Number

Explanation

• Integers, decimals and fractions can be ordered once they are written in a comparable form.
• Convert fractions to decimals, or use a common denominator, then place the values on a number line: values increase from left to right.
• This is especially important for negatives, because the number with the greater distance below zero is smaller; for example, $-1.7<-\dfrac{3}{2}$ since $-1.7<-1.5$.
• Use $=$ for equal values, $\neq$ for unequal values, and read $\leq$ or $\geq$ as allowing equality.
• In an exam, show the conversion that justifies each comparison.

Explanation

• The four operations must work with positive and negative integers, decimals, proper and improper fractions, and mixed numbers. For written decimal addition or subtraction, align decimal points so digits of equal place value share a column.
• When adding fractions, create a common denominator; when multiplying, cancel common factors where useful.
• Convert a mixed number to an improper fraction before multiplication or division, and divide by a fraction by multiplying by its reciprocal.
• Track signs separately: equal signs give a positive product or quotient and different signs give a negative one.
• Examiners award method marks for these conversions even if the final arithmetic slips.

Explanation

• Inverse operations undo one another: addition pairs with subtraction, multiplication with division, and squaring with square root when the required value is non-negative. Use these relationships to check calculations or reverse a sequence of operations.
• For a written calculation, follow conventional priority: brackets first; then powers and roots; then multiplication and division; then addition and subtraction.
• Operations at the same priority are completed from left to right.
• A reciprocal is also an inverse operation for multiplication: the reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$.
• Examiners expect each priority stage to be visible when the question carries more than one mark.

Explanation

• A prime number has exactly two positive factors, $1$ and itself; $1$ is not prime. Every integer greater than $1$ has a unique prime factorisation apart from the order of its factors.
• Produce it with a factor tree or repeated division, then write repeated factors using powers. For two or more numbers, the highest common factor uses every shared prime with its smallest exponent.
• The lowest common multiple uses every prime present with its largest exponent.
• This method avoids double-counting factors and also supports problems about making a product into a square or cube.
• Examiners require the factorisation to contain primes only.

Explanation

• A systematic listing strategy fixes one choice and cycles through every permitted second choice before moving on. Use a clear order, such as increasing digits or alphabetical letters, so you can see that no result is missing or repeated.
• When order matters, AB and BA are different outcomes; when order does not matter, list each pair only once.
• A table or tree can organise several stages.
• Higher tier: if one task can be completed in $m$ ways and a following independent choice in $n$ ways, the product rule gives $m\times n$ combined outcomes.
• Examiners award completeness only when the structure of the list is evident.

Explanation

• A positive integer power represents repeated multiplication: $a^n$ contains $n$ factors equal to $a$.
• An associated root reverses that power, so $\sqrt{81}=9$, $\sqrt[3]{125}=5$, and $\sqrt[4]{16}=2$.
• Learn common powers of $2$, $3$, $4$ and $5$ because they make exact roots quickly recognisable.
• Higher tier: to estimate a root that is not exact, place its radicand between nearby known powers; for example, $3^3<40<4^3$ shows $3<\sqrt[3]{40}<4$.
• In an exam, distinguish an exact evaluation from an estimate and clearly state the bounding powers used.

Explanation

• Integer indices include positive, zero and negative powers. For non-zero $a$, $a^0=1$ and $a^{-n}=\dfrac{1}{a^n}$; the negative index creates a reciprocal, not a negative answer.
• Roots reverse powers, including odd roots of negative numbers such as $\sqrt[3]{-216}=-6$.
• Higher-tier questions also use fractional indices: $a^{1/n}=\sqrt[n]{a}$ and $a^{m/n}=\sqrt[n]{a^m}$.
• Foundation questions stay with roots and integer indices.
• Choose the form that makes evaluation simplest, keep brackets around a negative base, and show the reciprocal or root step very clearly before giving the final value.

Explanation

• An exact answer keeps fractions, multiples of $\pi$ and, on Higher tier, surds rather than replacing them with rounded decimals. Combine exact fractions with a common denominator and leave circle results as a coefficient of $\pi$.
• Higher-tier surds are simplified by extracting square factors: $\sqrt{12}=\sqrt{4\times3}=2\sqrt3$.
• Like surds can then be collected, and a denominator such as $\sqrt5$ is rationalised by multiplying numerator and denominator by $\sqrt5$.
• Foundation questions do not require surd manipulation, but do require exact fractions and multiples of $\pi$.
• When the command says “exact”, no rounded decimal should appear in the final answer.

