Junior Math 3

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Unit 1: Indices, Standard Form and Surds
Unit overview By the end of this unit, students can use zero and negative indices, write large and small numbers in standard form and calculate with them, tell rational from irrational numbers, change recurring decimals to fractions, and simplify and calculate with surds. The theme is "Very big, very small, exactly right": the distance to the Sun, the width of a hair, red blood cells, planets and square gardens. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and tables and number lines throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. The lessons and Section A of the test are non-calculator; Section B of the test allows a calculator. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Indices and the laws of indices including zero and negative indices, standard form, and the real number system: rational and irrational numbers (JHS 3) • Cambridge Lower Secondary (Stage 9): Zero and negative indices, standard form for large and small numbers, and rational and irrational numbers including surds (Checkpoint Paper 1 and Paper 2) • UK Key Stage 3 (Year 9) and GCSE prep: Index laws with negative and zero powers, standard form calculations, recurring decimals to fractions, and simplifying surds (GCSE Higher number) • US Grade 8 (Common Core): Integer exponents (8.EE.1), scientific notation (8.EE.3–4), and rational and irrational numbers (8.NS.1–2) Students will be able to: • use the zero index and negative indices, including with fractions and powers of 10 • write large numbers in standard form, and change standard form back to ordinary numbers • write small numbers in standard form, and multiply, divide and add numbers in standard form • decide whether a number is rational or irrational, and write recurring decimals as fractions • simplify surds, and add, subtract, multiply and divide them All questions and problems are original and written for this course. British spelling is used (US: "exponent" for "index", "scientific notation" for "standard form"). GCSE prep: recurring decimals to fractions and surds are GCSE Higher topics.

  • Lesson 1 · Zero and Negative Indices: What Is 2⁻³?
  • Lesson 1 Exit Quiz: Zero and Negative Indices
  • Lesson 2 · Standard Form for Large Numbers: Writing the Distance to the Sun
  • Lesson 2 Exit Quiz: Standard Form: Large Numbers
  • Lesson 3 · Standard Form for Small Numbers and Calculations: The Width of a Hair
  • Lesson 3 Exit Quiz: Standard Form: Small Numbers
  • Lesson 4 · Rational and Irrational Numbers: Decimals That Never End
  • Lesson 4 Exit Quiz: Rational and Irrational Numbers
  • Lesson 5 · Simplifying Surds: √50 Is Really 5√2
  • Lesson 5 Exit Quiz: Simplifying Surds
  • Year 3 Unit 1 Test: Indices, Standard Form and Surds (Section A)

Unit 2: Percentages, Finance, Proportion and Bounds
Unit overview By the end of this unit, students can work back to an original amount from a percentage change, calculate compound interest and depreciation, convert between currencies, use formulae for direct and inverse proportion, and write error intervals and use bounds. The theme is "Money and measurement": sales receipts, savings accounts, school trips abroad, painters, falling stones and vegetable patches. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and tables and number lines throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. Lesson 2 (compound interest) allows a calculator; the other lessons and Section A of the test are non-calculator, and Section B of the test allows a calculator. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Percentages, profit and loss, simple and compound interest, depreciation, foreign exchange, and direct and inverse proportion (JHS 3) • Cambridge Lower Secondary (Stage 9): Reverse percentages, compound percentage change, direct and inverse proportion, and rounding and bounds (Checkpoint Paper 1 and Paper 2) • UK Key Stage 3 (Year 9) and GCSE prep: Reverse percentages, compound interest and depreciation, currency conversion, proportion formulae (y ∝ x, y ∝ x², y ∝ 1/x) and error intervals (GCSE Foundation and Higher) • US Grade 7 and 8 (Common Core): Multistep ratio and percent problems including interest and fees (7.RP.3), and proportional relationships (8.EE.5) Students will be able to: • find the original amount after a percentage increase or decrease • calculate compound interest and depreciation over several years, and compare them with simple interest • convert between currencies using exchange rates, including fees • set up and use formulae for direct and inverse proportion, including squares and roots • write error intervals, find upper and lower bounds, and use bounds in calculations All questions and problems are original and written for this course. British spelling is used. Currencies include the dollar ($), euro (€), pound (£), naira (₦), cedi (GH₵) and yen (¥); exchange rates are made up for practice. GCSE prep: proportion formulae and bounds in calculations are GCSE Higher topics.

