Geometry

Coordinates and Graphs — Plotting points on a Cartesian plane
Coordinates and Graphs — Drawing straight line graphs by generating tables of values

By the end of the lesson, the learner should be able to:
- identify the x-axis, y-axis, origin, and four quadrants of a Cartesian plane;
- correctly plot given points using their x- and y-coordinates;
- appreciate the use of the Cartesian plane as a tool for locating positions.

In groups, learners are guided to:
- Draw two perpendicular axes on graph paper, mark equal intervals, and label x and y directions; discuss the terms x-axis, y-axis, and origin
- Plot a range of points including points in all four quadrants, on the axes, and at the origin: e.g. A(2,5), B(–4,–1), C(6,–3), D(0,–5), E(3,0)
- Write the coordinates of given plotted points by reading the x- and y-values from the axes

How do we draw graphs of straight lines?

- Mentor Mathematics Grade 9 pg. 166–168
- Graph paper
- Ruler and pencil
- Digital devices
- Mentor Mathematics Grade 9 pg. 168–170

- Oral questions - Observation - Written exercises

Geometry

Coordinates and Graphs — Drawing parallel lines; establishing that parallel lines have equal gradients
Coordinates and Graphs — Drawing perpendicular lines; establishing that m₁ × m₂ = –1

By the end of the lesson, the learner should be able to:
- draw two or more parallel lines on the same Cartesian plane;
- calculate their gradients and establish that parallel lines have equal gradients (m₁ = m₂);
- find the equation of a line parallel to a given line and passing through a given point.

In groups, learners are guided to:
- Generate tables of values for y = 2x + 1 and y – 2x = 3; draw them on the same Cartesian plane and observe they are parallel
- Calculate the gradient of each line and verify m₁ = m₂; draw three parallel lines and confirm all three have the same gradient
- Find equations of parallel lines: e.g. parallel to y = ½x – 4 passing through P(6,–1); determine value of k in parallel-line problems

How do we use gradients to identify parallel lines?

- Mentor Mathematics Grade 9 pg. 170–174
- Graph paper
- Ruler
- Digital devices / internet (YouTube link)
- Mentor Mathematics Grade 9 pg. 174–179
- Ruler and protractor
- Digital devices

- Written tests - Oral questions - Observation

Geometry

Coordinates and Graphs — Applying straight-line graphs, parallel and perpendicular lines to real-life and mixed problems

By the end of the lesson, the learner should be able to:
- solve mixed problems on equations of lines, parallel lines, and perpendicular lines;
- determine unknown constants in line equations using parallelism or perpendicularity conditions;
- apply graphs of straight lines in real-life situations such as Integrated Science experiments.

In groups, learners are guided to:
- Plot points and draw three lines on the same plane; determine their equations from the graph
- Solve combined problems: find equation of L₁ parallel to y = 2x + 3 through P(2,6); find the gradient and equation of L₂ perpendicular to L₁ at P
- Find value of a in y = 3x + 2 and ay + x = 7 which are perpendicular; find value of m in line through A(2,1) and B(4,m) perpendicular to 3y = 5 – 2x
- Discuss: how Integrated Science uses straight-line graphs for experimental data

How do we apply graphs of straight lines in real-life situations?

- Mentor Mathematics Grade 9 pg. 166–179 (revision)
- Graph paper
- Revision exercise sheets
- Digital devices

- Written assessment - Oral questions - Peer assessment

Geometry

Scale Drawing — Compass bearings (Nθ°E/W, Sθ°E/W) and true bearings (3-digit clockwise notation)
Scale Drawing — Determining compass and true bearings of one point from another using a protractor; back bearings

By the end of the lesson, the learner should be able to:
- draw a compass rose and identify all 8 cardinal directions;
- state the compass bearing of one point from another in the format Nθ°E, Nθ°W, Sθ°E, or Sθ°W;
- state the same direction as a true bearing in 3-digit notation and convert between the two forms.

In groups, learners are guided to:
- Draw a compass rose; locate North, South, East, West, NE, SE, SW, NW and state the true bearing of each
- Follow steps: draw compass at reference point A, join A to B, measure the acute angle from the north/south line, and state the compass bearing (e.g. N60°E)
- Convert between compass and true bearing: S 45°E = 135°; true bearing 228° = S 48°W; practise expressing directions in both forms

How do we use scale drawing in real life?

