Junior Math 2

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Course Content

Unit 1: Negatives, Powers, Roots and Accuracy
Unit overview By the end of this unit, students can calculate with negative numbers using the correct order of operations, use the index laws, find and estimate square and cube roots, round to decimal places and significant figures, and estimate calculations to check answers. The theme is "Precise enough?": freezers, golf scores, computer memory, gardens, city populations and school trips. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and number lines and tables throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Integers and the four operations, indices and the laws of indices, square and cube roots, and approximation to decimal places and significant figures (JHS 2) • Cambridge Lower Secondary (Stage 8): Calculating with integers, using the index laws for multiplication and division, estimating square and cube roots, rounding to significant figures and estimating calculations (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Using the four operations with negative numbers, index notation and laws, roots, rounding to decimal places and significant figures, and estimating answers; GCSE Foundation number • US Grade 7 and 8 (Common Core): Operations with rational numbers including negatives (7.NS.1–3), and square and cube roots (8.EE.2) Students will be able to: • add, subtract, multiply and divide negative numbers, using the order of operations • use the index laws to multiply and divide powers, find a power of a power, and use the zero index • find square and cube roots, including of negative numbers, and estimate roots between whole numbers • round numbers to a given number of decimal places or significant figures • estimate the answer to a calculation and use it to check an exact answer All questions and problems are original and written for this course. British spelling is used (US: "exponent" for "index").

  • Lesson 1 · Calculating With Negative Numbers: Order of Operations With a Twist
  • Lesson 1 Exit Quiz: Calculating With Negatives
  • Lesson 2 · Index Laws: Add the Powers, Don’t Multiply Them
  • Lesson 2 Exit Quiz: Index Laws
  • Lesson 3 · Square Roots and Cube Roots: Between Which Two Whole Numbers?
  • Lesson 3 Exit Quiz: Square and Cube Roots
  • Lesson 4 · Decimal Places and Significant Figures: How Precise Is Precise Enough?
  • Lesson 4 Exit Quiz: Decimal Places and Significant Figures
  • Lesson 5 · Estimating Calculations: Check It Before You Trust It
  • Lesson 5 Exit Quiz: Estimating Calculations
  • Year 2 Unit 1 Test: Negatives, Powers, Roots and Accuracy (Section A)

Unit 2: Fractions, Percentages and Simple Interest
Unit overview By the end of this unit, students can calculate with mixed numbers using all four operations, convert fractions to terminating and recurring decimals, use multipliers for percentage increases and decreases, find percentage change, profit and loss, and calculate simple interest. The theme is "Money and more": recipes, paint, taxi fares, sales, market stalls and savings accounts. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and tables and division diagrams throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Operations on fractions, percentages, profit and loss, discount, and simple interest (a regular Paper 2 topic in JHS 2) • Cambridge Lower Secondary (Stage 8): Calculating with mixed numbers, recognising terminating and recurring decimals, and using percentage increase and decrease, including multipliers (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): The four operations with mixed numbers, recurring decimals, percentage change with multipliers, and simple interest; GCSE Foundation number and ratio • US Grade 7 (Common Core): Converting rational numbers to terminating or repeating decimals (7.NS.2d), and multistep percentage problems including tax, markups, markdowns and simple interest (7.RP.3, 7.EE.3) Students will be able to: • add, subtract, multiply and divide mixed numbers • convert fractions to decimals and tell whether a decimal terminates or recurs, using dot notation • increase and decrease amounts by a percentage using multipliers, including repeated changes • find percentage change and percentage profit or loss, and the selling price for a given profit • calculate simple interest, the total amount, and the rate or time All questions and problems are original and written for this course. British spelling is used (US: "repeating decimal" for "recurring decimal").

