Week 10 Model/Build Studio: How Low Can the Cost per Shirt Go?

Part 1 — The Big Picture

This week you learned that a rational function, one polynomial divided by another, can level off toward a horizontal line called a horizontal asymptote, and that you can find that line by comparing degrees. An average cost is exactly that kind of function: a fixed setup cost is spread over more and more items, so the cost per item falls but never drops below what each item itself costs. In this Studio you take a real-looking cost log for a club's T-shirt orders, build the total-cost model from the numbers, check it by substitution, turn it into an average-cost function, simplify it one step per line, and read its asymptotes on a graph in Desmos or GeoGebra. Then you check a chatbot's account of the same costs and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From five orders in the club's T-shirt log, what is the total-cost model C(x) = mx + b, what is the average cost per shirt A(x) = C(x) ⁄ x, what level does that average approach as orders grow, and at what order size does it match a vendor that charges 9 dollars a shirt with no setup fee?

Workshop data: Club T-shirt order log (five orders)

Source: A workshop-only cost log for a student club's custom T-shirt orders, printed in full in the first scaffold table: the number of shirts in each order and what the order cost in dollars.

• Rows: 5 orders; the first scaffold table lists all 5, and the second scaffold table has 3 rows, one for each question you answer
• Columns: number of shirts in the order (x) and total cost of the order in dollars
• Order sizes: 10, 20, 40, 60, and 120 shirts
• Total costs: 330, 360, 420, 480, and 660 dollars

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 order sizes 10, 20, 40, 60, 120 in the x column and the five total costs 330, 360, 420, 480, 660 in the y column. Tool: Desmos (free) or GeoGebra. Check: Five dots appear that lie along one straight, rising line.
2. In your notebook, build the model C(x) = mx + b. Find the slope m from the first two orders as the change in cost over the change in shirts, one step per line. Then substitute x = 10 and the cost 330 to find b. Write the finished model and say in words what m and b mean for the club. Tool: A notebook or document for your written work. Check: A linear model with a cost per shirt and a one-time setup fee.
3. Verify by substitution: put each of the five order sizes into your model, compute the cost step by step, and compare it with the logged cost. Record each check in the first scaffold table. Tool: A notebook or document for your written work. Check: Every substituted cost equals the logged cost.
4. Write the average cost per shirt as A(x) = C(x) ⁄ x. Rewrite it by dividing each term of the numerator by x, one step per line, and use the rewritten form to fill in the average-cost column of the first scaffold table. Check one average by dividing the logged cost by the number of shirts. Tool: A notebook or document for your written work. Check: The averages fall as the orders grow, and each matches the logged cost divided by the order size.
5. Compare the degrees of the numerator and the denominator of A(x) to find its horizontal asymptote, and find the input that makes the denominator 0 to name its vertical asymptote. Record both in the second scaffold table with what each means for the club. Tool: A notebook or document for your written work. Check: One horizontal asymptote and one vertical asymptote, and only one of them matters for real orders.
6. Back in the graphing calculator, type y = (3x+300)/x on a new line, then type your horizontal asymptote as a flat line and y = 9 on two more lines. Zoom out to the right, then click the place where the curve meets the line y = 9. Verify that point by substitution in your notebook and record it in the second scaffold table. Tool: Desmos (free) or GeoGebra. Check: The curve settles toward the flat asymptote line without touching it, and it meets the line y = 9 at exactly one point to the right of the vertical axis.

Fallback route

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

Plot the five orders from the cost log by hand on graph paper with shirts across the bottom and cost up the side, and draw the straight line through them. On a second sheet, plot the five average costs, sketch a smooth curve through them, and draw the flat asymptote line and the line at 9 dollars with a ruler. Complete both scaffold tables, every analysis question, and the critique step from your hand graphs and your notebook; submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Check your model against every order in the log, so you know it fits all five costs and not only the two you used to build it, then turn each cost into a cost per shirt.

Student scaffold table 1

Shirts x	Logged total cost (dollars)	Your model's cost, worked step by step	Match? (yes or no)	Average cost per shirt (dollars)
10	330
20	360
40	420
60	480
120	660

Answer each question about the average-cost function and say what it means for the club, so the formula and the real orders are compared side by side.

Student scaffold table 2

Question	Your work	Result	What it means for the club
What level does the average approach?
Where is the vertical asymptote?
When does the average equal 9 dollars?

Part 5 — Analysis Questions

1. Show the steps that find m and b, and write the finished model C(x). Then show, one step per line, how A(x) = C(x) ⁄ x is rewritten by dividing each term of the numerator by x, and use it to compute the average cost for all five orders.
2. In plain words, explain what the horizontal asymptote of A(x) says about the club's cost per shirt as orders grow, and why the average can get close to that level but never reach it.
3. Describe what the graph in step 6 shows near the vertical axis. Name the vertical asymptote, and explain why that part of the graph does not matter for any real order the club could place.
4. Use your table and the point where the curve meets y = 9 to explain which plan is cheaper per shirt for an order of 40 shirts and for an order of 60 shirts, and verify the meeting point by substitution.
5. Choose one and answer it in 3–4 sentences. (a) The model assumes every shirt costs the same 3 dollars to print. Explain how a bulk discount or a price increase would change the average-cost curve and the level it approaches, and which of your answers you would trust least. (b) The log records only five orders. Explain what the model assumes about order sizes the club never placed, and how your substitution checks help or fail to help.

