Week 6 Model/Build Studio: Which Polynomial Fits the Rug Chart?

Part 1 — The Big Picture

This week you learned to build a polynomial by multiplying and to take it apart by factoring, and to check every move by expanding or by plugging in a number. A rug shop's size chart is exactly that kind of problem: every rug is x + 5 feet long and x − 2 feet wide, so its area is a product of two binomials. In this Studio you build the area polynomial from that product, check it against five measured rugs by substitution, see the product and the expanded form land on the same curve in Desmos or GeoGebra, factor the polynomial back, and factor a double rug by pulling out the greatest common factor first. Then you check a chatbot's account of the same chart and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From the rug shop's size chart, which polynomial gives the area of every rug, does it match all five measured areas when you substitute, and does factoring it back recover the two side lengths?

Workshop data: Rug shop size chart (five sizes)

Source: A workshop-only size chart for one rug shop, printed in full in the second scaffold table: for each size number x, the rug is x + 5 feet long and x − 2 feet wide, and the chart lists its area in square feet.

• Rows: 5 rug sizes; the second scaffold table lists all 5, the first has 5 rows of FOIL steps, and the third has 2 rows of expressions to factor
• Columns: size number (x) and the measured area in square feet
• Size numbers: 3, 4, 5, 6, and 7
• Areas: 8, 18, 30, 44, and 60 square feet

Part 3 — Materials & Procedure

Tools and materials

• Desmos (free) or GeoGebra
• A notebook or document for your written work

Ways of working

• build-a-model-from-real-numbers
• apply-the-exponent-rules-step-by-step
• verify-by-substitution

Procedure

1. Open the free Desmos graphing calculator at desmos.com/calculator, or GeoGebra's graphing calculator. Click the plus sign at the top left and choose table. Type the five size numbers 3, 4, 5, 6, 7 in the x column and the five areas 8, 18, 30, 44, 60 in the y column. Tool: Desmos (free) or GeoGebra. Check: Five dots appear, rising from left to right and getting steeper as they climb.
2. In your notebook, multiply (x + 5)(x − 2) with FOIL, one product per line: first, outer, inner, and last. Use the product rule for x · x. Then combine the like terms and write the area polynomial. Record each step in the first scaffold table. Tool: A notebook or document for your written work. Check: Four products and one combined polynomial with three terms.
3. Back in the graphing calculator, type y = (x + 5)(x − 2) on a new line, then type your expanded polynomial from step 2 on another new line, using the caret key for the exponent. Tool: Desmos (free) or GeoGebra. Check: The two curves lie exactly on top of each other and pass through all five dots.
4. Verify by substitution: for each of the five sizes, put the size number into your expanded polynomial, compute the area step by step, and compare it with the chart. Record each check in the second scaffold table. Tool: A notebook or document for your written work. Check: Every substituted area equals the measured area in the chart.
5. Factor your expanded polynomial back into two binomials by finding two numbers whose product is the constant term and whose sum is the middle coefficient. Then factor the double-rug area 2x² + 6x − 20 completely, pulling out the greatest common factor first. Record both in the third scaffold table, and expand each answer to check it. Tool: A notebook or document for your written work. Check: Factoring recovers the rug's two side lengths, and the double rug factors as a number times those same two binomials.
6. In the graphing calculator, type y = 2x^2 + 6x − 20 and your fully factored form of it on two new lines. Tool: Desmos (free) or GeoGebra. Check: The two new curves lie on top of each other, and each point on them is twice as high as the single-rug curve at the same x.

Fallback route

When to use it: Use this route if no graphing calculator or internet connection is available.

Plot the five rows of the size chart by hand on graph paper with the size number across the bottom and square feet up the side, and sketch a smooth curve through them. Complete all three scaffold tables, every analysis question, and the critique step from your hand graph and your notebook; submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Build the area polynomial one FOIL product at a time, so every term can be checked on its own.

Student scaffold table 1

FOIL step	Product written out	Result
First: x · x
Outer: x · (−2)
Inner: 5 · x
Last: 5 · (−2)
Combined polynomial

Check your polynomial against every rug in the chart, so you know it fits all five sizes and not only the one you tried first.

