Maths for Year 7

🔢 Sequences Patterns, term-to-term rules, linear and non-linear

Patterns are everywhere: the seats in the rows of a theatre, the bricks in a growing wall, the house numbers down one side of a street (1, 3, 5, 7…). In this chapter you will continue patterns of shapes and numbers, find the rule that makes them, tell linear sequences from non-linear ones and draw sequences on a graph.

Practice: Chapter test: sequences · Sequences: key words

• 🔷 Sequences and patterns Continuing a sequence of numbers or shapes

  Build a row of squares from matchsticks and count the sticks: 4, 7, 10… Every new square needs the same number of extra sticks, so you can say how many sticks a long row needs without building it. A list of numbers or shapes that follows a rule like this is called a sequence.

  Practice: Sequences and patterns

• ➕ Term-to-term rules How to get from one term to the next

  A term-to-term rule works like a recipe: it tells you how to turn one term into the next. Give someone the first term and the rule, and they can carry on the sequence for as long as they like, without ever seeing it.

  Practice: Term-to-term rules

• 📈 Linear and non-linear sequences Going up by the same amount, or not

  Save £5 every week and your savings go up by the same amount each time: £5, £10, £15, £20. Pass a secret to 2 friends, who each pass it to 2 more, and the number who know it grows faster and faster: 1, 2, 4, 8, 16. The first is a linear sequence; the second is non-linear.

  Practice: Linear and non-linear sequences

• 📊 Sequences on a graph Plotting the terms and spotting a straight line

  A table of positions and terms can become a picture. Plot each term on a grid and a linear sequence lines up as straight as a ruler, while a non-linear one bends into a curve. One look at the graph tells you what kind of sequence it is.

  Practice: Sequences on a graph

🔤 Algebraic notation Function machines, expressions and substitution

Algebra is maths with letters. A letter stands for a number you don't know yet, or for any number at all, so one short line like 3n + 1 can describe a whole pattern. In this chapter you will use function machines, write expressions with letters, put numbers in place of the letters and use rules with letters to make…

Practice: Chapter test: algebraic notation · Algebraic notation: key words

• ⚙️ Function machines Inputs, outputs and inverse machines

  A function machine works like a vending machine for numbers: put a number in, the machine does something to it, and a new number comes out. And if you know what came out, you can run the machine backwards to find what went in.

  Practice: Function machines

• 🔤 Using letters for numbers 3a, a + 3, a², a ÷ 2: writing expressions

  You have a bag of marbles but haven't counted them. You can't write the number down, but you can give it a name: n. Three bags like it hold 3n marbles, whatever n turns out to be. In algebra a letter stands for a number, and there is a short, fixed way of writing it.

  Practice: Using letters for numbers

• 🔁 Substitution Putting numbers in place of letters

  A formula is only useful once you put real numbers into it. A cinema charges £6 a ticket plus a £2 booking fee, so n tickets cost 6n + 2 pounds. For 3 tickets, swap n for 3: 6 × 3 + 2 = £20. Swapping each letter for its number and working out the answer is called substitution.

  Practice: Substitution

• ✏️ Writing expressions from words Turning a sentence into algebra

  "Five more than my age", "twice as many as last week", "shared equally between four": everyday sentences are full of hidden maths. Turning the words into an expression makes them short, and then you can work with them.

  Practice: Writing expressions from words

• 🧮 Sequences from expressions Using a rule like 3n + 1 to make a sequence

  With a term-to-term rule you must work out every term before the one you want. A position-to-term rule like 3n + 1 jumps straight there: put in the position, get the term. Want the 100th term? It's one calculation.

  Practice: Sequences from expressions

⚖️ Equality and equivalence Equations, solving them and like terms

An equation is like a balance: whatever is on one side of the equals sign is worth exactly the same as the other side. In this chapter you will see what = really means, solve equations by doing the same to both sides, collect like terms to make expressions simpler, and meet the sign ≡ for expressions that are always…

Practice: Chapter test: equality and equivalence · Equality and equivalence: key words

• ⚖️ Equations and expressions What the equals sign really means

  At primary school the equals sign often seemed to mean "and the answer is". It means more than that: both sides have the same value. 6 + 4 = 3 + 7 is just as true as 6 + 4 = 10, because both sides are worth 10.

