Algebra

Matrices — Identifying and representing matrices in different situations
Matrices — Determining the order of a matrix

By the end of the lesson, the learner should be able to:
- identify a matrix from a table of information in different situations;
- represent data from real-life tables as a matrix;
- reflect on the use of matrices in everyday life.

In groups, learners are guided to:
- Discuss real-life tables such as football league tables, travel schedules, and shopping lists and count their rows and columns
- Study the Kape Furniture Company production tables for 2022 and 2023 and represent each as a matrix
- Arrange items in rows and columns and discuss how to write a matrix using correct notation

How do we use matrices in real-life situations?

- Mentor Mathematics Grade 9 pg. 38–40
- Football league tables / travel schedules
- Squared paper
- Digital devices
- Mentor Mathematics Grade 9 pg. 40–42

- Oral questions - Observation - Written exercises

Algebra

Matrices — Determining the position of items in a matrix
Matrices — Determining compatibility of matrices for addition and subtraction

By the end of the lesson, the learner should be able to:
- determine the position of any element in a matrix using row and column notation (aᵢⱼ);
- locate specific elements when given their row-column position;
- show interest in the systematic organisation of data in matrices.

In groups, learners are guided to:
- Discuss and identify the position of each element in terms of its row and column
- Use notation aᵢⱼ to describe element positions (e.g. a₁₂ = element in row 1, column 2)
- Solve exercises identifying and locating elements in matrices of various orders

How do we locate a specific element in a matrix?

- Mentor Mathematics Grade 9 pg. 42–43
- Matrix position charts
- Digital devices
- Mentor Mathematics Grade 9 pg. 43–44
- Compatibility charts

- Oral questions - Written exercises - Observation

Algebra

Matrices — Addition of matrices
Matrices — Subtraction of matrices

By the end of the lesson, the learner should be able to:
- add compatible matrices by adding corresponding elements;
- solve real-life problems involving addition of matrices;
- show interest in using matrices to organise and combine data.

In groups, learners are guided to:
- Represent first-leg and second-leg football results as matrices and add them to get season totals
- Add matrices by adding elements in the same position
- Solve exercises on addition of matrices and find unknowns in matrix addition equations

How do we add matrices to combine real-life data?

- Mentor Mathematics Grade 9 pg. 44–45
- Football league tables
- Digital devices / internet (YouTube link)
- Mentor Mathematics Grade 9 pg. 45–47
- Matrix exercise cards
- Digital devices

- Written assignments - Oral questions - Observation

Algebra

Matrices — End-of-sub-strand written assessment

By the end of the lesson, the learner should be able to:
- consolidate all matrix concepts (identification, order, position, compatibility, addition, subtraction);
- solve a variety of matrix problems accurately;
- appreciate the importance of matrices in organising and processing data.

In groups, learners are guided to:
- Complete a variety of revision exercises on all matrix concepts
- Solve mixed problems: represent tables as matrices, state order, locate elements, add and subtract matrices
- Complete a short end-of-sub-strand written assessment

How do we use matrices to solve real-life problems?

- Mentor Mathematics Grade 9 pg. 38–47
- Assessment papers
- Digital devices

- Written assessment - Oral questions - Observation

Algebra

Equations of a Straight Line — Identifying gradient; gradient as a measure of steepness

By the end of the lesson, the learner should be able to:
- identify the gradient of a straight line in real-life situations;
- describe gradient as the ratio of vertical distance to horizontal distance;
- appreciate gradient as a measure of steepness in everyday environments.

In groups, learners are guided to:
- Discuss steepness from the immediate environment: ladders, stairs, hills, and ramps
- Use an adjustable ladder inclined at different positions on a wall to demonstrate change in steepness
- Observe and climb stairs or hills and relate the experience to gradient
- Discuss positive, negative, zero, and undefined gradients from real-life examples

How do we use gradient or steepness in our daily activities?

- Mentor Mathematics Grade 9 pg. 48–50
- Adjustable ladder (practical)
- Gradient/slope diagrams
- Digital devices

- Oral questions - Observation - Written exercises

Algebra

Equations of a Straight Line — Calculating the gradient of a line from two known points (m = (y₂−y₁)/(x₂−x₁))

By the end of the lesson, the learner should be able to:
- derive and apply the formula m = (y₂ − y₁)/(x₂ − x₁) to calculate gradient;
- calculate the gradient of a line from two given points;
- identify whether the gradient is positive, negative, zero, or undefined.

