Week 15 Model/Build Studio: When Will the Tracer Fall to Its Target?

Part 1 — The Big Picture

This week you learned to solve for an unknown that sits in an exponent: match the bases when you can, take a logarithm of both sides when you cannot, and check every answer. A decaying sample is exactly that kind of problem: the mass is a power of one half, and the time you want is in the exponent. In this Studio you take an invented weighing log, made up for practice, build the decay model from the readings, rewrite it with the exponent rules, verify it against every reading by substitution, see it in Desmos or GeoGebra, and use it to find when the sample reaches two target masses. Then you check a chatbot's account of the same log and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From five weighings, what is the decay model A = A₀ · (1⁄2)^(t⁄h), how can it be written as a power of 2, when does the sample reach 20 grams and 100 grams, and does a larger starting sample change how long it takes to fall to one thirty-second of its size?

Workshop data: Tracer weighing log (five readings, invented for practice)

Source: An invented, workshop-only weighing log for a lab that tracks a decaying tracer sample, made up for practice (not real measurements) and printed in full in the first scaffold table: the time in hours and the mass in grams at each weighing.

• Rows: 5 weighings; the first scaffold table lists all 5, and the second scaffold table has 3 rows, one for each question you solve
• Columns: time in hours (t) and mass in grams (A)
• Times: 0, 4, 8, 12, and 16 hours
• Masses: 640, 320, 160, 80, and 40 grams

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, 4, 8, 12, 16 in the x column and the five masses 640, 320, 160, 80, 40 in the y column. Tool: Desmos (free) or GeoGebra. Check: Five dots appear that fall from left to right, dropping by less each time.
2. In your notebook, build the model A = A₀ · (1⁄2)^(t⁄h). Read A₀ from the 0-hour reading, find h as the number of hours between one reading and the reading that is half as large, and write the finished model. Tool: A notebook or document for your written work. Check: A model with a number for A₀ and a number for h.
3. Apply the exponent rules step by step: write 1⁄2 as 2⁻¹, then use the power rule to write the model as A₀ times a single power of 2. Record each step on its own line. Tool: A notebook or document for your written work. Check: The same model written with a single power of 2 whose exponent is negative.
4. Verify by substitution: put each of the five times into your model, compute the mass step by step, and compare it with the logged mass. Record each check in the first scaffold table. Tool: A notebook or document for your written work. Check: Every substituted mass equals the logged mass.
5. Solve three questions and record them in the second scaffold table. Set the model equal to 20 and solve by matching bases. Set it equal to 100, isolate the power, and solve by taking ln of both sides, giving the exact answer and a decimal rounded to two places. Then find how long a 1,280-gram sample following the same halving rule takes to fall to 40 grams. Tool: A notebook or document for your written work. Check: Three times, each with its equation and a substitution check.
6. Back in the graphing calculator, type y = 640*(1/2)^(x/4) on a new line, then type y = 20 and y = 100 on two more lines. Click each place where the curve meets one of the flat lines. Tool: Desmos (free) or GeoGebra. Check: The curve passes through all five dots and meets each flat line exactly once.

Fallback route

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

Plot the five readings from the weighing log by hand on graph paper with time across the bottom and mass up the side, sketch a smooth curve through them, and draw the flat lines at 20 grams and 100 grams 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 weighings and not only the ones you used to build it.

Student scaffold table 1

Time t (hours)	Logged mass (grams)	Your model's mass, worked step by step	Match? (yes or no)
0	640
4	320
8	160
12	80
16	40

Answer each question about the model and record how you solved it, so the method and the check sit side by side.

Student scaffold table 2

Question	Equation	Method (match bases or take a logarithm)	Time (hours)	Substitution check
When does the 640-gram sample reach 20 grams?
When does the 640-gram sample reach 100 grams?
When does a 1,280-gram sample reach 40 grams?

Part 5 — Analysis Questions

1. Show how you read A₀ and h from the log and write the finished model. Then show the exponent-rule steps that turn (1⁄2)^(t⁄4) into a single power of 2, every step on its own line.
2. Compare your times for the 640-gram sample to reach 20 grams and the 1,280-gram sample to reach 40 grams. What do you notice, and how does the algebra explain it when you divide both sides by the starting mass?
3. Explain why the 20-gram question can be solved by matching bases but the 100-gram question needs a logarithm of both sides. Name the two whole-number times that trap the 100-gram answer and the logged masses that prove it.
4. Describe where the curve in step 6 meets the lines y = 20 and y = 100, and explain how each crossing agrees with the times in your second scaffold table.
5. Choose one and answer it in 3–4 sentences. (a) The log has only five readings, each taken every 4 hours. Explain how a reading taken at 6 hours could confirm or challenge your model, and which of your answers you would trust least. (b) A real scale rounds each mass. Explain how a reading of 161 grams instead of 160 grams at 8 hours would affect your check, and why one reading slightly off does not by itself mean the halving model 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 15 Model/Build Studio tables above. They come from an invented practice log, made up for this Studio, of a tracer sample weighed every 4 hours, and the readings are 640 grams at 0 hours, 320 grams at 4 hours, 160 grams at 8 hours, 80 grams at 12 hours, and 40 grams at 16 hours. Please do these things. First, build the model A = A₀ · (1⁄2)^(t⁄h), where h is the number of hours it takes the mass to halve, and check it at all five readings. Second, rewrite (1⁄2)^(t⁄4) as a power of 2 using the exponent rules. Third, solve for the time when the sample reaches 20 grams, and for the time when it reaches 100 grams, giving the exact answer and a decimal rounded to two places. Fourth, find how long a 1,280-gram sample of the same tracer takes to fall to 40 grams. Then answer eight short review questions: evaluate −2⁴ and (−2)⁴; simplify 2³ · 2² and (2³)²; evaluate 2⁻³; say whether √2 and 0.125 are rational or irrational; simplify (x² − 9) ⁄ (x + 3); solve log₂(x) + log₂(x − 4) = 5 and check every candidate; say whether log₂(8 + 8) equals log₂(8) + log₂(8); solve −3x + 2 ≤ 14 written in words as negative 3x plus 2 is at most 14; and give the slope and the y-intercept of y = 4x − 3.