  Practice: Equations and expressions

• ➖ Solving equations with + and − Doing the same to both sides

  If x + 5 = 12, you can probably see that x is 7. This page gives you a method that still works when the numbers are big, awkward or decimal: undo what was done to x, and do the same to both sides so the equation stays balanced.

  Practice: Solving equations with + and −

• ✖️ Solving equations with × and ÷ Undoing a multiplication or a division

  Three identical parcels weigh 21 kg together. How heavy is one? You divide by 3 and get 7 kg. Without noticing, you have just solved the equation 3x = 21: you undid a multiplication with a division.

  Practice: Solving equations with × and ÷

• 🧺 Like terms Collecting like terms to simplify

  Tip a jar of coins on the table and sort them: the 10p coins in one pile, the 2p coins in another. Then you count each pile. Collecting like terms in algebra works the same way: put the same kind of terms together, then add them up.

  Practice: Like terms

• 🟰 Equivalence and identities When two expressions are always equal: ≡

  2a + 3a and 5a look different, but put in any number for a and they come out the same. Expressions that are always equal, whatever the letters stand for, are called equivalent, and they have their own sign: ≡.

  Practice: Equivalence and identities

🏷️ Place value and ordering Integers and decimals, rounding, median and range

Every number you will ever write is built from just ten digits, 0 to 9. What makes a 7 mean seven, seventy or seven thousandths is the column it sits in: its place value. In this chapter you will read and write numbers from billions down to thousandths, put them in order and read scales, round them sensibly, and use…

Practice: Chapter test: place value and ordering · Place value and ordering: key words

• 🔟 Place value in integers Ones to billions

  A big football stadium holds 60,000 people, a big city has 2,000,000 and more than 8,000,000,000 people live on Earth. To read numbers like these you need to know what each digit is worth. That depends on where it stands in the number: its place value.

  Practice: Place value in integers

• 🔸 Place value in decimals Tenths, hundredths and thousandths

  A bag of sugar weighs 1.5 kg, a can holds 0.33 litres and the fastest 100 m ever run took 9.58 seconds. Decimals are numbers with parts smaller than one. The same place value idea works after the decimal point: each column is worth ten times less than the one before it.

  Practice: Place value in decimals

• 📏 Ordering numbers and number lines Comparing with < and >, and reading scales

  Who ran faster? Which phone is cheaper? Is it colder today than yesterday? You compare numbers all the time. A number line shows numbers in order, and the same skill lets you read a ruler, a measuring jug or a set of kitchen scales.

  Practice: Ordering numbers and number lines

• 🎯 Rounding To the nearest 10, 100, whole number or decimal place

  The news says "about 40,000 fans" were at the match, when the exact number was 38,612. Rounding swaps a number for a simpler one close to it, so it is easier to say, remember and work with. The sign ≈ means "is approximately equal to": 38,612 ≈ 40,000.

  Practice: Rounding

• 📉 Median and range The middle value and the spread of a list

  Ali scored 6, 7, 7, 8 and 7 points in five games; Ben scored 1, 13, 7, 2 and 12. Both usually score about 7, but Ben is much less predictable. To describe a set of numbers (data) you need a typical value, such as the median, and a measure of how spread out the values are: the range.

  Practice: Median and range

🍕 Fractions, decimals and percentages Three ways to write the same amount

Half price, 50% off, 0.5 litres: three ways of writing the same amount. Fractions, decimals and percentages turn up everywhere, from recipes and test scores to sales and surveys. In this chapter you will see what a fraction really means, find equivalent fractions, switch between all three forms and read and draw pie…

Practice: Chapter test: fractions, decimals and percentages · Fractions, decimals and percentages: key words

• 🍕 Understanding fractions Parts of a whole and fractions on a number line

  Cut a pizza into 8 equal slices and eat 3 of them: you have eaten three eighths of the pizza. A fraction describes a part of a whole, and the whole must be split into equal parts.

  Practice: Understanding fractions

• 🟰 Equivalent fractions and simplifying Same value, different numbers

  Cut a cake in half and take one half, or cut it into quarters and take two: you get exactly the same amount of cake. 12 and 24 are equivalent fractions: different numbers, the same value.

