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National Institute of Education

Mathematics Lesson Plan on Bearings

Cai Simin

054354C23

PGDE (Secondary)

QCM 520 Mathematics I

Asst. Prof. Toh Tin Lam

24th October 2005

Table of Contents

Introduction 1

Overview 1

Concept Map 2

Lesson Plan 3

References 8

Appendix A: Bearings Puzzle 9

Appendix B: MRT & LRT System Map 14

Introduction

Subject : Mathematics 4017

Topic : (21) Bearings

Class : Secondary 3 Express

Ability : Average to High

Duration : 70 minutes

Overview

The topic on Bearings requires good foundation in geometry and trigonometry and it is an application of the knowledge learnt previously.

Students should be shown and be appreciative of the real-life applications of this topic, and the most obvious application is navigation through map reading. When students see the relevance of the subject they are reading to everyday context, and its interdisciplinary application, they are more motivated to learn (Santrock, 2004). In this case, students are expected to apply the map-reading skills they have learnt in Geography to the designed activities.

Activity One, a game of puzzle, is designed to encourage students to discuss math strategies with others (Carpenter, Lindquist, Matthews and Silver, 1983). That it is a game itself will motivate students also (Santrock, 2004), as they may find the drill and practice method too boring. The activity is also pitched at a higher level so that the higher achieving students will find it a challenge as well.

I expect students to face difficulties in reading the diagrams when the Northward direction is not vertically upwards, the “standard image” (Lee, 2005). They may not identify the parallel lines and the angles formed. Students may also face difficulties when given a word problem, and expected to draw their own diagrams.

Concept Map

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Lesson Plan

Learning Aids and Resources

1. Learning Environment

• Lesson is to be conducted in the classroom.

• An Overhead Projector to present the prepared worksheets.

• A Whiteboard for the teacher to conduct the lesson.

• Students should have with them protractors, rulers and glue.

2. Visual Aids

• Compasses and Maps

3. Information Technology

• The usage of Overhead Projector to aid students to complete Activity 1.

4. Worksheets

• Activity 1: Bearings Puzzle

• Activity 2: Travel around Singapore

5. References

• The students should have with them their textbook, Exploring Mathematics: Special/Express 3A by Sin Kwai Meng, edited by Wong Khoon Yoong.

Pre-requisite Knowledge and Skills

Students should already have the following knowledge or skills prior to the lesson:

1. Relating the 4 directions, North, South, East and West,

2. Usage of a compass to read a map,

3. Calculating real distance from a map using scales,

4. Apply properties of regular polygons, in particular, hexagons,

5. Identify alternate and corresponding angles formed by parallel lines,

6. Apply properties of isosceles triangles,

7. Identify the shortest distance from a point to a line is the perpendicular distance,

8. Recall and apply sine rule, and

9. Recall and apply cosine rule.

Specific Instructional Objectives

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

1. Recall the 3 rules: the bearing of a point P, from a reference point O, is measured from North in a clockwise direction, and it is always given in a three-digit number,

2. Interpret bearings from diagrams,

3. Applying the 3 rules to solve real-life problems, for example, reading a map, and

4. Complete the assignment given.

New Concepts and Terms

Bearings, Reference Point

Lesson Outline

1. Explain to students why there is a need for bearings

2. State the 3 rules

3. Make the distinction between measuring an angle in the clockwise or anticlockwise direction

4. Go through simple examples in textbook

5. Execution of Activity one

6. Explain Activity one

7. Execution of Activity two

8. Assigning of Homework

|Teaching / Learning Activities |Materials |Duration |

|Introduction | | |

| | | |

|Pass around compasses and maps in the class. Ask the students if they remember from |Compasses, |1 min |

|their Geography lessons how to read maps as well as the function of the compass. |Maps | |

|Students should be able to answer that the compass gives one the direction, and the | | |

|maps will also give the distance between locations. | | |

| | | |

|Using the whiteboard, recall with the students the 4 main directions, North, South, | | |

|East and West, and they are 2 perpendicular lines intersecting at a reference point O.| | |

|Recall also the directions North-East, North-West, South-East and South-West. | |1 min |

| | | |

|Ask the students if they think knowing these general directions are enough to make | | |

|accurate measurements. Explain to the students that these general directions are not | | |

|accurate enough. 3-figure bearings allow more accuracy, for example, when navigating | | |

|on the sea. Therefore, they will be learning a new topic, “Bearings”, which has the | | |

|real-life application for navigating. Inform students that they will need to | |1 min |

|incorporate all the knowledge from the previous chapters such as sine rule and cosine | | |

|rule, in order to solve the questions. | | |

|Lesson Development | | |

| | | |

|Inform the students there are 3 rules to remember when using bearings. | |8 min |

