teaching-notes — Mathematics (Bearings)

MATHEMATICS GRADE 10: BEARINGS AND SCALE DRAWING KEY DEFINITIONS • Bearing: The direction of one point relative to another. It is always measured clockwise from the North line. • Cardinal Points: The four main points of a compass: North (N), East (E), South (S), and West (W). Intermediate points include Northeast (NE), Southeast (SE), Southwest (SW), and Northwest (NW). • True Bearing: A bearing measured clockwise from the North line, expressed in three digits (e.g., 045°, 180°, 270°). If the angle is less than 100°, it is preceded by a zero. • Compass Bearing (or Conventional Bearing): A bearing measured from the North or South line towards East or West, expressed as an angle between 0° and 90° (e.g., N30°E, S60°W). • North Line: A vertical line pointing upwards, representing the direction of North. It is the reference line for all bearing measurements. • Scale: The ratio between the distance on a drawing (or map) and the actual distance on the ground. It can be expressed as a ratio (e.g., 1:100), a fraction, or a statement (e.g., 1 cm represents 10 m). • Scale Drawing: A drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (the scale). DETAILED CONTENT Bearing and scale drawing are essential tools for representing position and direction, with wide applications in navigation, mapping, and construction. Understanding these concepts allows for accurate communication of locations and distances. 1. REPRESENTING POSITION AND DIRECTION To represent position and direction, we use cardinal points and bearings. Cardinal Points The four cardinal points (North, East, South, West) form the fundamental basis of direction. Intermediate directions like Northeast, Southeast, Southwest, and Northwest provide more precision. When drawing, North is always assumed to be at the top of the page unless otherwise specified.

THE EIGHT CARDINAL POINTS

True Bearings True bearings provide a precise way to state a direction using a three-digit number. • Measurement: Always measured clockwise from the North line. • Notation: Always written with three digits. For example, East is 090°, South is 180°, West is 270°. If the angle is less than 100°, a leading zero is used (e.g., 45° is written as 045°). • Drawing: To draw a true bearing, first draw a North line at the starting point. Then, use a protractor to measure the angle clockwise from the North line. Example True Bearings: • 000° or 360° for North • 045° for Northeast • 090° for East • 135° for Southeast • 180° for South • 225° for Southwest • 270° for West • 315° for Northwest

TRUE BEARINGS

Compass Bearings Compass bearings provide direction relative to the nearest North or South line, then specifying whether to turn East or West. • Measurement: Measured from either the North (N) or South (S) line. The angle is always acute (between 0° and 90°). • Notation: Expressed as N or S, followed by the angle, then E or W. For example, N30°E means 30° East of North. S45°W means 45° West of South. • Drawing: Draw a North-South line. Then, measure the specified angle from either North or South towards East or West. Example Compass Bearings: • N0°E or N (North) • N45°E (Northeast) • N90°E or E (East) • S45°E (Southeast) • S0°E or S (South) • S45°W (Southwest) • N90°W or W (West) • N45°W (Northwest) 2. SCALE DRAWING TECHNIQUES Scale drawing is the process of creating a representation of a real-world object or area, where all dimensions are proportionally reduced or enlarged according to a specific scale. This is crucial for planning, design, and navigation. Understanding Scale Scale is a ratio that relates a distance on a drawing to the corresponding actual distance. It can be expressed in several ways: • Ratio Scale: 1:100 or 1:100,000. This means 1 unit on the drawing represents 100 or 100,000 of the same units in reality. For example, 1 cm on the map represents 100 cm (or 1 m) in reality. • Verbal Scale (Statement Scale): "1 cm represents 10 km". This directly states the relationship between units. • Graphic Scale (Bar Scale): A line marked with divisions that represent actual distances. This is useful because it remains accurate even if the drawing is resized. Calculations with Scale The key to scale drawing is the relationship:

Scale Formula

Formula

Scale = Distance on DrawingActual Distance

Note: Ensure units are consistent when using ratio scales.