In groups, learners are guided to:
- Study graphs and count vertical and horizontal steps between two points on a line
- Derive the gradient formula using change in y ÷ change in x
- Calculate gradients from given pairs of coordinates and from graphs
- Classify lines as having positive, negative, zero, or undefined gradients

How do we calculate the gradient of a line passing through two points?

- Mentor Mathematics Grade 9 pg. 50–52
- Graph paper / Cartesian plane charts
- Digital devices / internet (YouTube link)

- Written assignments - Oral questions - Observation

Algebra

Equations of a Straight Line — Determining the equation of a straight line given two points

By the end of the lesson, the learner should be able to:
- find the gradient of a line from two given points;
- derive the equation of a straight line given two points using the gradient formula;
- apply the method to find equations for sides of triangles and parallelograms.

In groups, learners are guided to:
- Follow step-by-step procedure: find gradient, pick one point plus P(x,y), set up gradient equation, cross-multiply and simplify
- Find equations of lines passing through various pairs of coordinates
- Find equations of lines AB, BC, AC for a triangle with given vertices

How do we determine the equation of a line when we know two points on it?

- Mentor Mathematics Grade 9 pg. 52–55
- Graph paper
- Worked example charts
- Digital devices

- Written assignments - Oral questions - Observation

Algebra

Equations of a Straight Line — Determining the equation of a straight line from a known point and gradient

By the end of the lesson, the learner should be able to:
- find the equation of a straight line given one point and the gradient;
- set up the gradient equation and simplify to get the line equation;
- show interest in applying the method to different point-gradient combinations.

In groups, learners are guided to:
- Follow step-by-step procedure: use given point B(x₁,y₁) and point C(x,y), form gradient ratio equal to given m, cross-multiply and simplify
- Solve exercises given various points and gradients (including fractional and negative gradients)
- Verify answers by substituting the given point back into the derived equation

How do we find the equation of a line when we know one point and its gradient?

- Mentor Mathematics Grade 9 pg. 55–57
- Worked example charts
- Graph paper
- Digital devices

- Written assignments - Oral questions - Observation

Algebra

Equations of a Straight Line — Expressing the equation of a straight line in the form y = mx + c

By the end of the lesson, the learner should be able to:
- rearrange any linear equation into the form y = mx + c;
- identify the gradient m and y-intercept c directly from the equation;
- appreciate the usefulness of the y = mx + c form in describing a line.

In groups, learners are guided to:
- Rearrange equations step by step: isolate y by moving x-terms to the right and dividing by the coefficient of y
- Identify gradient and y-intercept from equations already in y = mx + c form
- Complete tables matching equations, gradients, and y-intercepts
- Convert equations such as 4y + 3x − 2 = 0 into y = mx + c

How do we rewrite the equation of a straight line in the form y = mx + c?

- Mentor Mathematics Grade 9 pg. 57–59
- Equation rearrangement charts
- Digital devices

- Written tests - Oral questions - Observation

Algebra

Equations of a Straight Line — Interpreting the gradient m and y-intercept c in y = mx + c

By the end of the lesson, the learner should be able to:
- interpret m as the gradient and c as the y-intercept in the equation y = mx + c;
- complete tables of values and draw straight-line graphs from equations;
- recognise the use of equations of straight lines in real-life situations.

In groups, learners are guided to:
- Study the equation y = 2x + 4: discuss the coefficient of x as the gradient and the constant as the y-intercept
- Complete tables of values for given equations and use them to draw lines on a Cartesian plane
- Interpret real-life linear relationships using gradient and intercept
- Use digital devices or other resources to show different hills and relate to gradient

How does the equation y = mx + c describe a straight line?

- Mentor Mathematics Grade 9 pg. 59–61
- Graph paper / Cartesian plane
- Digital devices

- Written assignments - Oral questions - Observation

Algebra

Equations of a Straight Line — Determining the x-intercept and y-intercept

By the end of the lesson, the learner should be able to:
- determine the x-intercept by substituting y = 0 in a line equation;
- determine the y-intercept by substituting x = 0 in a line equation;
- use intercepts to draw straight-line graphs on a Cartesian plane.