Unit 3: Ratio, Proportion, Scale and Rates
Unit overview By the end of this unit, students can solve ratio problems when only one share or the difference is known, recognise and use direct and inverse proportion, use map scales and scale drawings, and calculate with compound measures such as speed, density and rates of pay. The theme is "How things compare": sweets, smoothies, taxis, painters, maps, cheetahs and weekend jobs. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and bar models, graphs, maps and tables throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Ratio and proportion, direct and inverse proportion, rates, scale drawing and map reading (JHS 2) • Cambridge Lower Secondary (Stage 8): Solving ratio problems, direct and inverse proportion, scale drawings, and compound measures including speed (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Ratio problems, direct and inverse proportion, scale factors, maps and scale drawings, and compound units such as speed, density and unit pricing; GCSE Foundation ratio and proportion • US Grade 7 (Common Core): Unit rates (7.RP.1), recognising and representing proportional relationships with tables, graphs and y = kx (7.RP.2), and scale drawings (7.G.1) Students will be able to: • solve ratio problems using one known share, the difference or the total, including three-part ratios • recognise direct proportion from tables, graphs and equations, and use the unitary method • solve inverse proportion problems by finding the constant product • use map scales and scale drawings to find real and drawn lengths, and write a scale as a ratio • calculate speed, distance, time, density and rates of pay, and convert between units All questions and problems are original and written for this course. British spelling is used (US: "metre" is "meter").

Unit 4: Expanding, Factorising and Formulae
Unit overview By the end of this unit, students can expand brackets with negative terms and letters outside, factorise expressions fully, use the index laws with algebraic terms, substitute positive and negative values into expressions and formulae, and change the subject of a formula. The theme is "Algebra that works": mental arithmetic tricks, gardens, cubes, thrown balls, temperatures and phone plans. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and area models and step tables throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Expansion of brackets, factorisation by common factors, substitution, and change of subject of a formula (JHS 2; a regular Paper 1 and Paper 2 topic) • Cambridge Lower Secondary (Stage 8): Expanding and factorising, simplifying with index laws, substituting into formulae, and changing the subject of simple formulae (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Multiplying a single term over a bracket, taking out common factors, simplifying with index laws, substituting into formulae and rearranging formulae; GCSE Foundation algebra • US Grade 7 (Common Core): Applying properties of operations to expand and factor linear expressions (7.EE.1–2), and solving and rearranging equations in context (7.EE.4) Students will be able to: • expand single brackets with negative numbers or letters outside, and expand and simplify • factorise expressions fully by taking out the highest common factor • simplify products and quotients of algebraic terms using the index laws • substitute positive and negative numbers into expressions and formulae, including powers • change the subject of a formula with one or two steps All questions and problems are original and written for this course. British spelling is used (US: "factor" for "factorise").

Unit 5: Equations and Inequalities
Unit overview By the end of this unit, students can solve linear equations with brackets, with the unknown on both sides and with fractions, form equations from words, angles, perimeters and ages, and solve linear inequalities and show them on a number line. The theme is "Finding the unknown": number puzzles, phone plans, candles, restaurant bills, fields and lifts. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and balances, step tables and number lines throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Linear equations in one variable, including brackets and fractions, word problems, and linear inequalities with number-line solutions (JHS 2; a regular Paper 1 and Paper 2 topic) • Cambridge Lower Secondary (Stage 8): Constructing and solving linear equations, including the unknown on both sides and brackets, and solving simple linear inequalities (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Solving linear equations with the unknown on both sides, forming equations from problems, and solving and representing inequalities; GCSE Foundation algebra • US Grade 7 (Common Core): Solving word problems with equations and inequalities of the form px + q = r and px + q > r (7.EE.4), and solving linear equations with the variable on both sides (8.EE.7) Students will be able to: • solve linear equations with brackets, including negative and fractional answers • solve equations with the unknown on both sides • solve equations involving fractions by clearing the denominators • form and solve equations from number puzzles, shapes, angles and ages • solve linear inequalities, show them on a number line and list integer solutions All questions and problems are original and written for this course. British spelling is used.

