Week 3 Model/Build Studio: The Dog-Run Function

Part 1 — The Big Picture

This week you learned that a function gives each input exactly one output, that function notation names that output, that a rule only accepts certain inputs, and that two functions can be chained so one feeds the next. In this Studio you build a function from a real design problem: the area of a dog run made from 20 meters of fencing. You fill in its table, graph it in Desmos or GeoGebra, decide which inputs make sense, chain it with a mulch-cost function, check every value by substitution, and then catch a chatbot's algebra slips on the same problems.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: For a dog run built from 20 meters of fencing, how does the run's area depend on the length of one side, which side lengths make sense, and what does the mulch cost when the area function feeds the cost function?

Workshop data: Animal shelter dog-run model

Source: A workshop-only model you build yourself: an animal shelter fences a rectangular dog run with 20 meters of fencing, mulch costs 3 dollars per square meter, and a gate adds a fixed 40 dollars. The model gives the run's area and cost for each whole-meter side length.

• Rows: 11 whole-meter side lengths, x = 0 through x = 10, one row per side length in the first scaffold table
• Model: With 20 meters of fencing, the two sides of the run are x and 10 − x meters, so the area is x(10 − x) square meters
• Prices: Mulch costs 3 dollars per square meter, and the gate costs a fixed 40 dollars

Part 3 — Materials & Procedure

Tools and materials

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

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, and start a blank graph. Tool: Desmos (free) or GeoGebra. Check: An empty expression list appears beside an empty graph.
2. Type A(x)=x(10-x) on the first line. On the next eleven lines type A(0), A(1), and so on up to A(10). Copy each value into the first scaffold table, and check each one by hand as one side times the other side. Tool: Desmos (free) or GeoGebra. Check: Each line shows a whole number, the values rise and then fall, and your hand product matches the tool on every row.
3. On a new line type y=A(x) to graph the model. Then type x=4 on its own line to draw a vertical line, and change the 4 to 2, 6, and 8 in turn. Tool: Desmos (free) or GeoGebra. Check: The curve is an upside-down U, and every vertical line you draw crosses it exactly once.
4. Type A(-2) and A(12) on two new lines, and record both values in the second scaffold table. Then show A(−2) by hand with −2 in parentheses: 10(−2) − (−2)². Tool: Desmos (free) or GeoGebra. Check: Both lines show a negative number, which no real run can have.
5. Type S(a)=3a for the mulch cost. On new lines type S(A(4)) and A(S(4)), and record both in the third scaffold table with a hand check that starts from the inside. Tool: Desmos (free) or GeoGebra. Check: The two lines show different values; only one of them is a mulch cost in dollars.
6. Type T(x)=S(A(x))+40 for the total cost of mulch and the gate, then type T(4) and T(5). Record both values in the third scaffold table and check each by substitution, one step per line. Tool: Desmos (free) or GeoGebra. Check: Each tool value matches your hand check, and T(5) is larger than T(4).

Fallback route

When to use it: Use this route if neither Desmos nor GeoGebra is available on your device.

Fill in every table by hand, multiplying one side by the other for each area and using a basic calculator for the checks. Sketch the graph on paper from your first table, draw the vertical lines with a ruler, answer every analysis question and complete the critique step the same way, and submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Build the model side length by side length, because each output should equal one side times the other side, and that product is what the function promises.

Student scaffold table 1

Side x (meters)	Other side 10 − x (meters)	Area A(x) by hand	Area from Desmos or GeoGebra
0	10
1	9
2	8
3	7
4	6
5	5
6	4
7	3
8	2
9	1
10	0

Test the model against the week's function checks, because a rule is only useful once you know which inputs it can honestly accept.

Student scaffold table 2

Check	What you did	What you saw	Your conclusion
Do two side lengths share an area?	Compare A(4) and A(6)
Vertical line test	Draw x = 2, 4, 6, and 8 across y = A(x)
A negative input	Evaluate A(−2) with the tool and by hand
An input past the fencing	Evaluate A(12) with the tool
Practical domain and range	Read them from your table and graph

Chain and combine the functions, because the order of a composition decides whether the answer even has the right units.

Student scaffold table 3

Expression	Inner value	Value from the tool	Hand check, inside first
S(A(4))
A(S(4))
T(4) = S(A(4)) + 40
T(5) = S(A(5)) + 40

Part 5 — Analysis Questions

1. Using your first table, which side length gives the largest area, and what is that area? What is the total cost T for that run? Show the substitution.
2. In your first table, which pairs of side lengths share an area? Does that sharing stop A from being a function? Use the vertical line test from your graph to support your answer.
3. The formula accepts x = −2 and x = 12, yet your tool shows a negative area for each. Explain the difference between the domain of the formula and the practical domain of the dog run, and state the practical domain and the range in interval notation.
4. Explain why S(A(4)) is a mulch cost in dollars but A(S(4)) is not. Name the inner function and the outer function in each, and say what goes wrong in A(S(4)).
5. Choose one and answer it in 3–4 sentences: (a) a real shelter might not use all 20 meters of fencing for a rectangle, or might put one side against a wall — name one way the model x(10 − x) would change and why; or (b) you checked each value with a tool and by hand — say which check you would trust if the two disagreed, and why.