| | | |

|Bearing of a point P, from a reference point O | | |

|Demonstrate on the board to remind students that when they read a direction from the | | |

|compass, they require 2 points on the map: one as the reference point and the other is| | |

|the direction where they want to head. If P is south of O, then with reference to P, O| | |

|is north of it. | | |

| | | |

|Therefore, students must be clear where the measurement of the bearing is FROM, or the| | |

|reference point. Explain to students that it is very obvious from the questions where | | |

|the reference point is. | | |

| | | |

|Bearing is measured at North in a clockwise direction | | |

|After knowing there is a reference point, draw the North pole at the point. Tell | | |

|student that just like the compass, one needs to know where the North pole is in order| | |

|to get a direction. Therefore, when they solve problems, they will have to draw a | | |

|North pole at the reference point to remind them that the measurement will be done | | |

|from that point. | | |

| | | |

|Then inform them that the measurement is read in a clockwise direction from the North | | |

|pole. Remind students that this is different from what they have learnt from | | |

|trigonometry where the angle is read in an anti-clockwise direction. | | |

| | | |

|Going back to the example on the board, write down “The bearing of P from O = 180°”. | | |

| | | |

|Bearing is always given in a 3-digit figure | | |

|Inform students that although measuring the angles is very straight forward, there is | | |

|something different from merely measuring from a reference point. When writing down | | |

|the bearing of a point with reference to another point, it always will be a 3-digit | | |

|number. For example, | | |

| | | |

|[pic] | | |

| | | |

|The bearing of A from O = 090°. | | |

| | | |

|Share with the students that the reason why there is a need for 3-digit is for | | |

|convention. For example, in war, when a fire order is being given, it is common | | |

|practice to give 3 digits. If one were to say “1, 1, 1” it is understood that it is | | |

|111°. However, if the target is at 30°, and one says “3, 0”, the other party will | | |

|waste time translating it to 30° as he will be expecting a third digit. Therefore, by | | |

|convention, bearings are given in 3 digits. This example will allow the students to | | |

|see how relevant the topic is to everyday context also. | | |

| | | |

|Write on the corner of the board the 3 things to note when using bearings. Remind | | |

|students to refer to them whenever they solve problems. | | |

| | | |

|Remind students that reference point is very important. In the same example, the | | |

|bearing of O from A = 270°. | | |

| | | |

|Ask students what is the maximum value of a bearing they think there possibly is. Most| | |

|students will say 360°. Tell them that it is correct, but it is also acceptable to say| |1 min |

|000°. Therefore, the bearing of O from P = 000° or 360°. | | |

| | | |

|Refer to Skill Practice 7A, Question 1. Allow students to discuss with their partners | | |

|for the various answers. Ask for volunteers for the answers and check for any | | |

|misconception. | | |

| | | |

|Refer to Skill Practice 7A, Question 3. Work through the question with them, showing | | |

|them the proper presentation of the solution. Remind students to draw the North pole | | |

|where the reference point is. Make sure the parallel lines are drawn long enough so | | |

|that students are able to make to see the alternate and corresponding angles formed by| | |

|the parallel lines. | | |

| | | |

|Execute Activity One. Refer to the instructions in Appendix A. This activity also | | |

|reconciles their knowledge of geometric properties in regular polygons. | | |

| | | |

|Discuss the outcomes of the activity with the students. Ask them if they had any | | |

|problems with arranging the triangles, if there were more than 1 possible arrangement.| | |

|For example, the position of Point C with reference to Point B. Therefore, bearings | |1 min |

|only tell us the position of a point with respect to another, but not the distance. | | |

|One cannot be sure where the exact position is unless the distance is given. | | |

| | | |

|Refer to Skill Practice 7B, Question 1. Explain the term “B is due east of A” on the | | |

|board. This question allows the student to revise their cosine rule, and the time to | |5 min |

|check for misconceptions. Students may press “cos 0.8910°” in their calculator, | | |

|instead of substituting the value 0.8910 in the cosine rule equation. | | |

| | | |

|Give an example where the North pole is not pointing directly upwards. Students may | | |

|face problems. Advise them to rotate the book so that they can see north pole in the | |5 min |

|upward direction again. There is no change to the approach to the questions. | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| |Activity One |15 min |

| | | |

| | | |

| | | |

| | |5 min |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | |5 min |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | |2 min |

|Consolidation | | |

| | | |

|Execute Activity Two. This brings in the application of bearing to everyday life, in |Activity Two |15 min |

|Singapore’s context. MRT and LRT route map is easily recognised as most students are | | |

|familiar with public transport system, and reinforce the 4 directions on a compass. | | |