Figure: Formula for calculating scale

This formula can be rearranged to find any of the three components: • Actual Distance = Distance on DrawingScale • Distance on Drawing = Scale × Actual Distance Steps for Scale Drawing: 1. Choose a suitable scale: This depends on the size of the object/area and the size of the paper. For example, if drawing a 100m field on A4 paper, 1 cm : 10 m (1:1000) might be suitable. 2. Draw the North line: Always start by establishing the North direction. 3. Plot the starting point: Mark the initial position. 4. Draw lines for each leg of the journey/object: * Measure the bearing from the North line using a protractor. * Calculate the length of the line segment on the drawing using the chosen scale (Distance on Drawing = Scale × Actual Distance). * Draw the line to the calculated length. 5. Repeat for subsequent points: If there are multiple segments, draw a new North line at each turning point and measure the new bearing from that North line. Applications in Real Life: • Navigation: Pilots, sailors, and hikers use bearings and maps with scales to determine their position and plot courses. • Town Planning: Architects and urban planners use scale drawings to design buildings and city layouts. • Construction: Engineers use scale drawings (blueprints) to guide the construction of bridges, roads, and buildings. • Land Surveying: Surveyors use bearings and distances to map land boundaries.

SCALE DRAWING OF A JOURNEY

COMPARISON TABLE

Comparison of True Bearings and Compass Bearings

Feature	True Bearing	Compass Bearing
Reference Line	Always from the North line	From North (N) or South (S) line
Direction of Measurement	Clockwise only	Towards East (E) or West (W)
Number of Digits	Always three digits (e.g., 045°, 180°)	Angle between 0° and 90° (e.g., N45°E, S30°W)
Uniqueness	Each direction has a unique 3-digit number.	Can be ambiguous without N/S and E/W.

Figure: Key differences between true and compass bearings

LEARNING ACTIVITIES 1. Compass Rose Drawing: Students draw a large compass rose, marking all 8 cardinal points and their corresponding true bearings and compass bearings. 2. "Treasure Hunt" (Classroom/School Grounds): Provide students with a starting point and a series of bearing and distance instructions (e.g., "From the flagpole, walk 50 m on a bearing of 090° to find the first clue. From there, walk 30 m on a bearing of S45°W..."). Students use compasses and measuring tapes to navigate. 3. Map Interpretation: Provide local Zambian maps (e.g., Lusaka city map, Kafue National Park map) with a scale. Ask students to: * Find the true bearing from one landmark to another. * Calculate the actual distance between two points using the map's scale. * Draw the route for a short journey on the map. 4. Scale Model Project: Students choose a simple object (e.g., a classroom, a phone, a house plan) and create a scale drawing of it, clearly stating the scale used and labelling dimensions. WORKED EXAMPLES Worked Example 1: Drawing a Path with Bearings and Scale A student walks 2 km from point A on a true bearing of 060° to point B. From point B, they walk 3 km on a true bearing of 150° to point C. Using a scale of 1 cm to 500 m, draw the path of the student.

Solution

Given:	Journey 1: Actual distance AB = 2 km, Bearing = 060° Journey 2: Actual distance BC = 3 km, Bearing = 150° Scale = 1 cm : 500 m
Find:	Scale drawing of the path ABC.
Convert Scale:	1 km = 1000 m 500 m = 5001000 km = 0.5 km So, scale is 1 cm : 0.5 km
Calculate Scaled Distances:	Length AB on drawing = Actual Distance AB ÷ 0.5 km/cm = 2 km ÷ 0.5 km/cm = 4 cm Length BC on drawing = Actual Distance BC ÷ 0.5 km/cm = 3 km ÷ 0.5 km/cm = 6 cm
Drawing Steps:	1. Mark point A. Draw a North line at A. 2. From A, measure 060° clockwise from the North line. Draw a line 4 cm long to mark point B. 3. At point B, draw a new North line. 4. From B, measure 150° clockwise from the new North line. Draw a line 6 cm long to mark point C.
Answer:	[A neatly drawn diagram showing points A, B, C with North lines, bearings 060° and 150°, and segment lengths 4 cm and 6 cm respectively, with the scale clearly stated.]