- Mentor Mathematics Grade 9 pg. 180–183
- Protractors and rulers
- Compass direction diagrams
- Graph paper
- Mentor Mathematics Grade 9 pg. 183–188
- Graph paper
- Maps and compass diagrams
- Digital devices

- Oral questions - Written exercises - Observation

Geometry

Scale Drawing — Making scale drawings from bearing-and-distance data involving 2–3 points
Scale Drawing — Complex scale drawing problems involving 3–4 bearing-and-distance points

By the end of the lesson, the learner should be able to:
- choose a suitable scale and make an accurate scale drawing from bearing-and-distance information;
- read off unknown distances and bearings from the completed scale diagram;
- appreciate the use of scale drawing in navigation and real-life problem solving.

In groups, learners are guided to:
- Make a scale drawing: point B is 400 m due East of A; C is 500 m on a bearing of 135° from B — use 1 cm : 100 m; find bearing of D from A, bearing of B from D, and distance AC
- Draw three schools: B is 3 600 m from A on bearing 075°; C is 4 800 m from B on bearing 165° — find distance AC and bearing of C from A
- Solve problems involving two ships, a coast guard, and a prison watch tower

How do we locate a point using bearing and distance?

- Mentor Mathematics Grade 9 pg. 186–191
- Protractors and rulers
- Graph paper
- Digital devices
- Mentor Mathematics Grade 9 pg. 188–192

- Written tests - Oral questions - Observation

Geometry

Scale Drawing — Identifying and determining angles of elevation by accurate scale drawing

By the end of the lesson, the learner should be able to:
- define angle of elevation as the angle by which the line of sight is raised from the horizontal to see an object above;
- make accurate scale drawings to determine angles of elevation;
- calculate heights and horizontal distances from scale drawings involving angles of elevation.

In groups, learners are guided to:
- Walk to the assembly ground; observe the top of the flagpost; discuss line of sight and the angle of elevation
- Make scale drawings: observer 80 m from an object 40 m high — choose scale 1 cm : 10 m, draw, and measure angle of elevation = 27°
- Solve: tower shadow 35 m long, height 15 m — find angle of elevation; Kirigo's eye at 172 cm views top of a post 8 m away at 60° — find the height of the post

How do we determine the angle of elevation using scale drawing?

- Mentor Mathematics Grade 9 pg. 191–196
- Graph paper
- Protractors and rulers
- Digital devices

- Written tests - Oral questions - Observation

Geometry

Scale Drawing — Identifying and determining angles of elevation by accurate scale drawing

By the end of the lesson, the learner should be able to:
- define angle of elevation as the angle by which the line of sight is raised from the horizontal to see an object above;
- make accurate scale drawings to determine angles of elevation;
- calculate heights and horizontal distances from scale drawings involving angles of elevation.

In groups, learners are guided to:
- Walk to the assembly ground; observe the top of the flagpost; discuss line of sight and the angle of elevation
- Make scale drawings: observer 80 m from an object 40 m high — choose scale 1 cm : 10 m, draw, and measure angle of elevation = 27°
- Solve: tower shadow 35 m long, height 15 m — find angle of elevation; Kirigo's eye at 172 cm views top of a post 8 m away at 60° — find the height of the post

How do we determine the angle of elevation using scale drawing?

- Mentor Mathematics Grade 9 pg. 191–196
- Graph paper
- Protractors and rulers
- Digital devices

- Written tests - Oral questions - Observation

Geometry

Scale Drawing — Identifying and determining angles of depression by accurate scale drawing

By the end of the lesson, the learner should be able to:
- define angle of depression as the angle by which the line of sight is lowered from the horizontal to see an object below;
- make accurate scale drawings to determine angles of depression;
- determine horizontal distances and heights from scale drawings involving angles of depression.

In groups, learners are guided to:
- Discuss the difference between elevation and depression using sketches; establish that the angle of depression equals the alternate angle of elevation
- Make scale drawing: eagle at height 600 m, horizontal distance 960 m to chick — find angle of depression = 32°
- Solve: boat 310 m from a 60 m lighthouse; plane at altitude 8 km between two airports with angles of depression 25° and 31° — find distance between airports and shortest distance to airport B

How do we determine the angle of depression using scale drawing?

