Week 11 Argument Workshop: Checking the Numbers in an Ad for a Hearing-Check App

Part 1 — The Big Picture

An ad can line up four numbers, each one true, and still leave a reader with the wrong picture. Whether it deserves trust depends on what each number measures and out of how many. This week you learned to find the conclusion a number is offered to support, to report a change between two percents in percentage points and as a percent change, to give every change in risk both ways with its base named, to count the people behind a screening claim, and to read a graph's axis before its shape. In this workshop you lay out a short ad for a hearing-check app, check each of its numbers, build a short argument of your own with its strongest objection and a reply, and then catch a chatbot making the slips this week warns about.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: In an illustrative ad for a hearing-check app, which of the ad's numbers, read correctly, support its conclusion that every adult in town should use the app, and what does a flag from the app really tell one adult?

Workshop data: Seven sentences and an appendix from an illustrative ad for a hearing-check app

Source: An illustrative ad, written for this workshop and printed in full in Part 4 with its appendix printed here; it comes from no real person, company, app, town, or clinic.

• Rows: 7 sentences, one row each in the sorting table, plus an appendix describing 3,000 adults, set out in a frequency table of 3 rows
• Columns: sentence, role in the argument, and the check to run
• Source text: the full ad: Every adult in town should use our free hearing-check app. Since the app launched, the share of adults who had a hearing check this year rose from 20 percent to 32 percent, up 60 percent. Adults who play loud music through earbuds every day face a far higher risk of hearing loss by middle age, 7 in 1,000 of them against 4 in 1,000 of other adults, so earbuds cause hearing loss. The app is 90 percent accurate, so if it flags you, you almost certainly have hearing loss. Our bar chart of user satisfaction shows our app's 88 towering over the rival app's 80. An audiologist objects: a phone check in a noisy room can miss real hearing loss or flag healthy ears, so anyone flagged or cleared should get a professional hearing test. The numbers speak for themselves: the app will protect the town's hearing.
• Appendix: Appendix to the ad: of 3,000 adults who tried the app last month, 1 in 60 had hearing loss; the app flagged 45 of them, and it also flagged 4 percent of the adults whose hearing was fine. The satisfaction chart's vertical axis starts at 78.

Part 3 — Materials & Procedure

Tools and materials

• Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them

Ways of working

• map-the-argument
• test-the-reasoning
• build-and-defend

Procedure

1. Read the whole ad in the first table of Part 4 twice: once for what it proposes, and once sentence by sentence. Then read the appendix line printed beneath the dataset description in Part 2. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: You can say in one sentence what the ad asks every adult in town to do.
2. In the Role column of the sorting table, label each sentence as the main conclusion, a reported number, a causal claim, an accuracy claim, a chart claim, the objection, or a restated conclusion, and in the last column name the check you will run, if any. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: Exactly one row is labelled the main conclusion, exactly one is labelled the objection, and every row has a role.
3. Type the figures from S2 and S3 into a free spreadsheet, one per cell. In separate cells, subtract the old share from the new one, then divide that difference by the old share and multiply by 100; do the same for the two risks per 1,000. Copy each figure into the numbers table. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: Each change in the numbers table appears both ways, and each matches the same step worked by hand.
4. Fill in the frequency table from the appendix, one count at a time: how many of the 3,000 adults had hearing loss, how many did not, how many in each group the app flagged, and the totals. Add each row and each column to check that the counts agree. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: Every row and column of the frequency table adds up, and the total of all four inner cells is 3,000.
5. Fill in the chart table for S5: measure each bar from the axis start in the appendix, compare the drawn heights, and then work the real gap between the two ratings as a percent of the rival's rating. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: The chart table gives both drawn heights, how many times taller one bar looks, and the real gap in words.
6. Write your own short argument on the ad's question in the build table: a conclusion, two premises that each bear on it, the strongest objection someone could raise stated as its holders would state it, and your reply. You may argue for or against adopting the app; the side you take is your choice. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: Your build table has a conclusion, two premises, a fair objection, and a reply, and every number in it carries its base.
7. Paste your completed tables into one approved chatbot, then paste the critique prompt from Part 6. Check each of its replies against your tables, quote at least two statements, and correct them with evidence. If you take the equivalent route, mark up the printed flawed analysis in Part 6 instead. Tool: Pen and paper or any free diagramming tool for argument maps and Venn diagrams; a free spreadsheet (Google Sheets, or any free spreadsheet) for the frequency, risk, and two-way tables; the Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy for named concepts; and one approved chatbot used as a sparring partner whose arguments the student maps and tests before trusting them. Check: Your critique quotes at least two statements, corrects each with evidence, and names one thing that was right.

Fallback route

When to use it: Use this route if no spreadsheet is available.