Unit 3: Double Brackets and Quadratics
Unit overview By the end of this unit, students can expand double brackets and squared brackets, factorise quadratic expressions including the difference of two squares, and solve quadratic equations by factorising. The theme is "Area in disguise": photo frames, patios, floor tiles, garden paths and picture frames, where every quadratic is an area. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and area grids and tables throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Expansion of algebraic expressions and factorisation, including the difference of two squares (JHS 3) • Cambridge Lower Secondary (Stage 9): Expanding the product of two linear expressions, and factorising (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 9) and GCSE prep: Expanding double brackets, factorising quadratics x² + bx + c, the difference of two squares, and solving quadratic equations by factorising (GCSE Foundation and Higher) • US Grade 8 and Algebra I (Common Core): Goes beyond Grade 8 and prepares for Algebra I: seeing structure in expressions (A-SSE.2) and solving quadratics by factoring (A-REI.4b) Students will be able to: • expand and simplify the product of two brackets, including negatives and coefficients • expand squared brackets and recognise perfect squares • factorise quadratic expressions of the form x² + bx + c, taking out common factors first • factorise a difference of two squares and use it for mental calculations • solve quadratic equations by factorising, and reject solutions that do not fit a problem All questions and problems are original and written for this course. British spelling is used (US: "factor" for "factorise"). GCSE prep: the whole unit is GCSE content.

Unit 4: Simultaneous Equations, Inequalities and Formulae
Unit overview By the end of this unit, students can solve linear equations with fractions and brackets, solve pairs of simultaneous equations by elimination, substitution and graphs, solve linear inequalities and list integer solutions, and rearrange formulae involving squares, roots and fractions. The theme is "Two clues, one answer": shared taxi fares, school shops, phone plans, gyms, lifts, vans and pendulums. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and tables, graphs and number lines throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Linear equations and inequalities in one variable, simultaneous linear equations, and change of subject of a formula (JHS 3) • Cambridge Lower Secondary (Stage 9): Equations with fractions and brackets, simultaneous equations, inequalities, and changing the subject of a formula (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 9) and GCSE prep: Solving equations with fractions, simultaneous equations algebraically and graphically, linear inequalities with integer solutions, and changing the subject (GCSE Foundation and Higher) • US Grade 8 (Common Core): Solving linear equations with rational coefficients (8.EE.7b) and pairs of simultaneous linear equations algebraically and by graphing (8.EE.8) Students will be able to: • solve linear equations with fractions and brackets on one or both sides • solve simultaneous equations by elimination, including multiplying one or both equations • solve simultaneous equations by substitution and by reading where two lines cross • solve linear and double inequalities, reversing the sign when needed, and list integer solutions • rearrange formulae involving squares, roots and fractions, including the subject appearing twice All questions and problems are original and written for this course. British spelling is used (US: "system of equations" for "simultaneous equations"). GCSE prep: changing the subject when it appears twice is a GCSE Higher skill.