In groups, learners are guided to:
- Discuss the meaning of x-intercept (where the line crosses the x-axis, y = 0) and y-intercept (where the line crosses the y-axis, x = 0)
- Draw lines y = x+2, y = x+5, y = x−3 and record their intercepts in a table
- Calculate x and y-intercepts for various equations including 3x + 2y = 12

How do we find where a straight line crosses the axes?

- Mentor Mathematics Grade 9 pg. 61–63
- Graph paper / Cartesian plane
- Digital devices

- Written exercises - Oral questions - Observation

Algebra

Equations of a Straight Line — End-of-sub-strand written assessment

By the end of the lesson, the learner should be able to:
- demonstrate mastery of all straight-line concepts;
- solve a variety of problems accurately;
- appreciate the application of straight-line equations in everyday contexts.

In groups, learners are guided to:
- Complete assessment exercises covering gradient, equation of a line (two points and point-gradient), y = mx + c form, interpretation, and intercepts
- Correct and discuss solutions as a class

How do we apply equations of straight lines to solve problems?

- Mentor Mathematics Grade 9 pg. 48–64
- Assessment papers
- Graph paper
- Digital devices

- Written assessment - Oral questions - Observation

Algebra

Equations of a Straight Line — End-of-sub-strand written assessment

By the end of the lesson, the learner should be able to:
- demonstrate mastery of all straight-line concepts;
- solve a variety of problems accurately;
- appreciate the application of straight-line equations in everyday contexts.

In groups, learners are guided to:
- Complete assessment exercises covering gradient, equation of a line (two points and point-gradient), y = mx + c form, interpretation, and intercepts
- Correct and discuss solutions as a class

How do we apply equations of straight lines to solve problems?

- Mentor Mathematics Grade 9 pg. 48–64
- Assessment papers
- Graph paper
- Digital devices

- Written assessment - Oral questions - Observation

Algebra

Linear Inequalities — Solving linear inequalities in one unknown

By the end of the lesson, the learner should be able to:
- form linear inequalities from real-life statements;
- solve linear inequalities in one unknown using inverse operations;
- note that dividing or multiplying by a negative number reverses the inequality sign.

In groups, learners are guided to:
- Discuss real-life inequality statements (e.g. "a number multiplied by 4 minus 5 is greater than the number multiplied by 3 plus 2") and form inequalities
- Solve inequalities by adding, subtracting, multiplying, or dividing both sides — noting the sign reversal rule for negatives
- Solve exercises involving fractional and compound inequalities

How do we solve linear inequalities in one unknown?

- Mentor Mathematics Grade 9 pg. 65–67
- Inequality statement cards
- Digital devices / internet (YouTube link)

- Oral questions - Written exercises - Observation

Algebra

Linear Inequalities — Representing linear inequalities in one unknown on a number line and on a Cartesian plane

By the end of the lesson, the learner should be able to:
- represent linear inequalities in one unknown on a graph;
- use a continuous line for ≤ or ≥ and a broken line for < or >;
- shade the unwanted region to indicate the solution set.

In groups, learners are guided to:
- Draw a number line and represent simple inequalities such as x < 4 and x ≥ −3
- Follow steps: treat inequality as an equation, draw the boundary line (broken or continuous), test a point, shade unwanted region
- Solve inequalities and represent the solution on a graph (e.g. 2y − 4 ≤ 2)

How do we represent inequalities in one unknown on a graph?

- Mentor Mathematics Grade 9 pg. 67–70
- Graph paper / Cartesian plane
- Digital devices

- Written assignments - Oral questions - Observation

Algebra

Linear Inequalities — Representing linear inequalities in two unknowns graphically

By the end of the lesson, the learner should be able to:
- generate a table of values and draw the boundary line for a linear inequality in two unknowns;
- test a point to determine which region satisfies the inequality;
- shade the unwanted region and identify the required region.

In groups, learners are guided to:
- Generate a table of values for the line y = 3x and draw the boundary (continuous line for ≤); test point (1,1) — is 1 ≤ 3(1)? — to identify the required region
- Represent inequalities such as y > 2x − 2 using a broken boundary line, test a point, and shade the unwanted region
- Identify inequalities represented by given shaded graphs

How do we represent a linear inequality in two unknowns on a graph?

- Mentor Mathematics Grade 9 pg. 70–72
- Graph paper / Cartesian plane
- Digital devices

- Written tests - Oral questions - Observation

Algebra

Linear Inequalities — End-of-sub-strand written assessment

By the end of the lesson, the learner should be able to:
- consolidate all linear inequality concepts: solving, one-variable graphs, two-variable graphs, and real-life applications;
- solve a variety of inequality problems accurately;
- appreciate the use of linear inequalities in real-life decision-making.

