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

• Written methods score for the method, then for the answer. In a decimal division such as $818.4\div1.2$, multiply both numbers by the same power of $10$ so the divisor is an integer, divide, then place the decimal point by comparing the size of the numbers: $818.4\div1.2$ is a little less than $818$, so $682$ is sensible and $68.2$ is not.
• For fraction addition or subtraction, write both fractions over a common denominator and keep the answer in the form requested, usually a mixed number. For multiplication or division of mixed numbers, convert every mixed number to an improper fraction first; to divide, multiply by the reciprocal of the second fraction only.
• A show-that question must display each stage, including the unsimplified product.
• Track signs separately: like signs give a positive result, unlike signs a negative one.
• Cancelling common factors before multiplying keeps the arithmetic small.

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 has exactly two factors, so $1$ is not prime. Every integer above $1$ is a product of primes in exactly one way, which is why the prime factorisation is a reliable route to the highest common factor and lowest common multiple.
• Find it with a factor tree or repeated division by primes, then write the answer as a product using index notation: $500=2^2\times5^3$. A list of primes without multiplication signs is not a product and drops the accuracy mark.
• For two numbers, a Venn diagram of their prime factors puts the shared primes in the overlap: the HCF is the product of the overlap and the LCM is the product of everything in the diagram.
• Given numbers already in index form, the HCF takes the lower power of each prime and the LCM the higher.
• Read the question twice, because HCF and LCM are routinely swapped.

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

• A positive index moves the decimal point right, so $5.3\times10^4=53\,000$; a negative index moves it left, so $7.4\times10^{-5}=0.000\,074$. To multiply, multiply the $A$ values and add the indices; to divide, divide the $A$ values and subtract the indices; then normalise, because $24\times10^{-2}$ is not standard form until it becomes $2.4\times10^{-1}$.
• To add or subtract, write both numbers with the same power of ten first, or convert to ordinary numbers: $9.7\times10^6+2.45\times10^7=0.97\times10^7+2.45\times10^7=3.42\times10^7$.
• Adding the $A$ values when the powers differ is the standard error and scores nothing.
• A method mark is available for a correct value in a non-standard form such as $34.2\times10^6$, and a final answer left as an ordinary number when standard form was asked for loses the last mark.
• To order numbers, put them all in the same form first.

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

• To find $\dfrac{a}{b}$ of an amount, divide by $b$ and multiply by $a$, or multiply by the fraction directly: $\dfrac{2}{7}$ of $420$ is $420\times2\div7=120$. A percentage becomes a decimal or fraction operator, $35\%$ of $420$ being $420\times0.35=147$.
• A percentage change is one multiplier: a $15\%$ decrease multiplies by $0.85$, an increase of $15\%$ by $1.15$.
• When several operators apply, each acts on the amount that remains, not on the original: “$\dfrac{3}{8}$ of the reduced price” means multiply the reduced price by $\dfrac{3}{8}$.
• Problems that mix a fraction, a percentage and a ratio are worked one step at a time, subtracting the shares already taken before splitting the remainder, and the scheme gives a process mark for each share.
• A correct answer with no working scores nothing on these questions.

Explanation

• Work in one unit at a time and convert before calculating: $1$ kg $=1000$ g, $1$ km $=1000$ m, $1$ litre $=1000$ cm$^3$, and money in either pounds or pence throughout, converting at the end.
• Time is the trap: $1.25$ hours is $1$ hour $15$ minutes, not $1$ hour $25$ minutes, and $2.75$ hours is $2$ hours $45$ minutes, so a time from $\text{distance}\div\text{speed}$ must be converted to hours and minutes when the question asks for it.
• A rate written per $100$ km or per kilogram is scaled to the actual distance or mass first.
• Round money to the nearest penny only at the end.
• Check the size of the answer against common sense; a journey time of hundreds of minutes for a few miles signals a wrong operation.

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.