High School Math Year 1

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Unit 1 · Number Systems: The Secret Code Behind Every Calculation
Unit overview This first unit of High School Maths builds the number skills that every later topic depends on. By the end of it, students can classify numbers as natural, integer, rational or irrational, calculate accurately with negative numbers and the order of operations, find prime factorisations, HCFs and LCMs, calculate fluently with fractions, decimals and recurring decimals, use standard form and significant figures, and work in number bases. The unit has five lessons of 10–15 minutes, each with ten practice questions and a six-question exit quiz, and a unit test with 15 multiple-choice questions and five worked problems. All questions are original, and every answer has been checked by computer. These skills appear in every upper-secondary maths course and exam: • WASSCE Core Mathematics (WAEC): Number and numeration: number bases, fractions and decimals, approximation and standard form, HCF and LCM, real numbers • Cambridge IGCSE Mathematics (0580, Extended): Number: types of number, order of operations, fractions and decimals, standard form, accuracy and estimation • GCSE Mathematics (UK, Higher): Number: prime factorisation, HCF and LCM, fractions, recurring decimals to fractions, standard form, estimation • US Algebra I (Common Core) and the SAT: The real number system (N-RN), operations with rational numbers, scientific notation; SAT Problem-Solving and Data Analysis foundations Students will be able to: • classify numbers and calculate with negatives using the correct order of operations • write numbers as products of primes and find HCFs and LCMs, choosing the right one in word problems • add, subtract, multiply and divide fractions and mixed numbers, and convert recurring decimals to fractions • write and calculate with numbers in standard form, round to decimal places and significant figures, and estimate • convert between number bases and calculate in bases other than ten Calculators are not used in the lessons or in Section A, matching non-calculator practice; Section B allows a calculator.

  • Lesson 1 · Types of Number and the Order of Operations: Where Does √2 Belong?
  • Lesson 1 Exit Quiz: Types of Number
  • Lesson 2 · Prime Factors, HCF and LCM: When Will the Buses Meet Again?
  • Lesson 2 Exit Quiz: HCF and LCM
  • Lesson 3 · Fractions, Decimals and Mixed Numbers: Getting Every Step Right
  • Lesson 3 Exit Quiz: Fractions and Decimals
  • Lesson 4 · Standard Form, Significant Figures and Estimation: Very Big, Very Small, Roughly Right
  • Lesson 4 Exit Quiz: Standard Form
  • Lesson 5 · Number Bases: Counting in Twos, Fives and Beyond
  • Lesson 5 Exit Quiz: Number Bases
  • Unit 1 Test: Number Systems (Section A)

Unit 2 · Ratio, Proportion and Percentages: Master the Maths of Money and Fair Shares
Unit overview This unit develops the multiplicative reasoning behind ratio, proportion and percentages, the skills students use most often outside the classroom. By the end of it, students can simplify ratios and share quantities in a given ratio, solve direct and inverse proportion problems, calculate percentages and percentage changes with multipliers, work backwards to an original amount and combine successive percentage changes, and solve problems on simple interest, discount, commission, profit and loss. The unit has five lessons of 10–15 minutes, each with ten practice questions and a six-question exit quiz, and a unit test with 15 multiple-choice questions and five worked problems. All questions are original, and every answer has been checked by computer. These skills appear in every upper-secondary maths course and exam: • WASSCE Core Mathematics (WAEC): Ratio, proportion and rates; percentages; simple interest, profit and loss, discount and commission • Cambridge IGCSE Mathematics (0580, Extended): Number: ratio and proportion, rates, percentages including reverse percentages and percentage change, simple interest • GCSE Mathematics (UK, Higher): Ratio, proportion and rates of change: sharing in a ratio, direct and inverse proportion, percentage change, reverse percentages, simple interest • US Algebra I (Common Core) and the SAT: Ratios and proportional relationships, percent problems and unit rates; SAT Problem-Solving and Data Analysis (ratios, rates, proportions and percentages) Students will be able to: • simplify ratios, share quantities in a given ratio and combine ratios of the form a : b and b : c • recognise and solve direct and inverse proportion problems, including best buys and recipes • find percentages of amounts, write one quantity as a percentage of another, and use multipliers for percentage change • find an original amount after a percentage change and combine successive percentage changes • solve problems on simple interest, discount, commission, and percentage profit and loss Lessons 1 and 2 and Section A are completed without a calculator; Lessons 3 to 5 and Section B allow a calculator, matching the real-world numbers used.