Worked Example: Drawing a path using bearings and scale

Worked Example 2: Calculating Actual Distance from a Map On a map with a scale of 1:50,000, the distance between two villages, Kanyama and Chawama, is measured as 8 cm. Calculate the actual distance between the two villages in kilometres.

Solution

Given:	Scale = 1:50,000 \| Distance on map = 8 cm
Find:	Actual distance in kilometres.
Interpret Scale:	1 cm on map represents 50,000 cm in reality.
Convert Units:	1 km = 1000 m = 100,000 cm So, 50,000 cm = 50,000100,000 km = 0.5 km Therefore, 1 cm represents 0.5 km.
Calculate Actual Distance:	Actual Distance = Distance on map × (Actual distance per cm) Actual Distance = 8 cm × 0.5 km/cm
Answer:	Actual Distance = 4 km

Worked Example: Calculating actual distance from a map

ASSESSMENT QUESTIONS 1. A boat sails from a harbour H on a true bearing of 075° for 60 km to a point P. From P, it then sails on a true bearing of 160° for 80 km to a point Q. * (a) Using a scale of 1 cm to 10 km, draw an accurate diagram to show the path of the boat. * (b) From your diagram, find: * (i) The actual distance from H to Q. * (ii) The true bearing of Q from H. 2. A map has a scale of 1:25,000. * (a) If the actual distance between two towns is 5 km, what would be the distance between them on the map in centimetres? * (b) If a forest on the map is represented by a rectangle of length 4 cm and width 3 cm, calculate the actual area of the forest in square kilometres. 3. An aeroplane flies from City A to City B on a compass bearing of N50°E. It then flies from City B to City C on a true bearing of 290°. * (a) Sketch a diagram showing the relative positions of City A, City B, and City C. Clearly mark the North lines and the given bearings. * (b) If the distance from City A to City B is 400 km and the distance from City B to City C is 300 km, describe how you would use a scale drawing to find the distance and bearing of City C from City A. (You do not need to draw it, just describe the steps). COMMON DIFFICULTIES • Incorrect North Line: Students often forget to draw a new North line at each turning point in a journey, or they draw it incorrectly (not parallel to the initial North line). • Measuring Bearings: Confusion between true bearings (clockwise from North) and compass bearings (from N/S towards E/W). Incorrectly measuring angles with a protractor (e.g., measuring anti-clockwise or from the wrong reference). • Scale Conversion Errors: Misinterpreting ratio scales (e.g., 1:50,000 means 1 unit on map is 50,000 units in reality, not 1 cm is 50,000 km). Errors in converting units (e.g., cm to km). • Accuracy in Drawing: Not using a sharp pencil, accurate ruler, and protractor can lead to significant errors in scale drawings. • Back Bearings: Forgetting that the bearing from B to A is different from the bearing from A to B (it's the forward bearing ± 180°). QUICK REFERENCE

Quick Reference: Bearings and Scale Drawing

True Bearing Rules	• Always 3 digits (e.g., 045°) • Measured clockwise from North (0°) • North = 000°, East = 090°, South = 180°, West = 270°
Compass Bearing Rules	• From North or South, towards East or West • Angle between 0° and 90° (e.g., N30°E, S60°W)
Scale Formula	Scale = Distance on DrawingActual Distance (Ensure consistent units before calculation)
Drawing Tips	• Draw North line at every turning point. • Use protractor for angles, ruler for lengths. • Convert actual distances to drawing distances using scale.

Figure: Summary of key rules and formulas for bearings and scale drawing

END OF TEACHING NOTES

teaching-notes — Mathematics (Bearings)

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