Unit 5: Functions and Graphs
Unit overview By the end of this unit, students can use function notation, find inverse and composite functions, find the equation of a line through two points, work with parallel lines and graphical solutions, draw and interpret quadratic graphs, and read rates of change from real-life graphs. The theme is "Graphs that tell stories": cinema prices, growing seedlings, burning candles, fountains, thrown balls, journeys and filling tanks. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and graphs, tables and function machines throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Linear relations, gradient, drawing and reading graphs of linear and quadratic relations (JHS 3) • Cambridge Lower Secondary (Stage 9): Functions, equations of straight lines, parallel lines, quadratic graphs and real-life graphs (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 9) and GCSE prep: Function notation, finding the equation of a line, parallel lines, graphical solutions, quadratic graphs and rates of change (GCSE Foundation and Higher) • US Grade 8 (Common Core): Defining, evaluating and comparing functions, and using functions to model relationships (8.F.1–5) Students will be able to: • use function notation, including negative inputs and solving f(x) = k • find inverse and composite functions • find the equation of a straight line from its gradient and a point, or from two points • identify parallel lines and solve equations using the points where graphs meet • draw quadratic graphs and find roots, the line of symmetry and the turning point • interpret the gradient of a real-life graph as a rate of change, including speed All questions and problems are original and written for this course. British spelling is used. GCSE prep: inverse and composite functions are GCSE Higher topics.

Unit 6: Pythagoras’ Theorem and Trigonometry
Unit overview By the end of this unit, students can use Pythagoras' theorem to find any side of a right-angled triangle, solve problems with ladders, screens, journeys and coordinates, and use sin, cos and tan to find missing sides and angles. The theme is "Measuring what you cannot reach": rope-stretchers, poles, kites, ladders, treasure maps, trees and wheelchair ramps. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and labelled triangle diagrams throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. Calculators are allowed in the lessons and in Section B of the test. Section A of the test is non-calculator, so it uses whole-number answers, exact square roots and the exact values sin 30°, cos 60° and tan 45°. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Pythagoras' theorem and its use in problems, including ladders and distances (JHS 3) • Cambridge Lower Secondary (Stage 9): Pythagoras' theorem in two dimensions, including coordinates (Checkpoint Papers 1 and 2) • UK Key Stage 3 (Year 9) and GCSE prep: Pythagoras' theorem and right-angled trigonometry for sides and angles, including exact values (GCSE Foundation and Higher) • US Grade 8 (Common Core): Using Pythagoras' theorem and its converse, including distance between points (8.G.6–8). Trigonometry goes beyond Grade 8 and prepares for high-school geometry. Students will be able to: • identify the hypotenuse and use Pythagoras' theorem to find the longest side • find a shorter side, including heights of isosceles and equilateral triangles • test whether a triangle is right-angled • solve problems with ladders, screens, journeys and the distance between two points • name the opposite, adjacent and hypotenuse and use sin, cos and tan to find a missing side • use inverse sin, cos and tan to find a missing angle All questions and problems are original and written for this course. British spelling is used. Students need a scientific calculator set to degrees for the lessons and Section B.

Unit 7: Transformations, Vectors and Similarity
Unit overview By the end of this unit, students can enlarge shapes by fractional scale factors from any centre, combine transformations and describe the result as one move, add, subtract and scale column vectors, find the magnitude of a vector, and use scale factors to find missing lengths in similar shapes. The theme is "Same shape, different place": shrinking photos, model cars, stage lighting, floor robots, drones and shadows. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and grids, arrows and diagrams throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Vectors in the plane, including magnitude, and transformations on the coordinate plane (JHS 3) • Cambridge Lower Secondary (Stage 9): Enlargement with fractional scale factors, combined transformations and similar shapes (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 9) and GCSE prep: Column vectors, vector arithmetic, enlargement, combined transformations and similar shapes (GCSE Foundation and Higher) • US Grade 8 (Common Core): Understanding congruence and similarity through transformations, dilations and sequences of transformations (8.G.3–5) Students will be able to: • enlarge a shape by a fractional scale factor from any centre, and find the scale factor and centre • carry out combined transformations and describe the single transformation that does the same job • add, subtract and multiply column vectors • find the magnitude of a vector and the distance between two points as a vector • decide whether shapes are similar and find missing lengths, including with parallel lines All questions and problems are original and written for this course. British spelling is used. GCSE prep: the magnitude of a vector and similar triangles with parallel lines are GCSE Higher topics.

