Lesson Note: Angles of Elevation and Depression

Subject: Mathematics

Class: JSS 2

Term: Second Term

Week: 8

Age: 11 – 14 years

Duration: 40 minutes

Topic: Angles of Elevation and Depression

Sub-topic: Understanding and Applying Angles of Elevation and Depression

Behavioral Objectives

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

1. Define angles of elevation and depression.
2. Explain how angles of elevation and depression are measured.
3. Solve mathematical problems involving angles of elevation and depression.
4. Apply angles of elevation and depression to real-life situations.

Keywords

• Angle of Elevation – The angle formed when an observer looks up at an object.
• Angle of Depression – The angle formed when an observer looks down at an object.
• Protractor – A tool used to measure angles.
• Clinometer – An instrument for measuring angles of elevation and depression.
• Trigonometry – A branch of mathematics used in calculating angles and distances.

Set Induction (Attention Grabber)

Ask students:

• Have you ever looked up at an airplane in the sky?
• Have you looked down from a tall building?
• Did you notice the angle between your line of sight and the ground?

Explain that these observations relate to angles of elevation and depression, which are important in navigation, architecture, and engineering.

Entry Behavior

Students are expected to have basic knowledge of angles and how to measure them using a protractor.

Learning Resources and Materials

• Protractors
• Clinometers
• Chart showing angles of elevation and depression
• Illustrations of real-life examples

Building Background/Connection to Prior Knowledge

• Recall the Pythagorean Theorem and how it is used to find the length of sides in right-angled triangles.
• Review how angles are measured using a protractor.

Embedded Core Skills

• Critical Thinking: Solving real-world problems.
• Numeracy: Using trigonometric ratios for calculations.
• Collaboration: Group work in solving problems.

Lesson Content

Definition of Angles of Elevation and Depression

1. Angle of Elevation: The angle between the horizontal line of sight and the line of sight to an object above the observer.

   • Example: Looking up at the top of a tree or a tall building.

2. Angle of Depression: The angle between the horizontal line of sight and the line of sight to an object below the observer.

   • Example: Looking down from a bridge at a boat on the water.

How to Measure Angles of Elevation and Depression

• Use a protractor or a clinometer to measure these angles.
• The horizontal line is always the reference line.
• The angle is measured from the horizontal to the line of sight.

Real-Life Applications of Angles of Elevation and Depression

1. Navigation: Pilots use angles of elevation to determine their altitude.
2. Surveying: Engineers use these angles to measure building heights.
3. Sports: Golfers and archers use elevation angles to determine the best shot.
4. Photography: Photographers adjust angles to capture the best perspective.

Solving Problems Involving Angles of Elevation and Depression

• Example 1: A man is standing 50 meters away from a building. If the angle of elevation to the top of the building is 45°, find the height of the building.

  Solution: Using tan θ = opposite/adjacent,

  • tan 45° = height / 50
  • 1 = height / 50
  • Height = 50 meters

• Example 2: A boy looks down from a tower of height 20 meters at a car. If the angle of depression is 30°, find the distance of the car from the base of the tower.

  Solution: Using tan θ = opposite/adjacent,

  • tan 30° = 20 / distance
  • Distance = 20 / tan 30° ≈ 34.64 meters

Evaluation Questions

1. What is an angle of elevation?
   a) The angle formed when an observer looks down at an object
   b) The angle formed when an observer looks up at an object
   c) The angle formed between two intersecting lines
   d) The angle formed between a straight line and a curve

2. What is an angle of depression?
   a) The angle formed when an observer looks down at an object
   b) The angle formed when an observer looks up at an object
   c) The angle formed between two intersecting lines
   d) The angle formed between a straight line and a curve

3. How is an angle of elevation measured?
   a) With a ruler
   b) With a compass
   c) With a protractor
   d) With a calculator

4. In which field is the concept of angles of elevation and depression used?
   a) Biology
   b) Mathematics
   c) History
   d) Literature

5. What is a clinometer used for?
   a) Measuring angles of elevation and depression
   b) Measuring the area of a triangle
   c) Measuring the volume of a solid
   d) Measuring the length of a curve

6. What type of angle is exactly 90°?
   a) Acute angle
   b) Right angle
   c) Obtuse angle
   d) Reflex angle

7. If the angle of elevation to the top of a building is 60° and the observer is 40 meters away, find the height of the building.
   a) 23 meters
   b) 40 meters
   c) 69 meters
   d) 80 meters

Class Activity Discussion (FAQs)

1. What is the main difference between angles of elevation and depression?

   • The angle of elevation is measured when looking up, while the angle of depression is measured when looking down.

2. Why is it important to learn about these angles?

   • They are used in navigation, construction, and real-life calculations.

3. Which instrument is used to measure angles of elevation and depression?

   • A clinometer or a protractor.

4. What type of triangle is formed when solving problems related to these angles?

   • A right-angled triangle.

5. Can angles of elevation and depression be the same?

   • Yes, in certain cases due to the property of alternate angles in parallel lines.

Assessment (Short Answer Questions)

1. Define an angle of elevation.
2. Define an angle of depression.
3. Name two real-life applications of angles of elevation and depression.
4. Explain how to measure an angle of elevation.
5. What is the difference between a right angle and an obtuse angle?

Conclusion

• The teacher will review key points and correct misconceptions.
• Students will be encouraged to identify angles of elevation and depression in their surroundings.