Mathematics - Senior Secondary 1 - Deductive proofs (I)

Deductive proofs (I)

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Term: 3rd Term

Week: 1
Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Deductive Proofs (I)
Sub-topic:
i. Types and properties of triangles
ii. Proofs that the sum of angles in a triangle is 180°, and that an exterior angle equals the sum of the two interior opposite angles.

 

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

  1. Identify different types of triangles based on sides and angles.
  2. State and apply the properties of triangles.
  3. Explain the format for carrying out deductive proofs in geometry.
  4. Prove that the sum of interior angles in a triangle is 180°.
  5. Prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

 

INSTRUCTIONAL TECHNIQUES:
• Question and answer
• Guided demonstration
• Group work
• Practical illustration
• Use of diagrams and logical reasoning

 

INSTRUCTIONAL MATERIALS:
• Cardboard paper
• Cutout models of triangles
• Protractors
• Rulers
• Graph sheets
• Whiteboard and markers
• Worksheets for practice

 

PERIOD 1 & 2: Introduction to Deductive Proofs and Triangle Properties

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of deductive proofs in geometry and explains its importance. Introduces types of triangles: scalene, isosceles, equilateral, acute, right, and obtuse triangles.

Students listen, take notes, and ask questions.

Step 2 - Properties of Triangles

Explains and displays key properties of triangles. Demonstrates properties using cutout models and real examples.

Students observe, take notes, and identify properties in triangle cutouts.

Step 3 - Format of Deductive Proofs

Explains and writes out the structure of a geometric proof: Given, To Prove, Construction (if any), Proof, and Conclusion.

Students copy the format and discuss with teacher and peers.

NOTE ON BOARD:
• Types of triangles: Scalene, Isosceles, Equilateral, Right, Acute, Obtuse
• Format of a geometric proof: Given → To Prove → Construction → Proof → Conclusion

 

EVALUATION (5 exercises):

  1. Name three types of triangles based on angles.
  2. Name three types of triangles based on sides.
  3. State the standard format for deductive proofs in geometry.
  4. Explain the meaning of “construction” in deductive proofs.
  5. Why is it important to follow a logical structure in a geometric proof?

CLASSWORK (5 questions):

  1. Identify the types of triangles in a set of cutout models.
  2. Write down and explain the properties of an isosceles triangle.
  3. State the format of a deductive proof.
  4. Match given triangle diagrams to their type and properties.
  5. Explain how a triangle is different from other polygons.

ASSIGNMENT (5 tasks):

  1. Find real-life objects shaped like different types of triangles.
  2. Draw and label different types of triangles in your notebook.
  3. Write out the format of deductive proofs in your own words.
  4. Write three properties of any triangle of your choice.
  5. Research and explain why triangle properties are important in construction.

 

PERIOD 3 & 4: Proof of Angle Sum and Exterior Angle Theorems

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Recaps triangle types and proof format. Introduces the theorem: “Sum of interior angles in a triangle is 180°.”

Students recall and respond to recap questions.

Step 2 - Proof of Angle Sum Theorem

Leads students through proving the sum of interior angles is 180° using parallel lines and alternate angles. Uses diagrams.

Students observe, follow the proof steps, and write them down.

Step 3 - Exterior Angle Theorem

Demonstrates that the exterior angle is equal to the sum of the two interior opposite angles with diagrams and explanation.

Students draw the diagrams and follow the logical proof on their worksheets.

Step 4 - Practice

Guides students through multiple examples. Monitors as they work in pairs to prove the theorems.

Students construct their own proofs and compare with peers.

NOTE ON BOARD:
• The sum of the interior angles of any triangle = 180°
• Exterior angle = sum of the two opposite interior angles

 

EVALUATION (5 exercises):

  1. State the sum of the interior angles of a triangle.
  2. Prove that angle A + angle B + angle C = 180°.
  3. What is an exterior angle of a triangle?
  4. State the exterior angle theorem.
  5. Use a diagram to prove the exterior angle theorem.

CLASSWORK (5 questions):

  1. Prove that the sum of the angles in triangle XYZ is 180°.
  2. Using a diagram, prove that the exterior angle at vertex X equals the sum of the two opposite interior angles.
  3. Construct triangle ABC and verify its interior angles using a protractor.
  4. Write down the steps in proving the angle sum theorem.
  5. Define the term “exterior angle” in your own words.

ASSIGNMENT (5 tasks):

  1. Construct three different triangles and verify that their interior angles sum up to 180°.
  2. Prove the exterior angle theorem using any triangle.
  3. Write a short explanation of how the angle sum property can help in solving unknown angles.
  4. Use cardboard cutouts to demonstrate the angle sum theorem to a friend or sibling.
  5. Research where the angle sum theorem is used in real life.

 

PERIOD 5: Conclusion and Review

PRESENTATION:
• Review the two key theorems and their proofs using diagrams and recap questions.
• Lead students in a quick group activity where each group proves one of the theorems on cardboard.
• Encourage students to ask any questions they still have.
• Provide feedback on previous assignments and classwork.

 

EVALUATION:

  1. Oral quiz on triangle properties and proof structure.
  2. Collect and assess students’ classwork and diagrams.

Provide feedback and correct misconceptions where necessary.