Week 11 Model/Build Studio: How Many Riders Shared the Bus?

Part 1 — The Big Picture

This week you learned to work with fractions that have algebra on the top and the bottom: to find the inputs they forbid, to combine them over a least common denominator, and to solve equations that contain them while checking every candidate. A shared cost is exactly that kind of fraction: when a fixed total is split among n people, each share is the total divided by n. In this Studio you take an invented bus-charter log, made up for practice, build the share model from it, combine two shares into one fraction, solve an equation about the group's size, check every answer by substitution, see the answers on a graph in Desmos or GeoGebra, and decide which answers the setting allows. Then you check a chatbot's account of the same charter and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From five share readings, what is the bus-charter model C(n) = k ⁄ n, how much does each share drop when 4 more riders join, how many riders did the club have if that drop is 15 dollars, and how many riders make each share 48 dollars?

Workshop data: Bus-charter share log (five readings, invented for practice)

Source: An invented, workshop-only log of a youth soccer club's bus charter, made up for practice (not real records) and printed in full in the first scaffold table: the number of riders sharing the charter and each rider's equal share in dollars.

• Rows: 5 readings; the first scaffold table lists all 5, and the second scaffold table has 3 rows, one for each question you answer
• Columns: number of riders (n) and each rider's share in dollars
• Riders: 8, 9, 10, 12, and 16 riders
• Cost per rider: 90, 80, 72, 60, and 45 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 rider counts 8, 9, 10, 12, 16 in the x column and the five shares 90, 80, 72, 60, 45 in the y column. Tool: Desmos (free) or GeoGebra. Check: Five dots appear that fall from left to right, steeply at first and then more slowly.
2. In your notebook, build the model C(n) = k ⁄ n. Multiply one share by its number of riders to find k, then repeat with a second reading to confirm it. Write the finished model and name the one input it forbids. Tool: A notebook or document for your written work. Check: A model of the form k ⁄ n with one number for k that every reading agrees with.
3. Verify by substitution: put each of the five rider counts into your model, divide step by step, and compare the result with the logged share. Record each check in the first scaffold table. Tool: A notebook or document for your written work. Check: Every substituted share equals the logged share.
4. Write the drop in each share when 4 more riders join, 720 ⁄ n − 720 ⁄ (n + 4), as one fraction. Build the least common denominator n(n + 4), rewrite both fractions over it, and subtract the whole second numerator. Write every step on its own line, then check your fraction at n = 12 against the two logged shares. Tool: A notebook or document for your written work. Check: A single fraction whose value at n = 12 equals the difference of the logged shares for 12 and 16 riders.
5. Set your fraction equal to 15 and solve: name the excluded values, clear the denominators, rearrange to a quadratic equal to 0, and factor. Then solve 720 ⁄ n = 48. Record both results, with the answers the setting keeps, in the second scaffold table. Tool: A notebook or document for your written work. Check: The drop equation has two candidates, one of them negative; the 48-dollar equation has one whole-number answer.
6. Back in the graphing calculator, type y = 720/x - 720/(x + 4) on a new line, then type y = 15 on another line. Click each place where the curve meets the flat line, and zoom out to see both. Tool: Desmos (free) or GeoGebra. Check: The curve meets the line y = 15 at two places: one at a positive rider count and one at a negative input left of the excluded values.

Fallback route

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

Plot the five readings from the share log by hand on graph paper with riders across the bottom and the share up the side, sketch a smooth curve through them, and draw the flat line at 15 dollars on a second sketch of the drop fraction using its values at 8, 12, and 16 riders. 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 reading in the log, so you know it fits all five shares and not only the two you used to build it.

Student scaffold table 1

Riders n	Logged share (dollars)	Your model's share, worked step by step	Match? (yes or no)
8	90
9	80
10	72
12	60
16	45

Answer each question about the charter and decide which answers the setting keeps, so the algebra and the real club are compared side by side.

Student scaffold table 2

Question	Equation or single fraction	Excluded values	Candidates	Riders the setting keeps, and why
What is the drop in each share when 4 more riders join?
How many riders if the drop is 15 dollars?
How many riders make each share 48 dollars?