- Mentor Mathematics Grade 9 pg. 196–201
- Graph paper
- Protractors and rulers
- Digital devices

- Written assignments - Oral questions - Observation

Geometry

Scale Drawing — Describing land boundaries using bearing and distance from a reference point

By the end of the lesson, the learner should be able to:
- describe the boundary points of a piece of land using bearing and distance from an external reference point;
- construct a scale drawing of the land from a bearing-and-distance table;
- appreciate the use of scale drawing in real-life land surveying.

In groups, learners are guided to:
- Take triangular piece of land PQR; join each vertex to reference point O; measure the bearing and distance for each vertex and record in a table
- Reconstruct the scale drawing of the farm from the bearing-and-distance data
- Solve: quadrilateral farm boundaries described from an external point with A(334°, 225 m), F(352°, 305 m), G(27°, 335 m), H(64°, 335 m)
- Discuss careers in scale drawing and surveying with parents or guardians

How do we use bearing and distance to describe and draw a piece of land?

- Mentor Mathematics Grade 9 pg. 200–203
- Protractors and rulers
- Graph paper
- Maps
- Digital devices

- Written exercises - Oral questions - Observation

Geometry

Similarity and Enlargement — Identifying similar figures; properties: proportional corresponding sides and equal corresponding angles

By the end of the lesson, the learner should be able to:
- identify similar figures by verifying that corresponding sides are proportional and corresponding angles are equal;
- state similar triangles in the correct vertex order;
- appreciate the occurrence of similar shapes in the environment.

In groups, learners are guided to:
- Collect objects from the school environment; sort by similarity and state reasons for grouping
- Verify triangle similarity: compare side ratios DE/AB = EF/BC = DF/AC; or measure corresponding angles with a protractor
- Prove similarity of rectangles (R: 8 cm × 6 cm and T: 12 cm × 9 cm), equilateral triangles, and squares; explain why all equilateral triangles are similar

What are similar objects?

- Mentor Mathematics Grade 9 pg. 205–209
- Ruler and protractor
- Cut-out shapes
- Digital devices

- Oral questions - Written exercises - Observation

Geometry

Similarity and Enlargement — Drawing similar figures by scaling all sides by the same ratio; equal corresponding angles

By the end of the lesson, the learner should be able to:
- draw a figure similar to a given figure by multiplying all sides by the same scale ratio;
- ensure all corresponding angles remain equal in the drawn figure;
- apply similar figures to real-life contexts such as plots and photographs.

In groups, learners are guided to:
- Use the side ratio as a scale: draw rhombus PQRS similar to ABCD (sides 3 cm) but twice as big — sides 6 cm, same angles 70° and 110°
- Draw similar triangles (e.g. right-angled triangle 3-4-5 scaled to QR = 6 cm), rectangles (AB = 8 cm, BC = 6 cm scaled to QR = 9 cm), and parallelograms
- Solve real-life problems: drawing similar rectangular plots of land; two rectangular photographs with given corresponding dimensions

How do we draw a figure similar to a given one?

- Mentor Mathematics Grade 9 pg. 209–213
- Ruler, protractor, and pair of compasses
- Graph paper
- Digital devices

- Written assignments - Oral questions - Observation

Geometry

Similarity and Enlargement — Drawing similar figures by scaling all sides by the same ratio; equal corresponding angles

By the end of the lesson, the learner should be able to:
- draw a figure similar to a given figure by multiplying all sides by the same scale ratio;
- ensure all corresponding angles remain equal in the drawn figure;
- apply similar figures to real-life contexts such as plots and photographs.

In groups, learners are guided to:
- Use the side ratio as a scale: draw rhombus PQRS similar to ABCD (sides 3 cm) but twice as big — sides 6 cm, same angles 70° and 110°
- Draw similar triangles (e.g. right-angled triangle 3-4-5 scaled to QR = 6 cm), rectangles (AB = 8 cm, BC = 6 cm scaled to QR = 9 cm), and parallelograms
- Solve real-life problems: drawing similar rectangular plots of land; two rectangular photographs with given corresponding dimensions

How do we draw a figure similar to a given one?

- Mentor Mathematics Grade 9 pg. 209–213
- Ruler, protractor, and pair of compasses
- Graph paper
- Digital devices

- Written assignments - Oral questions - Observation

Geometry

Similarity and Enlargement — Identifying an enlargement; locating the centre; determining positive and negative scale factors

By the end of the lesson, the learner should be able to:
- identify a transformation as an enlargement by verifying that image-distance/object-distance from a fixed point is constant;
- locate the centre of enlargement by extending lines through corresponding vertices;
- distinguish between positive and negative scale factors based on the relative positions of object and image.