Unit 3 · Indices and Surds: Tame Giant Powers and Tricky Roots
Unit overview This unit gives students full control of powers and roots, a skill used throughout algebra, exponential growth and trigonometry. By the end of it, students can apply the laws of indices to numbers and algebraic expressions, work with zero, negative and fractional indices, solve simple index equations, simplify and calculate with surds, and rationalise denominators, all without a calculator. The unit has five lessons of 10–15 minutes, each with ten practice questions and a six-question exit quiz, and a unit test with 15 multiple-choice questions and five worked problems. All questions are original, and every answer has been checked by computer. These skills appear in every upper-secondary maths course and exam: • WASSCE Core Mathematics (WAEC): Number and numeration: laws of indices, zero, negative and fractional indices, simple equations involving indices, and surds (simplifying, adding, multiplying and rationalising) • Cambridge IGCSE Mathematics (0580, Extended): Number: indices, including zero, negative and fractional indices; surds, including simplifying and rationalising the denominator • GCSE Mathematics (UK, Higher): Number and algebra: laws of indices, zero, negative and fractional indices; simplifying surds and rationalising denominators • US Algebra I (Common Core) and the SAT: Properties of exponents extended to rational exponents (N-RN.1, N-RN.2), rational and irrational numbers (N-RN.3); SAT Advanced Math: equivalent expressions with exponents and radicals Students will be able to: • use the laws of indices to multiply, divide and raise powers, including expressions with coefficients • evaluate and simplify expressions with zero and negative indices, giving answers with positive indices • evaluate fractional indices and solve simple index equations by writing both sides with the same base • simplify surds, and add, subtract, multiply and expand expressions containing surds • rationalise denominators of the forms a/√b and a/(b + √c), giving answers in exact form Calculators are not used in the lessons or in Section A, matching non-calculator practice; Section B allows a calculator, but every answer must be given in exact form.

Unit 4 · Algebraic Expressions: Crack the Language of Algebra
Unit overview This unit builds the core algebra skills that every later topic depends on. By the end of it, students can substitute positive, negative and fractional values into expressions and formulae, expand and simplify expressions with single brackets, factorise by taking out the highest common factor and by grouping, simplify, add, subtract, multiply and divide algebraic fractions, and form expressions and formulae from words. The unit has five lessons of 10–15 minutes, each with ten practice questions and a six-question exit quiz, and a unit test with 15 multiple-choice questions and five worked problems. All questions are original, and every answer has been checked by computer. These skills appear in every upper-secondary maths course and exam: • WASSCE Core Mathematics (WAEC): Algebraic processes: substitution, simplification and expansion, factorisation, algebraic fractions, forming expressions and formulae from word problems • Cambridge IGCSE Mathematics (0580, Extended): Algebra and graphs: substitution, expanding and simplifying, factorising, algebraic fractions, constructing expressions and formulae • GCSE Mathematics (UK, Higher): Algebra: notation and substitution, expanding single brackets, factorising, simplifying and combining algebraic fractions, forming expressions and formulae • US Algebra I (Common Core) and the SAT: Seeing structure in expressions (A-SSE), arithmetic with polynomials and rational expressions (A-APR), creating expressions and equations (A-CED); SAT Heart of Algebra Students will be able to: • substitute positive, negative and fractional values into expressions and formulae accurately • collect like terms and expand and simplify expressions with single brackets, including negative multipliers • factorise expressions completely by taking out the highest common factor, and factorise four terms by grouping • simplify algebraic fractions, and add, subtract, multiply and divide them • form expressions and formulae from words, including consecutive integers, perimeter, cost and age problems Calculators are not used in the lessons or in Section A, matching non-calculator practice; Section B allows a calculator.