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 10 Model/Build Studio tables above. They come from a student club's T-shirt order log: printing 10 shirts cost 330 dollars, 20 shirts cost 360 dollars, 40 shirts cost 420 dollars, 60 shirts cost 480 dollars, and 120 shirts cost 660 dollars. Please do these things. First, write the total cost as a linear function C(x) = mx + b of the number of shirts x, and say what m and b mean for the club. Second, write the average cost per shirt A(x) = C(x) ⁄ x, simplify it, and find A(40) and A(120). Third, give the horizontal asymptote of A(x) and what it means for the club, and say whether A(x) has a vertical asymptote. Fourth, another vendor charges 9 dollars per shirt with no setup fee; for how many shirts do the two plans cost the same per shirt? Then answer six short review questions: give the degree, the leading coefficient, and the end behavior of −2x⁴ + 3x² − 1; evaluate −2⁴ and (−2)⁴; simplify 2³ · 2⁴ and (2²)³; say whether 2⁻² is a negative number; say whether √2 and √16 are rational or irrational; and solve −2x + 1 < 9. Last, answer two preview questions from later in the course: solve x ⁄ (x − 2) = 2 ⁄ (x − 2); and say whether log(x + 4) can be rewritten as log(x) + log(4).

Test the response for these likely failures:

• Chatbots sometimes swap the two numbers in the cost model, calling 300 the cost per shirt and 3 the setup fee, or read the slope from a single order as 330 ⁄ 10 = 33.
• Chatbots sometimes cancel the x in (3x + 300) ⁄ x as if it were a common factor, writing 3 + 300 = 303 or 300 for the average cost.
• Chatbots sometimes call 4 the leading coefficient of −2x⁴ + 3x² − 1 and −2 its degree, or read the ends from the 3x² term and say both ends rise.
• Chatbots sometimes clear the denominators in x ⁄ (x − 2) = 2 ⁄ (x − 2) and report x = 2 as the solution.
• Chatbots sometimes evaluate −2⁴ as 16, treating the minus sign as part of the base.
• Chatbots sometimes multiply exponents when multiplying powers, writing 2³ · 2⁴ = 2¹², or add them for a power of a power, writing (2²)³ = 2⁵.
• 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 forget to reverse the sign when dividing by a negative, solving −2x + 1 < 9 as x < −4.
• 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 club's costs, 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 cost model is C(x) = 300x + 3, because the bigger number is the slope. The average cost is (3x + 300) ⁄ x = 3 + 300 = 303 dollars a shirt, since the x cancels. The horizontal asymptote is y = 300, so the average cost levels off at 300 dollars, and the graph has no vertical asymptote. The two vendors never cost the same per shirt. Review: −2x⁴ + 3x² − 1 has leading coefficient 4 and both ends rise because 3x² is positive; −2⁴ = 16; 2³ · 2⁴ = 2¹²; 2⁻² = −4; √2 is rational because it is 1.4142; and −2x + 1 < 9 gives x < −4. Preview: x ⁄ (x − 2) = 2 ⁄ (x − 2) has the solution x = 2; and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (the slope and the y-intercept swapped, a term cancelled instead of a factor, the wrong horizontal asymptote, a missed vertical asymptote, a missed equal-cost point, a coefficient read as an exponent, a negated power, a product rule and power rule mix-up, a negative exponent read as a negative number, an irrational number called rational, an inequality sign not reversed, an excluded value kept as a solution, 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 two preview sentences, say what the course will check later. This route assesses the same objective (Objective 7 — analyze polynomial and rational functions (end behavior, zeros and multiplicity, domain, and asymptotes), and simplify and solve rational and radical expressions and equations, including expressions with rational exponents, checking for extraneous solutions) 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

• Both completed scaffold tables
• A screenshot of your graph with the average-cost curve, the asymptote line, and the line at 9 dollars, 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 club's T-shirt cost log: two completed tables (five substitution checks with the average cost per shirt, and three questions about the average-cost function with what each answer means for the club), the cost model C(x) = mx + b with its numbers and the rewritten average-cost function, a graph of the average-cost curve with its asymptote line and the line at 9 dollars or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same costs, 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 cost log	m and b are found from the log step by step, the finished model has the correct numbers with their meanings stated, and all five substitution checks match the log.	The model is written, but one number is found without its steps or with one arithmetic slip, its meaning is missing, or only some orders are checked.	No model is built, or the numbers are not drawn from the cost log.
Rewriting and evaluating the average cost	A(x) = C(x) ⁄ x is rewritten by dividing each term of the numerator by x with every step shown, and all five averages are correct and checked against the log.	The rewrite skips a step or cancels incorrectly, or some averages are missing or contain an arithmetic slip.	No average-cost function is written, or no averages are computed.
Reading the asymptotes in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: the horizontal asymptote is found by comparing degrees and read as a level the average approaches but never reaches, the vertical asymptote is named and set aside as outside real orders, the meeting point with 9 dollars is verified by substitution, and a real limit of the model is explained.	The questions are attempted, but an asymptote is misread, the average is said to reach its level, the meeting point is unverified, or the limit is missing.	The analysis questions are missing or do not address the cost log.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the average-cost curve with its asymptote line and the line at 9 dollars, with 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 cost, average, and asymptote in the tables comes from your own calculations on the printed cost log, never from memory or a chatbot.
• The model's two numbers are found from the log, one step per line.
• Every one of the five substitution checks is written out step by step.
• The average-cost function is rewritten by dividing each term by x, never by cancelling a term.
• Every asymptote comes with a sentence about what it means for real orders.
• 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.