  Practice: Equivalent fractions and simplifying

• 🔸 Fractions and decimals Tenths, hundredths and converting between them

  A recipe asks for 34 of a kilogram of flour and the scales show 0.75 kg: the same amount. A decimal is another way of writing a fraction whose denominator is 10, 100, 1000 and so on, and any fraction can be turned into a decimal.

  Practice: Fractions and decimals

• 💯 Percentages Out of a hundred: converting between all three

  "40% off", "battery 85%", "92% in the test". Per cent means "out of a hundred" (from the Latin per centum). A percentage is a fraction with a denominator of 100, written with the sign %. Because everything is out of the same 100, percentages make amounts easy to compare.

  Practice: Percentages

• 🥧 Pie charts Reading and drawing a pie chart

  A pie chart is a circle cut into slices, one for each group. It shows at a glance how a whole is shared out: how a class travels to school, or which fruit is the favourite. The bigger the slice, the bigger the fraction of the whole.

  Practice: Pie charts

➕ Addition and subtraction Mental and written methods, perimeter and tables

You add and subtract every day: checking your change in a shop, working out how long until the bus comes, or how many more points your team needs. In this chapter you make your mental methods quicker and your written methods reliable, for whole numbers and decimals. Then you use them to find perimeters, read tables…

Practice: Chapter test: addition and subtraction · Addition and subtraction: key words

• 🧠 Mental methods Partitioning, rounding and adjusting, number bonds

  You don't need a pen to work out 245 + 99 or the change from £10. Good mental methods split numbers into easy parts, or round to a friendly number and then adjust. Choosing the right trick turns a hard sum into an easy one.

  Practice: Mental methods

• 🧾 Column addition and subtraction Whole numbers and decimals, lined up by place value

  Some calculations are too long to do in your head, like 3486 + 2758 or 8 − 2.35. Write the numbers in columns so that digits with the same place value sit under each other, then work from right to left, one column at a time.

  Practice: Column addition and subtraction

• 📐 Perimeter The distance all the way around a shape

  To put a fence round a garden or a border round a noticeboard, you need to know the distance all the way round its edge. That distance is the perimeter, and you find it by adding up the lengths of all the sides.

  Practice: Perimeter

• 📊 Tables, bar charts and frequency trees Reading and completing data

  A school survey, a bus timetable, the weather through the day: information often comes in tables and charts. To answer questions about it you mostly add and subtract, and you often have to work out a number that is missing.

  Practice: Tables, bar charts and frequency trees

• 🧩 Solving problems Bar models for adding and subtracting

  Word problems hide a calculation inside a story. A bar model is a quick drawing that shows which numbers are the parts and which is the whole, so you can see whether to add or subtract before you start.

  Practice: Solving problems

✖️ Multiplication and division Methods, area, the mean and the order of operations

Multiplying and dividing turn up everywhere: scaling up a recipe, sharing a bill, working out how much carpet a room needs or your average score in a game. In this chapter you build fast, reliable methods for whole numbers and decimals, learn the order to do a long calculation in, find areas of shapes and work out the…

Practice: Chapter test: multiplication and division · Multiplication and division: key words

• 🔟 Multiplying and dividing by 10, 100 and 1000 Digits move, the decimal point stays

  A metre is 100 centimetres, a kilogram is 1000 grams and a box might hold 10 pencils. Multiplying and dividing by 10, 100 and 1000 is the quickest calculation there is, once you see what happens to the digits.

  Practice: Multiplying and dividing by 10, 100 and 1000

• 🧾 Written methods Long multiplication and short division

  You know 7 × 8 straight away, but 243 × 36 or 852 ÷ 6 need a written method. Written methods break a big calculation into times-table facts and keep track of the place values for you.

  Practice: Written methods

• 🔸 Multiplying and dividing decimals Working with decimal numbers

  Three tins of soup at £1.35 each, a 2.4 m plank cut into equal pieces, a recipe for 1.5 times as many people: real calculations are full of decimals. The trick is to calculate with whole numbers, then put the decimal point back in the right place.