Work every change in the numbers table and every count in the frequency table by hand, with each subtraction, division, and multiplication written out on paper. Then complete every other table, analysis question, and the critique step, and submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

The ad comes first: give each of its seven sentences a role before you test anything.

Student scaffold table 1

Sentence	Role in the argument	Check to run, if any
S1. Every adult in town should use our free hearing-check app.
S2. Since the app launched, the share of adults who had a hearing check this year rose from 20 percent to 32 percent, up 60 percent.
S3. Adults who play loud music through earbuds every day face a far higher risk of hearing loss by middle age, 7 in 1,000 of them against 4 in 1,000 of other adults, so earbuds cause hearing loss.
S4. The app is 90 percent accurate, so if it flags you, you almost certainly have hearing loss.
S5. Our bar chart of user satisfaction shows our app's 88 towering over the rival app's 80.
S6. An audiologist objects: a phone check in a noisy room can miss real hearing loss or flag healthy ears, so anyone flagged or cleared should get a professional hearing test.
S7. The numbers speak for themselves: the app will protect the town's hearing.

Report each change in the ad both ways so no single number carries the whole story.

Student scaffold table 2

Claim	Figures in the ad	Plain difference	Change relative to the start
S2, the share of adults with a hearing check	20 percent to 32 percent
S3, the risk of hearing loss by middle age	4 in 1,000 to 7 in 1,000

Count the 3,000 adults in the appendix so you can see what a flag from the app means.

Student scaffold table 3

Group of adults	Flagged by the app	Not flagged	Total
Had hearing loss
Hearing was fine
All adults			3,000

Read the satisfaction chart's axis before its shape.

Student scaffold table 4

Quantity	Your figure	What it means in words
Height of our app's bar above the axis start
Height of the rival app's bar above the axis start
How many times taller our app's bar looks
Real gap between the two ratings, as a percent of the rival's rating

Build your own short argument on the same question and defend it.

Student scaffold table 5

Part	Your words
Your conclusion
Your first premise
Your second premise
The strongest objection, stated fairly
Your reply

Part 5 — Analysis Questions

1. Using your numbers table, give S2's change in percentage points and as a percent change, and give S3's change in risk as an absolute change per 1,000 and as a relative change in percent. Show the arithmetic.
2. Using your frequency table, how many of the 3,000 adults had hearing loss, how many adults were flagged in all, and how many of the flagged adults had hearing loss? Say in one plain sentence what this means for S4.
3. Explain what is wrong with the step in S3 from the two risks to "so earbuds cause hearing loss", naming one alternative you would check, and explain what the chart in S5 hides, naming the graph trick and giving the drawn bar heights and the real gap.
4. Name the ad's main conclusion and the sentences offered as premises for it, and say whether the numbers, read correctly, give it real support. Then compare your frequency table with the chapter's screening table: which group plays the role of the people without the condition, and why does it matter here?
5. Choose one and answer it in 3–4 sentences. (a) Name two things the checks in this workshop cannot tell you about whether the town should adopt the app, and say what you would need to find out for each. (b) Describe a number you have seen in an ad, a headline, or an app, and say which of this week's checks it most needed and what the fairer version would say.

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 11 Argument Workshop tables above. They test an illustrative ad for a hearing-check app. Please do eight things. First, tell me whether the ad's reasoning from S2 to S7 is valid. Second, describe the change in S2 in one sentence. Third, describe the change in risk in S3 in one sentence, and say whether S3 shows that earbuds cause hearing loss. Fourth, using the appendix, tell me what a flag from the app means for one adult. Fifth, tell me what the bar chart in S5 shows about the two apps. Sixth, tell me whether the audiologist's objection in S6 is a fallacy. Seventh, write a neutral one-paragraph summary of the ad and the objection, and tell me what research says about hearing-check apps. Eighth, tell me whether the town should adopt the app.

Test the response for these likely failures:

• Chatbots often call the ad's reasoning valid, or reason that if the app works more adults get checked, more adults got checked, so the app works.
• Chatbots often call the audiologist's objection in S6 an attack on the app's makers, or dismiss it because an audiologist sells hearing tests.
• Chatbots often summarize the objection as the audiologist telling people not to check their hearing.
• Chatbots often agree with S3 that earbuds cause hearing loss because daily loud-music listeners have the higher rate.
• Chatbots often describe S2 as a rise of 60 percentage points, or describe S3 as a percent rise in risk with no base.
• Chatbots often add that studies show hearing-check apps are as good as a clinic, or invent a share of adults with hearing loss.
• Chatbots often trust the 90 percent accuracy figure because the app's maker states it, or treat the ad itself as an expert source.
• Chatbots often repeat that a flagged adult almost certainly has hearing loss, or say that the ad and the audiologist are equally right about everything.
• Chatbots often accept a single story of one flagged user who later learned of real hearing loss as proof that the app works.
• Chatbots often write a neutral summary that calls the app lifesaving or calls the audiologist protective of her business.

