Week 13 Model/Build Studio: Which Culture Is Doubling?

Part 1 — The Big Picture

This week you learned that a line adds the same amount at every step while an exponential multiplies by the same factor, that an exponential function f(x) = a · bˣ is read from its initial value a and its growth factor b, and that a negative input gives a small positive output. A lab culture is exactly that kind of question: does it gain a fixed number of cells each hour, or does it double? In this Studio you take an invented hourly log of two lab cultures, made up for practice, decide which one is linear and which is exponential, build a model for each from the logged counts, use the exponent rules to show what one hour of growth does, verify every model value by substitution, see both on a graph in Desmos or GeoGebra, and answer questions about the past and the future of each culture. 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 four hourly counts of two cultures, which culture grows by a constant ratio and which by a constant difference, what model fits each, what does each predict at hour 5, and how many cells did Culture A have one hour before counting began?

Workshop data: Two-culture cell-count log (four hourly counts each, invented for practice)

Source: An invented, workshop-only cell-count log for a biology lab, made up for practice (not real counts) and printed in full in the first two scaffold tables: the number of cells in two cultures, A and B, counted once an hour for four hours.

• Rows: 4 hourly counts for each of 2 cultures; the first two scaffold tables list all 4 counts for one culture each, and the third scaffold table has 4 rows, one for each question you answer
• Columns: hour t and cell count, for Culture A and for Culture B
• Culture A counts: 150, 300, 600, and 1,200 cells at hours 0, 1, 2, and 3
• Culture B counts: 150, 450, 750, and 1,050 cells at hours 0, 1, 2, and 3

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 hours 0, 1, 2, 3 in the x column and Culture A's counts 150, 300, 600, 1200 in the y column. Add a second table with the same hours and Culture B's counts 150, 450, 750, 1050. Tool: Desmos (free) or GeoGebra. Check: Two sets of four dots appear, both starting at 150; one set bends upward more and more steeply, and the other rises in a straight line.
2. In your notebook, test each culture's log. For each culture, divide every count by the count one hour earlier, and subtract every count from the count one hour later. Record the ratios and the differences in the first two scaffold tables. Tool: A notebook or document for your written work. Check: One culture shows the same ratio every hour, and the other shows the same difference every hour.
3. Build a model for each culture. For the culture with a constant ratio, write f(t) = a · bᵗ with a as the count at hour 0 and b as the ratio. For the culture with a constant difference, write g(t) = 150 + dt with d as the difference. Write each model on its own line. Tool: A notebook or document for your written work. Check: An exponential model with a number for a and for b, and a linear model with a number for d.
4. Verify by substitution: put each hour 0, 1, 2, 3 into each model, compute the count step by step with the exponent first, and compare it with the logged count. Record each check in the first two scaffold tables. Tool: A notebook or document for your written work. Check: Every substituted count equals the logged count for its culture.
5. Apply the exponent rules step by step: rewrite your exponential model at hour t + 1 using the product rule, so that it becomes a number times 2ᵗ, and compare it with the model at hour t. Then use both models to answer the questions in the third scaffold table, showing each step. Tool: A notebook or document for your written work. Check: The rewritten model has the form of a number times 2ᵗ, and every row of the third scaffold table is filled in with its steps.
6. Back in the graphing calculator, type y = 150*2^x and y = 150 + 300x on two new lines. Zoom out to the left of the y-axis, then find the stretch of hours where the two graphs cross. Tool: Desmos (free) or GeoGebra. Check: Each graph passes through its four dots; the curve flattens toward the x-axis on the left without touching it, and the two graphs meet at (0, 150), where both cultures start, then cross again between hour 2 and hour 3.

Fallback route

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

Plot both cultures' counts by hand on graph paper with the hour across the bottom and the cell count up the side, sketch a smooth curve through Culture A's dots and a straight line through Culture B's dots, and mark where they cross. Complete all three 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

Test Culture A's log and check your model against every count, so you know the model fits all four hours and not only the first.

Student scaffold table 1

Hour t	Culture A count	Ratio to the hour before	Your model's count, worked step by step	Match? (yes or no)
0	150
1	300
2	600
3	1,200

Test Culture B's log the same way, so the adding pattern and the multiplying pattern can be compared side by side.

Student scaffold table 2

Hour t	Culture B count	Difference from the hour before	Your model's count, worked step by step	Match? (yes or no)
0	150
1	450
2	750
3	1,050

Answer each question from your models with every step shown, so each prediction can be checked by someone else.

Student scaffold table 3

Question	Model used	Work, step by step	Answer
Culture A at hour 5
Culture B at hour 5
Culture A at hour −1
Can Culture A's count ever be 0?