Test the response for these likely failures:

• Chatbots often multiply 640 by 1⁄2 first and write the model as 320^(t⁄4), letting the exponent act on the coefficient.
• Chatbots sometimes rewrite (1⁄2)^(t⁄4) as −2^(t⁄4), or evaluate 2⁻³ as −8.
• Chatbots sometimes add exponents in (2⁻¹)^(t⁄4) or in (2³)², or multiply them in 2³ · 2².
• Chatbots often solve log₂(x) + log₂(x − 4) = 5 to x = 8 or x = −4 and keep both, skipping the domain check.
• Chatbots sometimes split log₂(8 + 8) into log₂(8) + log₂(8) = 6.
• Chatbots sometimes evaluate −2⁴ as 16, treating the minus sign as part of the base.
• Chatbots sometimes call 0.125 irrational because it has three decimal places, or call √2 rational because a calculator shows 1.4142.
• Chatbots sometimes cancel terms instead of factors, crossing out the 9 and the 3 or the x in (x² − 9) ⁄ (x + 3).
• Chatbots sometimes forget to reverse the sign when dividing by a negative, solving negative 3x plus 2 is at most 14 as x is at most −4.
• Chatbots sometimes swap the two numbers in y = 4x − 3, calling −3 the slope.

Write 3–5 sentences: name what the chatbot got right about the tracer log, 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 sample starts at 640 grams and halves every 4 hours, so A = 320^(t⁄4). Rewriting, (1⁄2)^(t⁄4) = −2^(t⁄4). For 20 grams, 20 ⁄ 640 = 1⁄32, so t ⁄ 4 = 32 and t = 128 hours. For 100 grams, I took ln of the left side and log of the right side and got t = 4 · log(6.4) ⁄ ln 2, about 4.65 hours. A 1,280-gram sample is twice as big, so it takes twice as long to fall to 40 grams. Review: −2⁴ = 16; 2³ · 2² = 2⁶; 2⁻³ = −8; 0.125 is irrational because it has three decimal places; (x² − 9) ⁄ (x + 3) = x − 3 because the 9 and the 3 cancel; log₂(x) + log₂(x − 4) = 5 gives x = 8 or x = −4, and both work; log₂(8 + 8) = log₂(8) + log₂(8) = 6; negative 3x plus 2 is at most 14 gives x is at most −4; and in y = 4x − 3 the slope is −3." Mark it up: quote each flawed sentence, name what is wrong (the coefficient merged into the base, a negative exponent read as a negative number, a power rule and product rule mix-up, the exponent read as a multiplier, ln and log mixed in one solution, the starting amount assumed to change the time, a negated power, a rational number called irrational, terms cancelled instead of factors even where the final form happens to match, an extraneous candidate kept, the log of a sum split, an inequality sign not reversed, or the slope and the y-intercept swapped), cite the evidence from your tables or the course rule, and write a corrected sentence beneath it. This route assesses the same objectives (Objective 8 and Objective 2) 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 lab's tracer weighing log: two completed tables (five substitution checks, and three questions with their equations, methods, times, and checks), the decay model A = A₀ · (1⁄2)^(t⁄h) with its numbers and its power-of-2 form, a graph of the curve with the flat lines at 20 and 100 grams or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same log, 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 weighing log	A₀ and the halving time are read from the log with the steps shown, the finished model has the correct numbers, the power-of-2 form is reached with named exponent rules, and all five substitution checks match the log.	The model is written, but a number is given without its steps or with one arithmetic slip, the power-of-2 form is missing a step, or only some readings are checked.	No model is built, or its numbers are not drawn from the weighing log.
Solving and verifying the exponential equations	The 20-gram equation is solved by matching bases, the 100-gram equation is solved by isolating the power and taking a logarithm of both sides with an exact answer and a decimal check, the 1,280-gram question is solved, and every time is checked by substitution.	One equation is missing, or a time is given without steps or without a substitution check.	No equation is solved, or times are stated without any method.
Reading the answers in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: the equal times are explained by dividing out the starting mass, the choice between matching bases and a logarithm is justified with the trapping readings, the graph is matched to the algebra, and a real limit of the model is explained.	The questions are attempted, but the starting mass is said to change the time, the method choice is not justified, the graph is not tied to the algebra, or the limit is missing.	The analysis questions are missing or do not address the weighing 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 20 and 100 grams, 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 and mass in the tables comes from your own calculations on the printed weighing log, never from memory or a chatbot.
• The model's starting mass and halving time are read from the log, one step per line.
• The power-of-2 form is reached with named exponent rules, one step per line.
• Every one of the five substitution checks is written out step by step.
• Each solved time is substituted back into the model, and the 100-gram time is shown to sit between the two logged readings that trap it.
• 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.