (c) 8 pts — full credit for both solutions by factoring (NOT by dividing by x); 0 if only x=4 given (missing x=0 from dividing by x).
FRESH VARIANT: "(a) x² − 8x + 15 = 0 (b) (3x + 2)(x − 4) = 0 (c) 2x² − 10x = 0". Answers: (a) (x−3)(x−5)=0 → x=3 or x=5; (b) x=−2/3 or x=4; (c) 2x(x−5)=0 → x=0 or x=5. Same rubric.

──────────── PROBLEM 2 (26 points) — Square root property & completing the square ────────────
SHOW ME: "Solve each equation. Show every step. (a) Use the square root property: (x + 2)² = 25 (b) Use completing the square: x² − 4x − 12 = 0"
VETTED ANSWER: (a) x+2=±5 → x=−2+5=3 or x=−2−5=−7. Check: (3+2)²=25 ✓, (−7+2)²=25 ✓. (b) x²−4x=12 → add (−4/2)²=4 to both sides → x²−4x+4=16 → (x−2)²=16 → x−2=±4 → x=6 or x=−2. Check: 36−24−12=0 ✓, 4+8−12=0 ✓.
RUBRIC: (a) 12 pts — full credit for both solutions with ± shown; 6 for correct method but only one solution (dropped ±); 3 if set up correctly but arithmetic error. (b) 14 pts — 4 for correctly moving constant; 4 for adding correct (b/2)² to BOTH sides; 3 for correct perfect-square form; 3 for both final solutions.
FRESH VARIANT: "(a) (x − 3)² = 36 (b) x² + 6x − 7 = 0 by completing the square". Answers: (a) x−3=±6 → x=9 or x=−3; (b) x²+6x=7 → add 9 → (x+3)²=16 → x+3=±4 → x=1 or x=−7. Same rubric.

──────────── PROBLEM 3 (24 points) — Quadratic formula & discriminant ────────────
SHOW ME: "For each equation: (i) compute the discriminant and state what it tells you; (ii) solve if real solutions exist (use the quadratic formula for part a). (a) 2x² + 3x − 2 = 0 (b) x² + 3x + 4 = 0"
VETTED ANSWER: (a) a=2,b=3,c=−2. Discriminant=9+16=25 (positive → two real solutions). x=(−3±5)/4 → x=1/2 or x=−2. Check: 2(1/4)+3(1/2)−2=1/2+3/2−2=0 ✓. (b) a=1,b=3,c=4. Discriminant=9−16=−7 (negative → no real solutions). Stop here.
RUBRIC: (a) 12 pts — 4 for correct discriminant value (25); 4 for correct formula setup; 4 for both correct simplified solutions. (b) 12 pts — 6 for correct discriminant (−7); 6 for correctly concluding no real solutions exist (do NOT penalize for stopping; DO penalize if they continue to give "solutions").
FRESH VARIANT: "(a) 3x² − 5x − 2 = 0 (b) x² − 2x + 6 = 0". Answers: (a) disc=25+24=49; x=(5±7)/6 → x=2 or x=−1/3; (b) disc=4−24=−20 → no real solutions. Same rubric.

──────────── PROBLEM 4 (26 points) — Application: projectile ────────────
SHOW ME: "A ball is launched upward from the top of a 80-foot building with an initial velocity of 64 ft/s. Its height (in feet) above the ground t seconds after launch is given by h(t) = −16t² + 64t + 80. (a) Set up the equation to find when the ball hits the ground (h = 0). Show the standard form. (b) Solve the equation using the quadratic formula. Show the discriminant. (c) Interpret your answers: which value of t makes physical sense, and why?"
VETTED ANSWER: (a) −16t²+64t+80=0; divide by −16: t²−4t−5=0 (or keep original form — either is fine). (b) Using t²−4t−5=0: a=1,b=−4,c=−5. Disc=16+20=36. t=(4±6)/2 → t=5 or t=−1. (c) t=5 seconds makes physical sense (positive time); t=−1 is before the launch — discard it. The ball hits the ground 5 seconds after launch.
RUBRIC: (a) 6 pts — correct equation setup, zero on right. (b) 12 pts — 3 for correct discriminant; 5 for correct formula application; 4 for both values of t. (c) 8 pts — 4 for identifying t=5 as physical; 4 for clear explanation of why t=−1 is discarded (negative time / before launch).
FRESH VARIANT: "A rectangular garden has a length that is 4 feet more than its width, x. Its area is 77 square feet. (a) Set up the equation (standard form, zero on right). (b) Solve by the quadratic formula; show the discriminant. (c) Interpret: which value of x makes physical sense?" Answers: (a) x(x+4)=77 → x²+4x−77=0. (b) disc=16+308=324; x=(−4±18)/2 → x=7 or x=−11. (c) x=7 feet (positive width); x=−11 discarded (negative length). 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: 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 against the rubric, state the score, say what I got right, TEACH the gap, OFFER A RE-ATTEMPT with the FRESH VARIANT if I want to raise my score. Set this problem's score to my BEST attempt.
- If I ask about the material, answer briefly, then return. Off-topic: one friendly sentence, then back to the problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY — don't inflate; don't lowball. Grade only against the vetted key above.

