Week 4 Model/Build Studio: What Does a Phone Plan's Line Say?

Part 1 — The Big Picture

This week you learned that a straight line has two numbers that tell its whole story: the slope, which is how fast it climbs, and the y-intercept, which is where it starts. A phone plan that charges a flat monthly fee plus a price for each gigabyte is exactly that kind of line. In this Studio you take five real-looking monthly bills, build the line that fits them, check it against every bill by substitution, see it in Desmos or GeoGebra, and decide what a rival plan's line says. Then you check a chatbot's account of the same bills and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From five monthly bills, what are the phone plan's rate per gigabyte and its flat fee, does the line y = mx + b you build match every bill, and is a rival plan that always costs 12 dollars less a parallel line?

Workshop data: Phone plan bill log (five months)

Source: A workshop-only bill log for one phone plan, printed in full in the first scaffold table: the gigabytes of data used and the bill in dollars for five months.

• Rows: 5 months of bills; the first scaffold table has 3 rows of slope pairs and the second lists all 5 months
• Columns: gigabytes of data used (x) and the monthly bill in dollars (y)
• Data used: 2, 3, 5, 6, and 8 GB
• Bills: 30, 35, 45, 50, and 60 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 data values 2, 3, 5, 6, 8 in the x column and the five bills 30, 35, 45, 50, 60 in the y column. Tool: Desmos (free) or GeoGebra. Check: Five dots appear, rising from lower left to upper right in a straight row.
2. In your notebook, pick two rows from the bill log and compute the slope as rise over run, with the bills on top: (second bill minus first bill) divided by (second GB minus first GB). Then repeat with a different pair of rows. Write both calculations in the first scaffold table. Tool: A notebook or document for your written work. Check: Both pairs of rows give the same slope, measured in dollars per gigabyte.
3. Find the flat fee b by substituting one row and your slope into y = mx + b and solving for b, one step per line. Write the finished line y = mx + b with your numbers. Tool: A notebook or document for your written work. Check: A single line y = mx + b with a slope and a y-intercept in dollars.
4. Back in the graphing calculator, type y = m x + b on a new line and accept the sliders offered for m and b. Drag the sliders until the line passes through all five dots. Then type your own line from step 3 on another new line and check that it lands on the same dots. Tool: Desmos (free) or GeoGebra. Check: Your typed line passes through every one of the five dots, and the slider values match your m and b.
5. Verify by substitution: for each of the five rows, put the GB value into your line, compute the bill step by step, and compare it with the bill in the log. Record each check in the second scaffold table. Tool: A notebook or document for your written work. Check: Every substituted bill equals the logged bill.
6. A rival plan always bills 12 dollars less than this plan for the same data. Write its line, type it into the graphing calculator, and click the point where each line crosses the y-axis to read both y-intercepts. Tool: Desmos (free) or GeoGebra. Check: The rival line runs beside the first line at the same steepness and crosses the y-axis 12 dollars lower.

Fallback route

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

Plot the five rows from the bill log by hand on graph paper with GB across the bottom and dollars up the side, draw the straight line through them with a ruler, and read where it crosses the y-axis. Complete both 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

Use two different pairs of rows from the bill log to find the slope twice, so you can see that every pair gives the same rate.

Student scaffold table 1

Pair of rows used	Rise (change in bill)	Run (change in GB)	Slope, rise over run (dollars per GB)
2 GB and 3 GB
2 GB and 8 GB
Your own choice of pair

Check your line against every month in the log, so you know the model fits all five bills and not only the two you used.

