Mathematics - Senior Secondary 1 - Deductive proofs (II)

Deductive proofs (II)

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Term: 3rd Term

Week: 2
Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Deductive Proofs (II)
SUB-TOPICS:
i. Similar and congruent triangles
ii. Isosceles and equilateral triangles

 

SPECIFIC OBJECTIVES:

By the end of the lesson, students should be able to:

  1. Distinguish between similar and congruent triangles.
  2. Identify properties of isosceles and equilateral triangles.
  3. Apply deductive reasoning to prove triangle congruence and similarity.
  4. Prove basic triangle properties using axioms and known theorems.
  5. Demonstrate understanding of angles in polygons and parallel lines using triangle-based proofs.

 

INSTRUCTIONAL TECHNIQUES:

  • Demonstration
  • Question and answer
  • Guided discovery
  • Discussion
  • Collaborative learning

 

INSTRUCTIONAL MATERIALS:

  • Cut-out paper triangles (congruent and similar)
  • Rulers and protractors
  • Chart of triangle types
  • Whiteboard and markers
  • Parallel lines diagram
  • Plane shapes (polygon models)

 

PERIOD 1 & 2: Introduction to Similar and Congruent Triangles

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Reviews triangle basics. Introduces the concept of deductive proof and its importance in geometry.

Students listen and take notes.

Step 2 - Similar vs. Congruent Triangles

Explains and compares similar and congruent triangles using paper models. Demonstrates SSS, SAS, ASA, and RHS congruence rules.

Students participate by comparing triangle pairs and listing properties.

Step 3 - Deductive Proof of Congruence

Demonstrates step-by-step proof of triangle congruence using known theorems and axioms.

Students copy the proofs and identify the axioms used.

Step 4 - Practical Activity

Uses cut-out triangles and rulers to identify congruent parts and prove relationships.

Students carry out pair-work to determine congruency or similarity.

Step 5 - Summary

Summarizes key points about triangle congruence and introduces isosceles and equilateral triangle properties.

Students ask questions and copy summary notes.

 

NOTE ON BOARD:

  • Congruent triangles: SSS, SAS, ASA, RHS
  • Similar triangles: Same shape, different size, proportional sides
  • Isosceles triangle: Two equal sides and two equal angles
  • Equilateral triangle: All sides and angles equal

 

EVALUATION (5 exercises):

  1. What is the difference between similar and congruent triangles?
  2. State the SSS congruence rule.
  3. Identify whether two given triangles are congruent or similar.
  4. Draw an isosceles triangle and mark equal parts.
  5. What property makes a triangle equilateral?

 

CLASSWORK (5 questions):

  1. Use a protractor to verify two angles in a given isosceles triangle.
  2. Identify and mark corresponding parts in two similar triangles.
  3. Prove two triangles are congruent using the ASA rule.
  4. Draw and label an equilateral triangle.
  5. Explain why all equilateral triangles are also isosceles.

ASSIGNMENT (5 tasks):

  1. Create two sets of congruent triangles from cardboard and label corresponding parts.
  2. Research real-life examples of similar triangles in construction.
  3. Write three different congruence rules and draw examples.
  4. Describe how to use deductive reasoning to prove two triangles are similar.
  5. Illustrate an isosceles triangle and describe its line of symmetry.

 

PERIOD 3 & 4: Properties of Isosceles and Equilateral Triangles

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Review

Revisits triangle congruence and introduces angle relationships in special triangles.

Students review notes and engage in Q&A.

Step 2 - Isosceles Triangle Proof

Demonstrates proof that base angles in an isosceles triangle are equal using a compass and straightedge.

Students replicate the proof using their tools.

Step 3 - Equilateral Triangle Properties

Proves all internal angles in an equilateral triangle are 60° using angle sum property.

Students follow the demonstration and verify using a protractor.

Step 4 - Real-life Examples

Shows where isosceles and equilateral triangles appear in nature and architecture.

Students list other examples they can identify.

Step 5 - Group Activity

Divides class into groups to prove specific triangle properties using paper models.

Students collaborate and present their findings.

 

NOTE ON BOARD:

  • Isosceles triangle: Equal base angles
  • Equilateral triangle: All angles = 60°
  • Triangle angle sum = 180°

 

EVALUATION (5 exercises):

  1. Prove that base angles of an isosceles triangle are equal.
  2. What are the properties of an equilateral triangle?
  3. What is the sum of angles in any triangle?
  4. Draw and label a triangle with two equal sides.
  5. Explain why all equilateral triangles are congruent.

 

CLASSWORK (5 questions):

  1. Prove a triangle is isosceles using side and angle properties.
  2. Draw and measure the angles of an equilateral triangle.
  3. Use paper models to demonstrate triangle symmetry.
  4. Identify real-world structures that use equilateral triangles.
  5. How can you use deductive reasoning to classify a triangle?

 

ASSIGNMENT (5 tasks):

  1. Cut out an isosceles triangle and label the equal angles.
  2. Measure all sides and angles of an equilateral triangle drawn by you.
  3. Write a short explanation of triangle symmetry.
  4. Use a triangle to explain the angle sum property.
  5. Design a triangular logo using properties of isosceles and equilateral triangles.

 

PERIOD 5: Conclusion and Review

PRESENTATION:

  • Recap key concepts: congruent vs. similar triangles, special triangle properties.
  • Class discussion: How deductive proofs help in understanding geometry.
  • Students solve a few short triangle proof questions on the board.
  • Address lingering questions and misconceptions.

 

EVALUATION:

  1. Quick-fire quiz on triangle properties.
  2. Review correctness of constructions and logic in student proofs.
  3. Provide personalized feedback and corrections.