  Practice: Multiplying and dividing decimals

• 🧮 The order of operations Brackets first: BIDMAS

  What is 2 + 3 × 4? Some people say 20 and some say 14. If everyone worked in their own order, the same calculation would give different answers, so mathematicians agree on one order. The answer is 14.

  Practice: The order of operations

• 🟦 Area of rectangles, triangles and parallelograms Squares inside a shape, and the formulas

  How much carpet does a bedroom need, or how many tiles will cover a wall? You need the area: the amount of flat space a shape covers. Area is measured by counting squares, and for rectangles, triangles and parallelograms there are formulas that count them for you.

  Practice: Area of rectangles, triangles and parallelograms

• ⚖️ The mean Share it out equally: the mean average

  In four spelling tests you scored 3, 7, 4 and 6. What did you score on average? The mean answers it: if every test had the same score, and the total stayed the same, what would that score be?

  Practice: The mean

💷 Fractions and percentages of amounts Finding a part, and finding the whole

"A third off", "save 25%", "three quarters of the class walk to school": fractions and percentages of amounts are everywhere, from shop windows to test results. In this chapter you will find a fraction or a percentage of any amount, work backwards from a part to the whole, and use both to solve real problems, such as…

Practice: Chapter test: fractions and percentages of amounts · Fractions and percentages of amounts: key words

• 🍫 A fraction of an amount Divide by the bottom, multiply by the top

  There are 30 pupils in a class and 23 of them walk to school. How many is that? To find a fraction of an amount, you split the amount into equal parts and then take some of those parts.

  Practice: A fraction of an amount

• 🧩 Finding the whole When you know a part and its fraction

  Sam spent 38 of his savings on a game that cost £21. How much did he have to start with? This time you know a part and what fraction it is, and you work backwards to find the whole.

  Practice: Finding the whole

• 💯 A percentage of an amount 10%, 1%, 50% and building from them

  A jacket is 30% off. A pupil scores 85% in a test. A drink is 5% fruit juice. Per cent means "out of a hundred", so 30% is 30100. To find a percentage of an amount quickly, start from a few easy percentages and build up.

  Practice: A percentage of an amount

• 🛍️ Fraction and percentage problems Discounts, scores and comparing

  Sales, test results, recipes and football crowds all mix fractions and percentages. The skill is choosing what to work out, doing it one step at a time, and checking that the answer makes sense.

  Practice: Fraction and percentage problems

🌡️ Directed numbers Negative numbers: ordering and calculating

A winter night of −5 °C, the lift going down to floor −2 in a car park, a bank account in the red: numbers can go below zero. In this chapter you will put negative numbers in order, add, subtract, multiply and divide with them, and use them in longer calculations, in algebra and in equations.

Practice: Chapter test: directed numbers · Directed numbers: key words

• 🌡️ Negative numbers and ordering Below zero on the number line

  On a winter morning in Britain it might be −3 °C. In Oymyakon, a village in Siberia, the temperature can fall below −50 °C. Which is colder? Numbers below zero are called negative numbers, and a number line puts them all in order.

  Practice: Negative numbers and ordering

• ↔️ Adding and subtracting negatives Moves on a number line; minus a minus

  It is −3 °C and it gets 5 degrees warmer: the temperature is now 2 °C. Adding and subtracting with negative numbers are moves along the number line, and two models, jumps and counters, show why the rules work.

  Practice: Adding and subtracting negatives

• ✖️ Multiplying and dividing negatives Same signs positive, different signs negative

  What is 3 × (−2)? It means three lots of −2: −2, −4, −6. Multiplying and dividing with negative numbers works just as with positive numbers. The only new thing is deciding whether the answer is positive or negative.

  Practice: Multiplying and dividing negatives

• 🧮 The order of operations with negatives Longer calculations with negative numbers

  In 5 + 3 × (−4), what do you do first? Negative numbers follow the same order of operations as all other numbers. Get the order right first, then use the sign rules at each step.

  Practice: The order of operations with negatives

• 🔤 Substitution and equations with negatives Putting negative numbers into algebra

  A formula turns degrees Celsius into degrees Fahrenheit: F = 1.8C + 32. What is −10 °C in Fahrenheit? Putting a negative number in place of a letter is still substitution, and the same rules let you solve equations whose answers are negative.

