Maths · 1 Number

1.6 The four operations

Before you start

Times tables up to $12 \times 12$ , and multiplying and dividing whole numbers without a calculator.
Place value in decimals: tenths, hundredths and thousandths.
Equivalent fractions, and simplifying a fraction by dividing the top and the bottom by a common factor.
Finding the lowest common multiple (LCM) of two numbers, and the reciprocal of a fraction, from 1.1.
Reading a number line that includes negative numbers, e.g. $- 5$ is less than $- 2$ .
What an index means: $3^{2} = 3 \times 3$ .

Syllabus objectives covered (1)

Use the four operations for calculations with integers, fractions and decimals, including correct ordering of operations and use of brackets.

Part 1 of 8

The order of operations

When a calculation has more than one operation, the order you do them in changes the answer. In $2 + 3 \times 4$ , adding first gives $5 \times 4 = 20$ , but multiplying first gives $2 + 12 = 14$ . Everyone uses one agreed order, so a calculation has only one answer. Here it is $14$ .
The agreed order is Brackets, then Indices (powers and roots), then Division and Multiplication, then Addition and Subtraction. The initials spell BIDMAS. Some books say BODMAS, where O stands for 'orders'. Orders and indices mean the same thing.
$Brackets \to Indices \to \times and \div \to + and -$
The trap is in the name. D comes before M in BIDMAS, but division does not come before multiplication. They have the same rank, so you do them in the order they are written, from left to right.
$12 \div 4 \times 3 = 3 \times 3 = 9 not 12 \div 12 = 1$
Addition and subtraction also share one rank, so you also work them from left to right. $10 - 4 + 3 = 6 + 3 = 9$ . If you add first, you get $10 - 7 = 3$ , which is wrong.
Inside a bracket, the same order applies. When one bracket is inside another, work out the inside one first.
$2 \times (3 + {(8 - 6)}^{2}) = 2 \times (3 + 2^{2}) = 2 \times (3 + 4) = 2 \times 7 = 14$
A fraction bar acts as a bracket around the whole top and a bracket around the whole bottom. Work out the top completely and the bottom completely, then divide.
$\frac{6 + 9}{8 - 3} = \frac{15}{5} = 3$
Do one operation per line, and copy the rest of the line without change. It is slower to write but much harder to get wrong, and each line is working that can earn a method mark.
$18 - 2 \times 3^{2} = 18 - 2 \times 9 = 18 - 18 = 0$

Part 2 of 8

Brackets: writing them in, and typing them in

Brackets change the agreed order. $2 + 3 \times 4 = 14$ , but $(2 + 3) \times 4 = 5 \times 4 = 20$ .
A question can give you a calculation and an answer, and ask you to write in brackets to make it true. Work it out as written first. Then try each place a pair of brackets can go, and work out each version.
For example, $6 + 4 \times 5 - 2$ is $24$ as written. $(6 + 4) \times 5 - 2 = 48$ and $6 + 4 \times (5 - 2) = 18$ . With two pairs, $(6 + 4) \times (5 - 2) = 30$ . A pair around $4 \times 5$ changes nothing, because that multiplication is done first anyway.
When you have found a version that works, work it out once more with the brackets in place. This confirms that the brackets are in the right place.
A scientific calculator follows BIDMAS, but it only sees what you type. It cannot see a fraction bar unless you use its fraction key. To work out $\frac{7.2 + 4.8}{2.5 \times 1.6}$ in one line, put a bracket around the top and a bracket around the bottom.
$(7.2 + 4.8) \div (2.5 \times 1.6) = 12 \div 4 = 3$
Without those brackets, the calculator works out $7.2 + 4.8 \div 2.5 \times 1.6$ . It divides and multiplies first, which gives $7.2 + 3.072 = 10.272$ . That answer is wrong, and the calculator gives no warning.
A safe habit: work out the top and the bottom separately, write both values down, then divide. If you make a slip in the last step, the values you wrote down are still correct working.

