Week 14 Model/Build Studio: How Many Days Until the Download Count Hits the Target?

Part 1 — The Big Picture

This week you learned that a logarithm is the exponent you need, that the log graph is the exponential graph with its points swapped, and that the product, quotient, and power properties let you rewrite a log. A count that doubles every day is an exponential, and asking when it reaches a target is asking for an exponent. In this Studio you take an invented download log for a library app, made up for practice, build the doubling model from the numbers, solve it for the day with a logarithm, rewrite that answer with the quotient property, verify every step by substitution, see the model and its inverse as mirror images in Desmos or GeoGebra, and use it to predict two future days. 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 six days of downloads, what is the doubling model D = a · 2ⁿ, what does it look like solved for the day n, and on which days do the downloads reach 640 and 2560?

Workshop data: E-book download log (six days, invented for practice)

Source: An invented, workshop-only download log for a library's e-book app, made up for practice (not real download counts) and printed in full in the first scaffold table: the day number and the total downloads of a new title on that day.

• Rows: 6 logged days; the first scaffold table lists all 6, and the second scaffold table has 3 rows, one for each question you answer
• Columns: day number (n) and downloads (D)
• Days: 0, 1, 2, 3, 4, and 5
• Downloads: 5, 10, 20, 40, 80, and 160

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 six days 0, 1, 2, 3, 4, 5 in the x column and the six download counts 5, 10, 20, 40, 80, 160 in the y column. Tool: Desmos (free) or GeoGebra. Check: Six dots appear that rise from left to right, each one twice as high as the one before.
2. In your notebook, build the model D = a · 2ⁿ. Read a from day 0, find the growth factor by dividing each day's count by the day before, and write the finished model. Tool: A notebook or document for your written work. Check: A model with a number for a and a growth factor of 2 that every ratio agrees with.
3. Verify by substitution: put each of the six days into your model, compute the downloads step by step, and compare with the log. Record each check in the first scaffold table. Tool: A notebook or document for your written work. Check: Every substituted count equals the logged count.
4. Solve your model for n, one step per line: divide both sides by a, rewrite the power statement as a logarithm, and then use the quotient property to write n as a difference of two base-2 logs. Check your formula at D = 160 by computing both versions. Tool: A notebook or document for your written work. Check: A formula for n in two equivalent forms that both give day 5 at 160 downloads.
5. Use your formula to find the day on which the downloads reach 640 and the day on which they reach 2560. Divide by a first, name the power of 2 that is left, and read off the exponent. Substitute each day back into the model. Record both results in the second scaffold table. Tool: A notebook or document for your written work. Check: Two whole-number days, each confirmed by substituting it into D = a · 2ⁿ.
6. Back in the graphing calculator, type y = 5*2^x on a new line, then y = log_2(x/5) and y = x on two more lines. Click the dot at day 3 and then the point on the log curve at x = 40. Tool: Desmos (free) or GeoGebra. Check: The exponential curve passes through all six dots, and the log curve passes through the swapped point, mirror-imaged across the line y = x.

Fallback route

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

Plot the six days from the download log by hand on graph paper with days across the bottom and downloads up the side, sketch a smooth curve through them, then plot the swapped points with downloads across the bottom and days up the side, sketch that curve, and draw the line y = x 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 day in the log, so you know it fits all six days and not only the one you used to build it.

Student scaffold table 1

Day n	Logged downloads D	Your model's count, worked step by step	Match? (yes or no)
0	5
1	10
2	20
3	40
4	80
5	160

Answer each question with the inverse of your model and confirm it by substitution, so the logarithm and the doubling are compared side by side.

Student scaffold table 2

Question	Power of 2 after dividing by a	Day n as a logarithm	Day n as a number	Substitution check
Day for 160 downloads (a check on the log)
Day for 640 downloads
Day for 2560 downloads

Part 5 — Analysis Questions

1. Show the steps that solve D = a · 2ⁿ for n, every step on its own line, and write n first as one base-2 log and then, with the quotient property, as a difference of two base-2 logs. Show that both forms give 5 at D = 160.
2. Compare the day for 640 downloads with the day for 2560 downloads. How many times as many downloads is the second target, and how many more days does it take? Explain how the logarithm produces that pattern.
3. Explain in plain words why log₂(640 ⁄ 5) is a difference of logs and not a quotient of logs, and why log₂(160 + 5) cannot be split at all. Use a number check for each claim.
4. Describe what the graph in step 6 shows about the dot at day 3 and the point on the log curve at x = 40, and explain how the line y = x and the domain of the log curve agree with your formula for n.
5. Choose one and answer it in 3–4 sentences. (a) Downloads cannot double forever. Explain what would eventually slow the growth and which of your two predictions you would trust least. (b) The app logged one total per day. Explain how counting at a different time of day could change the numbers, 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 14 Model/Build Studio tables above. They come from an invented practice log, made up for this Studio, of a library e-book app's downloads of a new title over six days: 5 downloads on day 0, 10 on day 1, 20 on day 2, 40 on day 3, 80 on day 4, and 160 on day 5. Please do these things. First, write the model D = a · 2ⁿ for the downloads D after n days, and say what a and 2 mean. Second, solve that model for n, showing each step, and rewrite your answer with the quotient property of logarithms. Third, find the day on which the downloads reach 640, and the day on which they reach 2560. Fourth, say whether log₂(160 + 5) equals log₂(160) + log₂(5), and whether log₂(640 ⁄ 5) equals log₂(640) ⁄ log₂(5). Then answer seven short review questions: evaluate −2⁴ and (−2)⁴; simplify 2³ · 2² and (2³)²; evaluate 2⁻³; say whether log₂(8) and √2 are rational or irrational; say whether 5 · 2³ equals 10³; solve 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 x = −2 solves √(x + 3) = x + 1.