  Practice: Substitution and equations with negatives

➗ Adding and subtracting fractions Common denominators and mixed numbers

Half a pizza and a third of another: how much pizza is that altogether? Adding and subtracting fractions is easy when the pieces are the same size. This chapter shows how to make them the same size when they are not, and how to add and subtract mixed numbers like 234.

Practice: Chapter test: adding and subtracting fractions · Adding and subtracting fractions: key words

• 🍰 Same denominator Add the tops, keep the bottom

  A pizza is cut into 8 slices. You eat 2 and your friend eats 3: together you have eaten 5 eighths of the pizza. When the denominators are the same, the pieces are the same size, so you just count them.

  Practice: Same denominator

• 🔀 Different denominators Find a common denominator first

  What is a half plus a third? The pieces are different sizes, so you can't just count them. First cut both into pieces of the same size. The new denominator they share is called a common denominator.

  Practice: Different denominators

• 🥧 Mixed numbers and improper fractions Converting between them

  A recipe needs 234 cups of flour, but your scoop holds only a quarter of a cup. How many scoops do you need? A number like 234 is a mixed number: a whole number and a fraction. Writing it as an improper fraction, 114, answers the question at once: 11 scoops.

  Practice: Mixed numbers and improper fractions

• 🧮 Adding and subtracting mixed numbers Whole numbers and fractions together

  A plank is 314 m long and you saw off 134 m. How much is left? To add and subtract mixed numbers, you can work with the whole numbers and the fractions separately, or turn everything into improper fractions first.

  Practice: Adding and subtracting mixed numbers

📐 Lines, angles and shapes Geometric notation, measuring and drawing

Builders, map makers, engineers and game designers all describe shapes in the same exact language: points, lines, angles and sides. In this chapter you learn that language, measure and draw angles with a protractor, and sort triangles and other polygons by their properties.

Practice: Chapter test: Lines, angles and shapes · Lines, angles and shapes: key words

• 📍 Points, lines and notation Line segments, parallel and perpendicular, labelling

  A builder's plan, a map and a maths diagram all have to say exactly which line or corner they mean. Geometry uses capital letters for points and a few simple marks on diagrams, so everyone reads a drawing in the same way.

  Practice: Points, lines and notation

• 📐 Types of angles Acute, right, obtuse, straight and reflex

  Open a door, turn a key or steer a bike and you make a turn. An angle measures an amount of turn, in degrees (°). Once you know the types of angle, you can tell roughly how big an angle is at a glance.

  Practice: Types of angles

• 🧭 Measuring and drawing angles Using a protractor

  A protractor measures and draws angles. Its two scales trip a lot of people up, but one habit gets the right scale every time: start from the 0 that sits on one of the arms.

  Practice: Measuring and drawing angles

• 🔺 Triangles Equilateral, isosceles, scalene and right-angled

  Push on the corner of a square frame made of four sticks and it squashes out of shape; push on a triangle and it keeps its shape. That is why bridges, cranes and roofs are full of triangles. Triangles are sorted by their sides and by their angles.

  Practice: Triangles

• ⬠ Quadrilaterals and other polygons Squares, rhombuses, trapeziums and more

  Floor tiles, road signs and honeycomb are all made of polygons. The name of a quadrilateral carries a whole list of facts: say "rhombus" and everyone knows its four sides are equal and its opposite sides are parallel.

  Practice: Quadrilaterals and other polygons

📏 Angle rules Angles on a line, at a point, in triangles and quadrilaterals

With a few angle facts you can work out an angle without measuring it: from a straight line, from two crossing lines, or from the corners of a triangle or a quadrilateral. Surveyors, builders and designers rely on these rules, and in maths you also learn to give the reason for every step.

Practice: Chapter test: Angle rules · Angle rules: key words

• ↩️ Angles on a line and around a point 180° and 360°

  Slices of a round pizza, the hands of a clock, roads meeting at a roundabout: when angles meet at one point they always fill the same amount of turn. That gives two rules for finding a missing angle without a protractor.