Unit 5 · Linear Equations and Inequalities: Solve for x Like a Pro
Unit overview This unit builds the core algebra skill of solving: finding the value, or the range of values, that makes a statement true. By the end of it, students can solve linear equations with brackets and unknowns on both sides, clear fractions and decimals from equations, turn word problems into equations and answer them in context, solve linear and double inequalities and show them on a number line, and change the subject of a formula, including formulae with squares, roots and the subject appearing twice. The unit has five lessons of 10–15 minutes, each with ten practice questions and a six-question exit quiz, and a unit test with 15 multiple-choice questions and five worked problems. All questions are original, and every answer has been checked by computer. These skills appear in every upper-secondary maths course and exam: • WASSCE Core Mathematics (WAEC): Algebraic processes: linear equations (including fractions), word problems leading to linear equations, linear inequalities in one variable, change of subject of formulae • Cambridge IGCSE Mathematics (0580, Extended): Algebra: constructing and solving linear equations, solving linear inequalities and representing them on a number line, rearranging formulae (including the subject appearing twice) • GCSE Mathematics (UK, Higher): Algebra: solving linear equations, forming equations from contexts, solving linear inequalities and showing them on a number line, changing the subject of a formula • US Algebra I (Common Core) and the SAT: Creating and solving equations and inequalities in one variable (A-CED, A-REI), rearranging formulas to highlight a quantity; SAT Algebra (linear equations and inequalities) Students will be able to: • solve linear equations with brackets and unknowns on both sides, and check solutions by substitution • solve equations containing fractions and decimals by multiplying every term by a suitable number • form linear equations from word problems on ages, perimeters, consecutive numbers, sharing and angles, and answer in context • solve linear and double inequalities, reversing the sign when multiplying or dividing by a negative, show them on a number line and list integer solutions • change the subject of a formula, including formulae with brackets, fractions, squares and square roots, and the subject appearing twice Calculators are not used in the lessons or in Section A, matching non-calculator practice; Section B allows a calculator.

Unit 6 · Sets and Logic: Crack Every Venn Diagram and Win Every Argument
Unit overview By the end of this unit, students can describe sets by listing and by a rule, use set notation (∈, ∉, n(A), U, ∅, ⊆, ⊂, A′), find unions, intersections and complements, solve two- and three-set Venn diagram problems (including problems with an unknown x and the smallest or largest possible overlap), find the truth values of compound statements and conditionals, write converses and contrapositives, use counter-examples and decide which deductions are valid. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Sets: notation, union, intersection, complement, and two- and three-set Venn diagram problems; Logical reasoning: simple and compound statements, negation, implication, converse, contrapositive and valid arguments (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Set language and notation (ℰ, ∅, A′, ∪, ∩, ⊆, n(A)) and Venn diagrams with two sets (Core) and three sets (Extended) • GCSE Mathematics (UK): Set notation, union, intersection and Venn diagrams (used in probability), and logical argument with counter-examples (Foundation, leading into Higher) • US Algebra I and the SAT: Sets and Venn diagrams for counting (S-CP.1); conditional statements, converse, inverse and contrapositive (Geometry reasoning); SAT Problem-Solving and Data Analysis Students will be able to: • describe sets by listing and by a rule, and use set notation, subsets and complements correctly • find unions and intersections, name and shade regions, and use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) • fill three-set Venn diagrams from the centre outwards, and form and solve equations for an unknown region • find the truth values of compound statements and conditionals, write converses and contrapositives, and use counter-examples and valid deductions • solve multi-step set problems involving "neither", "only", "exactly one", "at least one" and the smallest or largest overlap Calculator use: all five lessons and the whole unit test are non-calculator, matching the non-calculator papers of IGCSE and GCSE and the way WASSCE sets questions on sets and logic. All questions are original and written for this course. British spelling is used. The universal set is written U throughout (IGCSE papers write ℰ and some books write ξ).

