Geometry

Coordinates and Graphs — Plotting points on a Cartesian plane

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

- Oral questions - Observation - Written exercises

Geometry

Coordinates and Graphs — Drawing straight line graphs by generating tables of values
Coordinates and Graphs — Drawing parallel lines; establishing that parallel lines have equal gradients

By the end of the lesson, the learner should be able to:
- generate a table of values for a given linear equation;
- plot the points and join them to draw a straight line graph;
- determine the equation of a straight line from a given graph.

In groups, learners are guided to:
- Generate a table of values for y = 3x – 3 by substituting chosen x-values; plot the points and join them
- Draw straight line graphs for equations such as y = 2x + 4, y + 2x = 3, y = –2x + 3, x + y = 5, and 3y = 9x – 12
- Read equations from given straight line graphs drawn on a Cartesian plane

How do we interpret graphs of straight lines?

- Mentor Mathematics Grade 9 pg. 168–170
- Graph paper
- Ruler and pencil
- Digital devices
- Mentor Mathematics Grade 9 pg. 170–174
- Ruler
- Digital devices / internet (YouTube link)

- Written assignments - Oral questions - Observation

Geometry

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

By the end of the lesson, the learner should be able to:
- draw perpendicular line pairs on the same Cartesian plane;
- verify that the product of gradients of perpendicular lines equals –1;
- find the equation of a line perpendicular to a given line and passing through a given point.

In groups, learners are guided to:
- Draw y = –2x + 2 and y = ½x + 2; use a protractor to confirm the 90° angle; verify: (–2) × (½) = –1
- Draw y = ¼x + 5 and y = –4x – 2; confirm perpendicularity by measuring the angle
- Find gradient of line w perpendicular to m (y = ¾x – 4) at Q(4,–1); write the equation of w; solve exercises finding perpendicular line equations through given points

How do we use gradients to identify perpendicular lines?

- Mentor Mathematics Grade 9 pg. 174–179
- Graph paper
- Ruler and protractor
- Digital devices

- Written assignments - 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)

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

- Oral questions - Written exercises - Observation

Geometry

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:
- determine the compass and true bearing of one point from another using a protractor;
- determine back bearings from given forward bearings;
- solve problems involving bearings of multiple points from a single reference location.

In groups, learners are guided to:
- Study a map of Hekima village; use a protractor to determine the compass and true bearing of the dispensary, quarry, mosque, market, and school from the tower
- Determine: if bearing of Q from P = 120°, bearing of P from Q = 300°; practise finding back bearings for various cases
- Solve: find bearing of R from P, bearing of Q from R, and bearing of Q from P using the given three-point diagram

How do we determine the bearing of one point from another?

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

- Written assignments - Oral questions - 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 — 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 — 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 — 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 — 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

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 — 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

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

Data Handling and Probability

Data Interpretation (Grouped Data) — Determining appropriate class width; drawing frequency distribution tables
Data Interpretation (Grouped Data) — Identifying the modal frequency and modal class from a frequency distribution table

By the end of the lesson, the learner should be able to:
- determine the range and calculate an appropriate class width for a given data set;
- group raw data into classes and draw a frequency distribution table using tally marks;
- appreciate the importance of organising data into groups for easier interpretation.

- Have learners each choose a number between 1 and 100; find the range, determine an appropriate class width (5–12 classes) and form the classes
- Apply: masses of 40 Hekima Junior School learners (range = 28 kg; class width 5 gives 6 classes: 30–34, 35–39, …, 55–59)
- Tally the marks of 60 Tiifu Junior School learners (range = 76; class width 10 gives 8 classes) and complete the frequency distribution table
- Use digital devices or other resources to organise and represent grouped data

How do we interpret data?

- Mentor Mathematics Grade 9 pg. 224–229
- Graph paper and exercise books
- Digital devices
- Mentor Mathematics Grade 9 pg. 229–231
- Frequency distribution tables

- Oral questions - Observation - Written exercises

Data Handling and Probability

Data Interpretation (Grouped Data) — Calculating the mean of grouped data using midpoints (x̄ = Σfx ÷ Σf)

By the end of the lesson, the learner should be able to:
- find the midpoint of each class by averaging the class limits;
- calculate Σfx by multiplying each midpoint by its frequency;
- determine the mean using the formula x̄ = Σfx ÷ Σf and apply it to real-life data.