In groups, learners are guided to:
- Complete mixed exercises: solve inequalities in one unknown, represent on graphs, represent two-unknown inequalities graphically, solve word problems
- Complete a short end-of-sub-strand written assessment
- Correct and discuss solutions as a class

How do we apply linear inequalities to solve and represent real-life situations?

- Mentor Mathematics Grade 9 pg. 65–73
- Assessment papers
- Graph paper
- Digital devices

- Written assessment - Oral questions - Observation

Algebra

2.1–2.3 Algebra — Strand 2 consolidation: matrices, equations of a straight line, and linear inequalities

By the end of the lesson, the learner should be able to:
- review and consolidate all Strand 2 concepts across the three sub-strands;
- solve mixed problems involving matrices, straight-line equations, and linear inequalities;
- show confidence in applying algebra to solve real-life problems.

In groups, learners are guided to:
- Work through mixed strand revision exercises covering matrices, equations of straight lines, and linear inequalities
- Identify connections between topics: e.g. graphing lines relates to graphing inequalities
- Peer-review solutions and discuss common mistakes
- Use digital devices or graphing tools to verify graphs and equations

How do matrices, equations of straight lines, and linear inequalities connect to real-life problems?

- Mentor Mathematics Grade 9 pg. 38–73 (revision exercises)
- Graph paper
- Revision exercise sheets
- Digital devices

- Written tests - Oral questions - Peer assessment

Measurements

Area — Area of a regular pentagon

By the end of the lesson, the learner should be able to:
- identify the properties of a regular pentagon;
- calculate the area of a regular pentagon by dividing it into triangles from the centre;
- appreciate the use of area of polygons in real-life situations.

In groups, learners are guided to:
- Draw and cut out a regular pentagon, discuss its properties, and count the triangles formed when vertices are joined to the centre
- Derive: Area of pentagon = area of one triangle × 5
- Solve real-life problems: pizza box lids, road signs, and hexagonal tiles shaped as pentagons

How do we work out the area of different surfaces?

- Mentor Mathematics Grade 9 pg. 73–75
- Cut-outs of pentagons and hexagons
- Ruler and pair of compasses
- Digital devices

- Oral questions - Observation - Written exercises

Measurements

Area — Area of a regular hexagon
Area — Surface area of rectangular-based prisms (cuboids)

By the end of the lesson, the learner should be able to:
- identify the properties of a regular hexagon;
- calculate the area of a regular hexagon by dividing it into six equal triangles from the centre;
- apply area of a hexagon to real-life situations such as tiling.

In groups, learners are guided to:
- Trace and cut out a hexagon, measure the perpendicular height ON of one triangle, and calculate the area of all six triangles
- Derive: Area of hexagon = area of one triangle × 6
- Solve problems involving hexagonal trampolines, tiling areas, and road signs
- Explore ethno-math patterns in fabrics and structures involving hexagons

How do we work out the area of a hexagon?

- Mentor Mathematics Grade 9 pg. 73–76
- Cut-outs of hexagons
- Ruler and pair of compasses
- Digital devices
- Mentor Mathematics Grade 9 pg. 76–79
- Rectangular prism models
- Rulers and scissors

- Oral questions - Written exercises - Observation

Measurements

Area — Surface area of triangular-based prisms
Area — Surface area of square, rectangular, and triangular-based pyramids

By the end of the lesson, the learner should be able to:
- identify the five faces of a triangular prism and sketch its net;
- work out the surface area as: 2 × (area of triangle) + 3 × (area of rectangle);
- apply surface area of triangular prisms to real-life problems such as roofing and tent-making.

In groups, learners are guided to:
- Use a triangular prism model; cut along its edges to separate faces and identify the two triangular and three rectangular faces
- Calculate the area of each face and sum all five to get total surface area
- Solve real-life problems: roofs of conference halls, tents, greenhouse structures, and detergent packaging boxes

How do we determine the total surface area of a triangular prism?