Unit 7 · Straight-Line Graphs: Draw It, Measure It, Predict With It
Unit overview This unit turns algebra into pictures. By the end of it, students can plot points in all four quadrants and draw straight lines from tables of values, find and interpret the gradient of a line, read and write equations in the form y = mx + c (from a gradient and a point, or from two points), find the midpoint of a segment and the distance between two points, and use linear graphs to model real situations such as conversions, journeys and tariffs. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Coordinate geometry of straight lines: plotting points and drawing linear graphs, gradient, the equation of a straight line, midpoint and distance between two points; interpreting linear and travel graphs • Cambridge IGCSE Mathematics (0580): Coordinate geometry: drawing linear graphs, gradient, length and midpoint of a line segment, y = mx + c, parallel lines; real-life graphs including conversion and distance–time graphs (Core and Extended) • GCSE Mathematics (UK): Algebra: plotting linear graphs, gradient and intercept, the equation of a line through two points, parallel lines, real-life and distance–time graphs (Foundation, leading into Higher) • US Algebra I and the SAT: Linear functions: slope as a rate of change, slope-intercept form, interpreting slope and intercept in context (F-IF, F-LE, A-CED); distance and midpoint (G-GPE); SAT Algebra: linear equations and graphs Students will be able to: • plot points in all four quadrants and draw straight-line graphs from tables of values, including horizontal and vertical lines • find the gradient of a line from two points or from a graph, recognise zero and undefined gradients, and use equal gradients for parallel lines and points on one line • read the gradient and y-intercept from y = mx + c, rearrange equations into this form, and find the equation of a line from a gradient and a point or from two points • find the midpoint of a segment, a missing endpoint, and the distance between two points as a surd or to 1 decimal place • build and interpret linear models in context, including conversion graphs, distance–time graphs and the crossing point of two tariffs Calculator use: Lessons 1, 2, 3 and 5 are non-calculator, matching the non-calculator papers where these skills are mostly tested. Lesson 4 allows a calculator, for distances given to 1 decimal place; surd answers are still worked out by hand. In the unit test, Section A is non-calculator and only Question 4 of Section B allows a calculator. All questions are original and written for this course. British spelling is used (US: "slope" for "gradient").

Unit 8 · Angles and Polygons: The Hidden Rules Behind Every Shape and Direction
Unit overview By the end of this unit, students can use angles on a straight line, around a point and vertically opposite angles to form and solve equations, find corresponding, alternate and co-interior angles with parallel lines, use the angle properties of triangles and special quadrilaterals, find the interior and exterior angles of polygons, and work with three-figure bearings and back bearings, giving the reasons that examiners reward. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Geometry: angles at a point and on a straight line, angles formed by parallel lines and a transversal, angle properties of triangles, quadrilaterals and polygons, and bearings (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Geometry: angle properties of lines, parallel lines, triangles, quadrilaterals and polygons (interior and exterior angles), and three-figure bearings (Core and Extended) • GCSE Mathematics (UK): Geometry and measures: angle facts, alternate, corresponding and co-interior angles, angle sums of triangles and polygons, properties of special quadrilaterals, and bearings (Foundation, leading into Higher) • US Algebra I and the SAT: Forming and solving linear equations in one variable from angle facts (A-CED, A-REI); theorems about lines, angles and triangles (G-CO.9, G-CO.10); SAT Geometry and Trigonometry (lines, angles and triangles) Students will be able to: • use angles on a straight line, around a point, vertically opposite, complementary and supplementary angles to form and solve equations, giving reasons • find corresponding, alternate and co-interior angles with parallel lines, and show whether two lines are parallel • use the angle sum and exterior angle of a triangle, isosceles triangles, and the angle properties of special quadrilaterals • find the interior and exterior angles of polygons, and the number of sides of a regular polygon • use three-figure bearings and back bearings, with parallel north lines, to find angles and directions Calculator use: all five lessons and the whole unit test are non-calculator, matching the non-calculator papers of IGCSE and GCSE, where angle facts and bearings are mostly tested and every answer is a whole number or an exact decimal. All questions are original and written for this course. British spelling is used (US: "co-interior angles" are often called "same-side interior angles").