Unit 6: Sequences, Gradient and Straight-Line Graphs
Unit overview By the end of this unit, students can find and use the nth term of a linear sequence, recognise square, cube, triangle, Fibonacci and geometric sequences, find the gradient of a line, read and write equations in the form y = mx + c, and interpret distance–time graphs. The theme is "Patterns and slopes": matchsticks, bowling pins, handshakes, ramps, water slides, taxis, gyms and bike rides. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and term cards, coordinate grids and journey graphs throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Number patterns and the nth term, gradient of a straight line, linear relations and their graphs, and travel graphs (JHS 2; a regular Paper 1 and Paper 2 topic) • Cambridge Lower Secondary (Stage 8): The nth term of linear sequences, special sequences, gradient, linear functions in the form y = mx + c, and real-life graphs (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Generating sequences from the nth term, recognising special sequences, gradient and intercept, y = mx + c, and interpreting distance–time graphs; GCSE Foundation algebra • US Grade 7–8 (Common Core): Proportional relationships and the constant of proportionality on graphs (7.RP.2b–d), and slope as a rate of change, y = mx + b (8.EE.5–6) Students will be able to: • find the nth term of a linear sequence, use it to find any term, and decide whether a number is in the sequence • recognise and continue square, cube, triangle, Fibonacci and geometric sequences • find the gradient of a line from two points, a graph or a real-life slope • read the gradient and intercept from y = mx + c, write equations of lines, and spot parallel lines • interpret distance–time graphs and calculate speed and average speed All questions and problems are original and written for this course. British spelling is used (US: "slope" for "gradient" and y = mx + b).

Unit 7: Angles, Polygons, Bearings and Loci
Unit overview By the end of this unit, students can use the angle facts for parallel lines, find interior and exterior angles of polygons, use three-figure bearings, describe and construct simple loci, and combine angle facts with clear reasons. The theme is "Shapes and directions": railway tracks, street maps, floor tiles, ships, lighthouses, orienteering, goats on ropes and roof trusses. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and angle diagrams, compass roses and construction drawings throughout. No calculator is used; a ruler, protractor and compasses are useful. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Angles formed by parallel lines and a transversal, angles of polygons, bearings, and ruler-and-compass constructions and loci (JHS 2; a regular Paper 2 topic) • Cambridge Lower Secondary (Stage 8): Angles with parallel lines, interior and exterior angles of polygons, bearings, and constructions including bisectors (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Angle facts with parallel lines, angle sums of polygons, bearings, constructions and loci, and deriving and explaining angle facts; GCSE Foundation geometry • US Grade 7–8 (Common Core): Angle relationships to write and solve equations (7.G.5), constructing geometric shapes (7.G.2), and angles created when parallel lines are cut by a transversal, and the triangle angle sum (8.G.5) Students will be able to: • find corresponding, alternate, co-interior and vertically opposite angles, and give reasons • find the interior angle sum of any polygon and the interior and exterior angles of regular polygons • use and write three-figure bearings, including back bearings • describe loci and construct the perpendicular bisector of a line • combine angle facts in multi-step problems, with algebra, giving a reason for every step All questions and problems are original and written for this course. British spelling is used (US: "parallelogram" is the same; "co-interior angles" are called "same-side interior angles").