In groups, learners are guided to:
- Study object ABC and image A'B'C'; join AA', BB', CC' and extend to find centre O; verify OA'/OA = OB'/OB = OC'/OC
- Record scale factor; note that when object and image are on opposite sides of O the scale factor is negative (e.g. –2.5)
- Identify real-life examples of enlargement: magnifying lens, projectors, photocopiers, and maps
- State two features common to object and image under enlargement

How do we use enlargement in real-life situations?

- Mentor Mathematics Grade 9 pg. 213–217
- Ruler
- Squared/graph paper
- Digital devices

- Written exercises - Oral questions - Observation

Geometry

Similarity and Enlargement — Constructing enlarged images given centre O and scale factor k; determining and applying the linear scale factor

By the end of the lesson, the learner should be able to:
- construct the image of a figure under enlargement given centre O and scale factor k (positive and negative);
- calculate the linear scale factor (LSF) as image side ÷ corresponding object side;
- use LSF to find unknown sides and solve real-life problems involving similar figures.

In groups, learners are guided to:
- Enlarge triangle ABC with scale factor 2.5, centre O: join O to each vertex, extend the line, and mark OA' = 2.5 × OA; repeat for B and C; join A'B'C'
- Enlarge with negative scale factor –2: extend lines from each vertex through O to the other side and mark OA' = 2 × OA beyond O
- Calculate LSF for similar rectangles (14 cm → 7 cm: LSF = 0.5); use LSF to find unknown dimensions — e.g. QR of a similar rectangle; widths of similar farm plots

How do we determine and apply the linear scale factor of similar figures?

- Mentor Mathematics Grade 9 pg. 217–222
- Ruler and pair of compasses
- Graph paper
- Digital devices

- Written tests - Oral questions - Observation

Geometry

Trigonometry — Identifying hypotenuse, opposite, and adjacent sides relative to a given acute angle in a right-angled triangle

By the end of the lesson, the learner should be able to:
- identify the hypotenuse as the side opposite the right angle and the longest side;
- identify the opposite and adjacent sides with reference to any given acute angle;
- appreciate the relationship between the angles and sides of a right-angled triangle.

In groups, learners are guided to:
- Draw right-angled triangles in different orientations on graph paper; label hypotenuse, opposite, and adjacent for each marked angle
- Discuss how the labelling of opposite and adjacent sides changes when the reference angle changes (standing at A vs. standing at C)
- Identify opposite, adjacent, and hypotenuse for each marked angle in a variety of triangles drawn in different positions

What is the relationship between angles and sides in a right-angled triangle?

- Mentor Mathematics Grade 9 pg. 223–225
- Ruler and protractor
- Graph paper
- Digital devices

- Oral questions - Observation - Written exercises

Geometry

Trigonometry — Identifying and calculating tan θ = opp/adj; sin θ = opp/hyp; cos θ = adj/hyp

By the end of the lesson, the learner should be able to:
- define the three trigonometric ratios (tangent, sine, cosine) for an acute angle in a right-angled triangle;
- calculate the decimal value of each ratio from given side lengths;
- appreciate that the ratio remains constant for a fixed angle regardless of triangle size.

In groups, learners are guided to:
- Measure opposite, adjacent, and hypotenuse sides in three similar right-angled triangles drawn to coincide at vertex A; compute opp/adj, opp/hyp, and adj/hyp for each — note the constant values
- Define: tan θ = opp/adj; sin θ = opp/hyp; cos θ = adj/hyp and express as decimals
- Express each trig ratio as a fraction and as a decimal for various triangles with given side lengths

How do we express trigonometric ratios from a right-angled triangle?

- Mentor Mathematics Grade 9 pg. 225–232
- Ruler and protractor
- Graph paper
- Digital devices

- Written assignments - Oral questions - Observation

Geometry

Trigonometry — Reading tables of sine, cosine, and tangent for acute angles; using mean difference columns

By the end of the lesson, the learner should be able to:
- read the sine, cosine, and tangent of acute angles from mathematical tables including the mean difference (ADD/SUBTRACT) column;
- find an angle given its sine, cosine, or tangent from tables;
- note that the cosine table uses the SUBTRACT column (cosine decreases as angle increases).