Unit 9 · Perimeter and Area: Measure Any Shape, From Pizza Slices to Football Fields
Unit overview By the end of this unit, students can find the perimeter and area of rectangles, triangles, parallelograms and trapeziums and work back to a missing length, calculate the circumference and area of circles, semicircles and quarter circles (exactly in terms of π and to 3 significant figures), find arc lengths, sector areas and sector perimeters, handle composite shapes, shaded regions and rings, and convert between units of area, including hectares, to solve tiling, painting, fencing and map-scale problems. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions (no calculator) and Section B has five structured problems (calculator allowed) with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Mensuration: perimeters and areas of triangles, parallelograms, trapeziums and circles, lengths of arcs and areas of sectors, composite plane figures, units of area and costing problems • Cambridge IGCSE Mathematics (0580): Mensuration: units of measure, perimeter and area of plane shapes, circumference and area of circles, arc length and sector area (Extended), compound shapes • GCSE Mathematics (UK): Geometry and measures: area of triangles, parallelograms and trapezia, circles, arcs and sectors (including exact answers in terms of π), composite shapes, converting units of area (Foundation, leading into Higher) • US Algebra I and the SAT: Area and circumference of circles (7.G.B.4), arc length and sector area (G-C.B.5), modelling with geometric shapes (G-MG.A.1); SAT Geometry and Trigonometry Students will be able to: • find the perimeter and area of rectangles, triangles, parallelograms and trapeziums, and find a missing length from the area or the perimeter • calculate the circumference and area of circles, semicircles and quarter circles, in terms of π and to 3 significant figures, and find a radius from a circumference or an area • find arc lengths, sector areas and sector perimeters, and find an unknown angle or radius • find the areas and perimeters of composite shapes, shaded regions and rings made from rectangles, triangles and parts of circles • convert between mm², cm², m², km² and hectares, and solve tiling, painting, fencing and map-scale problems Calculator use: Lessons 1 and 5 are non-calculator, like the polygon and unit-conversion questions on the non-calculator papers. Lessons 2, 3 and 4 allow a calculator (use the π key), and every rounded answer gives the same result with π = 3.142. Section A of the test is non-calculator with circle answers in terms of π; Section B allows a calculator. All questions are original and written for this course. British spelling is used.

Unit 10 · Statistics I: Turn Raw Data into Real Answers
Unit overview By the end of this unit, students can classify data and collect it fairly, calculate and choose between the mean, median, mode and range, find averages from frequency tables, draw and interpret bar charts and pie charts, and estimate the mean of grouped data. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (45 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Statistics: collection and tabulation of data, frequency tables, bar charts and pie charts, and the mean, median and mode of ungrouped and grouped data (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Statistics: classifying and tabulating data, bar charts and pie charts, averages and range, averages from frequency tables, the estimated mean and modal class of grouped data (Core and Extended) • GCSE Mathematics (UK): Statistics: sampling and bias, frequency tables, bar charts and pie charts, averages and spread, estimating the mean from grouped data, comparing distributions (Foundation, leading into Higher) • US Algebra I and the SAT: Interpreting categorical and quantitative data (S-ID.1–3): measures of centre and spread; SAT Problem-Solving and Data Analysis (data collection, charts, mean and median) Students will be able to: • classify data as qualitative or quantitative, discrete or continuous, primary or secondary, and design fair surveys and samples • calculate the mean, median, mode and range, find a missing value from the mean, and choose the most suitable average • find the mean, median and mode from a frequency table, and find a missing frequency from the mean • draw and read bar charts and pie charts, and compare two pie charts using their totals • find the modal class and the class containing the median, and estimate the mean of grouped data using midpoints Calculator use: Lessons 1–4 and Section A of the unit test are non-calculator, with totals chosen so that the division is short (pie-chart totals divide 360, frequency totals such as 20, 25 and 40). Lesson 5 and Section B allow a calculator, as in the IGCSE and GCSE calculator papers and WASSCE Paper 2, where grouped-data estimates usually appear; every rounded answer states its accuracy. All questions and data sets are original and written for this course. British spelling is used.

Unit 11 · Simultaneous Equations: Two Clues, One Perfect Answer
Unit overview By the end of this unit, students can solve a pair of linear simultaneous equations by elimination and by substitution, solve them graphically and explain when there is no solution or infinitely many, form and solve simultaneous equations from word problems (tickets, coins, ages, digits and mixtures), and tidy harder systems with fractions, decimals and brackets before solving and checking them. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Simultaneous linear equations in two variables, by elimination, substitution and graphs, and word problems (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Solving simultaneous linear equations, forming them from context, and finding the intersection of two straight-line graphs (Core and Extended) • GCSE Mathematics (UK): Algebra: solving two linear simultaneous equations algebraically and graphically, and setting them up from a context (Foundation, leading into Higher) • US Algebra I and the SAT: Systems of linear equations (A-REI.5, A-REI.6, A-REI.11, A-CED.3), including systems with no solution or infinitely many, and SAT Heart of Algebra questions Students will be able to: • solve simultaneous equations by elimination, adding, subtracting or multiplying one or both equations first • solve simultaneous equations by substitution, rearranging first when no letter is on its own • solve simultaneous equations graphically, read or estimate the crossing point, and recognise parallel lines (no solution) and the same line (infinitely many) • form two equations from a word problem about tickets, coins, ages, digits or mixtures, solve them and answer in context • clear fractions, decimals and brackets before solving, and check and interpret solutions, including those that do not make sense in context Calculator use: all five lessons and the whole unit test are non-calculator, matching the non-calculator papers of IGCSE and GCSE, where simultaneous equations are usually set with friendly numbers. All questions are original and written for this course. British spelling is used (US: "systems of equations" for "simultaneous equations").