Test the response for these likely failures:

• Chatbots often split the log of a sum, writing log₂(160 + 5) = log₂(160) + log₂(5), or turn the log of a quotient into a quotient of logs, writing log₂(640 ⁄ 5) = log₂(640) ⁄ log₂(5).
• Chatbots sometimes merge the starting count into the base, writing D = 5 · 2ⁿ as D = 10ⁿ, or solve 5 · 2ⁿ = 640 by taking log₂(640) before dividing by 5.
• Chatbots sometimes multiply the exponents in 2³ · 2² to get 2⁶, or add them in (2³)² to get 2⁵.
• Chatbots sometimes say 2⁻³ is −8 or −6.
• Chatbots sometimes call every logarithm irrational, including log₂(8), 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 accept x = −2 as a solution of √(x + 3) = x + 1 because squaring both sides produced it.

Write 3–5 sentences: name what the chatbot got right about the download 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 log starts at 5 and doubles, so D = 10ⁿ. Solving D = 5 · 2ⁿ for n gives n = log₂(D) ⁄ log₂(5). For 640 downloads, n = log₂(640) because the 5 does not matter, and 640 is not a power of 2, so the day cannot be found exactly. The quotient property also lets me write log₂(160 + 5) = log₂(160) + log₂(5). Review: −2⁴ = 16; 2³ · 2² = 2⁶; 2⁻³ = −8; log₂(8) is irrational because every log is; 5 · 2³ = 10³; negative 2x plus 1 is at most 9 gives x is at most −4; in y = −2x + 5 the slope is 5; (x² − 9) ⁄ (x + 3) = x − 3 because the 9 and the 3 cancel; and x = −2 solves √(x + 3) = x + 1 because squaring produced it." Mark it up: quote each flawed sentence, name what is wrong (a starting count merged into the base, a log of a quotient turned into a quotient of logs, the starting count ignored before the exponent question, a log split over a sum, a negated power, a product rule and power rule mix-up, a negative exponent read as a negative number, a rational number called irrational, a coefficient multiplied into a base, 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 an extraneous candidate kept), 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 7) 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 exponential curve, the log curve, and the line y = x, 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 library app's download log: two completed tables (six substitution checks, and three questions with their powers of 2, logarithms, days, and checks), the doubling model D = a · 2ⁿ with its numbers and its inverse written as a base-2 log and as a difference of logs, a graph of the model and its inverse with the line y = x 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 inverting the model from the download log	The starting count and growth factor are read from the log, the finished model D = a · 2ⁿ has the correct numbers, it is solved for n step by step into a base-2 log, the quotient-property form is reached by the named property, and all six substitution checks match the log.	The model is written, but a number is found without its steps or with one arithmetic slip, the inverse is missing a step, or only some days are checked.	No model is built, or its numbers are not drawn from the download log.
Solving for the day and verifying it	Both target days are found by dividing by the starting count, naming the power of 2, and reading off the exponent, and each is substituted back into the model and confirmed; the check at 160 downloads agrees in both forms of the inverse.	One target day is missing, or a day is given without steps or without a substitution check.	No target day is found, or days are stated without any method.
Reading the answers in the setting	Analysis questions 2, 3, 4, and 5 are answered accurately: four times the downloads is tied to 2 more days, the quotient and sum claims are each settled by a number check, the graph's swapped points and domain are matched to the formula, and a real limit of the model is explained.	The questions are attempted, but the pattern is misread, a log is split over a sum, the graph is not tied to the formula, or the limit is missing.	The analysis questions are missing or do not address the download log.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the six days on the exponential curve, the log curve through the swapped points, and the line y = x, 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 count and day in the tables comes from your own calculations on the printed download log, never from memory or a chatbot.
• The model's starting count and growth factor are read from the log, and the growth factor is confirmed by every ratio of consecutive days.
• The model is solved for n one step per line, and the quotient-property form is reached by a named property.
• Every one of the six substitution checks is written out step by step.
• Each predicted day is substituted back into D = a · 2ⁿ and confirmed.
• 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.