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Geometry · Module 1 · Topic 1 · Lesson 2
Build Truth,
Build Truth,
Step by Step
Pictures can suggest. Examples can support. Formal reasoning explains why a geometric claim must hold.
Our question
How do mathematicians know a claim is guaranteed?
Before station 1
One claim. Two possible failures.
A conclusion can fail because its starting information is false, or because the conclusion does not follow from that information. This studio teaches how to tell the difference.
Reasoning hook
“The football field is wet, so it must have rained.”
Reveal another possibility
Sprinklers could have made the field wet. A true conclusion does not prove that one particular cause happened.
Your reasoning board
Six connected moves
Station 1 of 6
Teach · model · guide · practice · explain
Formal Reasoning Studio
Keep the language precise
Reasoning reference
- Conditional
- A promise in the form “If p, then q.”
- Hypothesis
- The starting condition, p. hipótesis
- Conclusion
- The result the condition guarantees, q.
- Counterexample
- One case with p true and q false. contraejemplo
- Postulate
- An accepted starting statement. postulado
- Theorem
- A statement established from accepted facts and logic.
- Proof
- A chain of statements, each licensed by a reason. prueba
- Great circle
- A sphere-plane intersection through the sphere's center.
Independent exit ticket
Can your reasoning stand without the picture?
Complete after station 6. Print or answer on your own paper.
- For “If two angles form a linear pair, then they are supplementary,” identify p and q.
- Give a counterexample to “If a quadrilateral has four equal sides, then it is a square.” Explain why p stays true while q fails.
- Explain the difference between a postulate and a theorem.
- Explain why two distinct great-circle lines cannot be parallel on a sphere.
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