Unit 12 · Variation: Discover How One Thing Changes Another
Unit overview By the end of this unit, students can write and use variation laws with the symbol ∝: direct variation y = kx, inverse variation y = k/x, variation with powers and roots (x², x³, √x and 1/x²), joint variation such as y = kxz and y = kx/z², and partial variation y = a + bx, finding the two constants from two conditions. They can also test a table of values to decide which law fits. The unit builds on direct and inverse proportion from Unit 2 and on straight-line graphs from Unit 7. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Variation: direct, inverse, joint and partial variation, finding the constant of variation and using the law (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Direct and inverse proportion, including proportion to squares, cubes and square roots, finding and using the formula (Extended) • GCSE Mathematics (UK): Ratio, proportion and rates of change: y = kx and y = k/x, proportion with powers and roots, and recognising proportion from tables and graphs (Foundation, leading into Higher) • US Algebra I and the SAT: Direct and inverse variation, linear models y = a + bx, and choosing the model that fits a table of values (SAT Problem-Solving and Data Analysis, Advanced Math) Students will be able to: • write a direct variation law y = kx, find k and use the formula to find y or x • write an inverse variation law y = k/x, find k = xy and use it, including percentage changes • use variation with x², x³, √x and 1/x², and describe the effect of doubling, halving or tripling x • write and use joint variation laws such as y = kxz and y = kx/z², including the effect of changing two quantities at once • find the constants in partial variation y = a + bx from two conditions, and test a table of values to choose the law that fits Calculator use: all five lessons and the whole unit test are non-calculator. The numbers are chosen so that every constant k is a whole number or a short decimal, and every square, cube and root is exact, matching the non-calculator papers of IGCSE and GCSE and WASSCE Paper 1, where variation is often tested. All questions are original and written for this course. British spelling is used.

Unit 13 · Pythagoras and Trigonometry: Measure What You Cannot Reach
Unit overview By the end of this unit, students can use Pythagoras' theorem to find the hypotenuse or a shorter side of a right-angled triangle, test whether a triangle is right-angled, solve problems with ladders, diagonals, isosceles triangles and coordinates, use sine, cosine and tangent (SOH CAH TOA) to find missing sides and angles, and apply trigonometry to angles of elevation and depression and to bearings. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Pythagoras' theorem, trigonometric ratios of acute angles, angles of elevation and depression, and bearings with right-angled triangles (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Pythagoras' theorem and right-angled triangle trigonometry, including angles of elevation and depression and bearings (Core and Extended) • GCSE Mathematics (UK): Geometry and measures: Pythagoras' theorem and trigonometric ratios in right-angled triangles, exact values of sin, cos and tan, and bearings (Foundation, leading into Higher) • US Algebra I and the SAT: Right-triangle trigonometry and the Pythagorean theorem (G-SRT.6, G-SRT.8); SAT Geometry and Trigonometry (right triangles, sine, cosine and tangent) Students will be able to: • find the hypotenuse of a right-angled triangle, exactly as a surd or to 3 significant figures • find a shorter side, decide whether a triangle is right-angled, and solve problems with ladders, diagonals, isosceles triangles and coordinates • choose sine, cosine or tangent to find a missing side, with the unknown on the top or the bottom • find a missing angle with the inverse functions, and use the exact values for 30°, 45° and 60° • solve problems with angles of elevation and depression and with bearings using right-angled triangles Calculator use: all five lessons allow a calculator, because most answers must be rounded (lengths to 3 significant figures, angles to 1 decimal place), as on the calculator papers of IGCSE, GCSE and WASSCE. Lessons 1 and 2 also include exact questions (Pythagorean triples and surds), and Section A of the unit test is non-calculator, using triples, surds and given trigonometry values; Section B allows a calculator. All questions are original and written for this course. British spelling is used.