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 3 Studio tables above. An animal shelter builds a rectangular dog run with 20 meters of fencing, so if one side is x meters the other side is 10 − x meters, and the area is A(x) = x(10 − x) square meters. Mulch costs 3 dollars per square meter, so S(a) = 3a dollars covers a square meters, and a gate adds a fixed 40 dollars. Please answer these, showing every step. 1. Evaluate A(−2) = 10(−2) − (−2)². 2. Evaluate the term 3x² at x = −2. 3. Find (S ∘ A)(4) and (A ∘ S)(4), and say which one is the mulch cost for a 4-meter side. 4. Is the side of a square run with area 20 square meters, √20, rational or irrational? Is √25? 5. In the total cost T = 3a + 40 for a run of area a, which number is the fixed cost and which is the cost per square meter? Then, as stretch and review questions: 6. Write x² · x³ and (x²)³ as single powers of x. 7. Write 5⁻² as a number. 8. Solve −3x > 9. 9. Simplify (x + 3) ⁄ x. 10. Is x = −2 a solution of √(x + 3) = x + 1? 11. Rewrite log(x + 4) as a sum of two logarithms.

Test the response for these likely failures:

• Chatbots often treat −(−2)² as +4, reporting A(−2) = −20 + 4 = −16.
• Chatbots sometimes square the coefficient too, reading 3x² at x = −2 as (3 · −2)² = 36.
• On the stretch question, chatbots sometimes swap the two numbers and call 3 the fixed cost and 40 the rate.
• Chatbots sometimes call every square root irrational, or call √20 rational because 20 is a whole number.
• On the review question, chatbots sometimes multiply the exponents for a product, writing x² · x³ = x⁶.
• On the review question, chatbots occasionally give 5⁻² = −25.
• On the review question, chatbots sometimes divide −3x > 9 by −3 and keep the sign, giving x > −3.
• On the stretch question, chatbots often cancel the x in (x + 3) ⁄ x and report 3.
• On the stretch question, chatbots often square both sides and accept every candidate without checking it.
• On the stretch question, chatbots sometimes split log(x + 4) into log(x) + log(4).

Write 3–5 sentences naming one thing the chatbot got right and at least two places it went wrong or overreached on questions 1–5, quoting its words and correcting each with evidence from your own tables or Desmos lines. For question 3, say whether it applied the inner function first. For questions 6–11, mark each answer as review or as one to check later in the course, and note any that disagree with your own work.

Equivalent route (no AI required)

No AI account is required for this Studio. This route assesses the same objective, Objective 3 (use function notation; determine the domain and range of a function; evaluate functions at numeric and algebraic inputs; and form sums, differences, products, quotients, and compositions of functions), with comparable effort, about the same 90 minutes, under the same rubric for the same credit. Instead of the chatbot, critique this flawed analysis, printed here in full: "A(−2) = 10(−2) − (−2)² = −20 + 4 = −16. The term 3x² at x = −2 is (3 · −2)² = 36. (S ∘ A)(4) means S(4) times A(4), which is 12 × 24 = 288. √20 and √25 are both irrational, because they are written with a root sign. In T = 3a + 40, the fixed cost is 3 and the cost per square meter is 40. Also, x² · x³ = x⁶; 5⁻² = −25; −3x > 9 gives x > −3; (x + 3) ⁄ x = 3 after cancelling the x; x = −2 solves √(x + 3) = x + 1; and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong, and correct it with evidence from your own tables, sorting the last six claims into review and later in the course. Submit the marked-up analysis in place of the chatbot report and share link.

Part 7 — What to Submit

• All three completed scaffold tables
• 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 Studio report on the dog-run model: three completed tables (the area for each whole-meter side length, five function checks, and four chained or combined values with hand checks), written answers to five analysis questions, and a critique of a chatbot's work on the same problems, 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 numbers	All eleven side lengths are filled in with the other side and the area, every area is checked by hand as one side times the other, and the tool values match.	The table is mostly complete, but one or two areas or hand checks are missing or wrong.	No model is built, or the values are invented rather than computed.
Evaluating step by step with correct notation	A(−2) is worked by hand with −2 in parentheses and reaches −24, every composition shows its inner value first, and T(4) and T(5) are each checked one step per line.	Most evaluations are correct, but one negative input is squared outside its parentheses, or one hand check is missing.	The evaluations are not shown, or negatives and compositions are handled incorrectly throughout.
Verifying by substitution and the function checks	The second table names the shared areas, reports the vertical line test, and explains both out-of-range inputs, and every tool value in the third table matches a hand substitution.	The checks are recorded, but a conclusion is missing or one tool value has no matching hand substitution.	No function check or substitution is shown.
Analysis, domain and range, and limitation	Questions 1 through 4 are answered with values from the tables, the practical domain and range are stated in interval notation, the inner and outer functions are named, and the chosen option of question 5 gives a specific, reasoned answer.	Most questions are answered, but one is missing or unsupported, or the domain and range are swapped.	The analysis questions are not answered.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors on questions 1–5 are quoted and corrected with evidence from the student's own tables, one correct point is acknowledged, and the review and stretch answers are sorted; 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 value in the tables comes from your own tool lines or your own arithmetic, never from memory or a chatbot.
• Each area in the first table equals one side times the other side.
• Every negative input is written in parentheses before it is squared.
• Every composition is checked from the inside out, with the inner value recorded.
• No value, 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 from your tables.