Student scaffold table 2

Size x	Measured area (square feet)	Your polynomial's area, worked step by step	Match? (yes or no)
3	8
4	18
5	30
6	44
7	60

Factor both expressions and expand each answer, so every factoring carries its own check.

Student scaffold table 3

Expression	First move (GCF or product-and-sum)	Factored form	Check by expanding
x² + 3x − 10
2x² + 6x − 20

Part 5 — Analysis Questions

1. Show the four FOIL products for (x + 5)(x − 2), the combined area polynomial, and the factoring of both x² + 3x − 10 and 2x² + 6x − 20, with each factoring expanded back to check it.
2. In plain words, explain where the middle term of your polynomial comes from and why the shortcut x² − 10, which multiplies only the first terms and the last terms, gives an impossible area for the size-3 rug.
3. Describe your substitution checks: did all five rows match your polynomial? Then subtract each area from the next one in the chart. Are those steps equal, as they would be for a straight line, or do they change?
4. Compare the factored form and the expanded form. Which one tells a rug cutter the two side lengths, which one tells a buyer the area at a glance, and how does the double rug's factored form show that it is exactly two single rugs?
5. Choose one and answer it in 3–4 sentences. (a) Try your polynomial at size 2 and at size 1. Explain where the model stops describing real rugs and why a width of x − 2 feet sets that limit. (b) Real rugs may have fringe or rounded corners. Explain how a measured area that is a little off would show up in your substitution table and whether it would mean your polynomial is wrong.

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I've pasted my completed Week 6 Model/Build Studio tables above. A rug shop's size chart lists rugs whose length is x + 5 feet and whose width is x − 2 feet, for the size numbers x = 3, 4, 5, 6, and 7, with areas of 8, 18, 30, 44, and 60 square feet. Please do these things. First, multiply (x + 5)(x − 2) to write the area as a polynomial, and check your polynomial against all five areas. Second, factor x² + 3x − 10, and factor the double-rug area 2x² + 6x − 20 completely. Third, say whether x² + 25 can be factored over the real numbers. Then answer seven short review questions from earlier weeks: evaluate −2⁴ and (−2)⁴; simplify x² · x³ and (x²)³; say whether 2⁻² is a negative number; say whether √2 and √16 are rational or irrational; evaluate 3x² and (3x)² at x = 2; solve −2x + 1 < 9; and name the slope and the y-intercept of y = −2x + 5. Last, answer three preview questions from later in the course: simplify (x² − 9) ⁄ (x + 3); solve √(x + 3) = x + 1; and say whether log(x + 4) can be rewritten as log(x) + log(4).

Test the response for these likely failures:

• Chatbots sometimes multiply only the first terms and the last terms, writing (x + 5)(x − 2) = x² − 10, or they slip a sign and write x² − 3x − 10.
• Chatbots sometimes stop at (2x + 10)(x − 2) or (2x − 4)(x + 5) for 2x² + 6x − 20, which multiply back correctly but still hide a common factor of 2.
• Chatbots sometimes factor x² + 25 as (x + 5)(x − 5) or as (x + 5)(x + 5).
• Chatbots sometimes swap the two numbers of a line, calling 5 the slope of y = −2x + 5.
• Chatbots sometimes evaluate −2⁴ as 16, treating the minus sign as part of the base.
• Chatbots sometimes multiply exponents when multiplying powers, writing x² · x³ = x⁶, or add them for a power of a power, writing (x²)³ = x⁵.
• Chatbots sometimes say 2⁻² is a negative number, such as −4.
• Chatbots sometimes call √2 rational because a calculator shows 1.4142, or call √16 irrational because it has a root sign.
• Chatbots sometimes square the coefficient too, treating 3x² as (3x)², so at x = 2 they give 36 instead of 12.
• Chatbots sometimes forget to reverse the sign when dividing by a negative, solving −2x + 1 < 9 as x < −4.
• Chatbots sometimes cancel terms instead of factors, for example crossing out the x in (x + 3) ⁄ x to get 3.
• Chatbots sometimes keep both candidates of √(x + 3) = x + 1 without checking them in the original equation.
• Chatbots sometimes split the log of a sum, writing log(x + 4) = log(x) + log(4).

Write 3–5 sentences: name what the chatbot got right about the rug polynomial, then quote at least two of its errors (or two from the printed flawed analysis on the equivalent route), name what is wrong with each, and correct each one with evidence from your own tables, your substitution checks, or the course rule it breaks.