In groups, learners are guided to:
- Study the structure of trig tables: x° column, 0.0–0.9 main columns, ADD/SUBTRACT difference columns
- Read values step by step: tan 5.8° = 0.1016; tan 6.37° = 0.1116 using ADD column; sin 62.6° = 0.8878; sin 33.47° = 0.5515
- Find an angle from its ratio: tan θ = 0.8571 → θ = 40.6°; sin x = 0.9639 → x = 74.56°; cos x = 0.1234 → x = 82.91° using SUBTRACT column for cosine

How do we use trigonometric tables to find ratios and angles?

- Mentor Mathematics Grade 9 pg. 232–240
- Mathematical trig tables (sin, cos, tan)
- Scientific calculators
- Digital devices

- Written tests - Oral questions - Observation

Geometry

Trigonometry — Reading tables of sine, cosine, and tangent for acute angles; using mean difference columns

By the end of the lesson, the learner should be able to:
- read the sine, cosine, and tangent of acute angles from mathematical tables including the mean difference (ADD/SUBTRACT) column;
- find an angle given its sine, cosine, or tangent from tables;
- note that the cosine table uses the SUBTRACT column (cosine decreases as angle increases).

In groups, learners are guided to:
- Study the structure of trig tables: x° column, 0.0–0.9 main columns, ADD/SUBTRACT difference columns
- Read values step by step: tan 5.8° = 0.1016; tan 6.37° = 0.1116 using ADD column; sin 62.6° = 0.8878; sin 33.47° = 0.5515
- Find an angle from its ratio: tan θ = 0.8571 → θ = 40.6°; sin x = 0.9639 → x = 74.56°; cos x = 0.1234 → x = 82.91° using SUBTRACT column for cosine

How do we use trigonometric tables to find ratios and angles?

- Mentor Mathematics Grade 9 pg. 232–240
- Mathematical trig tables (sin, cos, tan)
- Scientific calculators
- Digital devices

- Written tests - Oral questions - Observation

Geometry

Trigonometry — Using a scientific calculator to find trig ratios and inverse trig functions for acute angles

By the end of the lesson, the learner should be able to:
- use the sin, cos, and tan keys on a scientific calculator to find trig ratios of given angles;
- use the shift + sin/cos/tan keys to find an angle from its ratio (inverse trig);
- compare calculator results with table values and appreciate the efficiency of technology.

In groups, learners are guided to:
- Ensure the calculator is set to degree mode (D displayed at top); discuss calculator key sequences
- Calculate: sin 45° = 0.7071; cos 71° = 0.3256; tan 55° = 1.428 using the calculator — give answers to 4 significant figures
- Find angles: tan⁻¹(0.3764) = 20.63°; sin⁻¹(0.500) = 30°; cos⁻¹(0.9998) = 1.28° using shift + ratio key
- Use IT/digital devices or other resources to explore trig ratios

How do we use a calculator to find trigonometric ratios and angles?

- Mentor Mathematics Grade 9 pg. 240–242
- Scientific calculators
- Mathematical trig tables
- Digital devices

- Written assignments - Oral questions - Observation

Geometry

Trigonometry — Using sin, cos, and tan to calculate unknown sides and angles in right-angled triangles

By the end of the lesson, the learner should be able to:
- select the appropriate trigonometric ratio to find an unknown side given one side and one acute angle;
- use inverse trig to find an unknown angle given two sides;
- apply trig ratios to solve real-life problems involving right-angled triangles.

In groups, learners are guided to:
- Identify the correct ratio: use tan 42° = x/7 → x = 7 × tan 42° = 6.303; use cos 36° = x/7 → x = 7 × cos 36° = 5.663
- Find angle a: sin a = 10/18 = 0.5556 → a = sin⁻¹(0.5556) = 33.75°
- Solve: length of perpendicular height of an equilateral triangle of side 10 cm; length of diagonal of a square with perimeter 36 cm; sides of an equilateral triangle with perpendicular height 18 cm

How do we apply trigonometric ratios to calculate lengths and angles of right-angled triangles?

- Mentor Mathematics Grade 9 pg. 238–243
- Mathematical trig tables
- Scientific calculators
- Digital devices

- Written assessment - Oral questions - Observation