If–Then Clues: Conditional Reasoning for Detectives
- Grade band
- 8
- Duration
- 45 minutes
- Skill
- Valid and invalid conditional reasoning
- Standards
- RI.8.8 — Delineate and evaluate the argument and specific claims in a text, assessing whether the reasoning is sound and the evidence is relevant and sufficient; recognize when irrelevant evidence is introduced.
- MP.3 — Construct viable arguments and critique the reasoning of others.
- Materials
- Student worksheet (printable, included)
- Argument sort cards (printable, included)
- Pencils
Learning objectives
- Students will be able to distinguish the four ways of reasoning from an if–then statement and identify which two are valid (affirming the “if” part; denying the “then” part).
- Students will be able to evaluate an argument and explain why a true conclusion can still come from invalid reasoning.
- Students will be able to recognize irrelevant evidence introduced into an argument and explain why it doesn't support the claim.
Lesson procedure
- Hook (5 minutes) — I Do. Write: “If it rained, the courtyard is muddy.” Then: “The courtyard is muddy. So it rained.” Ask for a show of hands: convincing? Then ask what else could make a courtyard muddy (sprinklers, a burst pipe). The rule only works in one direction.
- The four moves (10 minutes) — I Do. Using the rain rule, build a four-row chart: it rained → muddy (valid); not muddy → didn't rain (valid — the contrapositive); muddy → it rained (invalid — the converse); didn't rain → not muddy (invalid — the inverse). Give students a test: a valid move works no matter what other causes might exist.
- Argument sort (10 minutes) — We Do. Students sort the eight argument cards (Part 3, or the smartboard sort) into Valid and Invalid, naming the move for each.
- Snack puzzle (12 minutes) — You Do. Students solve Part 1. Every deduction must be written as one of the valid moves — no “it just has to be.”
- Critique (8 minutes) — You Do / Closure. Students complete Part 2, evaluating two classmates' arguments. Discuss: Dev's conclusion is true — so why is his argument still marked wrong?
Printable worksheet
Part 1: The movie-night snack puzzle. Quinn, Rory, and Sage each ate a different snack: pretzels, grapes, or popcorn.
- If Quinn ate the popcorn, then Rory ate the grapes.
- Rory did not eat the grapes.
- If Quinn ate the grapes, then Rory ate the pretzels.
- Sage did not eat the popcorn.
| pretzels | grapes | popcorn | |
|---|---|---|---|
| Quinn | |||
| Rory | |||
| Sage |
Write your solving path. For each step that uses Clue 1 or Clue 3, name the move (for example, “denying the ‘then’ part of Clue 1”).
Part 2: Critique the reasoning.
1. Dev says: “Quinn ate the pretzels, not the grapes. Clue 3 says if Quinn ate the grapes, Rory ate the pretzels. Quinn didn't eat the grapes, so Rory didn't eat the pretzels.” Dev's conclusion is true. Is his reasoning valid? Name the move and explain.
2. Ella says: “Sage's favorite snack in the whole world is popcorn, so Sage must have eaten the popcorn.” Identify the problem with Ella's evidence, and point to the clue that settles the question.
Part 3: Argument sort cards. Cut apart and sort into Valid and Invalid. Name the move on the back of each card.
Rule A: If a suspect was at the bakery, then they have flour on their shoes.
Rule B: If the window was broken, then the alarm sounded.
Try it on your smartboard
Rule A: If a suspect was at the bakery, then they have flour on their shoes. Rule B: If the window was broken, then the alarm sounded. Drag each argument to Valid or Invalid — works with touch, a stylus, or a mouse.
Valid
Invalid
Background for the teacher
Every if–then statement (“if P, then Q”) supports exactly two valid conclusions. Knowing P is true lets you conclude Q (affirming the antecedent, or modus ponens). Knowing Q is false lets you conclude P is false (denying the consequent, or modus tollens) — this is the same as using the contrapositive, “if not Q, then not P,” which always means the same thing as the original. The two tempting moves are invalid: concluding P from Q (affirming the consequent — treating the converse as true) and concluding not-Q from not-P (denying the antecedent — treating the inverse as true). Both fail because Q might have other causes.
Students accept invalid conditional reasoning most readily when the conclusion happens to be true, which is why Dev's argument in Part 2 matters: his answer is right, but the method would give wrong answers in other puzzles. Ella's argument illustrates the separate problem of irrelevant evidence — a preference says nothing about what Sage actually ate, and Clue 4 contradicts the conclusion outright. In the puzzle itself, both conditional clues are only useful through the contrapositive, so students who only know how to “follow the arrow forward” will get stuck.
Differentiation
Support: Provide an arrow diagram for each rule (P → Q) with the valid moves drawn in green and invalid moves in red. Let students sort only the Rule A cards first.
Extension: Ask students to write a fifth clue in if–then form that could replace Clue 4 and still leave exactly one solution — then prove it.
ELL support: Pre-teach valid, conclusion, and therefore/so. Use the same sentence frame for every card: “If ___, then ___. We know ___. So ___.”
IEP/504 support: Reduce the sort to four cards (one of each move), allow oral explanations for Part 2, and provide the puzzle with Clue 1's contrapositive already written out beneath it.
Answer key (for teachers)
Part 1 — Puzzle. Quinn: pretzels · Rory: popcorn · Sage: grapes.
Path: Clue 2 denies the “then” part of Clue 1, so Quinn did not eat the popcorn (valid — contrapositive). Clue 4 says Sage didn't either, so Rory ate the popcorn. Then Rory didn't eat the pretzels, which denies the “then” part of Clue 3, so Quinn did not eat the grapes (valid — contrapositive). Quinn ate the pretzels; Sage ate the grapes.
Part 2. 1. Invalid — Dev denies the “if” part (the inverse). Not eating grapes doesn't, by Clue 3, tell us anything about Rory. His conclusion is true only because of other clues. 2. Irrelevant evidence: a favorite snack isn't evidence of what Sage ate. Clue 4 directly says Sage did not eat the popcorn.
Part 3 — Sort. Valid: Nico (affirming the “if”), Lou (denying the “then”), alarm did not sound → window not broken (denying the “then”), window broken → alarm sounded (affirming the “if”). Invalid: Pia and alarm sounded → window broken (affirming the “then” / converse); Ada and window not broken → no alarm (denying the “if” / inverse).
Digital extension (optional)
Students who finish early can try a Super Sleuth case in Sleuth Scouts. The game is built for younger players, so it also makes a good cross-age tutoring activity: older students can coach a younger buddy through a case while naming each valid move out loud.