Mathematics - Junior Secondary 3 - Angles of elevation and depression

TERM: 2^ND TERM

WEEK 9
Class: Junior Secondary School 3
Age: 14 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Angles of Elevation and Depression
Focus: Apply trigonometric ratios to find angles of elevation and depression

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

Define and distinguish between the angle of elevation and depression.
Identify the parts of a triangle involved in calculating angles of elevation and depression.
Use trigonometric ratios (sine, cosine, tangent) to find angles of elevation and depression.
Solve real-life problems involving angles of elevation and depression.
Apply the concept to situations like building heights, mountains, and towers.

INSTRUCTIONAL TECHNIQUES:
• Question and answer
• Guided demonstration
• Discussion
• Practical exercises and problem-solving
• Real-life application

INSTRUCTIONAL MATERIALS:
• Protractors
• Calculators
• Triangular-shaped objects (optional)
• Worksheets
• Whiteboard and marker

PERIOD 1 & 2: Introduction to Angles of Elevation and Depression

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 - Introduction	Introduces the concepts of angles of elevation and depression, explaining their uses in real life.	Pupils listen and ask questions for clarification.
Step 2 - Explanation	Defines the angle of elevation as the angle formed when looking upward at an object. The angle of depression is formed when looking downward at an object.	Pupils take notes and ask questions.
Step 3 - Demonstration	Uses a protractor to demonstrate the angles of elevation and depression in various scenarios.	Pupils observe the demonstration and practice identifying angles.
Step 4 - Practice	Guides pupils through an example, calculating both angles using trigonometric ratios.	Pupils practice with guidance from the teacher.

NOTE ON BOARD:

Angle of Elevation: The angle formed between the line of sight to an object and the horizontal line when looking upward.
Angle of Depression: The angle formed between the line of sight to an object and the horizontal line when looking downward.
Example: A person standing 30 meters from a building observes the top of the building at a 45° angle of elevation.
Formula for Trigonometric Ratios:

tan(θ) = opposite/adjacent
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse

EVALUATION (5 exercises):

What is the angle of elevation when looking at the top of a building 40 meters away and 30 meters high?
Calculate the angle of depression for a person on top of a 50-meter building, looking down at a point 40 meters away.
If the angle of elevation is 30°, find the height of a tree that is 25 meters away from the observer.
A plane is at an altitude of 1,000 meters. If the angle of depression from the plane to a point on the ground is 15°, calculate the distance from the plane to the point on the ground.
A building has an angle of depression of 20°. How high is the building if the distance from the observer is 60 meters?

CLASSWORK (5 questions):

Find the angle of depression if the building is 100 meters tall, and the observer is 200 meters away.
Calculate the angle of elevation of a tree that is 10 meters tall, observed from a distance of 30 meters.
The angle of depression from the top of a lighthouse is 25°. Find the height of the lighthouse if the distance from the observer is 60 meters.
A person observes the top of a building at a 40° angle of elevation, standing 15 meters away. Find the height of the building.
If a plane is flying 2,000 meters above the ground and an observer measures the angle of depression as 5°, calculate the horizontal distance from the plane to the observer.

ASSIGNMENT (5 tasks):

Calculate the angle of elevation if a tree is 12 meters tall and the observer is 20 meters away.
Determine the height of a mountain if the angle of elevation from a point 500 meters away is 25°.
Find the angle of depression from the top of a tower that is 80 meters high, observed from a distance of 100 meters.
A car is 50 meters from a building. The angle of elevation to the top of the building is 35°. Calculate the height of the building.
A plane flying at an altitude of 1,500 meters observes a point 1,000 meters away. What is the angle of depression?

PERIOD 3 & 4: Trigonometric Ratios for Elevation and Depression

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 - Introduction	Reviews trigonometric ratios and their role in calculating angles of elevation and depression.	Pupils recall the use of sine, cosine, and tangent ratios.
Step 2 - Explanation	Demonstrates using trigonometric ratios to solve for the angles of elevation and depression.	Pupils follow the example and ask questions for clarity.
Step 3 - Practice	Provides more complex examples for pupils to solve individually or in groups.	Pupils solve problems with support from the teacher.
Step 4 - Problem Solving	Sets additional problems to be solved on the board, emphasizing the real-world application.	Pupils complete the problems and present their solutions.