Unit 8: Transformations and Vectors
Unit overview By the end of this unit, students can reflect shapes in the axes and in lines such as y = x, rotate shapes about a point, translate shapes using column vectors, enlarge shapes from a centre with a given scale factor, and recognise congruent and similar shapes. The theme is "Moving shapes": mirrors and kaleidoscopes, clock hands and windmills, chess knights and warehouse robots, photos, posters and phone screens. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and coordinate-grid diagrams throughout. No calculator is used. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Reflection, rotation, translation and enlargement on the coordinate plane, and column vectors (JHS 2; a regular Paper 2 topic) • Cambridge Lower Secondary (Stage 8): Reflecting, rotating, translating and enlarging shapes, describing transformations, and congruence (Checkpoint Paper 1, no calculator) • UK Key Stage 3 (Year 8): Reflections, rotations, translations with column vectors, enlargements with a centre and scale factor, and congruence and similarity; GCSE Foundation geometry • US Grade 8 (Common Core): Properties of rotations, reflections and translations (8.G.1), congruence through rigid motions (8.G.2), describing transformations with coordinates (8.G.3), and similarity through dilations (8.G.4) Students will be able to: • reflect shapes in the axes, in y = x and y = −x, and in vertical and horizontal lines • rotate shapes through 90° and 180° about a point, and describe a rotation fully • translate shapes using column vectors, and find the vector between two points • enlarge shapes from a centre, and find a scale factor • recognise congruent and similar shapes, and find missing lengths in similar shapes All questions and problems are original and written for this course. British spelling is used (US: "dilation" for "enlargement"). In the CSV files and entry sheets, a column vector is written as (top over bottom), for example (3 over −2).

Unit 9: Circles, Area and Volume
Unit overview By the end of this unit, students can find the circumference and area of circles and parts of circles, find the volume and surface area of prisms and cylinders, and convert between units of area and volume. The theme is "Round things and containers": bicycle wheels, pizzas, ponds, tins, water troughs, gift boxes and swimming pools. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and circle, net and solid diagrams throughout. Lessons 1–4 are calculator lessons: each one also has BECE-style questions that use π = 22/7 with numbers that work out neatly, so students practise both ways. Lesson 5 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): Circumference and area of circles using π = 22/7, and the mensuration of prisms and cylinders: volume, surface area and capacity (JHS 2; a regular Paper 2 topic) • Cambridge Lower Secondary (Stage 8): Circumference and area of circles, volume and surface area of prisms and cylinders, and converting between units of area and volume (Checkpoint Papers 1 and 2) • UK Key Stage 3 (Year 8): Circumference and area of circles, volume of prisms and cylinders, surface area from nets, and compound units of area and volume; GCSE Foundation mensuration • US Grade 7–8 (Common Core): Formulas for the area and circumference of a circle (7.G.4), area, volume and surface area of prisms (7.G.6), and the volume of cylinders (8.G.9) Students will be able to: • find the circumference of a circle and of semicircles, exactly in terms of π or to 1 decimal place • find the area of a circle, semicircle and quarter circle, and work backwards from an area • find the volume of prisms and cylinders, and convert between cm³ and litres • find the surface area of cuboids, triangular prisms and cylinders • convert between units of area and volume, including litres and m³ All questions and problems are original and written for this course. British spelling is used (US: "meter" for "metre", "liter" for "litre").

Unit 10: Statistics: Sampling, Charts and Averages
Unit overview By the end of this unit, students can choose fair samples and spot bias, organise large data sets into grouped frequency tables and diagrams, draw and read pie charts, find averages from frequency tables, and compare data sets using stem-and-leaf diagrams. The theme is "What does the data say?": school surveys, test marks, homework times, favourite sports, football goals, traffic and plant heights. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and tables, frequency diagrams and pie charts throughout. The whole unit is non-calculator. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Collecting and presenting data, frequency tables, pie charts, and the mean, median and mode (JHS 2; pie charts and averages from tables are regular Paper 2 questions) • Cambridge Lower Secondary (Stage 8): Sampling and bias, grouped frequency tables and diagrams, pie charts, stem-and-leaf diagrams, and averages and range used to compare data sets (Checkpoint Papers 1 and 2) • UK Key Stage 3 (Year 8): Sampling, grouped frequency tables, frequency diagrams, pie charts, stem-and-leaf diagrams, and averages from frequency tables; GCSE Foundation statistics • US Grade 6–7 (Common Core): Random sampling and inferences about a population (7.SP.1–2), comparing two data sets by centre and spread (7.SP.3–4), and displaying and summarising data (6.SP.4–5) Students will be able to: • explain why a random sample is fairer, spot bias, and estimate for a population from a sample • group data into suitable classes, find the modal class, and draw and read frequency diagrams • work out the angles for a pie chart and read frequencies back from one • find the mode, median, mean and range from a frequency table, and estimate the mean of grouped data • draw and read stem-and-leaf diagrams and compare two data sets using an average and the range All questions and problems are original and written for this course. British spelling is used (US: "favorite" for "favourite").

