Week 5 · Model/Build Studio

Week 5 Model/Build Studio: Which Phone Plan Is Cheaper Past 6 GB?

College Algebra Free sample edition

Part 1 — The Big Picture

This week you learned that the solution of a system is the one pair of numbers that makes two equations true at once, the point where two lines cross. Choosing between two phone plans is exactly that kind of question. One plan has a low monthly fee and a high price per gigabyte; the other has a higher fee and a lower price. In this Studio you build each plan's line from five bills, solve the system to find where the plans cost the same, check that pair in both equations, see the crossing in Desmos or GeoGebra, and decide which plan is cheaper past 6 GB. Then you check a chatbot's account of the same plans and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From the two plans' bills, what line does each plan follow, at what amount of data do the plans cost the same, and which plan is cheaper for a customer who uses more than that amount?

Workshop data: Two phone plans' bills at five data amounts

Source: A workshop-only comparison of two phone plans, printed in full in the first scaffold table: the bill in dollars for each plan at five amounts of data.

Rows: 5 data amounts; the first scaffold table lists all 5, and the second has 4 rows of checks
Columns: gigabytes of data used (x), Plan A's bill in dollars, and Plan B's bill in dollars
Data used: 2, 4, 6, 8, and 10 GB
Plan A bills: 30, 40, 50, 60, and 70 dollars
Plan B bills: 38, 44, 50, 56, and 62 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

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 amounts 2, 4, 6, 8, 10 in the first column, Plan A's bills 30, 40, 50, 60, 70 in the second column, and Plan B's bills 38, 44, 50, 56, 62 in a third column. Tool: Desmos (free) or GeoGebra. Check: Two rows of five dots appear, each in a straight row, and the two rows share one dot.
In your notebook, use two rows of the table to compute each plan's slope as rise over run, with the bills on top. Then find each plan's monthly fee b by substituting one row and the slope into y = mx + b and solving for b, one step per line. Write both lines in the first scaffold table. Tool: A notebook or document for your written work. Check: Plan A and Plan B each have one slope in dollars per gigabyte and one fee in dollars.
Solve the system made by your two lines by substitution: both lines say what y equals, so set the right sides equal, collect the x-terms on one side and the numbers on the other, and divide. Then substitute your x into either line to find y. Write every step on its own line. Tool: A notebook or document for your written work. Check: One ordered pair (x, y): the amount of data and the bill at which the two plans cost the same.
Verify by substitution: put your pair into Plan A's line and into Plan B's line, one step per line, and compare each result with the y-value of your pair. Record both checks in the second scaffold table, then check two more rows of the bill table the same way. Tool: A notebook or document for your written work. Check: The pair works in both lines, and both extra rows match their plan's line.
Back in the graphing calculator, type your two lines on new lines, such as y = 5x + 20 written with your own numbers. Click the point where the two lines cross and read its coordinates. Then click each line at x = 10 to compare the two bills there. Tool: Desmos (free) or GeoGebra. Check: The crossing point matches your pair, and to the right of it the line for one plan stays below the other.
A Plan C bills 6 dollars more than Plan A for every amount of data. Write its line, type it into the graphing calculator beside Plan A, and try to solve the system of Plan A and Plan C by subtracting one equation from the other. Tool: Desmos (free) or GeoGebra. Check: The two lines run side by side with no crossing point, and subtracting leaves a statement with no x in it.

Fallback route

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

Plot both plans' five rows from the bill table by hand on graph paper with GB across the bottom and dollars up the side, draw each plan's straight line with a ruler, and read where the lines cross. 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

Build each plan's line from the bill table, so you can compare the two plans as equations and not only as lists of prices.

Student scaffold table 1
Data used (GB)	Plan A bill (dollars)	Plan B bill (dollars)
2	30	38
4	40	44
6	50	50
8	60	56
10	70	62

Check your crossing pair in both lines, and two more rows of the table, so you know the system's solution and your lines are right.