Equivalent route (no AI required)

No AI account is required for this Studio. Instead of the chatbot, critique this flawed analysis, printed here in full: "The rug area is (x + 5)(x − 2) = x² − 10, because you multiply the first terms and the last terms. At size 3 that gives −1, so the shop's chart must be wrong. x² + 3x − 10 factors as (x − 5)(x + 2). The double rug is 2x² + 6x − 20 = (2x + 10)(x − 2), which is fully factored. x² + 25 = (x + 5)(x − 5). Review: −2⁴ = 16; x² · x³ = x⁶; 2⁻² = −4; √2 is rational because it is 1.4142; at x = 2, 3x² = 36; −2x + 1 < 9 gives x < −4; and in y = −2x + 5 the slope is 5. Preview: (x + 3) ⁄ x simplifies to 3; √(x + 3) = x + 1 has the two solutions 1 and −2; and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (a dropped middle term, a sign slip in a factoring, a greatest common factor left inside, a sum of squares called factorable, a negated power, a product rule and power rule mix-up, a negative exponent read as a negative number, an irrational number called rational, a coefficient squared along with x, an inequality sign not reversed, the slope and the y-intercept swapped, a term cancelled instead of a factor, an extraneous solution kept, or the log of a sum split), cite the evidence from your tables or the course rule, and write a corrected sentence beneath it; for the three preview sentences, say what the course will check later. This route assesses the same objective (Objective 5 — add, subtract, and multiply polynomials, apply the special products, and factor polynomial expressions by the greatest common factor, trinomial patterns, the difference of squares, and grouping) as the chatbot critique, takes comparable effort (about 30 minutes and 200–300 words), is scored by the same scoring criteria for the same credit, and its own submission artifact is the marked-up flawed analysis, posted in place of the chatbot share link and report.

Part 7 — What to Submit

• All three completed scaffold tables
• A screenshot of your graph with the product and the expanded form, or a labelled hand-drawn graph
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Model/Build Studio report on a rug shop's size chart: three completed tables (the FOIL products, five substitution checks, and two factorings with their checks), the area polynomial for a rug that is x + 5 feet by x − 2 feet, a graph of the product and its expanded form or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same chart, or, on the equivalent route, a marked-up correction of the printed flawed analysis. 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. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Building the model from the size chart	All four FOIL products of (x + 5)(x − 2) are written out with their signs, the middle terms are combined, and the finished area polynomial is correct.	The polynomial is written, but one product is missing or has a sign slip, or the products are given without being combined.	No polynomial is built, or it is not drawn from the size chart's side lengths.
Verifying by substitution	All five sizes are substituted into the polynomial step by step, every result is compared with the measured area, and the match is stated for each row.	Two to four sizes are checked, or the checks are given as results without the steps.	No substitution check is shown.
Factoring step by step, greatest common factor first	Both expressions are factored with every step shown: the product-and-sum pair for x² + 3x − 10, and the greatest common factor pulled out before factoring 2x² + 6x − 20; each answer is expanded back to check it, and analysis questions 2, 4, and 5 are answered accurately.	The factorings are attempted, but a common factor is left inside, a sign is slipped, a check is missing, or one of analysis questions 2, 4, and 5 is missing or inaccurate.	No factoring is shown, or the analysis questions are missing.
Graph and completed tables	All three scaffold tables are complete, and a graph or labelled hand drawing shows the five chart points on the area curve, with the product and the expanded form shown to agree and the axes labelled.	The tables or the graph are partly complete, or the axes are unlabelled.	No tables and no graph are submitted.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors are quoted and corrected with evidence from the student's own tables, substitution checks, or the course rule, and one correct point is acknowledged; both routes earn this band equally.	On either route, errors are named but corrected without evidence, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.

Quality Gate

• Every product, area, and factoring in the tables comes from your own calculations on the printed size chart, never from memory or a chatbot.
• All four FOIL products are written out, with the middle terms combined into one term.
• Every one of the five substitution checks is written out step by step.
• Every factoring is expanded back to the expression it came from.
• The double rug's factoring pulls out the greatest common factor before factoring the trinomial.
• No number, rule, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence or the course rule.