Week 7 Model/Build Studio: When Does the Water Rocket Land?

Part 1 — The Big Picture

This week you learned four ways to solve an equation where the unknown is squared, and you learned that the discriminant tells you how many real answers to expect before you solve. A rocket's height is exactly that kind of equation: the height is a quadratic in time. In this Studio you take a real-looking height log for a water rocket, build its height model, solve three questions about its flight, 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 flight and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From five height readings, what is the water rocket's height model h(t) = −16t² + vt + h₀, when does the rocket land, when is it 96 feet high, and does it ever reach 120 feet?

Workshop data: Water-rocket height log (five readings)

Source: A workshop-only height log for one water rocket launched from a rooftop deck, printed in full in the first scaffold table: the time in seconds after launch and the height in feet above the ground.

• Rows: 5 height readings; the first scaffold table lists all 5, and the second scaffold table has 3 rows, one for each question you solve
• Columns: time after launch in seconds (t) and height above the ground in feet
• Times: 0, 1, 2, 3, and 4 seconds
• Heights: 64, 96, 96, 64, and 0 feet

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 times 0, 1, 2, 3, 4 in the x column and the five heights 64, 96, 96, 64, 0 in the y column. Tool: Desmos (free) or GeoGebra. Check: Five dots appear that rise, level off, and fall back to the horizontal axis at 4 seconds.
2. In your notebook, build the model h(t) = −16t² + vt + h₀. Substitute t = 0 and the height at 0 seconds to find h₀. Then substitute t = 1 and the height at 1 second, and solve for v one step per line. Write the finished model. Tool: A notebook or document for your written work. Check: A model of the form −16t² + vt + h₀ with numbers for v and h₀.
3. Verify by substitution: put each of the five times into your model, compute the height step by step, and compare it with the logged height. Record each check in the first scaffold table. Tool: A notebook or document for your written work. Check: Every substituted height equals the logged height.
4. Set your model equal to 0 and solve for the landing time two ways: divide every term by −16 and factor, then use the quadratic formula on the original equation. Write every step on its own line, and record the two roots and the time you keep in the second scaffold table. Tool: A notebook or document for your written work. Check: Both methods give the same two roots, and one of them is a time before the launch.
5. Set your model equal to 96 and solve. Then set it equal to 120, move every term to one side, and find the discriminant before you try to solve. Record both results in the second scaffold table. Tool: A notebook or document for your written work. Check: The 96-foot equation has two real roots, and the 120-foot equation has a negative discriminant.
6. Back in the graphing calculator, type y = -16x^2 + 48x + 64 on a new line, then type y = 96 and y = 120 on two more lines. Click each place where the curve meets the horizontal axis or one of the two flat lines. Tool: Desmos (free) or GeoGebra. Check: The curve meets the horizontal axis at two places, one of them left of the launch, meets the line y = 96 at two places, and never meets the line y = 120.

Fallback route

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

Plot the five readings from the height log by hand on graph paper with time across the bottom and height up the side, sketch a smooth curve through them, and draw the flat lines at 96 feet and 120 feet with a ruler. 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

Check your model against every reading in the log, so you know it fits all five heights and not only the two you used to build it.

Student scaffold table 1

Time t (seconds)	Logged height (feet)	Your model's height, worked step by step	Match? (yes or no)
0	64
1	96
2	96
3	64
4	0

Solve each question about the flight and decide which answers the setting keeps, so the algebra and the real rocket are compared side by side.

Student scaffold table 2

Question	Equation in standard form	Discriminant	Roots	Times the setting keeps, and why
When does it land?
When is it 96 feet high?
Does it reach 120 feet?

