Week 2 Model/Build Studio: Ranges, Tolerances, and the Flip

Part 1 — The Big Picture

This week you learned to find every number that makes a statement true: one principle for equations, the flip rule for inequalities, and absolute value as a distance. In this Studio you build those statements yourself from four everyday scenarios, solve each one step by step, and then prove each answer twice, once by substitution and once by graphing both sides in Desmos. Then you ask a chatbot the same questions and catch the places where it slips, including the classic missing flip.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: For a draining tank, a rideshare budget, a drone on a straight road, and a terrarium thermostat, what inequality or equation describes each rule, which values make it true, and how can you prove each answer two different ways?

Workshop data: Week 2 Studio scenarios: four ranges and tolerances

Source: Four short scenarios written for this Studio, each with whole-number values chosen so every answer can be checked by substitution: a draining water tank, a rideshare budget, a delivery drone on a straight road, and a terrarium thermostat.

• Rows: 4 scenarios, one per row of the first scaffold table
• Water tank: starts with 50 liters, loses 4 liters per hour, and must keep at least 18 liters
• Rideshare: 3 dollars to book plus 2 dollars per mile, with a budget of at most 21 dollars
• Delivery drone: depot at kilometer marker 6 on a straight road; the drone reports being exactly 5 kilometers from the depot
• Terrarium thermostat: must keep the temperature within 3 degrees Celsius of 30

Part 3 — Materials & Procedure

Tools and materials

• Desmos (free) or GeoGebra
• Paper and pencil for number lines
• 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. For each scenario in the first scaffold table, write the inequality or equation that describes the rule, using the letter shown (h for hours, m for miles, x for a kilometer marker, T for a temperature in degrees Celsius). Tool: A document or notebook for your written answers. Check: Four statements: two linear inequalities, one absolute-value equation, and one absolute-value inequality.
2. Solve each statement step by step, writing every step on its own line and naming the move (subtract, divide and flip, split, add to all three parts). Write each answer in interval notation, or as a list of values for the equation. Tool: Paper and pencil for number lines. Check: The tank and rideshare answers are rays ending at a whole number, the drone has two whole-number answers, and the thermostat answer is a closed middle interval.
3. Verify by substitution: put one value from inside each answer and each endpoint back into the original statement, and record whether each side makes the statement true or false in the second column of the second scaffold table. Tool: A document or notebook for your written answers. Check: Every inside value makes the statement true, and each endpoint behaves the way its bracket or parenthesis says.
4. Open Desmos at desmos.com/calculator, or GeoGebra. Type y = 50 - 4x on one line and y = 18 on the next, then click the point where they cross. Next type y = 3 + 2x and y = 21 and click their crossing. Tool: Desmos (free) or GeoGebra. Check: Each pair of lines crosses at exactly one point, and the x-value of each crossing matches the endpoint of your answer.
5. Type y = abs(x - 6) and y = 5, and click each point where the V-shaped graph meets the flat line. Then change the second line to y = -5 and look for any crossing. Tool: Desmos (free) or GeoGebra. Check: Two crossings for y = 5 whose x-values match your two drone answers, and no crossing at all for y = −5.
6. Type y = abs(x - 30) and y = 3. Find the stretch of x-values where the V sits on or below the flat line, and record both ends in the second scaffold table. Tool: Desmos (free) or GeoGebra. Check: The V sits on or below the line along one stretch, and its two ends match the endpoints of your thermostat interval.
7. Answer the analysis questions in your document, then paste your tables into an approved chatbot with the prompt in Part 6 and check its answers against the scripted checks. Tool: A document or notebook for your written answers. Check: You have written answers and a list of the chatbot's slips with corrections.

Fallback route

When to use it: Use this route if Desmos and GeoGebra are both unavailable, or you have no device for graphing.

Do every algebra and substitution step as written. In place of each Desmos reading, draw a number line by hand for each answer, shade the solution, and mark each endpoint with a filled dot if it is included or an open dot if it is not; label the dots in words. Submit the same items for the same credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Turn each everyday rule into algebra and solve it, so every answer can be checked against the scenario it came from.

Student scaffold table 1

Scenario	Statement you wrote	Solution (interval or values)	What the solution means in the scenario
Water tank: 50 liters, loses 4 per hour, at least 18 must remain
Rideshare: 3 dollars to book plus 2 per mile, budget at most 21 dollars
Drone: exactly 5 kilometers from the depot at marker 6
Thermostat: within 3 degrees Celsius of 30

Prove each answer two ways, so a slip in one method shows up in the other.

Student scaffold table 2

Scenario	Substitution check (inside value and endpoint)	Desmos reading (crossing points or stretch)	Do the two methods agree?
Water tank
Rideshare
Drone
Thermostat

Part 5 — Analysis Questions

1. Solve the water-tank inequality 50 − 4h ≥ 18 step by step. Name the exact step where the inequality sign flips and say why it flips there and nowhere else.
2. Describe in words what Desmos showed for y = abs(x − 6) against y = 5 and against y = −5: where the graphs met, how many times, and what each crossing tells you about the drone.
3. Explain why the thermostat answer is one closed middle interval, an AND, and why it uses brackets rather than parentheses. What would change in the statement and in the answer if the rule said more than 3 degrees away instead?
4. The rideshare and water-tank statements are both linear inequalities, but only one needs the flip. Compare the two solves and explain what in each statement decides whether a flip happens.
5. Choose one and answer it in 3–4 sentences. (a) The tank model assumes it loses exactly 4 liters every hour: how could real draining differ, and how much should you trust the endpoint you found? (b) The drone model treats the road as a straight line: what would change if the road curved, and does the algebra still answer the real question?