Part 3 of 8

Negative numbers: adding and subtracting

Think of a number line. Adding a positive number moves you to the right. Subtracting a positive number moves you to the left.
$- 3 + 5 = 2 2 - 7 = - 5 - 4 - 6 = - 10$
Adding a negative number is the same as subtracting. $5 + (- 8)$ means 'move 8 the other way', so it equals $5 - 8 = - 3$ .
Subtracting a negative number is the same as adding. Subtraction finds the gap between two numbers: $4 - (- 6)$ asks how far $4$ is above $- 6$ . From $- 6$ up to $0$ is 6, and from $0$ up to $4$ is 4, so the gap is $10$ .
$4 - (- 6) = 4 + 6 = 10$
So when two signs are side by side, replace them with one sign. Two signs that are the same become $+$ . Two signs that are different become $-$ .
$+ (+ a) = + a - (- a) = + a + (- a) = - a - (+ a) = - a$
In a longer calculation, first change each pair of signs to one sign. Then work from left to right.
$- 7 + 3 - (- 2) - 5 = - 7 + 3 + 2 - 5 = - 4 + 2 - 5 = - 2 - 5 = - 7$

Part 4 of 8

Negative numbers: multiplying, dividing and powers

To multiply or divide, work with the sizes first, then decide the sign. Same signs give a positive answer. Different signs give a negative answer.
$(- 6) \times 4 = - 24 (- 6) \times (- 4) = 24 - 20 \div 5 = - 4 (- 20) \div (- 5) = 4$
Why does a negative times a negative give a positive? Look at a pattern: $3 \times (- 4) = - 12$ , $2 \times (- 4) = - 8$ , $1 \times (- 4) = - 4$ , $0 \times (- 4) = 0$ . Each answer is 4 more than the one before. Continue the pattern one more step and you get $(- 1) \times (- 4) = 4$ .
When several numbers are multiplied, count the negative signs. An even number of negatives gives a positive answer. An odd number gives a negative answer.
$(- 2) \times (- 3) \times (- 5) = 6 \times (- 5) = - 30$
Use the 'same signs give positive' rule only for multiplying and dividing. For adding, use the number line: $- 3 + (- 4) = - 7$ , not $7$ .
Brackets matter with powers. ${(- 3)}^{2}$ means $(- 3) \times (- 3) = 9$ . But $- 3^{2}$ means 'the negative of $3^{2}$ ', because the index belongs only to the 3 and indices come before the minus sign. So $- 3^{2} = - 9$ .
${(- 3)}^{2} = 9 - 3^{2} = - (3^{2}) = - 9$
An even power of a negative number is positive, and an odd power is negative: ${(- 2)}^{4} = 16$ and ${(- 2)}^{3} = - 8$ .
On a calculator, type the bracket: ${(- 3)}^{2}$ . If you type the minus sign and then $3^{2}$ , the calculator squares only the 3 and shows $- 9$ .

Part 5 of 8

Temperature changes and other practical problems

Temperatures below zero are negative numbers. A rise in temperature is an addition. A fall is a subtraction.
From $- 5$ °C, a rise of 9 degrees gives $- 5 + 9 = 4$ °C. From $3$ °C, a fall of 7 degrees gives $3 - 7 = - 4$ °C.
The difference between two temperatures is the higher one minus the lower one. The difference between $6$ °C and $- 11$ °C is $6 - (- 11) = 6 + 11 = 17$ degrees.
$6 - (- 11) = 17$
graph diagram — not drawn yet
A vertical number line drawn like a thermometer scale from -15 to 10 in steps of 1, labelled every 5, with 0 marked clearly; on its right a bracket runs from -11 up to 6, split at 0 into a lower part labelled 11 and an upper part labelled 6.
The gap from −11 °C to 6 °C crosses zero: 11 degrees up to zero plus 6 degrees above zero makes 17 degrees, which is 6 − (−11).
Check by splitting the gap at zero. From $- 11$ up to $0$ is 11 degrees. From $0$ up to $6$ is 6 degrees. Together that is $11 + 6 = 17$ degrees. A difference is a size, so give it as a positive number.
Repeated changes use multiplication. A fall of $1.5$ degrees every hour for 4 hours is a fall of $4 \times 1.5 = 6$ degrees in total. Work out the total change first, then apply it once to the starting value.
Read the question for the word that tells you the operation. 'Rose by', 'increased by' or 'warmer by' means add. 'Fell by', 'dropped by' or 'colder by' means subtract. 'Difference' or 'how much warmer' means higher minus lower.
The same method works for anything measured above and below a zero: heights above and below sea level, floors above and below ground level, or a bank balance that goes below zero. Decide which direction is positive, write each value with its sign, then add or subtract.