Student scaffold table 2
What you check	Line used	Worked step by step	Match? (yes or no)
Your crossing pair	Plan A
Your crossing pair	Plan B
The 2 GB row	Plan B
The 10 GB row	Plan A

Part 5 — Analysis Questions

Show the rise-over-run and fee calculations for both plans and write both lines y = mx + b. Then show every step of the substitution that solves the system, and give the crossing pair.
In plain words, what do the two numbers in your crossing pair mean for a customer? Explain which plan is cheaper for someone who uses 10 GB, by how much, and why the crossing point is where the answer changes.
Describe your substitution checks: did your pair work in both lines, and did the two extra rows match? Say what it would mean if the pair worked in Plan A's line but not in Plan B's.
Write Plan C's line. Use the slopes, the y-intercepts, and the statement left after subtracting to explain whether Plan A and Plan C ever cost the same, and name the kind of system they make.
Choose one and answer it in 3–4 sentences. (a) Real plans often cap or slow data past some amount. Explain how a cap could change which plan is cheaper and what you would check before choosing a plan for a heavy-data month. (b) A customer's data use changes from month to month. Explain how you would use the crossing point to choose a plan for someone who uses between 4 and 9 GB, and what the two-line model leaves out.

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 5 Model/Build Studio tables above. They compare two phone plans' bills at 2, 4, 6, 8, and 10 GB of data: Plan A bills 30, 40, 50, 60, and 70 dollars, and Plan B bills 38, 44, 50, 56, and 62 dollars. Please do these things. First, write each plan's bill as a line y = mx + b, where x is gigabytes used and y is the bill in dollars, and say what each slope and each y-intercept mean for a customer. Second, solve the system made by the two lines, check the pair in both equations, and say for which amounts of data Plan B is cheaper. Third, a Plan C bills 6 dollars more than Plan A for the same data; say whether Plan A and Plan C ever cost the same. 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 3x + 32 < 5x + 20 for x. 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 a plan's two numbers, calling Plan B's 32-dollar monthly fee its rate per gigabyte, or they read a rate from one row alone, such as 38 ⁄ 2 = 19 dollars per gigabyte, which mixes the 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 collect the x-terms on the left and then forget to reverse the sign when dividing by a negative, solving 3x + 32 < 5x + 20 as x < 6.
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 two plans, 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: "Plan B's rate is 32 dollars per gigabyte and its fee is 3 dollars, because 32 is the bigger number, so Plan B is y = 32x + 3. Using one row, 38 ⁄ 2 = 19 dollars per gigabyte is Plan B's rate. The plans cost the same at (50, 6). Plan A and Plan C give 0 = 6 when you subtract, so they have infinitely many solutions. 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 3x + 32 < 5x + 20 gives x < 6. 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, the crossing pair written in the wrong order, a false leftover read as infinitely many solutions, 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 Plan A, Plan B, and Plan C, 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 comparing two phone plans: two completed tables (the bill table with each plan's slope and fee, and four substitution checks), both plan lines and their crossing pair, a graph of Plans A, B, and C or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same plans, 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 both lines and solving the system	Each plan's slope is computed as rise over run with the bills on top, each fee is found by solving y = mx + b step by step, both lines are correct, and the system is solved by substitution or elimination with every step shown.	Both lines are written, but one slope or fee is missing or inverted, or the system is solved without its steps or with one arithmetic slip.	No lines are built, or the numbers are not drawn from the bill table.
Verifying by substitution	The crossing pair is substituted into both lines step by step, the two extra rows are checked, and the match is stated for each check.	The pair is checked in only one line, or the checks are given as results without the steps.	No substitution check is shown.
Reading the crossing point and classifying Plan C	Analysis questions 2, 4, and 5 are answered accurately: the crossing pair is read as an amount of data and a bill, the cheaper plan past it is named with the difference at 10 GB, Plan C is called parallel to Plan A with no solution, and a real limit of the model is explained.	The questions are attempted, but the crossing pair is read as one number, the Plan C judgment lacks a reason, or the limit is missing.	The analysis questions are missing or do not address the two plans.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows Plans A and B crossing at the solution pair and Plan C beside Plan A, 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 table, never from memory or a chatbot.
Each slope is computed as rise over run with the bills on top.
The system is solved with every step written out, and the crossing pair is checked in both lines.
Each line names the slope as dollars per gigabyte and the y-intercept as the monthly fee, not the other way around.
The Plan C judgment rests on the slopes, the y-intercepts, and the statement left after subtracting.
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.

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