- Mentor Mathematics Grade 9 pg. 79–81
- Triangular prism models
- Rulers and scissors
- Digital devices / internet (YouTube link)
- Mentor Mathematics Grade 9 pg. 81–85
- Pyramid models
- Digital devices

- Written tests - Oral questions - Observation

Measurements

Area — Area of a sector; area of a segment of a circle
Area — Surface area of a cone (curved surface and total surface area)

By the end of the lesson, the learner should be able to:
- calculate the area of a sector using Area = (θ/360°) × πr²;
- calculate the area of a segment as: area of sector − area of triangle;
- apply sectors and segments to real-life problems such as windscreen wipers and pendulums.

In groups, learners are guided to:
- Draw a circle with a sector and a segment; make cut-outs and discuss their differences
- Derive and apply: Area of sector = (θ/360°) × πr²; find angle θ given area and radius
- Calculate area of segment = area of sector − ½bh; solve problems involving pendulums, bus wipers, and golf course models

How do we calculate the area of a sector and a segment?

- Mentor Mathematics Grade 9 pg. 85–91
- Circle cut-outs with sectors and segments
- Pair of compasses and ruler
- Digital devices
- Mentor Mathematics Grade 9 pg. 91–93
- Card paper and scissors
- Pair of compasses
- Scientific calculators

- Written exercises - Oral questions - Observation

Measurements

Volume of Solids — Volume of triangular-based prisms

By the end of the lesson, the learner should be able to:
- identify the cross-sectional area of a triangular prism;
- calculate the volume of a triangular prism using V = cross-sectional area × length;
- apply volume of triangular prisms to real-life problems such as greenhouses and concrete blocks.

In groups, learners are guided to:
- Collect prism-shaped objects; identify and discuss features of triangular prisms
- Calculate height of the triangular face using Pythagoras, find cross-sectional area, then multiply by length
- Solve problems: greenhouse volumes, concrete blocks, and loading company loaders

How do we determine the volume of different solids?

- Mentor Mathematics Grade 9 pg. 98–101
- Triangular prism models
- Rulers
- Digital devices

- Oral questions - Written exercises - Observation

Measurements

Volume of Solids — Volume of rectangular-based prisms (cuboids)

By the end of the lesson, the learner should be able to:
- calculate the volume of a rectangular prism using V = l × w × h;
- determine height or base area when volume is given;
- apply volume of rectangular prisms to real-life situations such as storage tanks, sandboxes, and water tanks.

In groups, learners are guided to:
- Collect rectangular containers from the locality; measure length, width, and height; calculate base area then multiply by height
- Solve problems: movie theatre popcorn containers, rectangular water tanks, and soap containers
- Determine height from given volume and base area

How do we use the volume of solids in real-life situations?

- Mentor Mathematics Grade 9 pg. 101–105
- Rectangular prism containers
- Rulers
- Digital devices

- Written assignments - Oral questions - Observation

Measurements

Volume of Solids — Volume of square, rectangular, and triangular-based pyramids

By the end of the lesson, the learner should be able to:
- calculate the volume of pyramids using V = ⅓ × base area × perpendicular height;
- determine a missing dimension when volume and other dimensions are given;
- apply volume of pyramids to real-life problems.

In groups, learners are guided to:
- Identify and name pyramids from their bases; use digital resources to explore the formula
- Apply V = ⅓ × base area × h to square, rectangular, and triangular-based pyramids
- Solve problems: parachutes, glass pyramids, crystals formed by two pyramids joined at the base

How do we calculate the volume of a pyramid?

- Mentor Mathematics Grade 9 pg. 105–109
- Pyramid models
- Digital devices / internet

- Written exercises - Oral questions - Observation

Measurements

Volume of Solids — Volume of a cone

By the end of the lesson, the learner should be able to:
- calculate the volume of a cone using V = ⅓πr²h;
- find a missing dimension (radius or height) when the volume is given;
- apply volume of a cone to real-life situations such as ice cream cups and party hats.

In groups, learners are guided to:
- Use digital resources to explore the formula for volume of a cone
- Apply V = ⅓πr²h to calculate volumes; rearrange to find radius or height
- Solve problems: ice cream vendor cones, hourglasses made of two identical cones, yoghurt cups, and birthday party hats

How do we work out the volume of a cone?

- Mentor Mathematics Grade 9 pg. 109–111
- Cone models
- Scientific calculators
- Digital devices

- Written tests - Oral questions - Observation

Measurements

Volume of Solids — Volume of a frustum (cone-based and pyramid-based)

By the end of the lesson, the learner should be able to:
- identify a frustum as the solid formed when the top of a cone or pyramid is cut off horizontally;
- calculate the volume of a frustum using: V = volume of big solid − volume of small solid;
- apply volume of frustum to real-life situations such as lampshades and water troughs.

