The Growing Contest
Two fertilizer brands, two trays of bean plants, and one question: is the difference REAL? Students plot both distributions, measure the gap between means in MADs, and learn that a difference only counts when it beats the spread.
Common Core standards covered
- 7.SP.B.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
- 7.SP.B.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
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Teacher-facing overview
Big idea: Assess the visual overlap of two numerical distributions, express the difference between centers as a multiple of a measure of variability, and draw informal comparative inferences about two populations.
Student target: “I can compare two distributions with dot plots, compute means and MADs, express the difference of means as a multiple of the MAD, and translate that multiple into a verdict about the populations.”
Success criteria: I can build and read paired dot plots for visual overlap. · I can compute a mean absolute deviation from a distance table. · I can express a gap between means in MAD units. · I can give a verdict: a many-MADs gap is meaningful; a fraction-of-a-MAD gap is noise.
Essential question
Tray A grew taller ON AVERAGE — but plants vary. When is a gap between groups big enough to mean something, and what do you measure it against?
Learning target: I can compare two distributions with dot plots, compute means and MADs, express the difference of means as a multiple of the MAD, and translate that multiple into a verdict about the populations.
Success criteria
- I can build and read paired dot plots for visual overlap.
- I can compute a mean absolute deviation from a distance table.
- I can express a gap between means in MAD units.
- I can give a verdict: a many-MADs gap is meaningful; a fraction-of-a-MAD gap is noise.
Why this matters
7.SP.B.3-4 is the heart of statistical thinking: differences only mean something relative to variability. The gap-in-MADs score is the standard's own construction (its example: 10 cm apart, about twice the MAD) and is the informal ancestor of every effect size and t-test students will meet later. Paired with the Poll's wobble, it completes the message: data answers comparisons with 'how many spreads apart,' never with bare means.
Students should already be able to
- Mean of a data set.
- Dot plots.
- Sampling wobble — The Lunch Line Poll.
Vocabulary to define explicitly
- Distribution overlap
How much two groups' dot plots share territory.
Example: Trays A and B barely touch — A's shortest beats most of B.
Not an example: Comparing single champions — one tall plant in B doesn't overlap-proof the trays.
🔒 3 more terms with examples and non-examples — Mean absolute deviation (MAD), Gap in MAD units, Informal inference — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up spread-hunting.
- Contest-sheet pipelines during floor work.
- The club ruling debate.
Summative — the graded evidence: The exit ticket (quotient and two verdicts), scored with the four-criterion rubric.
The task students work on
The garden club tests fertilizer brands. CONTEST 1: five plants per tray, heights in mm — Tray A (brand Alpha): 70, 72, 74, 76, 78. Tray B (brand Beta): 62, 64, 66, 68, 70. CONTEST 2, murkier: Tray C: 63, 67, 71, 75, 79 vs Tray D: 61, 65, 69, 73, 77. For each contest: dot plots on one axis, means, MADs, the gap in MAD units, and a verdict the club can act on. The club's rule of thumb on trial: a gap under one MAD is noise; two or more MADs starts to mean business.
Contest sheet — plots first, then the numbers.
| Contest | Means | MADs | Gap in MADs | Verdict |
|---|---|---|---|---|
| 1: A vs B | ||||
| 2: C vs D |
Anticipated student strategies
Eyes first, then the MAD as the formal ruler, then the quotient that arbitrates between them — perception calibrated by computation.
- Plot-first eyeballing. The student stacks both dot plots on one axis and judges overlap before computing.
The standard's 'visual overlap' verb. Share first — the eye's verdict gets tested by the numbers.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Draw out the missing ingredient: spread. If plants naturally vary by 20 mm, 8 is noise; if they vary by 2, 8 is a landslide. Today's tool measures gaps in units of natural variation.
Listen for: The Percent Report's baselines — same principle: the gap needs a denominator. Today's denominator is the MAD.
Post the pipeline: plot → means → MADs → gap ÷ MAD → verdict sentence. The standard's own benchmark quoted: its example calls TWO MADs a noticeable separation.
- MAD = average distance from the mean — the distance column does it.
- The quotient is the finding; the sentence is the deliverable.
Listen for: 8 mm only means something against how much plants naturally vary — the MAD converts millimeters into 'units of ordinary difference.'
🔒 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
- “A won because its mean is higher. Done.” Means alone can't carry a verdict — an 8 mm gap is huge against a 2.4 spread and nothing against a 20. Divide by the MAD first.
- “The tallest plant decides the contest.” — the teacher response is in the full lesson.
- “MAD is the range.” — the teacher response is in the full lesson.
- “0.4 MADs apart still means C is better — a win is a win.” — 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 doContest 1: Tray A (70, 72, 74, 76, 78) vs Tray B (62, 64, 66, 68, 70). Full pipeline and verdict.
🔒 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
- Group X: mean 40, Group Y: mean 34, both MADs 2. Gap in MAD units?
🔒 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: Dot-plot both trays and eyeball the overlap. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Compute both means and both MADs. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Divide the gap by the MAD — the separation score. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Judge a second, murkier contest. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: MAD by table (distance column, then average); extend with the murky pair and the how-many-MADs-is-enough 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 comparison with dot plots, means, MADs, the gap-in-MADs score, and a verdict in plain language.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.