Carnival Game Lab
The class inspects carnival games for fairness: equal-sector spinners get uniform models, a wobbly bottle-cap toss gets its model from data, and a ring toss whose observed wins crash below its advertised odds gets investigated. Probability from 0 to 1, with predictions on the line.
Common Core standards covered
- 7.SP.C.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
- 7.SP.C.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
- 7.SP.C.7Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
- 7.SP.C.7.aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
- 7.SP.C.7.bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on observed frequencies?
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Teacher-facing overview
Big idea: Express probabilities as numbers between 0 and 1, approximate probabilities from observed frequencies, and develop uniform and non-uniform probability models — comparing models to observations and explaining discrepancies.
Student target: “I can express any probability as a number from 0 to 1, build uniform models from equally likely outcomes and empirical models from frequencies, predict long-run results, and interrogate model-vs-data gaps.”
Success criteria: I can place events on the 0-to-1 scale and interpret positions. · I can build a uniform model and use it to predict repeated plays. · I can estimate a probability from observed frequency. · I can compare a model to data and propose sources of discrepancy.
Essential question
Every game promises odds. Which promises can you compute from symmetry, which must you measure — and what does it mean when the measurement disagrees?
Learning target: I can express any probability as a number from 0 to 1, build uniform models from equally likely outcomes and empirical models from frequencies, predict long-run results, and interrogate model-vs-data gaps.
Success criteria
- I can place events on the 0-to-1 scale and interpret positions.
- I can build a uniform model and use it to predict repeated plays.
- I can estimate a probability from observed frequency.
- I can compare a model to data and propose sources of discrepancy.
Why this matters
7.SP.C.5-7 assembles probability's grammar: the 0-1 scale (C.5), frequency as its estimator (C.6), and the two ways to build a model — symmetry or data (C.7a, C.7b) — plus the audit habit when they disagree. The carnival makes the epistemology honest: some games CAN be modeled a priori, some must be measured, and advertised odds are claims, not facts. Compound events (next lesson) and all of high-school statistics build on exactly this grammar.
Students should already be able to
- Fractions, decimals, percents interchangeably — Unit 6.
- Long-run thinking — The Lunch Line Poll's wobble.
- Proportional prediction — The Print Shop Test.
Vocabulary to define explicitly
- Probability (0 to 1)
A number from 0 (impossible) to 1 (certain) expressing likelihood; ½ is the coin-flip middle.
Example: P(win) = 0.2 on the fair five-sector spinner.
Not an example: 'Probably 3' — probabilities live in [0, 1]; a 3 is not a probability.
🔒 3 more terms with examples and non-examples — Uniform model, Relative frequency, Model vs observation — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The 0-1 line pins.
- Audit-sheet models during floor work.
- The ring-toss findings.
Summative — the graded evidence: The exit ticket (scale, uniform, frequency), scored with the four-criterion rubric.
The task students work on
The class audits the school carnival's games before opening night. THE SCALE: warm-up events pinned to the 0-1 line. THE SPINNER: five equal sectors, one is a winner — model it and predict wins in 600 plays. THE DIE GAME: win on a 3 or a 6 — model it and predict 600 rolls. THE CAP TOSS: a bottle cap is no coin — 60 test throws land open-up 42 times; build its model from the data. THE RING TOSS AUDIT: the poster claims a 1-in-2 win rate, but 60 test throws produced 12 wins — file a finding.
Audit sheet — model, prediction, and any discrepancies.
| Game | Model source | P(win) | Prediction / finding |
|---|---|---|---|
| Spinner (5 equal, 1 wins) | 600 plays → ? | ||
| Die game (win on 3 or 6) | 600 rolls → ? | ||
| Cap toss (42 up in 60) | |||
| Ring toss (claimed 1/2; 12 of 60) |
Anticipated student strategies
Symmetry first where it's honest, frequency where it isn't, prediction from both — and the audit when claim and data collide.
- Symmetry counting. The student counts equally likely outcomes and divides: 1/5, 2/6.
The uniform builder (C.7a). Demand the equal-likelihood justification, not just the count.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
1, 0, 0.5, and a debate — perfect. Language fixed: impossible is 0, certain is 1, coin-flip is ½, and everything else lives between. Numbers, not vibes.
Listen for: Probabilities are shares of certainty — you can't be more than certain or less than impossible. The scale is closed.
Board the two builders: C.7a symmetry (justify equal likelihood), C.7b frequency (divide, and report softness).
- Uniform models need a symmetry ARGUMENT, not a hope.
- Frequency models inherit the data's noise — say ≈, not =.
Listen for: The die's faces are interchangeable by design; the cap's shape favors one side by an amount no argument can compute. Symmetry is a property, not a default.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “120 wins in 600 means exactly 120.” P = 0.2 promises the long-run RATE, not a schedule. Roughly 120 — the wobble around it is normal breathing, not malfunction.
- “The cap is like a coin: 50-50.” — the teacher response is in the full lesson.
- “12 wins in 60 at a half-chance game is just bad luck.” — the teacher response is in the full lesson.
- “After five losses on the spinner, a win is due.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 4 independent.
- I doThe spinner: five equal sectors, one winning. Model, and the prediction for 600 plays.
🔒 5 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Place on the 0-1 line with a number: drawing a red card from a standard deck.
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Place events on the 0-to-1 probability line. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Build the uniform spinner model and predict 600 plays. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Model the cap toss from 60 observed throws. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Audit the ring toss: model vs observation. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: The 0-1 line anchors all judgments; extend with the discrepancy investigation and the is-it-loaded debate.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One uniform model with a long-run prediction, and one observed-frequency model, each with the probability stated as a number in [0, 1].
Mastery: The student completes the task accurately and explains the strategy with limited prompting.