Part 5 — Analysis Questions

1. Show how the ratios and the differences in your tables lead to each model, every step on its own line, and write both models. Then show the product-rule steps that rewrite the exponential model at hour t + 1 as a number times 2ᵗ.
2. From hour 0 to hour 3, by how many cells does Culture B grow, and by what factor does Culture A grow? Explain how the constant difference and the constant ratio produce those two results.
3. Find Culture A's count at hour −1 and check it against the log. Then explain in plain words why a negative input gives a small positive count, and why Culture A's model never reaches 0 cells.
4. Both graphs start at (0, 150). Describe where they cross again in step 6 and which culture is larger before and after that crossing. Explain how that picture agrees with your hour-5 predictions in the third scaffold table.
5. Choose one and answer it in 3–4 sentences. (a) A real culture runs out of food and space. Explain why Culture A's model cannot keep doubling forever and which of your predictions you would trust least. (b) Cell counts are estimates made from a small sample under a microscope. Explain how a counting error at one hour could change the model you built, 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 13 Model/Build Studio tables above. They come from an invented practice log, made up for this Studio, of two lab cell cultures counted once an hour. Culture A had 150, 300, 600, and 1,200 cells at hours 0, 1, 2, and 3. Culture B had 150, 450, 750, and 1,050 cells at the same hours. Please do these things. First, decide which culture grows by a constant ratio and which by a constant difference, and write a model for each, showing each step. Second, use your models to find each culture's count at hour 5. Third, use the exponent rules to rewrite 150 · 2^(t + 1) as a number times 2^t, and say what that shows about one hour of growth. Fourth, find Culture A's count at hour −1, one hour before counting began, and say whether its count can ever reach 0. Then answer seven short review questions: evaluate −2⁴ and (−2)⁴; simplify 2³ · 2⁴ and (2³)⁴; evaluate 2⁻³; say whether e and 2.25 are rational or irrational; evaluate 150 · 2³; give the slope and the y-intercept of y = 300x + 150; and solve −3x + 2 ≤ 11 written in words as negative 3x plus 2 is at most 11. Last, answer three more: simplify (x² − 9) ⁄ (x + 3), solve √(x + 3) = x + 1 and check every candidate, and say whether log(x + 4) can be rewritten as log(x) + log(4).

Test the response for these likely failures:

• Chatbots sometimes merge the coefficient into the base, writing Culture A's model 150 · 2ᵗ as 300ᵗ, or computing 150 · 2³ as 300³.
• Chatbots sometimes rewrite 150 · 2^(t + 1) as 150 · 2ᵗ + 2, or simplify 2³ · 2⁴ as 2¹² and (2³)⁴ as 2⁷.
• Chatbots sometimes report Culture A's count at hour −1 as −300 or as a negative number, or evaluate 2⁻³ as −8.
• Chatbots sometimes swap the two numbers in Culture B's line y = 300x + 150, calling 150 the slope.
• Chatbots sometimes call e rational because a calculator shows 2.71828, or call 2.25 irrational because it is a decimal.
• 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 3x plus 2 is at most 11 as x is at most −3.
• Chatbots sometimes cancel terms instead of factors, crossing out the 9 and the 3 or the x in (x² − 9) ⁄ (x + 3).
• Chatbots sometimes square √(x + 3) = x + 1 into x² + x − 2 = 0 and report both x = 1 and x = −2 as answers without checking them.
• 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 cultures, 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: "Culture A adds 150, 300, then 600 cells, so it is linear, and Culture B is the exponential one. Culture A's model is 300ᵗ, so at hour 3 it has 300³ cells. Rewriting, 150 · 2^(t + 1) = 150 · 2ᵗ + 2, so each hour adds 2 cells. One hour before counting began, Culture A had 150 · 2⁻¹ = −300 cells. Review: −2⁴ = 16; 2³ · 2⁴ = 2¹² and (2³)⁴ = 2⁷; 2⁻³ = −8; e is rational because a calculator shows 2.71828; in y = 300x + 150 the slope is 150; negative 3x plus 2 is at most 11 gives x is at most −3; (x² − 9) ⁄ (x + 3) = x − 3 because the 9 and the 3 cancel; and √(x + 3) = x + 1 has the two answers 1 and −2. Preview: log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (differences mistaken for ratios, the coefficient merged into the base, the product rule misused, a negative exponent read as a negative number, a negated power, a product rule and power rule mix-up, an irrational number called rational, the slope and the y-intercept swapped, an inequality sign not reversed, terms cancelled instead of factors even where the final form happens to match, an extraneous candidate 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 preview sentence, say what the course will check next week. This route assesses the same objectives (Objective 8 and Objective 3) 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

• All three completed scaffold tables
• A screenshot of your graph with both models and their dots, 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 two cell cultures: three completed tables (Culture A's ratios and four substitution checks, Culture B's differences and four substitution checks, and four questions with their models, work, and answers), the exponential and linear models with their numbers and the product-rule rewrite, a graph of both models with their dots 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 verifying the two models from the log	The constant ratio and the constant difference are found from the log, both models have the correct numbers, and all eight substitution checks are worked with the exponent first and match the log.	Both models are written, but a ratio or difference is found without its steps or with one arithmetic slip, or only some hours are checked.	No model is built, or the models' numbers are not drawn from the log.
Applying the exponent rules and evaluating the models	The product-rule rewrite reaches a number times 2ᵗ one step per line, the hour-5 predictions are correct for both models, and the hour −1 count is worked as a reciprocal and is positive.	The rewrite is missing a step, one prediction has an arithmetic slip, or the hour −1 count is given without the reciprocal step.	No exponent-rule work and no predictions are submitted.
Reading the models in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: the eightfold growth is tied to the ratio and the 900-cell growth to the difference, the positive hour −1 count and the never-zero count are explained, the crossing is matched to the predictions, and a real limit of the model is explained.	The questions are attempted, but a pattern is misread, the never-zero count is not explained, the graph is not tied to the predictions, or the limit is missing.	The analysis questions are missing or do not address the culture log.
Graph and completed tables	All three scaffold tables are complete, and a graph or labelled hand drawing shows both cultures' dots on their models with the crossing marked and 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 ratio, difference, count, and prediction in the tables comes from your own calculations on the printed culture log, never from memory or a chatbot.
• Each model's numbers are drawn from the log, with the ratio or the difference shown.
• Every one of the eight substitution checks is written out step by step, with the exponent evaluated before the multiplication.
• The product-rule rewrite of the exponential model is shown one step per line.
• The hour −1 count is worked as a reciprocal and is positive.
• 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.