Unit 8: Sectors, 3D Shapes and Compound Measures
Unit overview By the end of this unit, students can find arc lengths, sector areas and perimeters, find the volumes of pyramids, cones, spheres and composite solids, draw and read plans and elevations, use density and pressure, and solve capacity problems with litres, rates and depths. The theme is "Measuring the things we build": pizzas, clocks, ice creams, chocolate truffles, model buildings, icebergs, tanks and aquariums. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and diagrams throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. Calculators are allowed in Lessons 1, 2, 4 and 5 and in Section B of the test. Lesson 3 and Section A of the test are non-calculator, using π = 22/7 or answers in terms of π. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Mensuration of plane shapes and solids, including sectors, cones, pyramids and spheres, with π = 22/7 (JHS 3) • Cambridge Lower Secondary (Stage 9): Arcs and sectors, volumes of 3D shapes, plans and elevations, and compound measures (Checkpoint Papers 1 and 2) • UK Key Stage 3 (Year 9) and GCSE prep: Arc length and sector area, volumes of cones, pyramids and spheres, plans and elevations, density and pressure (GCSE Foundation and Higher) • US Grade 8 (Common Core): Volumes of cones, cylinders and spheres and their use in real-world problems (8.G.9) Students will be able to: • find the arc length, area and perimeter of a sector, including answers in terms of π • find the volume of pyramids, cones, spheres, hemispheres and composite solids • draw and interpret plans and elevations, including counting cubes from a plan • use density = mass ÷ volume and pressure = force ÷ area, with correct units • convert between cm³, litres and m³ and solve problems about filling, emptying and depth All questions and problems are original and written for this course. British spelling is used. Weights are given in newtons where pressure is involved, so students do not need to convert mass to weight.

Unit 9: Scatter Graphs, Two-Way Tables and Investigations
Unit overview By the end of this unit, students can draw and interpret scatter graphs, describe correlation, use lines of best fit to make predictions and judge when not to, read and complete two-way tables, estimate the mean from grouped data, and spot misleading graphs and biased investigations. The theme is "Data you can trust": heights and feet, revision and scores, journeys to school, homework times, adverts and surveys. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and graphs and tables throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Statistics: collecting, presenting and interpreting data, including frequency tables and averages from grouped data (JHS 3) • Cambridge Lower Secondary (Stage 9): Scatter graphs and correlation, grouped data, two-way tables and interpreting results (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 9) and GCSE prep: Correlation, lines of best fit, interpolation and extrapolation, two-way tables, estimated mean and misleading graphs (GCSE Foundation) • US Grade 8 (Common Core): Scatter plots, association, lines of best fit and two-way tables (8.SP.1–4) Students will be able to: • describe positive, negative and no correlation and its strength, and explain why correlation does not prove cause • draw a line of best fit, make predictions from it and judge whether they are reliable • read, complete and build two-way tables, and find probabilities and fractions from them • estimate the mean from a grouped frequency table and find the modal class and median class • identify misleading graphs and biased samples, and plan a fair investigation All questions and problems are original and written for this course. British spelling is used.

Unit 10: Probability: Combined Events and Set Notation
Unit overview By the end of this unit, students can count outcomes with the product rule, list sample spaces, draw and use tree diagrams for independent and dependent events, work out expected results, and use Venn diagrams and set notation to find probabilities. The theme is "Chance you can count": passwords and bike locks, coins and spinners, bags of sweets, school fair games and surveys. The unit has five lessons of 10–15 minutes, a unit test with four exam-style problems, and tree diagrams, grids and Venn diagrams throughout. Every practice and quiz question comes with a full step-by-step solution, and the practice set in each lesson rises from warm-up questions to exam-standard and challenge questions. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Probability of simple and combined events, and sets: Venn diagrams and set notation (JHS 3) • Cambridge Lower Secondary (Stage 9): Combined events, sample spaces, expected frequency and relative frequency (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 9) and GCSE prep: The product rule, tree diagrams, dependent events, set notation and Venn diagrams (GCSE Foundation and Higher) • US Grade 7 (Common Core): Probabilities of compound events using lists, tables and tree diagrams (7.SP.8) Students will be able to: • count outcomes with the product rule and list sample spaces systematically • draw tree diagrams and find probabilities of independent combined events • find probabilities when items are taken without replacement • work out expected frequencies and use relative frequency to estimate probability • use ξ, A ∩ B, A ∪ B and A′ and find probabilities from Venn diagrams All questions and problems are original and written for this course. British spelling is used.

