Week 12 Model/Build Studio: How Long Does the Sandbag Fall?

Part 1 — The Big Picture

This week you learned that a root undoes a power, that a fraction exponent is a root, and that squaring both sides of a radical equation can create a candidate that never worked. A falling object is exactly that kind of formula: its fall time is a square root of its drop height. In this Studio you take an invented drop-test log, made up for practice, build the fall-time model from one reading, rewrite it with a rational exponent using the exponent rules, verify it against every reading by substitution, see it on a graph in Desmos or GeoGebra, and use it to answer two questions, one of which has an extraneous candidate. Then you check a chatbot's account of the same test and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From four timed drops, what is the fall-time model t = √(h ⁄ k), how can it be written with a rational exponent, how high must a drop start for a 4-second fall, and is there any drop height for a fall of −2 seconds?

Workshop data: Sandbag drop-test log (four readings, invented for practice)

Source: An invented, workshop-only drop-test log for a stunt-rigging crew, made up for practice (not real measurements) and printed in full in the first scaffold table: the drop height in feet and the timed fall in seconds for a sandbag dropped from rest.

• Rows: 4 timed drops; the first scaffold table lists all 4, and the second scaffold table has 3 rows, one for each question you solve
• Columns: drop height in feet (h) and fall time in seconds (t)
• Heights: 36, 100, 196, and 324 feet
• Times: 1.5, 2.5, 3.5, and 4.5 seconds

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 four heights 36, 100, 196, 324 in the x column and the four times 1.5, 2.5, 3.5, 4.5 in the y column. Tool: Desmos (free) or GeoGebra. Check: Four dots appear that rise from left to right, climbing more slowly as the height grows.
2. In your notebook, build the model t = √(h ⁄ k). Substitute the 36-foot reading, so 1.5 = √(36 ⁄ k). Square both sides, then solve for k one step per line, and write the finished model. Tool: A notebook or document for your written work. Check: A model of the form t = √(h ⁄ k) with a number for k.
3. Apply the exponent rules step by step: write your model with a rational exponent, use the quotient rule to split the root of the fraction, and write it in the form t = c · h^(1⁄2). Record each step on its own line. Tool: A notebook or document for your written work. Check: The same model written as a number times h to the one-half power.
4. Verify by substitution: put each of the four heights into your model, compute the time step by step, and compare it with the logged time. Record each check in the first scaffold table. Tool: A notebook or document for your written work. Check: Every substituted time equals the logged time.
5. Set your model equal to 4 and solve for the drop height: isolate the root, square both sides, solve, and check. Then set the model equal to −2, square both sides, and check the height you get in the original equation. Record both results in the second scaffold table. Tool: A notebook or document for your written work. Check: The 4-second equation gives a height that checks, and the −2-second candidate fails its check.
6. Back in the graphing calculator, type y = sqrt(x/16) on a new line, then type y = 4 and y = -2 on two more lines. Click each place where the curve meets one of the two flat lines. Tool: Desmos (free) or GeoGebra. Check: The curve passes through all four dots, meets the line y = 4 once, and never meets the line y = −2.

Fallback route

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

Plot the four readings from the drop-test log by hand on graph paper with height across the bottom and time up the side, sketch a smooth curve through them and through the point at height 0 and time 0, and draw the flat lines at 4 seconds and −2 seconds 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 four drops and not only the one you used to build it.

Student scaffold table 1

Drop height h (feet)	Logged time (seconds)	Your model's time, worked step by step	Match? (yes or no)
36	1.5
100	2.5
196	3.5
324	4.5

Answer each question about the model and decide which candidates survive the check, so the algebra and the real drop are compared side by side.

Student scaffold table 2

Question	Radical equation	Squared equation	Candidate	Does it check, and why
What is k?
What height gives a 4-second fall?
What height gives a −2-second fall?

