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
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:
- 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
- Mentor Mathematics Grade 9 pg. 166–179 (revision)
- Revision exercise sheets

- Written assignments - Oral questions - Observation

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

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
- Mentor Mathematics Grade 9 pg. 186–191

- Oral questions - Written exercises - Observation

Geometry

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:
- make accurate scale drawings for problems involving three or four points with given bearings and distances;
- determine distances and bearings not given in the problem directly from the scale drawing;
- solve real-life navigation and positioning problems using scale drawing.

In groups, learners are guided to:
- Solve: three islands A, B, C — B is 50 km on bearing 035° from A; C is 60 km on bearing 135° from B; port D is 80 km due south of B — find bearing of C from A, bearing of D from C, and distance AC
- Solve: two houses 750 m apart, one due north of the other, observation point due west — find distances using scale drawing
- Solve: town A on bearing 050° from C, town B on bearing 020° from C; find distance A to C and bearing of B from A

How do we use scale drawing to solve multi-point navigation problems?

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

- Written assignments - Oral questions - Observation

Geometry

Scale Drawing — Identifying and determining angles of elevation by accurate scale drawing
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 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
- Mentor Mathematics Grade 9 pg. 196–201

- Written tests - 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 — 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
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:
- 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
- Mentor Mathematics Grade 9 pg. 223–225
- Ruler and protractor

- Written tests - 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 — 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
Data Handling and Probability

Trigonometry — Using sin, cos, and tan to calculate unknown sides and angles in right-angled triangles
Data Interpretation (Grouped Data) — Determining appropriate class width; drawing frequency distribution tables

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
- Mentor Mathematics Grade 9 pg. 224–229
- Graph paper and exercise books

- Written assessment - Oral questions - Observation

Data Handling and Probability

Data Interpretation (Grouped Data) — Identifying the modal frequency and modal class from a frequency distribution table
Data Interpretation (Grouped Data) — Calculating the mean of grouped data using midpoints (x̄ = Σfx ÷ Σf)
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 modal frequency as the highest frequency in a grouped data set;
- identify the modal class as the class with the highest frequency;
- apply the concept of modal class to real-life data sets such as school scores and goal tallies.

- Recall the meaning of mode from Grade 8 (most frequently occurring value); discuss how mode applies to grouped data
- From the frequency distribution table of 52 learners' masses, identify: modal frequency = 14; modal class = 45–49 kg
- Solve exercises: words read per minute, number of learners in schools, goals in netball matches, and mobile money agent data
- Recognise the modal class by identifying the class with the highest frequency from prepared tables

How do we identify the most common class in grouped data?

- Mentor Mathematics Grade 9 pg. 229–231
- Frequency distribution tables
- Digital devices
- Mentor Mathematics Grade 9 pg. 231–234
- Exercise books
- Scientific calculators
- Mentor Mathematics Grade 9 pg. 234–236

- 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
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:
- 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
- Mentor Mathematics Grade 9 pg. 224–238 (revision)
- Revision exercise sheets

- Written tests - Oral questions - Observation

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

- Written tests - Oral questions - Observation

Data Handling and Probability

Probability — Drawing tree diagrams to represent possible outcomes of a single-stage event
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:
- 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
- Mentor Mathematics Grade 9 pg. 239–255 (revision)
- Coins, dice, and coloured marbles/balls
- Revision exercise sheets

- Written exercises - Oral questions - Observation