In groups, learners are guided to:
- Use a model pyramid or cone; cut horizontally to form a frustum and a smaller solid
- Apply: Volume of frustum = volume of big cone/pyramid − volume of small cone/pyramid
- Solve problems: cone-shaped water containers, lampshades, and water troughs cut from a pyramid

How do we determine the volume of a frustum?

- Mentor Mathematics Grade 9 pg. 111–114
- Cone and pyramid models
- Scissors
- Digital devices

- Written assignments - Oral questions - Observation

Measurements

Volume of Solids — Volume of a frustum (cone-based and pyramid-based)

By the end of the lesson, the learner should be able to:
- identify a frustum as the solid formed when the top of a cone or pyramid is cut off horizontally;
- calculate the volume of a frustum using: V = volume of big solid − volume of small solid;
- apply volume of frustum to real-life situations such as lampshades and water troughs.

In groups, learners are guided to:
- Use a model pyramid or cone; cut horizontally to form a frustum and a smaller solid
- Apply: Volume of frustum = volume of big cone/pyramid − volume of small cone/pyramid
- Solve problems: cone-shaped water containers, lampshades, and water troughs cut from a pyramid

How do we determine the volume of a frustum?

- Mentor Mathematics Grade 9 pg. 111–114
- Cone and pyramid models
- Scissors
- Digital devices

- Written assignments - Oral questions - Observation

Measurements

Mass, Volume, Weight and Density — Converting units of mass

By the end of the lesson, the learner should be able to:
- identify and state the units of mass and their abbreviations;
- convert units of mass from one form to another (g, kg, mg, Hg, Dg, dg, cg, μg, Mg, tonnes);
- appreciate the importance of accurate mass measurement in everyday life.

In groups, learners are guided to:
- Discuss different instruments used in weighing: beam balance, electronic weighing machine, spring balance
- Take a walk outside class, collect objects of different sizes, weigh them, and express masses in kilograms
- Convert masses between units and record findings in a table

How do you weigh materials and objects?

- Mentor Mathematics Grade 9 pg. 115–117
- Beam balance / electronic weighing machine
- Objects of different sizes
- Digital devices

- Oral questions - Written exercises - Observation

Measurements

Mass, Volume, Weight and Density — Relationship between mass and weight; W = mg

By the end of the lesson, the learner should be able to:
- distinguish between mass (quantity of matter, kg) and weight (gravitational pull, N);
- calculate weight from mass using W = mg (g ≈ 10 N/kg);
- convert weight back to mass and apply to real-life situations.

In groups, learners are guided to:
- Use a beam balance and a spring balance to measure mass and weight of the same stones; record in a table and compare mass (kg) vs. weight (N)
- Derive the relationship: Weight = mass × gravitational force (10 N/kg)
- Solve problems: weight of a 97 kg object; mass from a given weight of 250 N; weight on a different planet

How does mass relate to weight?

- Mentor Mathematics Grade 9 pg. 117–119
- Beam balance and spring balance
- Stones of different sizes
- Digital devices

- Written tests - Oral questions - Observation

Measurements

Mass, Volume, Weight and Density — Density: concept, formula, and calculations (D = M/V)

By the end of the lesson, the learner should be able to:
- define density as mass per unit volume;
- calculate density, mass, or volume using the formula D = M/V;
- convert density between g/cm³ and kg/m³ and apply to real-life situations.

In groups, learners are guided to:
- Use a 100 cm³ container and a beam balance; fill with different substances (sand, water, gravel, soil), measure mass, and calculate mass ÷ volume
- Derive: Density = Mass ÷ Volume; rearrange to find mass or volume
- Convert: 1 g/cm³ = 1 000 kg/m³; solve problems using oil, paraffin, copper, mercury, and petrol densities

How do we determine the density of a substance?

- Mentor Mathematics Grade 9 pg. 119–122
- 100 cm³ containers
- Beam balance / electronic scale
- Sand, water, gravel, and soil
- Digital devices

- Written assignments - Oral questions - Observation

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
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:
- 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
- Mentor Mathematics Grade 9 pg. 180–183
- Protractors and rulers
- Compass direction diagrams
- Graph paper

- Written assessment - Oral questions - Peer assessment

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 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 — 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 — 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

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