Lesson 10 — Building Triangles: Possible and Impossible

Summary

Learners build triangles from given side lengths and discover the triangle inequality: a triangle closes only when the two shorter sides add to more than the longest. They classify triples of lengths as possible or impossible and state the condition in words.

Objectives

• Construct triangles from given conditions and tell when a triangle is impossible. (D05.S3.07.01)

Connection

A builder has three planks — 2, 3, and 5 metres — and wants a triangular frame. But 2 + 3 equals exactly 5: the two short planks just lie flat along the long one and never close into a peak. A triangle needs the two shorter sides to reach past the longest. This is why bridges are braced with triangles — they cannot collapse the way a rectangle can. For thousands of years builders have known this; the ancient Egyptian land-measurers (“rope-stretchers”) used knotted ropes to lay out shapes, and the 3-4-5 rope triangle has been used for right angles from Egypt to India to China. Shape has conditions, and knowing them tells you what can and cannot be built.

Materials

• Straws or strips in assorted lengths
• Triangle condition page
• Rulers or a marked string

Preparation

• Prepare strips or straws in the needed lengths (e.g., 2, 3, 4, 5, 6, 7, 8) per pair.
• Have worked examples ready: 3-4-5 → possible; 2-3-5 → impossible (2+3=5); 1-2-4 → impossible (1+2<4).
• Recall from Grade 6: classifying shapes by attributes.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) three lengths form a triangle only when the two shorter sides add to more than the longest (the triangle inequality); (2) if the sum equals the longest, the sides lie flat — no triangle; if it is less, they cannot reach — no triangle; and (3) this condition lets a learner predict before building. Because constructing and classifying is a foundational skill, use explicit instruction and worked examples first, then hands-on building, then the rule in words (Kirschner, Sweller & Clark, 2006). Watch for the slip of testing only one pair of sides instead of the two-shortest-vs-longest test. Retrieval: ask learners to recall shapes and their attributes from Grade 6 — a triangle is the rigid three-sided shape. The global lens: the 3-4-5 right-triangle rope trick appears across ancient Egypt, India, and China, and the triangle’s rigidity is used in bridges and roofs worldwide — geometry is a shared human tool. The egalitarian lens (docs/philosophy.md §4): a simple sum test lets anyone, with nothing but sticks, decide what can be built — the rule belongs to everyone. The critical-thinking lens (docs/philosophy.md §7): predicting before building, then checking, is honest reasoning.

Procedure

1. Gather (5 min). Some sets of sticks can close into a triangle and some cannot. Today you find the rule that tells them apart — before you even try.
2. Try to build (10 min). Take your strips. Try 3, 4, and 5: they close into a triangle. Now try 2, 3, and 5: the two short strips (2 and 3) lie flat along the 5 — no triangle. Now try 1, 2, and 4: the 1 and 2 cannot even reach — no triangle.
3. Discover the rule (10 min). Look at what worked: 3 + 4 = 7, which is more than 5. What failed: 2 + 3 = 5 (equal to 5) and 1 + 2 = 3 (less than 4). The rule: the two shorter sides must add to more than the longest side. This is the triangle inequality.
4. Worked example (10 min). Which triples form a triangle? 4, 5, 6 → 4 + 5 = 9 > 6, so yes. 3, 3, 6 → 3 + 3 = 6, equal to 6, so no (it lies flat). 2, 2, 5 → 2 + 2 = 4 < 5, so no.
5. Practice with a partner (10 min). Take your triangle condition page. For each triple, predict with the sum test, then build to check. Sort each as “possible” or “impossible” and say why. Trade and check each other.
6. Close (5 min). A triangle closes only if the two shorter sides add to more than the longest. Equal or less — no triangle. State the condition, predict, then build to confirm.

Differentiation

• Support: Use only whole-number triples with strips that can actually touch; describe the sum test aloud so learners with low vision can follow by listening.
• Extension: Given two side lengths, find the full range of possible values for the third side, and explain why triangles are the rigid shape used in bridges and roofs.

Assessment

• Formative (observation/performance): Can the learner predict (with the sum test) and then build to confirm whether a triple forms a triangle, stating the condition in words?
• Self-check: The learner asks, “Did I compare the two shorter sides’ sum to the longest side? Did I build to confirm my prediction, and can I say the triangle inequality in my own words?”

Home connection

At home, gather three sticks or strings of different lengths and test the rule: do the two shorter ones reach past the longest? Try three sets and predict each outcome before checking.

Resources

• The triangle inequality is standard in middle-grades geometry; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The note that the 3-4-5 rope triangle appears in ancient Egypt, India, and China is documented history — India’s rope/altar constructions and Pythagorean triples (MacTutor, “The Indian Sulbasutras,” S-292) and China’s gou-gu right-triangle rule (MacTutor, “Nine Chapters on the Mathematical Art,” S-289), with Egypt’s knotted-rope surveyors (harpedonaptai) attested in Greek accounts; treating all traditions as worthy is a value (docs/philosophy.md §4). The claim that geometry’s rules belong to everyone is a value commitment.