Mathematics - Senior Secondary 2 - Bearing

Bearing

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TERM: 2ND TERM

WEEK: 8
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Bearing
Focus: Definition and drawing of 4, 8 and 16 cardinal points; notation for bearings (cardinal and 3-digit); making sketches involving lengths and bearings; problem solving on lengths, angles and bearings.

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

  1. Define bearing and accurately draw the 4, 8 and 16-point compass rose.
  2. State the two types of bearing notation (cardinal and 3-digit) and give examples of each.
  3. Sketch word-problems on bearings, showing given lengths and angles.
  4. Solve bearing problems involving distances and angles using Pythagoras’ theorem, trigonometric ratios, and sine & cosine rules.

INSTRUCTIONAL TECHNIQUES:
• Lecture and demonstration
• Guided practice
• Group discussion
• Practical drawing exercises
• Problem-solving sessions

INSTRUCTIONAL MATERIALS:
• Compass-rose chart (4-, 8- and 16-point)
• Ruler, pencil, protractor
• Handouts with blank compass diagrams and grid paper
• Computer-assisted resources (GeoGebra applet or PowerPoint)
• Worksheets for drawing and problem-solving

 

PERIOD 1 & 2: Definition & Drawing of Cardinal Points

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

1 – Introduction

Defines bearing: the clockwise angle measured from North. Shows compass-rose chart with 4 points (N, E, S, W).

Listen, copy definition, ask questions.

2 – 8-Point

Adds NE, SE, SW, NW between the 4 cardinal points. Demonstrates spacing at 45° intervals.

Observe, note how angles subdivide the circle.

3 – 16-Point

Inserts NNE, ENE, ESE, SSE, SSW, WSW, WNW, NNW at 22½° intervals. Relates to clock face analogy.

Participate in placing intermediate points on blank diagram.

4 – Practice

Distributes handouts; guides students to draw 4-, 8-, then 16-point compass within time.

Draw diagrams, label all points; ask for clarification as needed.

NOTE ON BOARD:

  • 4-point: N, E, S, W (90° apart)
  • 8-point: add NE, SE, SW, NW (45° apart)
  • 16-point: add NNE, ENE, ESE, SSE, SSW, WSW, WNW, NNW (22½° apart)

EVALUATION (5 exercises):

  1. List the 4 cardinal points.
  2. At what angle (°) is SE from North?
  3. Name the points halfway between N and NE in a 16-point rose.
  4. Draw and label an 8-point compass rose.
  5. Give one real-life use of a 16-point compass.

CLASSWORK (5 questions):

  1. Draw a 4-point compass and write the angles.
  2. Draw an 8-point rose and mark 0°, 45°, 90°, … 315°.
  3. Extend your 8-point to a 16-point rose; label all 16.
  4. What is the bearing of WNW in degrees?
  5. Which two points lie at 180° and 202½°?

ASSIGNMENT (5 tasks):

  1. Research another navigation tool that uses a compass rose.
  2. Illustrate a 16-point compass in your notebook.
  3. List all 16 point-names in order, clockwise from North.
  4. Explain why 16 points might be more useful than 4 in surveying.
  5. Describe a scenario where intermediate compass points are essential.

 

PERIOD 3 & 4: Bearing Notation & Sketching

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

1 – Cardinal Notation

Explains “N 30° E” means 30° east of North, and “S 45° N” means 45° north of South. Gives examples.

Copy notation rules; convert given examples.

2 – 3-Digit Notation

Shows 3-digit system: measure clockwise from North; e.g. 075°, 350°.

Note that 000° ≡ N, 090° ≡ E, etc.; practice naming.

3 – Sketching Lines

Demonstrates on grid how to draw a line of given length at given bearing using protractor and ruler.

Follow along: draw at least two sample lines.

4 – Guided Practice

Gives mixed exercises: convert between cardinal & 3-digit, then sketch lines of length 5 cm at bearings N20°W, 145°, etc.

Work in pairs to convert and sketch; check each other’s work.

NOTE ON BOARD:

  • Cardinal: N θ E, S φ W, etc.
  • 3-digit: θ° (clockwise from N)
  • To sketch: mark North, measure θ° clockwise, draw line of given length.

EVALUATION (5 exercises):

  1. Convert N 60° W to 3-digit notation.
  2. Write 225° in cardinal form.
  3. Sketch a 6 cm line at bearing S 30° E.
  4. Sketch a 4 cm line at 300°.
  5. What is the bearing of E 15° N in 3-digit form?

CLASSWORK (5 questions):

  1. Convert 315° to cardinal notation.
  2. Convert S 20° W to degrees.
  3. Sketch 7 cm at 110°.
  4. Sketch 5 cm at N 40° E.
  5. Name the bearing 025° in cardinal form.

ASSIGNMENT (5 tasks):

  1. Convert five random cardinal notations to 3-digit.
  2. Convert five 3-digit bearings to cardinal.
  3. Sketch three lines of given lengths & bearings.
  4. Explain step-by-step how to use a protractor for bearings.
  5. Find and record a real map bearing example (e.g. hiking map).

 

PERIOD 5: Problem Solving with Bearings

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

1 – Method Intro

Reviews Pythagoras, trig ratios, sine & cosine rules for solving triangles.

Listen and recall relevant formulas.

2 – Example 1

Solves: “From A, walk 8 km at 045° to B, then 6 km at 135° to C. Find AC.” Uses cosine law.

Observe each step, ask clarifying questions.

3 – Example 2

Solves: “From P, walk 10 m at S 20° E to Q, then 7 m at S 70° W to R. Find PR.” Breaks into components, uses Pythagoras.

Take notes, check intermediate calculations.

4 – Guided Practice

Distributes problem sheet; students solve two given bearing-distance problems in pairs, with teacher support.

Work out answers, sketch diagrams, apply formulas.

NOTE ON BOARD:

  • Cosine rule: a² = b² + c² – 2bc cos A
  • Resolve into N–S, E–W components when bearings not symmetric.

WORKINGS FOR EXAMPLES:

  1. Example 1: ∠BÂC = difference of bearings = 135°–45° = 90° → right angle → AC = √(8²+6²) = 10 km.
  2. Example 2: Resolve Q→R into components; find displacement PR by √((10 sin 20° – 7 sin 70°)² + (10 cos 20° + 7 cos 70°)²).

EVALUATION (5 exercises):

  1. A→B: 5 km at 090°, B→C: 5 km at 180°; find AC.
  2. A→B: 12 m at N 30° E, B→C: 9 m at N 60° W; find AC.
  3. Use cosine rule to solve a triangle with sides 7, 10 and included angle 50°.
  4. Resolve a 15 km walk at 250° into components.
  5. Find bearing from C back to A in Example 1.

CLASSWORK (5 questions):

  1. From D, 6 km at 350° to E, then 8 km at 080° to F; find DF.
  2. From G, 9 m at S 45° W, then 4 m at E 30° N; find displacement.
  3. Solve triangle with sides 5, 7 and angle between 70° using cosine rule.
  4. Convert your answer in (3) to a bearing problem: sketch the position.
  5. Explain why resolving into components helps with non-right triangles.

ASSIGNMENT (5 tasks):

  1. Create and solve two self-devised bearing–distance problems.
  2. Explain, with diagram, how you would find a ship’s bearing using two landmarks.
  3. Find an online navigation problem and solve it step-by-step.
  4. Write a brief report on the use of bearings in aviation or maritime navigation.
  5. Reflect on which method (Pythagoras vs. cosine rule vs. components) you find most intuitive and why.