Week 11 — Cancel, Combine, Check: A Short Rational-Expression Report

Assignment 10 of the term — simplify, combine, and solve four short rational problems, explain what each answer means, and tell someone new to algebra why two pumps working together finish sooner than either alone.

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), and in Week 2 you cleared fractions with a least common denominator; this week you use both on fractions built from polynomials. 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 problems from this week: the week's quotient (x² − 6x + 5) ⁄ (x − 1), the difference x ⁄ (x² − 4) − 1 ⁄ (x − 2), the equation 1 ⁄ (x − 1) + 1 = 2 ⁄ (x² − 1), and a pump problem. In the pump problem, one pump alone empties a flooded basement in 12 hours and a second pump alone empties it in 24 hours, and both run at steady rates.

Work these four problems, showing your arithmetic:

1. Simplify the week's quotient (x² − 6x + 5) ⁄ (x − 1). Its graph has a hole at x = 1. What is the y-coordinate of the hole?
2. Subtract x ⁄ (x² − 4) − 1 ⁄ (x − 2) and write the difference as one simplified fraction. What does it equal at x = 6?
3. Solve 1 ⁄ (x − 1) + 1 = 2 ⁄ (x² − 1). Which candidate survives the check?
4. Two pumps empty a flooded basement. One alone takes 12 hours and the other alone takes 24 hours. How many hours do they take working together?

Beside each answer, add one brief interpretation: what that value shows and one use or limit of it. For problem 1, show the factoring and the cancel, write the restriction, and say why x = 1 gives a hole and not a zero. For problem 2, show the minus sign reaching every term of the second numerator, and check your value against the original expression at x = 6. For problem 3, name the excluded values first, say which candidate is extraneous and why, and check the one that survives. For problem 4, write each pump's rate, add the rates, and check that the two pumps' shares make one whole job.

Explain it to a non-expert

A housemate is helping you deal with the flooded basement and has never studied algebra. They think the two pumps together will take 36 hours, or maybe 18, the average. Write a 3–5 sentence message for your housemate: why adding or averaging the times cannot be right, how thinking about each pump's share per hour gives the time together, and one honest limit of the model, such as what it assumes about the pumps. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine two friends washing a big stack of dishes: one alone needs 30 minutes and the other alone needs 60 minutes. Model answer: "Working together can only be faster than the faster friend alone, so the answer has to be under 30 minutes. In one minute the first friend does one thirtieth of the stack and the second does one sixtieth, and together that is one twentieth of the stack each minute. So the two of them finish in 20 minutes. That assumes they never get in each other's way at one sink."

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

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for your housemate 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 rational expressions and equations: excluded values, simplifying by cancelling common factors (never terms), combining fractions over a least common denominator, solving rational equations and rejecting extraneous solutions, and work-rate problems. 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 week's quotient (x^2 - 6x + 5) / (x - 1), the difference x / (x^2 - 4) - 1 / (x - 2), the equation 1 / (x - 1) + 1 = 2 / (x^2 - 1), and two pumps that empty a flooded basement alone in 12 hours and 24 hours, at steady rates.
PROBLEM P1: Simplify the week's quotient (x² − 6x + 5) ⁄ (x − 1). Its graph has a hole at x = 1. What is the y-coordinate of the hole?
PROBLEM P2: Subtract x ⁄ (x² − 4) − 1 ⁄ (x − 2) and write the difference as one simplified fraction. What does it equal at x = 6?
PROBLEM P3: Solve 1 ⁄ (x − 1) + 1 = 2 ⁄ (x² − 1). Which candidate survives the check?
PROBLEM P4: Two pumps empty a flooded basement. One alone takes 12 hours and the other alone takes 24 hours. How many hours do they take working together?
After the four answers, I must briefly interpret each one: what that value shows and one use or limit of it; for P1, the factoring, the cancel, the restriction, and why x = 1 is a hole and not a zero; for P2, the minus sign reaching every term of the second numerator, with a check against the original at x = 6; for P3, the excluded values named first, the extraneous candidate rejected with its reason, and the survivor checked; for P4, each pump's rate, the rates added, and a check that the shares make one whole job. I must then write a 3-5 sentence message for a housemate who is new to algebra, explaining why adding or averaging the times cannot be right, how each pump's share per hour gives the time together, and one honest limit of the model. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct rational-expression computations (40 points), reasoning about restrictions, 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 hole y-coordinate, value of the difference at x = 6, surviving candidate, or time together = 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, such as a fraction in place of a decimal.
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, calling x = 1 a zero of the quotient does not count as the account; for P3, keeping x = 1 as a solution does not count as the account. For P4, BOTH an accurate explanation of the rates added, with the whole-job check, AND a final message that explains why the times do not add 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 housemate 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 rational-expression report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised rational-expression report with all four answers, their brief interpretations, and your message for your housemate — 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 11 skills — simplifying a rational expression by cancelling common factors and keeping its restriction, combining fractions over a least common denominator, solving a rational equation and rejecting an extraneous solution, and modeling a work-rate setting — 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 housemate. 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 rational-expression 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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