COMPLETION + REPORT. After 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 7 ASSIGNMENT — Four Methods, One Toolkit
Student: [name] | Date: ___
Problem 1 (Factoring & zero-product): a/24 — [one line]
Problem 2 (Square root property & completing the square): b/26 — [one line]
Problem 3 (Quadratic formula & discriminant): c/24 — [one line]
Problem 4 (Application — projectile): 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.
• Every answer is pre-computed and independently re-verified (w07_verify.py, PASS): P1 {x²−7x+10→{2,5}; (2x−1)(x+3)→{1/2,−3}; 3x²−12x→{0,4}}; P2 {(x+2)²=25→{3,−7}; x²−4x−12 complete-square→{6,−2}}; P3 {2x²+3x−2 disc=25→{1/2,−2}; x²+3x+4 disc=−7 no real}; P4 {−16t²+64t+80=0→t=5,−1 → t=5 physical}.

Canvas placement block

canvas_object    = Assignment
title            = "Week 7 Assignment — Four Methods, One Toolkit (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 Oct 18
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-7 assignment is the AI-coached, self-scored version in I-assignment-and-rubric-week-07.md. This file shows the same Week-7 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 6 (factoring, square root property, completing the square, quadratic formula, discriminant) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade

The Assignment

This week you added four solving methods to your algebra toolkit. In four problems, you'll show you can apply each method correctly, use the discriminant to predict solution types, and interpret your answers in a real-world context. 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.

Problem 1 — Factoring & zero-product property (24 pts).
Solve each equation by factoring. Show every step and verify at least one solution by substituting back.

(a) x² − 7x + 10 = 0
(b) (2x − 1)(x + 3) = 0
(c) 3x² − 12x = 0

Problem 2 — Square root property & completing the square (26 pts).
Solve each equation. Show every step.

(a) Use the square root property: (x + 2)² = 25
(b) Use completing the square: x² − 4x − 12 = 0

Problem 3 — Quadratic formula & discriminant (24 pts).
For each equation: (i) compute the discriminant and state what it tells you about the number of real solutions; (ii) solve if real solutions exist (use the quadratic formula for part a).

(a) 2x² + 3x − 2 = 0
(b) x² + 3x + 4 = 0

Problem 4 — Application: projectile (26 pts).
A ball is launched upward from the top of an 80-foot building with an initial velocity of 64 ft/s. Its height (in feet) above the ground t seconds after launch is given by h(t) = −16t² + 64t + 80.

(a) Set up the equation to find when the ball hits the ground (h = 0). Write it in standard form.
(b) Solve the equation using the quadratic formula. Show the discriminant.
(c) Interpret your answers: which value of t makes physical sense, and why?

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 rule, 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-07.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 — w07_verify.py, PASS.)

• Problem 1: (a) factors: (x−2)(x−5)=0 → x=2 or x=5 [check: 4−14+10=0 ✓]. (b) zero-product directly: x=1/2 or x=−3. (c) GCF: 3x(x−4)=0 → x=0 or x=4 (dividing by x loses x=0 — common error to watch).
• Problem 2: (a) x+2=±5 → x=3 or x=−7 [check: (3+2)²=25 ✓, (−7+2)²=25 ✓]. (b) x²−4x=12; add (2)²=4 → (x−2)²=16 → x−2=±4 → x=6 or x=−2 [check: 36−24−12=0 ✓].
• Problem 3: (a) disc=9+16=25; x=(−3±5)/4 → x=1/2 or x=−2. (b) disc=9−16=−7 (negative) → no real solutions.
• Problem 4: (a) −16t²+64t+80=0 (or divide by −16: t²−4t−5=0). (b) disc=16+20=36; t=(4±6)/2 → t=5 or t=−1. (c) t=5 seconds is physical (positive time after launch); t=−1 is discarded (negative time, before launch).

Canvas placement block

canvas_object    = Assignment
title            = "Week 7 Assignment — Four Methods, One Toolkit (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 Oct 18
published        = true
rubric_ref       = "week-07-assignment-rubric"
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"