Topic 01 · 36 lessons

Numbers

Every other branch of maths is built on numbers. Before you can solve an equation, measure an angle or work out interest on a loan, you need to be comfortable moving between whole numbers, fractions, decimals and percentages, and you need to know how the basic operations behave.

This section treats numbers as a toolkit. Each lesson takes one idea, explains why it works rather than just how, and finishes with examples you can check yourself.

Lessons in this topic

1.1Counting and Place ValueEvery number up to 100 is just some tens and some ones. Where a digit sits tells you what it is worth.
1.2Comparing and Ordering NumbersTo compare two numbers, look at the tens first and the ones second. The signs <, > and = write the answer down in one mark.
1.3Adding Within 20Adding is putting two groups together and counting the lot. A few tricks make it quick: count on, make ten, and use doubles.
1.4Subtracting Within 20Subtracting is taking some away, or finding the gap between two numbers. Both ways give the same answer, and adding checks it.
1.5Skip Counting in 2s, 5s and 10sSkip counting means counting in equal jumps instead of ones. Counting in 2s, 5s and 10s is a fast way to count groups and the first step towards times tables.
1.6Odd and Even NumbersAn even number makes pairs with nothing left over. An odd number always has one left over. The last digit tells you which, however big the number is.
1.7Place Value to a MillionEach place in a number is worth ten times the place to its right. That one rule lets you read and write numbers as large as a million.
1.8Rounding and EstimatingRounding swaps a number for a nearby, friendlier one. Estimating uses rounded numbers to get a quick answer that is close to the real one.
1.9Column AdditionWrite numbers one under the other so each place lines up, add the columns from right to left, and carry when a column makes ten or more.
1.10Column SubtractionLine up the digits, subtract column by column from the right, and when a digit is too small, exchange one from the column to its left.
1.11Multiplication: Equal Groups and ArraysMultiplication is a quick way to add equal groups. An array of rows and columns shows the same thing and also shows why the order does not matter.
1.12Multiplying Larger NumbersSplit a big number into tens and ones, multiply each part, and add the pieces. The column method does the same thing in a shorter layout.
1.13Division: Sharing and GroupingDivision splits a total into equal shares or equal groups. It is the opposite of multiplication, and sometimes there is a remainder.
1.14Short and Long Division with RemaindersShort division works through a number digit by digit, carrying what is left over. Chunking handles bigger divisors by taking away easy multiples.
1.15Factors and MultiplesFactors are the numbers that divide exactly into a number. Multiples are what you get by counting in steps of that number.
1.16Prime NumbersA prime has exactly two factors: 1 and itself. Every other whole number above 1 is built by multiplying primes.
1.17Highest Common Factor and Lowest Common MultipleThe HCF is the biggest number that divides both numbers. The LCM is the smallest number that both divide into. Listing or prime factors finds either.
1.18Fractions ExplainedA fraction names part of a whole: the bottom number says how many equal parts, the top number says how many of them you have.
1.19Equivalent Fractions and SimplifyingEquivalent fractions name the same amount. Multiply or divide the top and bottom by the same number to find them, and divide by the highest common factor to simplify.
1.20Comparing and Ordering FractionsTo compare fractions, make the parts the same size by writing them over a common denominator, then compare the numerators.
1.21Mixed Numbers and Improper FractionsAn improper fraction is one whole or more, and a mixed number shows the same amount as wholes plus a fraction. Division and multiplication switch between them.
1.22Adding and Subtracting FractionsFractions can only be added or subtracted when the parts are the same size. Write them over a common denominator, then add or subtract the numerators.
1.23Multiplying FractionsTo multiply fractions, multiply the numerators and multiply the denominators. Multiplying by a proper fraction takes a part of an amount, so the answer gets smaller.
1.24Dividing FractionsDividing by a fraction asks how many of those parts fit. The shortcut is to multiply by the reciprocal, which is the fraction turned upside down.
1.25Finding a Fraction of an AmountTo find a fraction of an amount, divide by the denominator to find one part, then multiply by the numerator to get the parts you need.
1.26Decimals ExplainedDecimals continue the place-value pattern to the right of the point: tenths, hundredths, thousandths and beyond.
1.27Comparing and Rounding DecimalsCompare decimals digit by digit from the left, after lining up the decimal points. To round, look at the next digit: 5 or more rounds up.
1.28Adding and Subtracting DecimalsLine up the decimal points so each place value sits under the same one, fill gaps with zeros, then add or subtract as with whole numbers.
1.29Multiplying and Dividing DecimalsMultiply decimals as whole numbers, then put the point back by counting decimal places. To divide by a decimal, scale both numbers until the divisor is whole.
1.30PercentagesPer cent means "out of one hundred". Percentages let us compare parts of different-sized wholes on a common scale.
1.31Fractions, Decimals and Percentages TogetherFractions, decimals and percentages are three ways to write the same amount. Convert them to one form to compare them.
1.32Ratios and Sharing in a RatioA ratio compares quantities. To share in a ratio, add the parts, find the value of one part, then multiply.
1.33Rates and Direct ProportionIn direct proportion, when one quantity doubles so does the other. Find the amount for one unit first, then scale up.
1.34Negative Numbers and Integer OperationsNegative numbers sit to the left of zero on the number line. Adding moves right, subtracting moves left, and two negatives multiplied make a positive.
1.35Order of OperationsWhen a calculation has more than one operation, a fixed order says which to do first: brackets, then powers, then multiply and divide, then add and subtract.
1.36Powers and RootsA power is repeated multiplication; a root runs it backwards. A handful of index laws do almost all the work.

The big ideas

a)One value, many forms: Three quarters, $0.75$ and $75\%$ are the same amount written three different ways. Choosing the most convenient form is half the skill.
b)Place value: In $507$ the $5$ means five hundreds because of where it sits. Decimals simply continue the same pattern to the right of the point.
c)Building blocks: Every whole number greater than $1$ is either prime or a product of primes in exactly one way. That fact powers fraction simplifying and modern encryption.
d)Repeated operations: Multiplication is repeated addition; powers are repeated multiplication. Roots and logarithms run those processes backwards.

Worked example

Which is bigger: 5/8 or 0.6?

Compare $\tfrac{5}{8}$ and $0.6$ without a calculator.

Convert the fraction to a decimal by dividing the top by the bottom: $5 \div 8 = 0.625$ .
Now both numbers are decimals: compare $0.625$ with $0.600$ .
Line up the digits. The tenths are equal ( $6$ ), but in the hundredths place $2 > 0$ .

Answer: $\tfrac{5}{8}$ is larger, by $0.025$ .

Where you will meet it

Discounts in shops, recipe scaling, splitting a bill, reading a nutrition label and comparing phone contracts all come down to confident work with fractions, decimals and percentages.

The wider field

Numbers also covers these ideas, which build on the lessons above:

Negative numbers
Factors and multiples
Ratios and proportion
Number bases such as binary and hexadecimal
Sequences and patterns