Two-Way Table Investigation
Students survey the class on two yes-or-no questions and use a two-way table with relative frequencies to decide whether the answers are actually related.
Common Core standards covered
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Teacher-facing overview
Big idea: Construct and interpret two-way tables of categorical data, using relative frequencies to describe possible association.
Student target: “I can organize categorical survey data in a two-way table, compute row relative frequencies, and use them to describe a possible association between the two variables.”
Success criteria: I can build a two-way table with correct row and column totals from raw responses. · I can compute relative frequencies BY ROW and say what each percentage means. · I can compare the right two percentages to judge association. · I can state a finding's limits: sample size, one class, and no causation.
Essential question
No numbers to plot — just yes/no answers. Can 'association' still be measured when the data is categories?
Learning target: I can organize categorical survey data in a two-way table, compute row relative frequencies, and use them to describe a possible association between the two variables.
Success criteria
- I can build a two-way table with correct row and column totals from raw responses.
- I can compute relative frequencies BY ROW and say what each percentage means.
- I can compare the right two percentages to judge association.
- I can state a finding's limits: sample size, one class, and no causation.
Why this matters
Most real questions are categorical — treated or not versus recovered or not, breakfast or not versus alert or not — and the two-way table is how medicine, polling and product testing measure connection without a single scatter plot. The row-relative-frequency move is the entire trick: raw counts mislead whenever groups differ in size, and dividing by the row total is what makes 6 fairly comparable with 8. It is also the year's last, quietest rerun of its oldest lesson: compare rates, not totals.
Students should already be able to
- Percentages from fractions (7.RP.A.3).
- Association language — the last two lessons.
- The rates-not-totals instinct from Unit Rate Racing.
Vocabulary to define explicitly
- Two-way table
A grid counting individuals by TWO categorical variables at once.
Example: Rows: on a sports team / not. Columns: plays an instrument / doesn't. Each student lands in exactly one cell.
Not an example: Two separate tally lists lose the pairing — the whole point is who answered WHAT TOGETHER.
🔒 3 more terms with examples and non-examples — Joint and marginal counts, Row relative frequency, Association (categorical) — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The warm-up trap: raw-count reasoning surfaced and corrected.
- Table cross-checks during fieldwork.
- Council sentences: percentages, direction, limitation — all three present.
Summative — the graded evidence: The exit ticket (breakfast table), scored with the four-criterion rubric.
The task students work on
The student council wants data for activity planning and your team got the survey job. Twenty-eight students answered two yes/no questions: 'Are you on a sports team?' and 'Do you play an instrument?' A second survey asked 'Do you have a pet?' and 'Do you do regular chores?' Build each table, run the row percentages, and report — in council-ready sentences — where the associations are, and what your data cannot claim.
Survey 1 counts — complete the totals, then the row percentages.
| Plays instrument | No instrument | Row total | % playing instrument | |
|---|---|---|---|---|
| On a sports team | 6 | 10 | ||
| Not on a team | 8 | 4 | ||
| Column total |
Anticipated student strategies
Let raw counts fail first, fix them with row division, then give the fix a benchmark. The arc IS the standard: association in categorical data is visible only through relative frequencies.
- Compare the raw cells. The student sees 6 athlete-musicians versus 8 non-athlete-musicians and calls it close.
The productive mistake — let it be voiced. The rows have different sizes (16 vs 12), so raw cells cannot be compared. Its failure motivates everything after.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Draw out the missing fact: how many athletes total? How many non-athletes? 16 versus 12 — the groups differ in size, so raw counts compare apples to a smaller bag of oranges. Name the fix to come: divide by the row.
Listen for: 'Out of how many?' — a count without its group size is not yet evidence.
State the reading precisely: among athletes, 37.5% play; among non-athletes, 66.7% do. Knowing the row changes the odds — that is association. Then the caution stack: 28 students, one class, and nothing here says sports CAUSE non-playing (schedules, interest, both?).
- Totals cross-check before any division: 16 + 12 = 14 + 14 = 28.
- Same-column percentages, different rows — that is the comparison that answers the question.
Listen for: Dividing by 28 answers 'what share of the CLASS is both' — a different question. The association question lives inside each row.
🔒 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
- “6 athletes vs 8 non-athletes play — about the same.” The rows have different sizes: 6 OF 16 versus 8 OF 12 is 37.5% versus 66.7%. Counts compare only after dividing by the group.
- “Divide by the grand total for percentages.” — the teacher response is in the full lesson.
- “An association in the table proves causation.” — the teacher response is in the full lesson.
- “Any difference in percentages is an association.” — the teacher response is in the full lesson.
- “Row and column percentages are interchangeable.” — 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 3 independent.
- I doComplete Survey 1's totals and compute the athletes' instrument percentage.
🔒 4 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
- Breakfast survey: 24 ate breakfast (18 felt alert); 16 skipped it (8 felt alert). Build the table with totals.
🔒 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: Choose two categorical questions and collect a response from every student. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Organize responses into a two-way table with row and column totals. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Calculate relative frequencies by row and interpret what they show. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Decide whether the data suggests association, and name a limitation of the survey. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Provide a partly completed table for support; extend by comparing row and column relative frequencies.
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: Students compute one relative frequency and write a claim about association supported by their table.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.