Part 5 — Analysis Questions

1. Show the substitutions that find h₀ and v, and write the finished model. Then show the factoring steps and the quadratic-formula steps for the landing equation, every step on its own line.
2. Your landing equation has two roots. In plain words, explain which one is the landing time and why the setting rules the other one out, even though it makes the equation true.
3. Describe what the second scaffold table shows for 96 feet. Why does that question keep both roots when the landing question keeps only one?
4. Use the discriminant of the 120-foot equation to explain whether the rocket ever reaches 120 feet, and say what the graph in step 6 shows that agrees with your answer.
5. Choose one and answer it in 3–4 sentences. (a) The model ignores air resistance. Explain how real air could change the landing time and which of your answers you would trust least. (b) The sensor logged only five whole-second readings. Explain what could happen between the readings that the log cannot show, 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 7 Model/Build Studio tables above. They come from a science club's water-rocket launch log: the rocket leaves a 64-foot rooftop deck, and a motion sensor recorded its height as 64 feet at 0 seconds, 96 feet at 1 second, 96 feet at 2 seconds, 64 feet at 3 seconds, and 0 feet at 4 seconds. Please do these things. First, the height follows h(t) = −16t² + vt + h₀; use the log to find v and h₀ and write the model. Second, solve −16t² + 48t + 64 = 0 by factoring and by the quadratic formula, and say when the rocket lands. Third, find every time the rocket is 96 feet high. Fourth, use the discriminant to decide whether the rocket ever reaches 120 feet. 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; and give the slope and the y-intercept of y = −2x + 5. Last, answer two preview questions from later in the course: simplify (x² − 9) ⁄ (x + 3); and say whether log(x + 4) can be rewritten as log(x) + log(4).

Test the response for these likely failures:

• Chatbots often report both roots of −16t² + 48t + 64 = 0, t = 4 and t = −1, as landing times, or drop one of the two real times the rocket is 96 feet high.
• Chatbots sometimes square the coefficient along with t, reading −16t² as (−16t)², or they read b as 48 but c as −64 by moving the sign.
• 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 cancel terms instead of factors, for example crossing out the x in (x + 3) ⁄ x to get 3.
• 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 rocket's flight, 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 h(t) = −16t² + 64t + 48, because the bigger number goes with t. Solving −16t² + 48t + 64 = 0 by dividing both sides by t gives −16t + 48 + 64 ⁄ t = 0, and the formula gives t = 4 and t = −1, so the rocket lands twice, at 4 seconds and at −1 second. The rocket is 96 feet high only at t = 1. For 120 feet the discriminant is −20, so the rocket reaches 120 feet at about t = 1.5. In −16t², the t is multiplied by −16 first and then squared, so at t = 1 the term is 256. Review: −2⁴ = 16; 2³ · 2⁴ = 2¹²; 2⁻² = −4; √2 is rational because it is 1.4142; −2x + 1 < 9 gives x < −4; and in y = −2x + 5 the slope is 5. Preview: (x + 3) ⁄ x simplifies to 3; and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (the starting height and the speed swapped, dividing by the variable, a rejected root kept as a landing time, a real time dropped, a negative discriminant treated as a real crossing, a coefficient squared along with t, 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 term cancelled instead of a factor, 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 two preview sentences, say what the course will check later. This route assesses the same objective (Objective 6 — solve quadratic equations by factoring, the square root property, completing the square, and the quadratic formula; use the discriminant to predict the number of real solutions; and analyze quadratic functions and their graphs, including the vertex, intercepts, and maximum or minimum value) 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 two flat 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 water rocket's height log: two completed tables (five substitution checks, and three flight questions with their roots and the answers the setting keeps), the height model h(t) = −16t² + vt + h₀ with its numbers, a graph of the curve with the flat lines at 96 and 120 feet or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same flight, 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 height log	h₀ and v are found by substituting the readings at 0 and 1 seconds step by step, the finished model has the correct numbers, and all five substitution checks match the log.	The model is written, but one number is found without its steps or with one arithmetic slip, or only some readings are checked.	No model is built, or the numbers are not drawn from the height log.
Solving and verifying the flight questions	The landing equation is solved by factoring and by the quadratic formula with every step shown and the two methods agreeing, the 96-foot equation is solved, and each kept root is checked by substitution.	One method or one question is missing, or roots are given without steps or without a substitution check.	No equation is solved, or roots are stated without any method.
Reading the answers in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: the root before the launch is rejected with a reason, both 96-foot times are kept, the negative discriminant is read as never reaching 120 feet, and a real limit of the model is explained.	The questions are attempted, but a rejected root is kept, a real time is dropped, the discriminant is misread, or the limit is missing.	The analysis questions are missing or do not address the height log.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the five readings on the curve with the flat lines at 96 and 120 feet, 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 height, root, and discriminant in the tables comes from your own calculations on the printed height log, never from memory or a chatbot.
• The model's two numbers are found by substitution, one step per line, from the log.
• Every one of the five substitution checks is written out step by step.
• The landing equation is solved by factoring and by the quadratic formula, and the two methods agree.
• Every kept or rejected root comes with a reason tied to the rocket's flight.
• 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.