Explanation

• Standard form writes a number as $A\times10^n$, where $1\leq A<10$ and $n$ is an integer. A positive index moves the decimal point right to make a large ordinary number; a negative index moves it left to make a small one.
• For multiplication, multiply the $A$ values and add the indices; for division, divide the $A$ values and subtract the indices.
• For addition or subtraction, first rewrite the terms with the same power of $10$.
• Finally normalise the coefficient so it lies in the required interval.
• Examiners can award method marks for correct index work before normalisation.

Explanation

• A terminating decimal has finitely many decimal places, so write it over the matching power of $10$ and simplify: $0.375=\dfrac{375}{1000}=\dfrac38$. Convert a fraction to a decimal by dividing its numerator by its denominator.
• On Higher tier, recurring decimals are also converted algebraically: name the decimal $x$, multiply by a power of $10$ so the repeating blocks align, subtract to remove the recurring part, and solve for $x$.
• A recurring dot or bar must cover the whole repeating block.
• Foundation questions use terminating decimal–fraction equivalents only.
• Examiners expect the resulting fraction in its simplest form.

Explanation

• Fractions and ratios describe the same partition in different forms. If a group is $\dfrac35$ of the whole, the remainder is $\dfrac25$, so the part-to-part ratio is $3:2$.
• Conversely, in a ratio $a:b$, the first share is $\dfrac{a}{a+b}$ of the whole and the second is $\dfrac{b}{a+b}$.
• In multi-step problems, first use the ratio to find each share, then apply any fraction operator to the relevant share.
• Keep the named order of the groups throughout.
• Examiners commonly award one mark for finding one ratio part and a later accuracy mark for the required fractional amount.

Explanation

• A fraction or percentage acts as an operator, meaning it multiplies a quantity. To find $\dfrac ab$ of an amount, divide by $b$ and multiply by $a$, or multiply directly by $\dfrac ab$.
• Convert a percentage to a fraction over $100$ or to its decimal multiplier; for example, $17.5\%=0.175$.
• A percentage decrease of $p\%$ leaves the multiplier $1-\dfrac p{100}$, while an increase uses $1+\dfrac p{100}$.
• When several operators occur, apply them to the correct intermediate amount in the stated order.
• Examiners award method marks for a correct multiplier even if the evaluation is wrong.

Explanation

• Standard units cover mass, length, time, money and other measures; compound measures combine units, such as kilometres per hour or grams per cubic centimetre. Choose units appropriate to the context, then convert all quantities to compatible units before calculating.
• Length factors are squared for area and cubed for volume: because $1\text{ m}=100\text{ cm}$, $1\text{ m}^2=10\,000\text{ cm}^2$.
• Time is base $60$, not base $100$, so $1.5$ hours is $1$ hour $30$ minutes.
• Give money to the nearest penny when appropriate.
• Examiners expect a numerical value and a correctly converted unit.

Explanation

• An estimate replaces the original inputs with nearby values that are easy to calculate, often by rounding each to one significant figure. Carry out the operation on those rounded inputs and use $\approx$, because the result is not exact.
• Estimation checks whether a calculator answer has a sensible size, sign and decimal position; it should be an independent calculation, not a rounding of the calculator display.
• Choose compatible approximations, especially for division, so mental arithmetic stays simple.
• If the exact answer differs by about a factor of $10$ or $100$, suspect a place-value entry error.
• Examiners require both the estimate and a clear comparison when asked to comment.

Explanation

• Decimal places count digits after the decimal point; significant figures begin at the first non-zero digit. Identify the deciding digit immediately after the required place: $5$ or more rounds up, while $4$ or less leaves the retained digit unchanged.
• Choose a degree of accuracy appropriate to the context, such as money to the nearest penny.
• A rounded value also represents an error interval.
• For rounding to a unit $u$, subtract and add $\dfrac u2$; include the lower boundary but exclude the upper boundary because that endpoint rounds to the next value.
• Truncation instead keeps values from the stated number up to the next truncation step.

Explanation

• A measured or rounded value stands for a range of possible true values. The lower and upper limits are usually half a rounding unit below and above the stated value, with the upper limit excluded.
• Use these limits to find a maximum possible error or to decide whether a claimed result is possible. For a sum, the smallest total uses all lower limits and the greatest possible total approaches all upper limits.
• Higher-tier bounds calculations also choose numerator and denominator limits deliberately, but Foundation questions can require interpreting accuracy and maximum error.
• Never treat rounded inputs as exact.
• Examiners expect the chosen limits to be written before the calculation.