  Practice: Angles on a line and around a point

• ✂️ Vertically opposite angles Crossing lines make equal pairs

  Open a pair of scissors and look where the blades cross: the two angles opposite each other are always the same size, however wide you open them. The same is true whenever two straight lines cross.

  Practice: Vertically opposite angles

• 🔺 Angles in a triangle They add up to 180°

  Draw any triangle you like, long and thin or neat and even, measure its three angles and add them up: you always get 180°. This one fact lets you find the missing angle of any triangle.

  Practice: Angles in a triangle

• 🔷 Angles in quadrilaterals They add up to 360°

  Draw a line from one corner of a quadrilateral to the opposite corner and it splits into two triangles. So its four angles add up to two triangles' worth, 360°, whatever its shape.

  Practice: Angles in quadrilaterals

🗺️ Coordinates and transformations Four quadrants, translations, reflections and rotations

Maps, computer screens, video games and sat navs all find places with two numbers: across and up. In this chapter you plot points in all four quadrants, move shapes by sliding, flipping and turning them, and look for symmetry in shapes and patterns.

Practice: Chapter test: Coordinates and transformations · Coordinates and transformations: key words

• 📍 Coordinates in four quadrants (x, y): along the corridor, then up the stairs

  In a game of battleships, "B4" tells you exactly which square to aim at. Coordinates work the same way on a grid: two numbers pin down one exact point, even to the left of zero or below it.

  Practice: Coordinates in four quadrants

• ➡️ Translations Sliding a shape without turning it

  Slide a book across a table without turning it: every corner moves the same distance in the same direction. In maths this slide is called a translation.

  Practice: Translations

• 🪞 Reflections Mirror images in a line

  Look in a mirror and raise your right hand: your reflection raises its left. A reflection in maths works the same way. It flips a shape over a mirror line, so the image faces the other way.

  Practice: Reflections

• 🔄 Rotations Turning a shape about a point

  The hands of a clock, a Ferris wheel and a door on its hinges all turn about a fixed point. A rotation turns a shape about a point called the centre of rotation.

  Practice: Rotations

• 🦋 Symmetry Lines of symmetry and rotational symmetry

  A butterfly, a snowflake and a road sign all look balanced. That balance is symmetry. A shape can have a mirror line through it, or look exactly the same after part of a turn, or both.

  Practice: Symmetry

🧠 Estimating and mental maths Significant figures, estimates and clever shortcuts

Good mathematicians don't reach for a calculator straight away. They round numbers to make a quick estimate, spot shortcuts that make a calculation easy to do in their head, and check an answer by undoing it. This chapter shows you how, so you can tell at a glance whether an answer is about right.

Practice: Chapter test: estimating and mental maths · Estimating and mental maths: key words

• 🎯 Rounding to significant figures 1 significant figure for quick estimates

  A news report says 48,000 people watched a match, even though the exact number was 47,612. Rounding keeps the size of a number but drops the detail you don't need. Rounding to one significant figure is the quickest way to get a number that is easy to work with, whether it is in the thousands or in the hundredths.

  Practice: Rounding to significant figures

• 🤔 Estimating answers Is my answer about right?

  You're at the till with three things that cost £3.89, £2.15 and £5.99. Is a £10 note enough? Round each price and add: £4 + £2 + £6 = £12, so no. An estimate is a rough answer worked out with easy numbers. It tells you what size the real answer should be.

  Practice: Estimating answers

• ⚡ Mental strategies Changing the order, splitting and doubling

  Can you work out 25 × 7 × 4 in your head? It looks hard, but 25 × 4 = 100, and 100 × 7 = 700. Changing the order, splitting numbers up, and doubling and halving turn hard calculations into easy ones. These shortcuts work because of a few rules that numbers always obey.

  Practice: Mental strategies

• ↩️ Checking with inverse operations Undo it to check it

  To check that you've locked the door, you try the handle. To check a calculation, you can undo it. An inverse operation is the opposite operation: it takes you back to the number you started with. If you don't get back there, there is a mistake somewhere.

  Practice: Checking with inverse operations

🎲 Sets and probability Venn diagrams, outcomes and how likely things are

Will it rain tomorrow? Who will win the match? What are the chances of rolling a six? Probability is the maths of chance: it measures how likely something is with a number from 0 to 1. First you'll sort things into sets with Venn diagrams, then list everything that could happen and work out probabilities as fractions.