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 2 Studio tables above. Please answer each question and show every step. (1) A water tank holds 50 liters and loses 4 liters per hour; at least 18 liters must remain. Solve 50 − 4h ≥ 18 and write the answer in interval notation. (2) Solve |x − 6| = 5. (3) Solve |x − 6| = −5. (4) Write the solution of |T − 30| ≤ 3 in interval notation. (5) In y = 50 − 4x, which number is the slope and which is the y-intercept, and what does each mean for the tank? As quick review from last week: (6) evaluate −2⁴ and (−2)⁴; (7) simplify x² · x³ and (x²)³; (8) rewrite 2⁻³ without a negative exponent; (9) is √16 rational or irrational; (10) evaluate 3x² at x = 2. As stretch questions for later in the course: (11) does (x + 3) ⁄ x simplify to 3; (12) is log(x + 4) equal to log(x) + log(4)?

Test the response for these likely failures:

• Chatbots often divide 50 − 4h ≥ 18 through by −4 without flipping, reporting h ≥ 8 instead of h ≤ 8, or write the interval with a bracket on infinity.
• Chatbots often split |x − 6| = −5 anyway and report x = 1 or x = 11, without checking either candidate in the original equation.
• Chatbots sometimes swap the roles in y = 50 − 4x, calling 50 the rate and −4 the starting amount.
• Chatbots sometimes square the minus sign in −2⁴ and report 16 for both expressions.
• Chatbots sometimes multiply the exponents in x² · x³ and report x⁶ for both expressions.
• Chatbots sometimes read 2⁻³ as a negative number such as −8.
• Chatbots sometimes call √16 irrational because it is written with a root sign.
• Chatbots sometimes read 3x² as (3x)² and report 36 at x = 2.
• Chatbots sometimes cancel the x in (x + 3) ⁄ x and report 3.
• 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, quoting its words and correcting each with a substitution check or a step from your own tables.

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 tank: 50 − 4h ≥ 18 gives −4h ≥ −32, so h ≥ 8, written [8, ∞]. The drone: |x − 6| = 5 gives x = 11 only. And |x − 6| = −5 gives x = 1 or x = 11. The thermostat: |T − 30| ≤ 3 means T ≤ 27 or T ≥ 33. In y = 50 − 4x, the slope is 50 and the y-intercept is −4. Review: −2⁴ = 16, x² · x³ = x⁶, 2⁻³ = −8, √16 is irrational because it has a root sign, and 3x² at x = 2 is 36. Stretch: (x + 3) ⁄ x = 3, and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (a missing flip, a bracket on infinity, a missing case, candidates kept without a check, the middle and the rays swapped, slope and intercept swapped, or a Week 1 exponent or number-set slip), back each correction with a substitution check or a step from your own tables, and write a corrected sentence beneath it; for the two stretch sentences, say what the course will check later. This route assesses the same objective (Objective 2 — solve linear equations and linear inequalities in one variable, including absolute-value equations and inequalities, and express solution sets in interval notation) 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
• Your Desmos screenshots or hand-drawn number lines with labelled endpoints
• 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 four everyday rules: two completed tables that turn each rule into an inequality or equation, solve it step by step, and prove each answer by substitution and by a Desmos graph or a hand-drawn number line; written answers to five analysis questions; and a critique of a chatbot's answers to the same questions, 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 models	All four scenarios are written as correct statements: two linear inequalities with the right direction, one absolute-value equation, and one absolute-value inequality, each using the scenario's own numbers.	Three of the four statements are correct, or a statement has the right structure but a wrong number or direction.	No statements are written, or they do not come from the scenarios.
Solving step by step	Every statement is solved with each step on its own line, the flip happens exactly at the division by a negative, the absolute-value equation is split into two cases, and every answer is written in correct interval notation or as a list.	The solves are present, but one step is skipped, one interval has a bracket or parenthesis error, or one case of the split is missing.	No solving steps are shown, or the answers appear without any work.
Verifying by substitution and by graph	Every answer is checked by substitution at an inside value and at each endpoint, and each check is matched to a Desmos reading or a labelled hand-drawn number line; hand-drawn number lines earn this band equally.	Substitution checks or graph readings are present for only some scenarios, or a check is done but not compared with the graph.	No substitution checks and no graph or number line are submitted.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors are quoted and corrected with a substitution check or a step from the student's own work, and one correct point is acknowledged; both routes earn this band equally.	On either route, errors are named but corrected without a check, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.
Reflection on what the model leaves out	The chosen reflection option is answered in 3–4 sentences that connect a specific assumption of the model to how much the algebraic answer can be trusted in the real scenario.	The reflection is answered but stays general or does not name a specific assumption.	No reflection is submitted.

Quality Gate

• Every statement is built from the scenario's own numbers, and every answer comes from your own algebra, never from a chatbot or memory.
• Every answer is checked by substitution at an inside value and at each endpoint.
• Every interval uses a bracket only on an included endpoint and always a parenthesis beside infinity.
• The Desmos reading or the hand-drawn number line agrees with the algebra for all four scenarios, or the disagreement is explained.
• No value, step, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with a substitution check or a step from your own work.