- Introduce the midpoint: e.g. midpoint of 0–4 = (0+4)/2 = 2; build an extended table with columns: Class / Midpoint (x) / Frequency (f) / fx
- Work through Example 4: trucks crossing a weighing bridge — Σf = 40, Σfx = 520, mean = 13 tonnes
- Solve: mean of marks of 40 learners using class width 10 (classes 20–29 to 80–89); number of daily calls at a customer care office; number of trees planted in 20 villages
- Use IT devices or other materials to verify mean calculations

How do we calculate the mean of grouped data?

- Mentor Mathematics Grade 9 pg. 231–234
- Exercise books
- Scientific calculators
- Digital devices

- Written assignments - Oral questions - Observation

Data Handling and Probability

Data Interpretation (Grouped Data) — Building cumulative frequency columns; identifying the median class

By the end of the lesson, the learner should be able to:
- define cumulative frequency and build a cumulative frequency column by successively adding frequencies;
- determine the median class by finding the class containing the N/2 position;
- identify the values of L, cfa, fm, and im needed in the median formula.

- Brainstorm the meaning of cumulative frequency; build the column by adding frequencies row by row: first cf = f₁; second cf = f₁ + f₂; and so on
- Use the grouped data of 50 learners (class 10–14 to 45–49) to build the cumulative frequency column and confirm the last cf = Σf = 50
- Identify the median class: N/2 = 40/2 = 20 → the class containing the 20th value is 50–59 (cf jumps from 17 to 23)
- Identify: L (lower class boundary of median class), cfa (cf of class above), fm (frequency of median class), im (class width)

How do we find the middle value of grouped data?

- Mentor Mathematics Grade 9 pg. 234–236
- Exercise books
- Digital devices

- Oral questions - Written exercises - Observation

Data Handling and Probability

Data Interpretation (Grouped Data) — Calculating the median using the formula: Median = L + [(N/2 − cfa) ÷ fm] × im

By the end of the lesson, the learner should be able to:
- apply the median formula using the values identified from the cumulative frequency table;
- correctly compute L as the average of the lower boundary of the median class and the upper boundary of the class above it;
- determine the median of grouped data from real-life situations and appreciate its use.

- Work through Example 5: 40 learners' Mathematics marks — median class 50–59; L = (49+50)/2 = 49.5; cfa = 17; fm = 6; im = 10 → Median = 49.5 + [(20−17)/6] × 10 = 54.5
- Work through Example 6: vaccination ages of 82 people — median class 11–15; L = 10.5; cfa = 29; fm = 16; im = 5 → Median = 14.25
- Solve: electricity units used by 70 customers; masses of 30 hospital patients; 400 m race times for 50 learners
- Use IT devices to verify median calculations

How do we calculate the median of grouped data?

- Mentor Mathematics Grade 9 pg. 236–238
- Exercise books
- Scientific calculators
- Digital devices

- Written tests - Oral questions - Observation

Data Handling and Probability

Data Interpretation (Grouped Data) — Mixed problems on class width, frequency tables, modal class, mean, and median

By the end of the lesson, the learner should be able to:
- solve mixed problems covering all grouped data concepts: class width, frequency tables, modal class, mean, and median;
- collect, organise, and interpret real-life data;
- appreciate data interpretation in real-life situations such as health, agriculture, and school performance.

In groups, learners are guided to:
- Collect real-life data: use distances from school or home to health facilities using different routes; organise into a frequency table, identify the modal class, calculate mean and median
- Work through comprehensive revision exercises involving full data sets from raw data to table to modal class to mean to median
- Discuss applications: Integrated Science data, Social Studies population data, Agricultural harvest records
- Use digital devices or other materials to search for and interpret real-life data sets

How do we use grouped data interpretation in real-life situations?

