Lesson 11 — Capstone: Lines of Fit, and Why Association Is Not Causation

Summary

The unit capstone completes the data strand and closes the year. Learners fit a line informally to data, evaluate the fit, use the line to predict, and — the honest heart of the lesson — explain that association does not prove causation. They then retrieve the unit’s ideas and revisit the wonders from Lesson 1.

Objectives

• Fit a line informally to data to model a relationship, evaluate the fit, and explain that association does not prove causation. (D05.S4.08.02)
• Synthesize the unit’s eight objectives into one view of number, function, shape, and data, and return to the Lesson 1 wonders. (D05.S1.08.01, D05.S1.08.02, D05.S2.08.01, D05.S2.08.02, D05.S3.08.01, D05.S3.08.02, D05.S4.08.01, D05.S4.08.02 — synthesis)

Connection

Ice-cream sales and drownings both rise in summer. Do ice creams cause drownings? No — a third thing, summer heat, raises both. This is the most important habit data can teach: two things moving together is a clue, not a proof. A line through the points lets you predict, but the line never proves why. Today you learn to fit that line honestly — and to say, with confidence, “they move together, but that does not mean one causes the other.”

Materials

• Line-of-fit sheet
• Unit map (from Lesson 1)

Preparation

• Copy the line-of-fit sheet; return each learner’s Lesson 1 unit map and wonders.
• Have worked examples ready: fit a line through study-hours vs. score points, predict a score at a new study time, and judge the fit by how near the points sit; then the ice-cream/drowning example.
• Retrieval: this capstone is itself a spaced, interleaved retrieval pass across the whole unit (Dunlosky et al. 2013, S-039) — it works because learners must recall, not re-read.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a line of best fit is drawn informally so about half the points sit on either side, and it models the relationship; (2) the fit is judged by how near the points lie to the line, and the line can then predict within the data’s range; (3) association does not prove causation — a common cause or coincidence may explain the pattern. Because fitting a line is a foundational skill, use explicit instruction and a worked example first (Kirschner, Sweller & Clark, 2006). This is a summative moment, but mastery is personal, not ranked — celebrate growth, never compare learners (philosophy §8). The critical- thinking lens is the lesson’s core: “correlation is not causation” protects people from false conclusions in medicine, policy, and daily life (S-355). The egalitarian lens: honest data reading lets anyone — not only an expert — question a claim that “this causes that” and ask who benefits from believing it. The environment lens: the same caution guards climate and health claims — a rising number does not by itself name the cause. Keep the four lenses explicit in the closing retrieval.

Procedure

1. Fit the line (15 min). On your line-of-fit sheet, draw a straight line through the points so about half sit above and half below. This is your line of best fit — an informal model of the relationship. Judge the fit: are the points near the line or far? Use the line to predict one new value within the data’s range.
2. Association is not causation (10 min). Ice-cream sales and drownings both rise in summer. Does ice cream cause drowning? No — summer heat raises both. Two things moving together is a clue, never a proof of cause.
3. Retrieve the unit (15 min). With a partner, recall the eight ideas without looking: rational vs. irrational; square roots on the line; scientific notation; functions and f(x); solving linear equations; systems by graphing; transformations; the Pythagorean theorem; scatter plots and lines of fit.
4. Revisit your wonders (7 min). Open your Lesson 1 map. Which wonders can you now answer? Which grew into new questions? Answer one you can, and keep one to wonder about.
5. Close (3 min). Share one idea you can now explain. Remember: number, function, shape, and data are one sense-making habit — and honest data reading never mistakes a clue for a cause.

Differentiation

• Support: Provide a line pre-drawn and guide judging the fit by counting points on each side; describe each step aloud for learners with low vision.
• Accessibility: Fit the line tactilely with a string or stick through raised dots or stones and describe the fit aloud for learners who are blind or have low vision; for learners with dyscalculia, keep the task qualitative (near/far, above/below) and center the causation argument rather than computation.
• Extension: Compare two candidate lines and argue which fits better (which leaves points farthest), and critique a real headline’s causal claim by naming a possible common cause.

Assessment

• Summative (portfolio + self): The completed line-of-fit sheet is the unit’s capstone artifact, kept in the portfolio, alongside the Lesson 1 map with answered wonders.
• Self-assessment: Learners mark one idea they can explain confidently and one they will keep learning — “not yet” is information, never a verdict.

Home connection

Share with someone at home the rule “association is not causation,” and find one real claim (in the news, an ad, or a conversation) where two things rise together — then ask together what else might explain it.

Resources

• On “correlation does not imply causation”: Wikipedia, “Correlation does not imply causation” (S-355). On spaced, interleaved retrieval as a capstone: Dunlosky et al. (2013), https://doi.org/10.1177/1529100612453266 (S-039). On explicit instruction for novice skills: Kirschner, Sweller & Clark (2006), S-011.