Student scaffold table 2

GB used (x)	Logged bill (dollars)	Your line's bill, worked step by step	Match? (yes or no)
2	30
3	35
5	45
6	50
8	60

Part 5 — Analysis Questions

1. Show the rise-over-run calculation for two different pairs of rows and the steps that find the flat fee b. Write the finished bill line y = mx + b.
2. In plain words, what does your slope mean for a customer, with units, and what does your y-intercept mean? Explain why the bill for 2 GB divided by 2 is not the rate per gigabyte.
3. Describe your substitution checks: did all five rows match your line? Say what it would have meant if one month had not matched.
4. Write the rival plan's line. Use the slopes and the y-intercepts to explain whether the two lines are parallel and whether the two plans could ever bill the same amount for the same data.
5. Choose one and answer it in 3–4 sentences. (a) Real phone plans often cap or slow data after some amount. Explain where your straight-line model could stop telling the truth and what you would check before using it to predict a bill for 20 GB. (b) A month's real bill might include a tax or a one-time charge. Explain how one such month would show up in your substitution table and whether it would mean your line 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 4 Model/Build Studio tables above. They come from a phone plan's bill log for five months: 2 GB cost 30 dollars, 3 GB cost 35 dollars, 5 GB cost 45 dollars, 6 GB cost 50 dollars, and 8 GB cost 60 dollars. Please do these things. First, write the monthly bill as a line y = mx + b, where x is gigabytes used and y is the bill in dollars, and say what the slope and the y-intercept each mean for a customer. Second, a rival plan's bill is always 12 dollars less than this plan's bill for the same data; write its line and say whether the two lines are parallel. 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; evaluate 3x² and (3x)² at x = 2; and solve −2x + 1 < 9. 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 often swap the two numbers in a price line, calling the 20-dollar starting fee the rate per gigabyte, or they read a slope from one row alone, such as 30 ⁄ 2 = 15 dollars per gigabyte, which mixes the flat fee into the rate.
• 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 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 bill line, 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 plan's rate is 20 dollars per gigabyte and its flat fee is 5 dollars, because 20 is the bigger number, so the bill is y = 20x + 5. Using one row, 30 ⁄ 2 = 15 dollars per gigabyte is the rate. The rival plan is y = 5x + 8, and since it starts lower it must be steeper, so the lines are not parallel. Review: −2⁴ = 16 and (−2)⁴ = 16; 2³ · 2⁴ = 2¹²; 2⁻² = −4; √2 is rational because it is 1.4142; at x = 2, 3x² = 36; and −2x + 1 < 9 gives x < −4. 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 (the slope and the y-intercept swapped, a rate read from one row, a parallel judgment that ignores the slopes, 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, 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 4 — analyze linear functions (slope, intercepts, and the forms of a line, including parallel and perpendicular lines) and solve systems of linear equations and inequalities in two variables) 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 both lines, 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 phone plan's bill log: two completed tables (two slope calculations, and five substitution checks), the bill line y = mx + b with its slope and flat fee, a graph of the plan and its rival or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same bills, 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 bill log	The slope is computed as rise over run from at least two pairs of rows with the bills on top, the flat fee is found by solving y = mx + b step by step, and the finished line has the correct slope and y-intercept.	The line is written, but one slope calculation is missing or inverted, or the flat fee is found without its steps or with one arithmetic slip.	No line is built, or the numbers are not drawn from the bill log.
Verifying by substitution	All five rows are substituted into the line step by step, every result is compared with the logged bill, and the match is stated for each row.	Two to four rows are checked, or the checks are given as results without the steps.	No substitution check is shown.
Reading slope, intercept, and the rival line	Analysis questions 2, 4, and 5 are answered accurately: the slope is read as dollars per gigabyte, the y-intercept as the flat fee, the rival line has the same slope and a lower y-intercept and is called parallel, and a real limit of the model is explained.	The questions are attempted, but the slope and the y-intercept are swapped, the parallel judgment lacks a reason, or the limit is missing.	The analysis questions are missing or do not address the bill log.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the five data values on the bill line and the rival line beside it, 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 slope, fee, and bill in the tables comes from your own calculations on the printed bill log, never from memory or a chatbot.
• Each slope is computed as rise over run with the bills on top, for at least two different pairs of rows.
• Every one of the five substitution checks is written out step by step.
• The finished line names the slope as dollars per gigabyte and the y-intercept as the flat fee, not the other way around.
• The rival plan's line and the parallel judgment rest on the two slopes and the two y-intercepts.
• 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.