|Make an estimation of 1:2000 for the scale. | | |

| | | |

|Work through at least one question with the students so that they understand the | | |

|instructions. At the same time, enlarge the map on the screen, as the resolution on | | |

|paper for the students may not be too good. | | |

| | | |

|Discuss the answers with the students. Allow for some slight errors. | | |

|Closure | | |

| | | |

|Recap the 3 rules with the students. Ask if they have any difficulties when doing the | |5 min |

|previous activity. | | |

| | | |

|Assign homework to the students: | | |

|Skill Practice 7A: Questions 7, 9 | | |

|Skill Practice 7B: Questions 5, 6, 8, 12, 13. | | |

| | | |

|Provide students with the URL of an online game which requires students to manoeuvre a| | |

|canoe through different gates using bearings. The | | |

|resource/bearings/canoebearings.html. | | |

References

Carpenter, T. P., Lindquist M. M., Matthews, W., & Silver, E. A. (1983). Results of the Third NAEP Mathematics Assessment: Secondary school. Mathematics Teachers, 76 (9), 652-659.

Lee, P. Y. (ed.) (2005). Teaching of Geometry. Teaching Secondary School Mathematics: A Resource Book. Singapore: McGraw Hill.

Ministry of Education: GCE ‘O’ Level Mathematics Syllabus (4017).

Santrock, J. W. (2004). Educational Psychology: Classroom Update. McGraw Hill: International Edition.

Sin, K. M. (2001), Wong, K. Y. (ed.). Exploring Mathematics: Special/Express 3A. Singapore: SNP Pan Pacific Publishing.

Activity One: Bearings Puzzle

Instructions to Teachers:

1. Every student is given a copy of the worksheet titled Activity One: Bearings Puzzle (Students’ Copy), and the 10 cut out triangles of Activity One: Bearings Puzzle (Completed copy).

2. Explain to the students that for each of the 10 congruent equilateral triangles, a point is located at its centroid. Recall some properties with the students:

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3. Using Activity One: Bearings Puzzle (Transparency), draw students’ attention to the 2 hexagons formed by the 10 points. Also, identify the location of point O.

4. There should be as little guidance as possible to the students, allowing them to exercise their own knowledge, and have fun with the activity. If students require assistance, refer to Activity One: Bearings Puzzle (Guide) to point out relevant angles to help them along.

Instructions to Students:

1. You are given 10 small congruent, equilateral triangles, each labelled with a point in the centre. The aim of this activity is to arrange these 10 triangles on the worksheet so that the bearings stated on all the small triangles hold true.

2. There is only one reference point, O, given to you. Subsequently, you are to follow the alphabetical order, from A to I, to arrange the remaining 9 triangles.

3. You are to use geometrical properties of triangles to solve the puzzle.

Activity One: Bearings Puzzle (Students’ Copy)

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Activity One: Bearings Puzzle (Transparency)

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Activity One: Bearings Puzzle (Guide)

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Activity One: Bearings Puzzle (Completed Copy)

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Activity Two: MRT and LRT System Map

Instructions to Students:

1. They are to calculate the distance separating two locations. They are also to find the bearings of

a) Boon Lay from Punggol,

b) Sembawang from Serangoon,

c) Changi Airport from Pasir Ris,

d) Orchard from Choa Chu Kang, and

e) Ang Mo Kio from Kallang.

2. They will draw a straight line connecting the two locations, then using a ruler and a protractor, find the distance separating them, as well as the bearings.

-----------------------

AB = BC = AC

OA = OB = OC = 2OX = 2OY = 2OZ

∡ABC = ∡BCA = ∡CAB = 60°

∡AXC = ∡BYC = ∡CZB = 90°

∡AOC = ∡BOC = ∡AOB = 120°

∡OAC = ∡OCA = ∡OCB = ∡OBC = ∡OBA = ∡OAB = 30°

N

N

N

N

Sine Rule

Cosine Rule

Solution of rig瑨愭杮敬⁤牴慩杮敬൳名楲⁧慲楴獯漠⁦扯畴敳愠杮敬൳匍湩ⱥ挠獯湩⁥湡⁤慴杮湥⁴景ht-angled triangles

Trig ratios of obtuse angles

Sine, cosine and tangent of acute angles

Pythagoras’ Theorem

Sec 3

Similar triangles

Ratio property of sides

Applications

Height, distance, angles of elevation and depression

Bearings

Sec 2

Angle Properties of polygons

Sec 1

Angle properties

Angles formed with common vertex with parallel lines

Angle properties of triangle

Plane & Solid Figures

Special triangles and quads

Names of polygons

O

P

P

O

P

O

A

N

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