EVALUATION (5 exercises):

A person is standing 10 meters from a building. If the angle of elevation to the top of the building is 30°, what is the height of the building?
From a point 60 meters away from a tower, the angle of depression is 45°. Calculate the height of the tower.
An observer standing at the bottom of a hill sees a point 50 meters away at an angle of elevation of 35°. How high is the hill?
If a person observes the top of a building at a 25° angle of elevation, and they are 40 meters away, what is the height of the building?
A plane is flying at an altitude of 1,500 meters. If the angle of depression is 10°, find the horizontal distance from the plane to the point on the ground.

CLASSWORK (5 questions):

Calculate the height of a building when the angle of elevation is 15°, and the distance from the observer is 50 meters.
An observer is 80 meters away from a lighthouse and sees the top at a 30° angle of elevation. Find the height of the lighthouse.
Find the angle of depression from the top of a 120-meter tower when an observer is 150 meters away from the base.
A tree has an angle of elevation of 20° from a point 40 meters away. Find its height.
Determine the angle of depression from a plane at 2,000 meters altitude when the horizontal distance from the point is 500 meters.

ASSIGNMENT (5 tasks):

A building is observed from a point 150 meters away. If the angle of elevation is 10°, calculate the height of the building.
Find the angle of elevation if a mountain is 1,200 meters high and the observer is 1,500 meters away.
Calculate the angle of depression from a 50-meter-high tower to a point 75 meters away.
A person observes the top of a tree at a 45° angle of elevation from a point 20 meters away. Find the height of the tree.
A plane is flying at 1,800 meters altitude. Find the angle of depression if the distance to the point on the ground is 1,000 meters.

PERIOD 5: Application to Real-Life Situations

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 - Introduction	Introduces real-life applications: buildings, mountains, and plane flights.	Pupils listen and ask questions.
Step 2 - Problem Solving	Provides real-life problems for pupils to solve using angles of elevation and depression.	Pupils work on problems individually and in pairs.
Step 3 - Review and Drill	Guides pupils through their solutions, emphasizing real-life relevance.	Pupils present their solutions and discuss.

EVALUATION (5 questions):

If the angle of depression from a plane flying at 2,000 meters altitude is 5°, what is the distance from the plane to the point on the ground?
A mountain is observed at a 25° angle of elevation from a point 100 meters away. What is the height of the mountain?
A tree is 15 meters away, and the angle of elevation is 30°. Find the height of the tree.
A person observes the top of a 90-meter building at a 35° angle of depression from 200 meters away. Calculate the height of the building.
A ship sails 500 meters from a lighthouse. The angle of depression from the top of the lighthouse to the ship is 12°. Find the height of the lighthouse.

CLASSWORK (5 tasks):

Find the height of a tower using the angle of elevation from a distance of 200 meters at 40°.
A person standing 100 meters away from a mountain sees the top at a 30° angle of elevation. What is the height of the mountain?
The angle of depression from the top of a 150-meter building is 10°. Find the distance from the base of the building.
A plane flying at an altitude of 1,800 meters observes a point 2,000 meters away. Find the angle of depression.
A lighthouse is 60 meters tall. If the angle of depression is 15°, calculate the distance from the top of the lighthouse to the ship below.

ASSIGNMENT (5 tasks):

Write a real-life problem involving angles of elevation and depression and solve it.
Find the height of a building when the angle of elevation is 20° and the distance from the observer is 75 meters.
Calculate the angle of depression when the altitude of a plane is 2,500 meters and the horizontal distance is 1,500 meters.
A mountain is observed at a 45° angle of elevation. The observer is 500 meters away. Calculate the height of the mountain.

Write a real-life problem related to angles of depression and elevation and solve it using trigonometric ratios.