Unit 11: Full Mock Exam
Unit overview This unit is a full mock of the final exam, in the three-paper pattern of GCSE, which also covers both BECE papers and both Checkpoint papers. Lesson 1 teaches exam technique: reading the question, showing working for method marks, rounding and checking, and managing time. Then come the three papers: Paper 1 is 40 multiple-choice questions (45 minutes, no calculator, marked automatically with a score for each strand); Paper 2 is 10 worked problems without a calculator and Paper 3 is 10 worked problems with a calculator (60 minutes and 40 marks each). Papers 2 and 3 are timed on the page; after submitting, students see the model solution and mark scheme under every question and tick the marks they earned, and the page adds up their total. Lesson 5 turns the results into a revision plan. Every Paper 1 question and every practice and quiz question comes with a full step-by-step solution. The mock is modelled on these exams: • BECE Mathematics (WAEC): Paper 1 objective questions and Paper 2 structured questions, without a calculator (Paper 1 and Paper 2 here) • Cambridge Lower Secondary (Stage 9): Checkpoint Paper 1 (no calculator) and Paper 2 (calculator), each 60 minutes • UK GCSE (Foundation and Higher): One non-calculator and two calculator papers; the mock uses one of each style for the worked papers • US Grade 8 (Common Core): End-of-year review of the number system, expressions and equations, functions, geometry, and statistics and probability Students will be able to: • read exam questions carefully, follow command words and give answers to the accuracy asked • show working that earns method marks, and check answers by estimating and using inverses • manage time across a timed paper, using the marks as a guide • complete three full mock papers covering all of Junior 3 • mark their own work against a mark scheme and turn their errors into a revision plan All questions and problems are original and written for this course. British spelling is used.

Unit 12: Final Revision
Unit overview This unit is the final revision before the exam. Four lessons revise the four strands of the course (number, algebra, geometry and measures, and statistics and probability), each built around the methods every exam tests and the traps that cost the most marks. The last lesson is about exam day itself: pacing a paper, using a calculator well, what to do when stuck, and a final game plan. Each lesson ends with a five-question exit quiz, and every practice and quiz question comes with a full step-by-step solution. There is no unit test: the mock exam in Unit 11 has already done that job. The calculator policy is mixed: a question that allows a calculator says so, and all the others are non-calculator. The revision covers the content of every exam the course prepares for: • BECE Mathematics (WAEC): Number, algebra, geometry and mensuration, statistics, probability and sets, without a calculator • Cambridge Lower Secondary (Stage 9): Checkpoint Paper 1 (no calculator) and Paper 2 (calculator) content across all four strands • UK Key Stage 3 (Year 9) and GCSE prep: Indices, surds, quadratics, simultaneous equations, trigonometry, tree diagrams and set notation (GCSE Foundation and Higher) • US Grade 8 (Common Core): The number system, expressions and equations, functions, geometry, and statistics and probability Students will be able to: • use indices, standard form, surds, percentages and bounds accurately • expand, factorise and solve linear, quadratic and simultaneous equations and inequalities • use Pythagoras, trigonometry, angle facts and area and volume formulae • analyse data and work out probabilities with tables, trees and Venn diagrams • pace an exam paper, use a calculator well and follow a final game plan All questions and problems are original and written for this course. British spelling is used.