- Mentor Mathematics Grade 9 pg. 224–238 (revision)
- Revision exercise sheets
- Scientific calculators
- Digital devices

- Written assessment - Oral questions - Peer assessment

Data Handling and Probability

Probability — Experiments involving equally likely outcomes; P(event) = favourable outcomes ÷ total outcomes
Probability — Determining the range of probability; P(certain event) = 1; P(impossible event) = 0; 0 ≤ P(A) ≤ 1; P(A') = 1 − P(A)

By the end of the lesson, the learner should be able to:
- identify equally likely outcomes in experiments such as tossing a coin or rolling a die;
- calculate the probability of a simple event using P = favourable outcomes ÷ total possible outcomes;
- appreciate that equally likely events have equal chances of occurring.

In groups, learners are guided to:
- Toss a coin repeatedly and record outcomes; discuss: is there a side that will always face up? Establish that head and tail are equally likely
- Roll a regular die; list all 6 equally likely outcomes; find P(5) = 1/6, P(3) = 1/6
- Solve: triangular pyramid (4 faces), basket with one red and one blue pen, five girls with tags 1–5, Sande's three pens (blue, black, red)
- Recall Grade 8 probability; discuss how prior learning connects to the current work

Why is probability important in real-life situations?

- Mentor Mathematics Grade 9 pg. 239–241
- Coins and dice
- Coloured pens / objects in a bag
- Digital devices
- Mentor Mathematics Grade 9 pg. 241–243

- Oral questions - Observation - Written exercises

Data Handling and Probability

Probability — Identifying mutually exclusive events; P(A or B) = P(A) + P(B) (addition law)

By the end of the lesson, the learner should be able to:
- define mutually exclusive events as events where the occurrence of one prevents the occurrence of the other;
- apply the addition law: P(A or B) = P(A) + P(B) for mutually exclusive events;
- identify mutually exclusive events in real-life situations and solve related problems.

In groups, learners are guided to:
- Toss a coin once; discuss: can head and tail both face up at the same time? Establish mutual exclusivity
- Identify real-life mutually exclusive events: at school or at home; lunch at home or at school; football or volleyball choice
- Roll a die: P(1 or 2) = 1/6 + 1/6 = 2/6; P(even number) = P(2) + P(4) + P(6) = 3/6; P(3 or 5 or 4) = 3/6
- Solve: cards numbered 1–9 (P(odd), P(prime), P(prime or even)); spinner numbered 1–8; word MUTUALLY written on separate cards

How do we calculate the probability of mutually exclusive events?

- Mentor Mathematics Grade 9 pg. 243–247
- Coins and dice
- Number cards
- Spinners
- Digital devices

- Written tests - Oral questions - Observation

Data Handling and Probability

Probability — Performing experiments involving independent events; P(A and B) = P(A) × P(B) (multiplication law)

By the end of the lesson, the learner should be able to:
- define independent events as events where the outcome of one does not affect the outcome of the other;
- apply the multiplication law: P(A and B) = P(A) × P(B) for independent events;
- solve real-life problems involving two or more independent events including with and without replacement.

- One learner holds a coin, another holds a die; toss simultaneously — discuss: does the coin outcome affect the die outcome? Establish independence
- Establish: the word "and" in probability means multiply: P(A and B) = P(A) × P(B)
- Work through: basket with 4 red and 3 green balls (with replacement) — P(red then green) = 4/7 × 3/7 = 12/49; P(all red) = 16/49
- Solve: pink and orange marbles without replacement; three learners hitting a target with different individual probabilities; coin-and-die combined experiments
- Apply to real life: rain and lateness; pen and ruler usage in class

How do we calculate the probability of independent events occurring together?

- Mentor Mathematics Grade 9 pg. 247–251
- Coins, dice, and coloured balls/marbles
- Digital devices

- Written assignments - Oral questions - Observation

Data Handling and Probability

Probability — Performing experiments involving independent events; P(A and B) = P(A) × P(B) (multiplication law)

By the end of the lesson, the learner should be able to:
- define independent events as events where the outcome of one does not affect the outcome of the other;
- apply the multiplication law: P(A and B) = P(A) × P(B) for independent events;
- solve real-life problems involving two or more independent events including with and without replacement.

