Week 7 — Two Answers, One Setting: A Short Quadratic Report

Assignment 7 of the term — solve four short quadratic equations, explain what each answer means, and tell someone new to algebra why an equation can give an answer the real situation throws out.

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 6 you factored polynomials such as x² − 6x + 5 = (x − 1)(x − 5); this week you set expressions like that equal to 0 and solve them four ways. 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 equations from this week: 3x² − 12x = 0, (x − 3)² = 16, the equation x² + 6x + 5 = 0, and a patio problem. The patio is rectangular, 3 feet longer than it is wide, and covers 40 square feet, so its width w satisfies w(w + 3) = 40, which is w² + 3w − 40 = 0 in standard form.

Work these four problems, showing your arithmetic:

1. Solve 3x² − 12x = 0 by factoring. What is its nonzero solution?
2. Solve (x − 3)² = 16 with the square root property. What is the smaller solution?
3. Completing the square turns x² + 6x + 5 = 0 into (x + 3)² = k. What is k?
4. A rectangular patio is 3 feet longer than it is wide, and its area is 40 square feet. Its width w satisfies w² + 3w − 40 = 0. What is its length, in feet?

Beside each answer, add one brief interpretation: what that value shows about the equation and one use or limit of it. For problem 1, name both solutions and say what dividing both sides by x would lose. For problem 2, say why there are two solutions and check both. For problem 3, say why the same number goes on both sides, then finish the solve and check it. For problem 4, say which solution of the width equation the setting rejects and why, and check the area.

Explain it to a non-expert

A neighbor is helping you lay out the patio and has never studied algebra. They saw your equation w² + 3w − 40 = 0 give two answers and are confused. Write a 3–5 sentence message for your neighbor: why a squared equation can give two answers, why only one of them can be the patio's width, and one honest limit of the equation, such as what it assumes about the patio's shape. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine a square picture frame that must enclose 64 square inches of photo, so its side s satisfies s² = 64. Model answer: "The equation asks for a number that times itself makes 64. Both 8 and negative 8 do that, because a negative times a negative is positive, so the math gives two answers. A side of a frame cannot be negative, so the frame's side is 8 inches. The equation assumes a perfect square opening, so a frame with a mat border would need a different equation."

Use that pattern for the patio, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for your neighbor 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 quadratic equations: standard form, factoring and the zero-product property, never dividing by the variable, the square root property, completing the square, the quadratic formula, and answers a setting rejects. 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 equations 3x^2 - 12x = 0, (x - 3)^2 = 16, and x^2 + 6x + 5 = 0, and a rectangular patio 3 feet longer than it is wide with an area of 40 square feet, so its width w satisfies w(w + 3) = 40, which is w^2 + 3w - 40 = 0.
PROBLEM P1: Solve 3x² − 12x = 0 by factoring. What is its nonzero solution?
PROBLEM P2: Solve (x − 3)² = 16 with the square root property. What is the smaller solution?
PROBLEM P3: Completing the square turns x² + 6x + 5 = 0 into (x + 3)² = k. What is k?
PROBLEM P4: A rectangular patio is 3 feet longer than it is wide, and its area is 40 square feet. Its width w satisfies w² + 3w − 40 = 0. What is its length, in feet?
After the four answers, I must briefly interpret each one: what that value shows about the equation and one use or limit of it; for P1, both solutions and what dividing both sides by x would lose; for P2, why there are two solutions, with both checked; for P3, why the same number goes on both sides, with the solve finished and checked; for P4, which solution of the width equation the setting rejects and why, with the area checked. I must then write a 3-5 sentence message for a neighbor who is new to algebra, explaining why a squared equation can give two answers, why only one can be the patio's width, and one honest limit of the equation. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct quadratic computations (40 points), reasoning about solutions, methods, and the setting (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 nonzero solution, smaller solution, value of k, or patio length = 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 about the equation plus a use or limit = 2; exactly one of the two = 1; neither = 0. For P1, naming only x = 4 without the lost x = 0 does not count as the account; for P2, a single root without the +/- does not count as the account. For P4, BOTH an accurate explanation of why the setting rejects w = -8, with the area check, AND a final message that explains the two answers and the choice with an honest limit = 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 neighbor 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 quadratic report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised quadratic report with all four answers, their brief interpretations, and your message for your neighbor — 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 7 skills — solving a quadratic equation by factoring without dividing by the variable, by the square root property, and by completing the square, and judging which solution a real setting allows — 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 neighbor. 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 quadratic 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.

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