Part 5 — Analysis Questions

1. Show how you found k and the excluded value, then show every step that turns 720 ⁄ n − 720 ⁄ (n + 4) into one fraction and every step that solves the drop equation, each on its own line.
2. Your drop equation has two candidates. In plain words, explain which one is the club's starting size and why the setting rules the other one out, and say whether that other candidate is an extraneous solution or a true solution the setting rejects.
3. Use your single fraction to find the drop for 8 riders and for 16 riders, and compare each with the logged shares. What do the two drops show about how much one extra group of 4 riders saves as the club grows?
4. Describe what the graph in step 6 shows near the inputs your model forbids and at the two crossings with y = 15, and connect each feature to your excluded values and your candidates.
5. Choose one and answer it in 3–4 sentences. (a) The model assumes the charter total stays the same however many riders come. Explain how a per-rider fee or a bigger bus could break that assumption and which of your answers you would trust least. (b) The log has only five readings. Explain what the model claims about rider counts the club never logged, 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 11 Model/Build Studio tables above. They come from an invented practice log for a youth soccer club's bus charter, made up for this Studio: the charter costs the same flat total however many riders share it, and the log lists each rider's equal share as 90 dollars with 8 riders, 80 dollars with 9 riders, 72 dollars with 10 riders, 60 dollars with 12 riders, and 45 dollars with 16 riders. Please do these things. First, the share follows C(n) = k ⁄ n; use the log to find k and write the model, and name the excluded value. Second, write the drop in each share when 4 more riders join, 720 ⁄ n − 720 ⁄ (n + 4), as one fraction over a least common denominator. Third, solve 720 ⁄ n − 720 ⁄ (n + 4) = 15 and say how many riders the club had at first. Fourth, solve x ⁄ (x − 2) = 2 ⁄ (x − 2), and simplify (x + 3) ⁄ x. Then answer six short review questions from earlier weeks: 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; solve −2x + 1 < 9; give the slope and the y-intercept of y = −2x + 5; and evaluate 3x² at x = 2. Last, answer one preview question from later in the course: say whether log(x + 4) can be rewritten as log(x) + log(4).

Test the response for these likely failures:

• Chatbots often call the candidate n = −16 an extraneous solution of the charter equation, or report x = 2 as the solution of x ⁄ (x − 2) = 2 ⁄ (x − 2).
• Chatbots sometimes cancel terms instead of factors, for example crossing out the x in (x + 3) ⁄ x to get 3, or cancelling the n in 2880 ⁄ (n(n + 4)) against a term.
• Chatbots sometimes square the coefficient along with x, evaluating 3x² at x = 2 as (3 · 2)² = 36.
• 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 swap the two numbers in y = −2x + 5, calling 5 the slope.
• 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 bus-charter model, 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 model is C(n) = 90n, because the share is 90 dollars for 8 riders. The excluded value is n = 720. The drop when 4 more riders join is 720 ⁄ n − 720 ⁄ (n + 4) = (720 − 720) ⁄ (n − (n + 4)) = 0 ⁄ (−4) = 0, so the share never drops. Solving 720 ⁄ n − 720 ⁄ (n + 4) = 15 gives n = 12 or n = −16, and −16 is an extraneous solution because it makes a denominator zero. x ⁄ (x − 2) = 2 ⁄ (x − 2) has the solution x = 2. (x + 3) ⁄ x simplifies to 3. Review: −2⁴ = 16; 2³ · 2⁴ = 2¹²; 2⁻² = −4; √2 is rational because it is 1.4142; −2x + 1 < 9 gives x < −4; in y = −2x + 5 the slope is 5; and 3x² at x = 2 is 36. Preview: log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (a model that does not match the log, a wrong excluded value, tops and bottoms combined instead of using a least common denominator, a candidate the setting rejects mislabelled as extraneous, an excluded value kept as a solution, a term cancelled instead of a factor, 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, the slope and the y-intercept swapped, a coefficient squared along with x, 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 preview sentence, say what the course will check later. This route assesses the same objectives (Objective 7 and Objective 5) 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 curve and the flat line, 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 bus-charter share log: two completed tables (five substitution checks, and three charter questions with their excluded values, candidates, and the answers the setting keeps), the share model C(n) = k ⁄ n with its number, the drop in each share written as one fraction, a graph of the drop curve with the flat line at 15 dollars or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same charter, 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 share log	k is found by multiplying a share by its rider count and confirmed with a second reading, the excluded value is named, and all five substitution checks match the log.	The model is written, but k is found without its steps or with one arithmetic slip, the excluded value is missing, or only some readings are checked.	No model is built, or the number is not drawn from the share log.
Combining and solving with rational expressions	The drop is written as one fraction over the least common denominator with the minus reaching the whole second numerator and checked at a logged rider count, the drop equation is cleared and factored with every step shown, and the 48-dollar equation is solved and checked by substitution.	One question is missing, or the fraction or a solution is given without steps or without a substitution check.	No fraction is combined and no equation is solved, or answers are stated without any method.
Reading the answers in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: the negative candidate is rejected by the setting and correctly described as a true solution rather than an extraneous one, the two drops are compared, the graph is tied to the excluded values and candidates, and a real limit of the model is explained.	The questions are attempted, but a rejected candidate is kept, it is mislabelled, a graph feature is misread, or the limit is missing.	The analysis questions are missing or do not address the share log.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the five readings and the drop curve meeting the flat line at 15 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 share, fraction, and candidate in the tables comes from your own calculations on the printed share log, never from memory or a chatbot.
• The model's number k is found from the log and confirmed with a second reading.
• Every one of the five substitution checks is written out step by step.
• The drop is written as one fraction over the least common denominator, with the minus sign reaching the whole second numerator, and checked at one logged rider count.
• Every kept or rejected candidate comes with a reason tied to the club or to an excluded value.
• 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.