- One learner holds a coin, another holds a die; toss simultaneously — discuss: does the coin outcome affect the die outcome? Establish independence
- Establish: the word "and" in probability means multiply: P(A and B) = P(A) × P(B)
- Work through: basket with 4 red and 3 green balls (with replacement) — P(red then green) = 4/7 × 3/7 = 12/49; P(all red) = 16/49
- Solve: pink and orange marbles without replacement; three learners hitting a target with different individual probabilities; coin-and-die combined experiments
- Apply to real life: rain and lateness; pen and ruler usage in class

How do we calculate the probability of independent events occurring together?

- Mentor Mathematics Grade 9 pg. 247–251
- Coins, dice, and coloured balls/marbles
- Digital devices

- Written assignments - Oral questions - Observation

Data Handling and Probability

Probability — Drawing tree diagrams to represent possible outcomes of a single-stage event

By the end of the lesson, the learner should be able to:
- draw a tree diagram to represent all possible outcomes of a single probability experiment;
- place correct probabilities on each branch ensuring branches from each node sum to 1;
- use tree diagrams to solve real-life probability problems involving single outcomes.

In groups, learners are guided to:
- Draw branches for a coin toss: label X (head) and Y (tail); place P(H) = 1/2 and P(T) = 1/2 on the branches; confirm the two branches sum to 1
- Draw tree diagram for Musau's arrival: P(late) = 40%, P(early) = 60% — two branches from a single starting point
- Draw tree diagram for a school presidential election: P(Salma) = 0.32, P(Kerubo) = 0.41, P(Nanjala) = 0.27 — three branches summing to 1
- Solve: basket with 3 yellow and 2 blue balls; school modes of transport (bus 0.25, motorcycle 0.38, walking); Judy's fruit basket (25 oranges, 28 mangoes, 17 avocados)

How do we use a tree diagram to show the outcomes of a probability experiment?

- Mentor Mathematics Grade 9 pg. 251–255
- Graph paper or blank paper
- Ruler and pencil
- Digital devices

- Written exercises - Oral questions - Observation

Data Handling and Probability

Probability — Mixed problems on equally likely outcomes, range, mutually exclusive events, independent events, and tree diagrams

By the end of the lesson, the learner should be able to:
- solve mixed problems covering all probability concepts: equally likely outcomes, range of probability, mutually exclusive events, independent events, and tree diagrams;
- apply probability to real-life decision-making situations;
- appreciate probability as a tool for predicting outcomes in real life while avoiding harmful gambling practices.

In groups, learners are guided to:
- Work through comprehensive revision exercises covering: simple probability, complementary events, addition law, multiplication law, and tree diagrams
- Solve real-life problems: weather forecasting (probability of rain and lateness); team selection (probability of a class captain); fruit distribution
- Discuss: how probability applies to real life — weather, sports outcomes, disease vaccines, business decisions
- Explore using digital devices or other resources to simulate and explore probability experiments

Why is probability important in real-life situations?

- Mentor Mathematics Grade 9 pg. 239–255 (revision)
- Coins, dice, and coloured marbles/balls
- Revision exercise sheets
- Digital devices

- Written assessment - Oral questions - Peer assessment

Data Handling and Probability

Probability — Mixed problems on equally likely outcomes, range, mutually exclusive events, independent events, and tree diagrams

By the end of the lesson, the learner should be able to:
- solve mixed problems covering all probability concepts: equally likely outcomes, range of probability, mutually exclusive events, independent events, and tree diagrams;
- apply probability to real-life decision-making situations;
- appreciate probability as a tool for predicting outcomes in real life while avoiding harmful gambling practices.

In groups, learners are guided to:
- Work through comprehensive revision exercises covering: simple probability, complementary events, addition law, multiplication law, and tree diagrams
- Solve real-life problems: weather forecasting (probability of rain and lateness); team selection (probability of a class captain); fruit distribution
- Discuss: how probability applies to real life — weather, sports outcomes, disease vaccines, business decisions
- Explore using digital devices or other resources to simulate and explore probability experiments

Why is probability important in real-life situations?

- Mentor Mathematics Grade 9 pg. 239–255 (revision)
- Coins, dice, and coloured marbles/balls
- Revision exercise sheets
- Digital devices

- Written assessment - Oral questions - Peer assessment