Practice: Chapter test: sets and probability · Sets and probability: key words

• ⭕ Sets and Venn diagrams Sorting into sets that overlap

  Think of the pupils in your class who play football and the ones who play tennis. Some play both, and some play neither. Maths has a neat way to sort things like this: put them into sets and draw a Venn diagram, where circles overlap to show what the sets have in common.

  Practice: Sets and Venn diagrams

• 📏 The probability scale From impossible to certain, from 0 to 1

  The weather forecast says there is a 70% chance of rain. A coin has a fifty-fifty chance of landing heads. Probability puts a number on how likely something is to happen: 0 means it is impossible, 1 means it is certain, and everything else is in between.

  Practice: The probability scale

• 📋 Listing outcomes Every result that could happen

  Before you can say how likely something is, you need to know everything that could happen. Roll one dice and there are 6 possible results; roll two and there are 36. Listing them in an organised way, so that none are missed and none are counted twice, is the first step in working out a probability.

  Practice: Listing outcomes

• 🎲 Calculating probability Successful outcomes ÷ all outcomes

  A bag holds 10 counters and 5 of them are blue. If you take one without looking, what is the chance it is blue? When every outcome is equally likely, a probability is simply a fraction: the outcomes you want, out of all the outcomes.

  Practice: Calculating probability

• 🙅 The probability of something not happening 1 minus the probability

  If the chance of rain tomorrow is 30%, what is the chance that it stays dry? It either rains or it doesn't, so between them the two must make up the whole 100%: the chance of no rain is 70%. This simple idea makes many probabilities quick to find.

  Practice: The probability of something not happening

🔑 Factors, multiples and primes Primes, squares, cubes, HCF and LCM

Some cicadas spend 13 or 17 years underground before they come out, and both are prime numbers. Primes are the building blocks of whole numbers: every whole number bigger than 1 is either a prime or can be made by multiplying primes together, in only one way. In this chapter you'll find factors and multiples, hunt for…

Practice: Chapter test: factors, multiples and primes · Factors, multiples and primes: key words

• 🧱 Factors and multiples What divides a number, and what it divides

  24 chairs can be set out in equal rows of 1, 2, 3, 4, 6, 8, 12 or 24, with none left over, but not in rows of 5 or 7. The numbers that divide 24 exactly are its factors, and 24 is a multiple of each of them.

  Practice: Factors and multiples

• 💎 Prime numbers Numbers with exactly two factors

  Try to set out 7 chairs in equal rows: you can only make one row of 7, or seven rows of 1. Numbers like 7, which can't be arranged in a rectangle any other way, are prime numbers. They are the building blocks of all whole numbers, and very large primes help to keep messages and payments secret on the internet.

  Practice: Prime numbers

• 🟪 Squares, cubes and roots 4², 2³, √49

  Arrange counters in squares and you need 1, 4, 9, 16, … of them: these are the square numbers. Stack little cubes into bigger cubes and you need 1, 8, 27, …: the cube numbers. Going back the other way, from 49 to 7 or from 27 to 3, is finding a root.

  Practice: Squares, cubes and roots

• 🌳 Prime factorisation Factor trees and products of primes

  Every whole number bigger than 1 is either a prime or can be made by multiplying primes together, and there is only one way to do it, like a building that can only be made from one set of bricks. Writing a number as a product of primes shows what it is built from, and makes finding the HCF and LCM easy.

  Practice: Prime factorisation

• 🤝 HCF and LCM Highest common factor and lowest common multiple

  Two buses leave the station together. One comes back every 12 minutes, the other every 18 minutes. When will they next leave together? That needs the lowest common multiple. Cutting two ribbons into the longest equal pieces needs the highest common factor. Both come straight from factors and multiples.

  Practice: HCF and LCM

🏁 Year 7 review Revision tips and the end-of-year test

You have reached the end of Year 7 maths. Before the end-of-year test, look back over the year: the big ideas of each chapter are below, with a link to go back to it. Then take the test at the bottom of this page.

Sections:

• How to revise 🧠
• The year in one table

Practice: End-of-year test: Maths