Unit 14 · Surface Area and Volume: Fill It, Wrap It, Pack It
Unit overview By the end of this unit, students can find the volume of cuboids and prisms (including triangular, trapezium and L-shaped prisms) and work back to a missing length, find surface areas from nets, calculate the volume and surface area of cylinders, use the exam formulas for pyramids, cones and spheres, and convert between mm³, cm³, m³ and litres to solve filling, rate and packing problems. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions (no calculator) and Section B has five structured problems (calculator allowed) with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Mensuration: surface areas and volumes of cuboids, prisms, cylinders, cones, pyramids and spheres, capacity, and practical problems • Cambridge IGCSE Mathematics (0580): Mensuration: volume and surface area of prisms and cylinders, pyramids, cones and spheres using given formulas, units of volume and capacity (Core and Extended) • GCSE Mathematics (UK): Geometry and measures: volume and surface area of cuboids, prisms and cylinders, cones, pyramids and spheres, converting units of volume and compound measures such as rates of flow (Foundation, leading into Higher) • US Algebra I and the SAT: Volume formulas for cylinders, pyramids, cones and spheres (G-GMD.A.3), modelling with solids (G-MG.A.1); SAT Geometry and Trigonometry Students will be able to: • find the volume of cuboids and prisms, including triangular, trapezium and L-shaped prisms, and find a missing length from the volume • draw and use nets to find the surface area of cuboids, open boxes and prisms, and work back from a surface area • calculate the volume, curved surface area and total surface area of cylinders, in terms of π and to 3 significant figures • use the formulas for pyramids, cones and spheres, including slant height by Pythagoras and composite solids • convert between mm³, cm³, m³ and litres, and solve filling, emptying, rate and packing problems Calculator use: Lessons 1, 2 and 5 are non-calculator, with friendly numbers. Lessons 3 and 4 allow a calculator (use the π key), and every rounded answer gives the same result with π = 3.142. Section A of the test is non-calculator with answers in terms of π; Section B allows a calculator. All questions are original and written for this course. British spelling is used. Lesson pages. Paste each shortcode in

Unit 15 · Probability: Predict the Chance of Anything
Unit overview By the end of this unit, students can place events on the probability scale and find the probability of a single event, estimate probabilities from experiments and predict expected outcomes, list sample spaces for two events, use tree diagrams for independent and dependent events, and find probabilities from Venn diagrams and two-way tables. The unit has five lessons of about 15 minutes, each ending with a five-question exit quiz, and an exam-style unit test: Section A has 18 multiple-choice questions and Section B has five structured problems with marks shown on every part, a model solution and an M/A/B mark scheme (44 marks, 70 minutes). 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. These skills map to the main upper-secondary courses and exams: • WASSCE Core Mathematics (WAEC): Probability: experimental and theoretical probability, sample spaces for two events, mutually exclusive and independent events, tree diagrams and probability from Venn diagrams (Paper 1 objectives and Paper 2 structured questions) • Cambridge IGCSE Mathematics (0580): Probability: the probability scale, relative and expected frequency, sample-space diagrams, tree diagrams with and without replacement, and probabilities from Venn diagrams and two-way tables (Core and Extended) • GCSE Mathematics (UK): Probability: the 0 to 1 scale, relative frequency and expected outcomes, sample-space diagrams, tree diagrams, Venn diagrams, two-way tables and the addition rule (Foundation, leading into Higher) • US Algebra I and the SAT: Probability rules (S-CP.1, S-CP.2, S-CP.7): sample spaces, unions, intersections and complements, independent events and the addition rule; SAT Problem-Solving and Data Analysis (probability from two-way tables) Students will be able to: • place events on the probability scale, find the probability of a single event and its complement, and find a missing probability or a missing total • estimate a probability from relative frequency, combine results, find an expected number and judge whether a dice, coin or spinner is fair • list the sample space for two events in a list or a grid and use it to find probabilities • draw and use tree diagrams for independent and dependent events, with and without replacement, including "at least one" problems • find probabilities from Venn diagrams and two-way tables, use the addition rule and the OR rule for mutually exclusive events Calculator use: all five lessons and the whole unit test are non-calculator. Probabilities are given as fractions in their simplest form, or as decimals where the question uses decimals, matching the non-calculator papers of IGCSE and GCSE, where most probability questions appear. All questions and data sets are original and written for this course. British spelling is used.