Part 5 — Analysis Questions

1. Show the squaring step that finds k from the 36-foot reading, every step on its own line, and write the finished model. Then show the exponent-rule steps that turn it into t = c · h^(1⁄2), and state c.
2. Compare the 36-foot drop with the 324-foot drop. How many times as high is the second drop, and how many times as long is its fall? Explain how the square root in the model produces that pattern.
3. Solve for the drop height that gives a 4-second fall and check it. Then explain in plain words why squaring √(h ⁄ 16) = −2 produces a height that is not a real answer, even though the squaring step was done correctly.
4. Describe what the graph in step 6 shows at the lines y = 4 and y = −2, and explain how each picture agrees with your algebra in the second scaffold table.
5. Choose one and answer it in 3–4 sentences. (a) The model ignores air resistance. Explain how real air could change the fall times for a light object and which of your answers you would trust least. (b) Suppose a real drop test like this one were timed by hand. Explain how a reaction delay of a fraction of a second could affect the k you found, 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 12 Model/Build Studio tables above. They come from an invented practice log for a stunt-rigging crew's drop test, made up for this Studio: a sandbag dropped from rest four times, with a logged 1.5-second fall from 36 feet, a 2.5-second fall from 100 feet, a 3.5-second fall from 196 feet, and a 4.5-second fall from 324 feet. Please do these things. First, the fall time follows t = √(h ⁄ k); use the 36-foot reading to find k, showing each step, and write the model. Second, rewrite the model with a rational exponent in the form t = c · h^(1⁄2) and say what c is. Third, find the drop height for a 4-second fall. Fourth, square both sides of √(h ⁄ 16) = −2 and say whether the height you get is a real answer. Then answer six short review questions: evaluate −2⁴ and (−2)⁴; simplify h^(1⁄2) · h^(1⁄2) and (h^(1⁄2))^(1⁄2); evaluate 16^(−1⁄2); say whether √2 and √(36 ⁄ 16) are rational or irrational; solve −2x + 1 ≤ 9 written in words as negative 2x plus 1 is at most 9; and give the slope and the y-intercept of y = −2x + 5. Last, answer two more: 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 square √(h ⁄ 16) = −2 into h ⁄ 16 = 4 and report h = 64 feet as a real drop height without substituting it back.
• Chatbots sometimes write t = √(h ⁄ 16) as t = 16h^(1⁄2) or t = h^(1⁄2) ⁄ 16, moving the 16 out from under the root without taking its root.
• Chatbots sometimes multiply the exponents in h^(1⁄2) · h^(1⁄2) to get h^(1⁄4), or add them in (h^(1⁄2))^(1⁄2) to get h¹.
• Chatbots sometimes say 16^(−1⁄2) is −4 or −1⁄4.
• Chatbots sometimes call √(36 ⁄ 16) irrational because it has a root sign, or call √2 rational because a calculator shows 1.4142.
• Chatbots sometimes evaluate −2⁴ as 16, treating the minus sign as part of the base.
• Chatbots sometimes forget to reverse the sign when dividing by a negative, solving negative 2x plus 1 is at most 9 as x is at most −4.
• Chatbots sometimes swap the two numbers in y = −2x + 5, calling 5 the slope.
• Chatbots sometimes cancel terms instead of factors, crossing out the 9 and the 3 or the x in (x² − 9) ⁄ (x + 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 drop test, 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: "From the 36-foot reading, 1.5 = √(36 ⁄ k), so 1.5 = 6 ⁄ k and k = 4. Rewriting, t = √(h ⁄ 16) = 16h^(1⁄2). A 4-second fall needs 4 = √(h ⁄ 16), so h = 64 feet. Squaring √(h ⁄ 16) = −2 gives h = 64, so a 64-foot drop takes −2 seconds as well. Review: −2⁴ = 16; h^(1⁄2) · h^(1⁄2) = h^(1⁄4); 16^(−1⁄2) = −4; √(36 ⁄ 16) is irrational because it has a root sign; negative 2x plus 1 is at most 9 gives x is at most −4; in y = −2x + 5 the slope is 5; and (x² − 9) ⁄ (x + 3) = x − 3 because the 9 and the 3 cancel. Preview: log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (the root taken of only the top before squaring, a 16 moved out of the root without its root, a radical equation squared on one side only, an extraneous candidate kept, a negated power, a product rule and power rule mix-up, a negative exponent read as a negative number, a rational number called irrational, an inequality sign not reversed, the slope and the y-intercept swapped, terms cancelled instead of factors even where the final form happens to match, 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 1) 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 crew's sandbag drop test: two completed tables (four substitution checks, and three questions with their radical equations, squared equations, and candidates), the fall-time model t = √(h ⁄ k) with its number and its rational-exponent form, a graph of the curve with the flat lines at 4 and −2 seconds or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same test, 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 and rewriting the model from the drop log	k is found by squaring both sides of the 36-foot reading step by step, the finished model has the correct number, the rational-exponent form t = c · h^(1⁄2) is reached with named exponent rules, and all four substitution checks match the log.	The model is written, but k is found without its steps or with one arithmetic slip, the rational-exponent form is missing a step, or only some readings are checked.	No model is built, or its number is not drawn from the drop log.
Solving and verifying the radical equations	The 4-second equation is solved by isolating the root, squaring, solving, and checking, and the −2-second equation is squared and its candidate is substituted back and rejected with the reason.	One equation is missing, or a candidate is given without steps or without a substitution check.	No radical equation is solved, or candidates are stated without any method.
Reading the answers in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: the nine-times-higher, three-times-longer pattern is explained by the square root, the extraneous candidate is rejected with a reason, the graph is matched to the algebra, and a real limit of the model is explained.	The questions are attempted, but the pattern is misread, an extraneous candidate is kept, the graph is not tied to the algebra, or the limit is missing.	The analysis questions are missing or do not address the drop log.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the four readings on the curve with the flat lines at 4 and −2 seconds, 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 time, height, and candidate in the tables comes from your own calculations on the printed drop-test log, never from memory or a chatbot.
• The number k is found by squaring both sides of the 36-foot reading, one step per line.
• The rational-exponent form is reached with named exponent rules, one step per line.
• Every one of the four substitution checks is written out step by step.
• Every candidate from a squared equation is substituted back into the original equation, and any extraneous candidate is rejected with its reason.
• 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.