The Earth as a Sphere

Grade 12 · Mathematics

Semester 1 | Period 3 | Week 14

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Subject: Mathematics

Semester: 1

Period: 3

Week: 14

WEEK 14

Class: Grade 12
Age: 17 years
Duration: 40 minutes (5 periods)
Subject: Mathematics
Topic: The Earth as a Sphere
Focus:

  • Describe the Earth as a sphere
  • Identification of the line of longitude (meridian), latitude, equator, North Pole, South Pole, small circle, great circle
  • Distance along the great circle
  • Radius of the parallel of latitudes
  • Distance along the parallel of latitudes
  • Mathematical problems related to the Earth as a sphere

SPECIFIC OBJECTIVES:

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

  1. Describe the Earth as a sphere.
  2. Identify key geographical lines such as the Equator, lines of longitude, and latitude, as well as the North and South Poles.
  3. Understand the concepts of small circles and great circles.
  4. Calculate distances along the great circle and parallel latitudes.
  5. Solve mathematical problems related to the Earth’s sphere.

 

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Discussion
  • Practical activity
  • Real-life connections

 

INSTRUCTIONAL MATERIALS:

  • Real globe and skeletal globe
  • Charts illustrating the Earth’s features
  • Mathematical problem charts
  • Markers and whiteboard

 

PERIOD 1 & 2: Introduction to the Earth as a Sphere

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Introduction

Teacher explains that the Earth is a sphere and that it rotates around an axis.

Students listen attentively and ask clarifying questions.

Step 2: Key Features

Teacher introduces key features of the Earth: Equator, North Pole, South Pole, lines of longitude, lines of latitude, small circles, and great circles.

Students observe and take notes on the globe, identifying features.

Step 3: Global Coordinates

Teacher demonstrates the role of latitude and longitude in determining locations on Earth.

Students interact with the globe and practice identifying coordinates.

Step 4: Analogy

Teacher uses the analogy of a ball to demonstrate the Earth as a sphere, comparing it to a globe.

Students discuss the analogy and share their understanding of how latitude and longitude work.

NOTE ON BOARD:

  • Earth’s Features:
    • Equator: 0° Latitude
    • Poles: 90° North and South Latitude
    • Lines of Longitude: Run from the North Pole to the South Pole (Meridians)
    • Great Circle: Any circle drawn on the surface of a sphere that divides it into two equal halves.
    • Small Circle: Any circle that does not divide the sphere into two equal halves.

EVALUATION (5 exercises):

  1. Identify the two poles of the Earth.
  2. What is the Equator?
  3. Define the term "meridian."
  4. What is the difference between a small circle and a great circle?
  5. Name a real-life application of using longitude and latitude.

CLASSWORK (5 questions):

  1. Locate the Equator on the globe.
  2. What is the angle of latitude at the North Pole?
  3. How are the lines of longitude different from lines of latitude?
  4. Define the term “parallel of latitude.”
  5. Where is the 0° Longitude located?

ASSIGNMENT (5 tasks):

  1. Find a map showing the lines of latitude and longitude.
  2. Draw and label the Earth’s poles, Equator, and the major lines of longitude.
  3. Research a location’s coordinates using latitude and longitude.
  4. Explain why the Earth is considered a sphere.
  5. Draw the concept of a great circle on a globe.

 

PERIOD 3 & 4: Distance along the Great Circle and Parallel of Latitudes

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Distance along Great Circle

Teacher explains the concept of distance along the great circle. Provides a formula and demonstrates using the globe.

Students take notes and follow the example.

Step 2: Radius of Parallel of Latitudes

Teacher demonstrates the radius of the parallel latitudes and explains how it decreases as the latitude moves towards the poles.

Students observe and note down the concept.

Step 3: Calculating Distance

Teacher demonstrates how to calculate the distance along the great circle using the formula:

d=θ×r

where d is distance, θ is the central angle, and r is the radius of the Earth.

Students practice calculating distance in pairs.
     
     

EVALUATION (5 exercises):

  1. Calculate the distance along the great circle between two points given their coordinates.
  2. Calculate the radius of a parallel of latitude at 30° N.
  3. How does the radius change as you move away from the Equator?
  4. What is the formula for calculating the distance along a great circle?
  5. What factors affect the distance along a parallel of latitude?

CLASSWORK (5 questions):

  1. Calculate the distance between two points on the Earth’s surface using their coordinates.
  2. What is the distance along the parallel of latitude at 45° N if the radius of Earth is 6400 km?
  3. Given two points on the Earth’s surface, calculate the distance along the great circle.
  4. What is the radius of a parallel of latitude at 0° (Equator)?
  5. How does the distance along a parallel of latitude compare to that along the great circle?

ASSIGNMENT (5 tasks):

  1. Research and calculate the distance between two major cities along the great circle.
  2. Explain why the distance along the great circle is always the shortest route between two points on Earth.
  3. Calculate the radius of the Earth based on the distance along a parallel of latitude at 60° N.
  4. Solve a problem involving the distance between two points using the formula.
  5. Explain the relationship between the radius of the Earth and the distance along the great circle.

 

PERIOD 5: Mathematical Problems on Earth as a Sphere

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Problem Solving

Teacher presents a series of mathematical problems based on the Earth as a sphere. These will include calculations for distance along the great circle, radius of parallel latitudes, and latitude-longitude problems.

Students solve the problems individually and ask questions if needed.

Step 2: Review of Concepts

Teacher revises the key concepts of distance along the great circle, parallel latitudes, and the radius of the Earth.

Students participate in the review session and clarify doubts.

EVALUATION (5 exercises):

  1. Solve a problem on calculating the distance between two points using the great circle distance formula.
  2. Find the distance between two cities along the parallel of latitude.
  3. Calculate the radius of a parallel at 60° latitude.
  4. Solve a problem on the change in radius along the Earth’s surface.
  5. Solve a problem on the distance between two coordinates using latitude and longitude.

CLASSWORK (5 questions):

  1. Calculate the great circle distance between two points on Earth.
  2. Solve for the radius of the parallel at 30° latitude.
  3. Find the distance along a parallel of latitude at 90° latitude.
  4. Given two coordinates, calculate the distance along the great circle.
  5. Solve a problem involving the radius of the Earth and distance between latitudes.

ASSIGNMENT (5 tasks):

  1. Solve the problem of finding the distance between two places using latitude and longitude.
  2. Calculate the distance between two points using the great circle method.
  3. Research the concept of "geodesic" and explain it.
  4. Solve problems involving the radius of Earth and distance along a parallel.
  5. Review and explain the concept of great and small circles.

 

Instructional Resources:

  • Real Globe
  • Skeletal Globe
  • Charts of Problems on Longitude and Latitude
  • Mathematical formula charts for distance and radius calculation