The Triangle Factory
The factory takes custom triangle orders — three sides, three angles, or a mix — and not every order can ship. Students draw with ruler and protractor to sort orders into exactly one triangle, infinitely many, or none at all, and discover the triangle inequality as quality control.
Common Core standards covered
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Teacher-facing overview
Big idea: Draw geometric shapes with given conditions, focusing on constructing triangles from three measures of angles or sides and noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Student target: “I can construct triangles from given side and angle conditions with ruler and protractor, and classify any order as determining a unique triangle, many triangles, or no triangle — with the reason.”
Success criteria: I can build a triangle from three sides when it exists. · I can apply the triangle inequality to reject impossible side orders. · I can explain why three angles (summing to 180°) give infinitely many similar triangles. · I can attempt an order honestly — my verdict comes from construction, not guessing.
Essential question
Three numbers walk into the factory claiming to be a triangle. When do they determine exactly one, when a whole family, and when nothing at all?
Learning target: I can construct triangles from given side and angle conditions with ruler and protractor, and classify any order as determining a unique triangle, many triangles, or no triangle — with the reason.
Success criteria
- I can build a triangle from three sides when it exists.
- I can apply the triangle inequality to reject impossible side orders.
- I can explain why three angles (summing to 180°) give infinitely many similar triangles.
- I can attempt an order honestly — my verdict comes from construction, not guessing.
Why this matters
7.G.A.2 is where drawing becomes reasoning: the verdicts one/many/none are the first taste of what 'determined' means in mathematics — the idea beneath congruence criteria in Grade 8 and proof in high school. The triangle inequality is the standard's built-in physics (arms must reach), and the three-angles family reconnects to this unit's scale factor: same shape, any k. The factory framing makes failure informative — a rejected order teaches more than a shipped one.
Students should already be able to
- Measuring with ruler and protractor.
- Angle sum of a triangle (180°) informally.
- Scale factor k — Theme Park Scale-Up.
Vocabulary to define explicitly
- Triangle inequality
Each side must be shorter than the other two combined — or the arms can't reach.
Example: 2-3-7 fails: 2 + 3 = 5 < 7. The short arms can't span the long side.
Not an example: 2-3-4 passes every check (2+3>4, 2+4>3, 3+4>2) — it ships.
🔒 3 more terms with examples and non-examples — Unique (determined), A family of triangles (many), No triangle (rejected) — 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 attempt honesty.
- Stamped verdicts during floor work — reasons especially.
- The policy draft.
Summative — the graded evidence: The exit ticket (two verdicts and the family-pinning insight), scored with the four-criterion rubric.
The task students work on
The Triangle Factory ships custom triangles. Today's order stack: #1 sides 2, 3, 7 (the customer insists). #2 sides 3, 4, 5. #3 sides 5, 5, 5. #4 angles 40°, 60°, 80° — 'any size is fine.' #5 sides 4, 4, 9. #6 sides 2, 3, 4. Each order gets an honest construction attempt and a stamped verdict: SHIPS ONE, SHIPS A FAMILY, or REJECTED — with the mathematical reason on the stamp.
Order stack — verdict after honest attempt.
| Order | Conditions | Attempt result | Verdict & reason |
|---|---|---|---|
| #1 | sides 2, 3, 7 | ||
| #2 | sides 3, 4, 5 | ||
| #3 | sides 5, 5, 5 | ||
| #4 | angles 40°, 60°, 80° | ||
| #5 | sides 4, 4, 9 | ||
| #6 | sides 2, 3, 4 |
Anticipated student strategies
Construction first so verdicts are earned, the inequality second as the pattern the failures share, and the angle chain last to open the family case.
- Base-and-arcs construction. The student draws the longest side as base and swings the two other lengths from its ends (compass or marked ruler); the intersection is the apex — or isn't.
The decisive method: the arcs meeting IS existence, meeting at one point (up to reflection) IS uniqueness. Share first.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
They can't — the arms together span only 5. Name the discovery: each side must be beaten by the other two combined. The factory's first quality rule, found by failure.
Listen for: Flat on the base they reach 5 of the 7 — sides 2, 3, 5 would collapse to a flat 'triangle': the boundary case, still rejected.
Stack tracings to verify. Contrast: existence (the arcs meet) vs uniqueness (they meet in only one place above the base). #3 (5-5-5) runs the same way — unique. #6 (2-3-4) closes low and wide — awkward but determined.
- Verdict language: ships one / ships a family / rejected.
- Reflections count as the same shipped triangle.
Listen for: Each arc fixes an exact distance from its endpoint; two circles cross at most once above the base. The geometry rations the meeting points.
🔒 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
- “Any three lengths make a triangle if you draw carefully.” Care can't stretch arms: 2 + 3 = 5 will never span 7. The inequality is physics, not skill — swing the arcs and watch.
- “Three angles determine a triangle just like three sides.” — the teacher response is in the full lesson.
- “The inequality only needs checking on the longest side.” — the teacher response is in the full lesson.
- “A very thin triangle like 2-3-4.9 is impossible.” — 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 doStamp order #1 (sides 2, 3, 7) and order #2 (sides 3, 4, 5) with verdicts and reasons.
🔒 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
- Sides 5, 6, 12 — verdict and reason.
🔒 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: Attempt the impossible order and find the rule it breaks. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Build the 3-4-5 order and test its uniqueness. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Build the all-angles order and discover the family. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Sort the full order stack: one, many, or none. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Straw or strip models make the inequality physical; extend with the two-sides-and-an-angle orders and the shipping-policy write-up.
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 order of each verdict (one/many/none), drawn or refuted with the reason named.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.