Write 3–5 sentences that quote at least two statements from the chatbot you used, say what is wrong with each and correct it with evidence from the ad, the appendix, your tables, or this week's chapter, and name one thing the chatbot got right.

Equivalent route (no AI required)

No AI account is required for this workshop. Instead of the chatbot, critique this flawed analysis of the ad, printed here in full: "The ad's main conclusion is S2, that hearing checks rose. Its reasoning is valid: if the app works, more adults get their hearing checked; more adults got their hearing checked; so the app works. The share of adults with a hearing check rose 60 percentage points. Daily earbud users nearly double their risk, which proves that earbuds cause hearing loss. Since the app's maker says it is 90 percent accurate, anyone the app flags certainly has hearing loss, and the town should certainly adopt it. One user was flagged and later learned of real hearing loss, so the app works. Studies show that hearing-check apps are as accurate as a clinic. The chart shows our app is several times better liked than the rival. S6 is an attack on the app's makers by an audiologist who wants people to stop checking their hearing. A neutral summary: this lifesaving app is opposed only by a professional protecting her income." Mark it up: quote each flawed sentence, name what is wrong (the wrong main conclusion, affirming the consequent called valid, a percent change written as percentage points, a risk with no base, a cause read into a pattern, a maker trusted about its own product, false certainty, one story treated as evidence, an invented study, a truncated chart read at face value, a fair objection called an attack and turned into a straw version, or a summary that takes a side), cite the evidence from the ad, the appendix, your tables, or this week's chapter, and write a corrected sentence beneath it. This route assesses the same objectives (Objective 1 and Objective 6) 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 five completed tables
• 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 Argument Workshop report on the hearing-check ad: five completed tables (the sentence sort, the numbers both ways, the frequency count, the chart check, and your own argument), written answers to five analysis questions, and a critique of a chatbot's evaluation of the same ad, 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
Laying the ad out and finding its conclusion	Every sentence has a fitting role, the main conclusion, the reported numbers, the causal claim, the accuracy claim, the chart claim, and the objection are placed correctly, and the sentences offered as premises for S1 are named.	The roles are mostly right, but a reported number is treated as the conclusion, the objection is treated as support, or the premises for S1 are not named.	No sorting table is recorded, or it is copied from a chatbot without any checking.
Reporting the changes both ways	S2's change is given in percentage points and as a percent change, S3's change is given as an absolute change per 1,000 and as a relative change, each with the arithmetic shown, and S3's leap from a pattern to a cause is named with one alternative to check.	The changes are given only one way, a percentage-point change is written as a percent, a risk is given without its base, or S3's causal leap is not addressed.	No numbers table is submitted.
Counting the people behind a flag	The frequency table is complete and adds up, the flagged adults are read as a count out of all flagged adults, base-rate neglect is named in S4, and the false alarms are explained from how rare hearing loss is in the group.	The table is partly complete or does not add up, or the count is right but S4's mistake is not named or explained.	No frequency table or reading of the flags is submitted.
Reading the chart's axis	Both bars are measured from the axis start, the apparent ratio is compared with the real gap worked as a percent, and the truncated axis is named.	The truncated axis is named but the bars are not measured, or the bars are measured without saying what the real gap is.	No chart check is submitted.
Building and defending your own argument	Your conclusion and two premises each bear on the question, every number you cite carries its base and is read correctly, the strongest objection is stated as fairly as its best defender would state it, and the reply engages it; either side earns this band.	The argument is present, but a premise leans on S4's certainty or a number without its base, the objection is weakened into a straw version, or the reply ignores it.	No argument of your own is submitted.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two statements are quoted and corrected with evidence from the ad, the appendix, the tables, or the chapter, and one correct statement 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.
Reflection on what a numbers check can show	The chosen reflection option is answered in 3–4 sentences that name a specific limit or a specific example and tie it to a check the student could run.	The reflection is answered but stays general or names nothing to check.	No reflection is submitted.

Quality Gate

• Every sentence in the sorting table has a role, exactly one is labelled the main conclusion, and exactly one is labelled the objection.
• Every change in the numbers table is given both ways, and every risk is written as a count out of 1,000.
• Every row and column of the frequency table adds up, and the flagged adults are read as a count out of all flagged adults, never as a rounded percent.
• The chart table measures both bars from the axis start before it judges the gap.
• The objection in S6 is stated fairly and treated as a point to weigh, not as a fallacy.
• Your own argument uses premises that bear on its conclusion, states the strongest objection fairly, and replies to it, and it takes either side without penalty.
• No statistic, study, survey figure, or quotation is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two statements with evidence from the ad, the appendix, the tables, or the chapter.