One invented case, worked all the way through: a robotics club sells T-shirts to raise 00. The modeling cycle step by step — a measurable question, assumptions written down, variables with units, cost, revenue and profit functions, break-even and the goal, a trend fitted to a week of sales with its residuals, validation against last year, an assumption tested, a price chosen with a quadratic, and a report that says how far the model can be trusted. The numbers are invented; the method is the point. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Shirts sell for $15 each. Sell n of them and 15n dollars come in.
Revenue is what comes in, cost is what goes out, and profit is only the difference. A fee paid once stays outside the n — and every model has to say which of the three you are reporting.
What people get wrong
⚠️People often think…
Selling n shirts at $15 with a $150 fee costs (150 + 6)n.
The $150 is paid once, not once per shirt. Cost is 150 + 6n. Ask of every number: does this repeat, or does it happen a single time?
A one-time fee stays outside the n.
⚠️People often think…
The answer is 83.33 shirts, so round to 83.
The goal was AT LEAST $600. 83 shirts falls short, so you round UP to 84. Rounding follows the question, not the decimal.
Round the way the question needs.
⚠️People often think…
The data keeps rising, so the model is exponential.
Rising is not enough. Check both: steady DIFFERENCES mean linear, steady RATIOS mean exponential. Pick by which one actually holds, not by how the picture feels.
Differences or ratios. Test, do not guess.
⚠️People often think…
The fit was excellent over days 1–5, so the day 20 prediction is solid.
A fit only speaks for the range it was built on. Day 20 is far outside days 1–5, and nothing in the data says the pattern kept going. Report the limit alongside the number.
A fit does not cover ground it never saw.
Name it, build it, compute, then check
1Watch one
Shirts sell at $15, cost $6 each, and setup is $150. How many to clear at least $600 profit?
Name the variable: n is the number of shirts sold.
Build the profit: 15n − (150 + 6n) = 9n − 150.
Set it against the goal: 9n − 150 ≥ 600, so 9n ≥ 750 and n ≥ 83.33.
Interpret it: you cannot sell a third of a shirt, and 83 falls short — so 84.
2Do one with me
Same fundraiser. Fill in the model.
The profit expression is 9n −
Shirts needed for $600 profit is
Steady ratios point to a model that is
💬One sentence, then you move on
Why does the $150 stay outside the n?
3Try one
Name the one thing a bare number is missing before it counts as a model.
I want a hint first
You assumed every shirt sells, the price never changes and the fee is paid once. What are those called?
💬Last one — then you're done here
Why is a good fit not a promise about the future?
Where this goes
Where this lives
Every budget, every estimate, every forecast you will ever be handed is somebody’s model. Knowing to ask what they assumed is the whole skill.
What this feeds
That closes the math wing. Every unit before this one was a piece of the same move: name it, build it, check it.
Name one assumption you would have to make to model your own week.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 Any modeling task — the routine
Turn the situation into a question with a number in it. List the assumptions you are choosing. Name the variables WITH units and say which one you control. Build the function from the assumptions (fixed amount + rate × n for a linear cost). Compute, then round the way the context demands. Say what each number means. Check one prediction against real data. Report the answer with its assumptions and where it stops applying. Our case: P(n) = 9n − 150, break-even 17, goal 84, validated by last year’s 70 shirts → $480.
A model is a set of assumptions turned into a function, and it is exactly as trustworthy as those assumptions inside the range of the data that checked them. The algebra is the easy part — every unit before this one taught it. The modeling is choosing the assumptions on purpose, picking the function family the data actually support, and saying out loud where the model stops. All numbers in this room are invented.
⚠️ Traps the test loves
Answering 83.33 shirts, or rounding to 83. The goal is “at least $600,” so round UP to 84.
Confusing revenue with profit. 15n came in; 150 + 6n went out; only 9n − 150 is yours.
Multiplying the fixed fee by n. The $150 is paid once: 150 + 6n, never (150 + 6)n.
Calling every rising data set exponential. Check the ratios; if the differences are steadier, it is linear.
Trusting an extrapolation because the fit was good. Day 20 is not in days 1–5.
Reporting a bare number. 84 without its assumptions and limits is not a model.
📐 The fundraiser model — the chart
Piece
Formula (invented numbers)
Worked
Cost
C(n) = 150 + 6n
C(50) = 450
Revenue
R(n) = 15n
R(50) = 750
Profit
P(n) = 9n − 150
P(50) = 300
Break-even
9n − 150 = 0
n = 16.67 → 17 shirts
Goal of $600
9n − 150 ≥ 600
n ≥ 83.33 → 84 shirts
Validation
P(70) vs. last year’s $480
630 − 150 = 480 ✓
Daily sales trend
S(d) ≈ 4d + 10 from 14, 18, 22, 25, 31
residuals 0, 0, 0, −1, +1
Best price
P(p) = (p − 6)(120 − 6p) − 150
vertex p = 13, profit 144
Pledges to sales
sales = 0.8 × pledges
84 sales ← 105 pledges
🎯 How the test will ask
A situation in words — “write a function for the cost / profit.” Fixed amount first, then rate × n.
“What does the slope mean? The intercept?” Answer in the units of the problem: dollars per shirt; loss with no sales.
“How many to break even / reach the goal?” Solve, then round in the direction the words demand.
A table of data — “which family fits, and what does it predict?” Differences, ratios, then a fit, then residuals.
“Is this prediction reasonable?” Inside the data or beyond it? Is there a ceiling the model ignores?
“Name an assumption and say how you would check it.” Pick one that would change the answer if wrong.
“Write a two-sentence report.” The number, the assumptions, the limit.
✅ Before the test, can you…
Recite the eight steps of the modeling cycle and say what each one produces?
Turn “fixed fee plus a cost per item” into a linear function, and revenue minus cost into profit?
Find break-even and a goal, and round each the right way?
Look at five data points and say linear, exponential, quadratic or flattening — with the evidence?
Compute residuals and judge a fit?
Explain why extrapolation is risky, with a concrete example like 5,120 shares in a school of 600?
Write a report that states the answer, the assumptions and the limits in three sentences?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a modeler on one invented case — a robotics club’s T-shirt fundraiser: put the modeling cycle in order, graph a week of sales and read the trend, sort the assumptions that hold from the ones that do not, then argue how far the model can be trusted. Every number is invented. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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