WJEC GCSE Mathematics and Numeracy

Place value and directed numbers

Each digit in a number is worth a power of ten that depends on its position. In $4\,805.27$ the 8 is worth 800, the 5 is worth 5 and the 7 is worth $\frac{7}{100}$. Whole numbers of any size are read and written in groups of three digits, so $3\,200\,450$ is three million, two hundred thousand, four hundred and fifty.

Directed numbers are numbers with a sign, positive or negative. On a number line the larger number is always the one further right, so $-2 > -7$, and ordering a list such as $-3,\ 4,\ -8,\ 0,\ -1$ gives $-8,\ -3,\ -1,\ 0,\ 4$.

Adding a negative number moves left and subtracting a negative number moves right, because subtracting takes away a debt.

When multiplying or dividing, two numbers with the same sign give a positive answer and two with different signs give a negative one.

Order of operations and inverse operations

A calculation with several operations is worked in a fixed order so that everyone gets the same answer: brackets first, then indices (powers and roots), then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar works like a bracket round the top and round the bottom.

Each operation has an inverse operation that undoes it: addition and subtraction undo each other, multiplication and division undo each other, and squaring and taking a square root undo each other. Inverses are used to check an answer (if $48 \div 6 = 8$ then $8 \times 6$ should give 48) and to work backwards.

A number machine shows a chain of operations. Take an input $x$ that is multiplied by 3 and then has 5 added. If the output is 26, work backwards using the inverses in reverse order: subtract 5 to get 21, then divide by 3 to get 7. The input was 7.

Rounding

To round to a decimal place, look at the digit one place further right: 5 or more rounds up, and below 5 leaves the digit as it is. Rounding 3.847 to 2 decimal places gives 3.85, and rounding 0.0638 to 2 decimal places gives 0.06. A whole number is rounded to the nearest 10, 100 or 1000 in the same way, so 6,450 to the nearest 100 is 6,500 and 6,449 is 6,400.

To round to significant figures, count from the first non-zero digit, since leading zeros only hold place value. Zeros at the end of a whole number that are needed to hold place value stay as zeros.

In the last row the zeros are kept because 3.0 and 3.00 show the level of accuracy given.

Some answers need rounding in a particular direction because of what the number means. If each minibus holds 16 people and 75 people need transport, $75 \div 16 = 4.6875$, and 5 minibuses are needed, so the answer is rounded up. If a ribbon costs £1.40 per metre and you have £10, $10 \div 1.40 = 7.14\ldots$, so you can buy 7 whole metres and the answer is rounded down.

Rounding part-way through a multi-step calculation introduces errors that grow, so keep the full calculator value (or at least two more figures than needed) until the last step, and round only the final answer. A final answer should also be given to a sensible accuracy for the context: a length measured in centimetres is not usually quoted to eight decimal places.

Estimating and checking

An estimate rounds every number to one significant figure so the calculation can be done in the head, and it shows whether an answer is about the right size. Dividing by a number less than 1 makes a result bigger, which is worth watching for.

The exact value is 1,213.1 to 1 decimal place, which agrees with the estimate. Other checks are to use the inverse operation, to repeat the calculation a different way, or to ask whether the answer is sensible (a person cannot be 17 metres tall).

Using a calculator

The calculator follows the order of operations, so brackets must be typed where a fraction bar or a grouped expression needs them. To find $\dfrac{3.2 + 4.8}{2.5 \times 1.6}$, the top and the bottom each need brackets, giving $(3.2 + 4.8) \div (2.5 \times 1.6) = 2$. Without brackets, the calculator would divide only by 2.5 and then multiply by 1.6.

Power keys find squares, cubes and other powers, and there are keys for square roots and cube roots, for entering numbers in standard form, and for the trigonometric functions. The answer should be checked against an estimate, because a slip in typing is the most common source of error.

Limits of accuracy

A measurement is never exact. A length given as 7.4 cm to the nearest tenth of a centimetre could really be anywhere that would round to 7.4, so the possible error is half a unit of the last place, here 0.05 cm.

Every rounded value has a lower bound, which is included, and an upper bound, which is the smallest number that would round up to the next value, so it is not included. For 7.4 cm the bounds are 7.35 and 7.45, and the value $x$ satisfies $7.35 \le x < 7.45$. In the same way a mass of 60 kg to the nearest 10 kg has bounds 55 and 65, and a number given as 2,300 to 2 significant figures has bounds 2,250 and 2,350.

A common upper bound is written as 7.45 even though 7.45 itself would round up, because any value just below it, such as 7.4499, rounds down. For calculations it is used as the upper bound.

Bounds in calculations

Higher tier only: combine the bounds so that the result is as large or as small as it can be.

The rectangle below has sides 7.4 cm and 3.2 cm, each to 1 decimal place. Its bounds are $7.35 \le l < 7.45$ and $3.15 \le w < 3.25$.

A cyclist travels 45 km, measured to the nearest kilometre, in 3.2 hours, measured to the nearest tenth of an hour. For the largest possible speed use the largest distance and the smallest time, and for the smallest speed the reverse.

Both round to 14 km/h to 2 significant figures, which suggests the speed can be stated as 14 km/h to that accuracy. If the bounds had rounded to different values the speed could not be given to that accuracy.