Unit 11: Probability and Sets
Unit overview By the end of this unit, students can find probabilities for equally likely outcomes, use the fact that probabilities add to 1, list sample spaces for two events, estimate probabilities from experiments, and solve problems with Venn diagrams. The theme is "What are the chances?": dice and board games, raffles, weather forecasts, school buses, coin experiments and club surveys. The unit has five lessons of 10–15 minutes, a unit test with three worked problems, and probability scales, sample-space tables and Venn diagrams throughout. The whole unit is non-calculator. These skills appear in every lower-secondary maths course and exam: • BECE Mathematics (WAEC): Probability of simple events, experimental probability, outcomes of two events, and sets problems solved with Venn diagrams (JHS 2; Venn diagram word problems are a regular Paper 2 question) • Cambridge Lower Secondary (Stage 8): Equally likely outcomes, P(not A), sample spaces for two events, and experimental probability compared with theory (Checkpoint Papers 1 and 2) • UK Key Stage 3 (Year 8): Probabilities summing to 1, sample spaces, relative frequency and expected outcomes, and Venn diagrams; GCSE Foundation probability • US Grade 7 (Common Core): Probability as a number from 0 to 1 (7.SP.5), relative frequency (7.SP.6), probability models (7.SP.7) and compound events using lists and tables (7.SP.8) Students will be able to: • find the probability of an event with equally likely outcomes, as a fraction, decimal or percentage • use P(not A) = 1 − P(A) and find a missing probability • list a sample space for two coins, a coin and a dice, or two dice, and use it to find probabilities • find relative frequencies and expected frequencies, and judge whether a coin or dice is fair • solve "both" and "neither" problems with a Venn diagram and find probabilities from it All questions and problems are original and written for this course. British spelling is used. "Dice" is used for both one and many (US: "die" and "dice").

Unit 12: Junior 2 Review and End-of-Year Exam
Unit overview This unit brings the whole of Year 2 Maths together. Four review lessons revisit number and proportion, algebra, geometry and measures, and statistics and probability, each built around the traps that cost the most marks. The unit ends with a full end-of-year exam: Paper 1 is 40 multiple-choice questions (45 minutes, no calculator, marked automatically) and Paper 2 is six structured problems (30 marks, Part 1 without and Part 2 with a calculator), with model solutions and a marking scheme. The exam is modelled on the papers students will meet later: • BECE Mathematics (WAEC): Paper 1 objective questions and Paper 2 structured questions across number, algebra, geometry, mensuration (with π = 22/7), statistics, probability and sets (JHS 2 content; Paper 1 and Part 1 of Paper 2 are good non-calculator BECE practice) • Cambridge Lower Secondary (Stage 8): Checkpoint-style Paper 1 (no calculator) and Paper 2 (calculator) covering number, algebra, geometry and measures, and statistics and probability • UK Key Stage 3 (Year 8): End-of-year assessment of the Year 8 programme of study, with non-calculator and calculator sections in the style of GCSE Foundation papers • US Grade 7–8 (Common Core): Coverage of ratio and proportional relationships, the number system, expressions and equations, geometry, and statistics and probability in a mixed review Students will be able to: • use powers, roots, significant figures, percentage multipliers, ratio and rates without slips • expand, factorise, solve equations and inequalities, and link nth terms and gradients to graphs • use angle facts, polygons, bearings, transformations and circle and volume formulae • choose the right statistics and probability tools for a question • complete a full end-of-year exam and use the results to plan revision All questions and problems are original and written for this course. British spelling is used.