Part 6 of 8

Fractions and mixed numbers: adding and subtracting

You can only add or subtract fractions that have the same denominator. Then you add or subtract the numerators and keep the denominator.
$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$
The denominator stays the same because it names the size of the pieces. Two sevenths plus three sevenths is five sevenths. The pieces are still sevenths.
If the denominators are different, change both fractions to a common denominator first. Use the LCM of the denominators, because it keeps the numbers smallest. For $\frac{5}{6} + \frac{3}{4}$ , the LCM of 6 and 4 is 12.
$\frac{5}{6} + \frac{3}{4} = \frac{10}{12} + \frac{9}{12} = \frac{19}{12}$
$\frac{19}{12}$ is an improper fraction: its numerator is bigger than its denominator. To write it as a mixed number, divide: $19 \div 12 = 1$ remainder $7$ . So $\frac{19}{12} = 1 \frac{7}{12}$ .
To change a mixed number into an improper fraction, multiply the whole number by the denominator, then add the numerator.
$2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{13}{5}$
With mixed numbers, change them to improper fractions first. This method works every time.
$3 \frac{1}{4} - 1 \frac{2}{3} = \frac{13}{4} - \frac{5}{3} = \frac{39}{12} - \frac{20}{12} = \frac{19}{12} = 1 \frac{7}{12}$
You can also subtract the whole numbers and the fractions separately. But in the example above, $\frac{1}{4} - \frac{2}{3}$ is negative, and a negative fraction part is easy to get wrong. Improper fractions avoid that problem.
Give the answer in its simplest form: divide the numerator and the denominator by their HCF, e.g. $\frac{10}{12} = \frac{5}{6}$ . If the question asks for a mixed number, give a mixed number.

Part 7 of 8

Fractions and mixed numbers: multiplying and dividing

To multiply fractions, multiply the numerators together and multiply the denominators together. You do not need a common denominator.
$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$
Cancel before you multiply, to keep the numbers small. A numerator and a denominator that share a factor can both be divided by it. In $\frac{6}{7} \times \frac{14}{9}$ , divide 6 and 9 by 3, and divide 14 and 7 by 7.
$\frac{6}{7} \times \frac{14}{9} = \frac{2}{1} \times \frac{2}{3} = \frac{4}{3} = 1 \frac{1}{3}$
To divide by a fraction, multiply by its reciprocal. Dividing by a number and multiplying by its reciprocal always give the same answer: $6 \div 2 = 3$ and $6 \times \frac{1}{2} = 3$ .
$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1 \frac{7}{8}$
Turn over only the fraction you are dividing by, which is the second one. The first fraction stays as it is.
With mixed numbers, change to improper fractions before you multiply or divide.
$1 \frac{1}{2} \times 2 \frac{2}{3} = \frac{3}{2} \times \frac{8}{3} = \frac{24}{6} = 4$
Do not multiply the whole numbers and the fractions separately. That gives $1 \times 2 + \frac{1}{2} \times \frac{2}{3} = 2 \frac{1}{3}$ , which is wrong. A mixed number is a sum, $1 \frac{1}{2} = 1 + \frac{1}{2}$ , and when you multiply two sums, every part must multiply every part.
Write a whole number as a fraction over 1. Negative fractions follow the same sign rules as negative integers.
$4 \div \frac{2}{3} = \frac{4}{1} \times \frac{3}{2} = 6 - \frac{2}{3} \times \frac{3}{4} = - \frac{6}{12} = - \frac{1}{2}$

Part 8 of 8

Decimals without a calculator

To add or subtract decimals, line up the decimal points, so that tenths are under tenths. Fill any empty places with zeros.
$7.4 - 2.68 = 7.40 - 2.68 = 4.72$
To multiply, ignore the decimal points and multiply the whole numbers. Then count the decimal places in the question, and give the answer the same total number of decimal places.
$0.6 \times 0.04 : 6 \times 4 = 24, 1 + 2 = 3 decimal places, answer 0.024$
Check that the size makes sense. $0.04$ is multiplied by $0.6$ , which is less than 1, so the answer must be less than $0.04$ . $0.024$ is less than $0.04$ , so it passes the check.
To divide by a decimal, change the calculation so that you divide by a whole number. Multiply both numbers by the same power of 10. This makes an equivalent fraction, so the answer does not change.
$7.2 \div 0.03 = \frac{7.2}{0.03} = \frac{720}{3} = 240$
When you divide a positive number by a positive number less than 1, the answer is bigger than the number you started with. $240$ is bigger than $7.2$ , as it must be.
If a calculation mixes fractions and decimals, change everything into one form. Fractions are better when a fraction such as $\frac{1}{3}$ has a recurring decimal: $0.25 + \frac{1}{3} = \frac{1}{4} + \frac{1}{3} = \frac{7}{12}$ . Give the final answer as a fraction or as a decimal, not as a mix of the two.

Worked examples

Example 1. Work out $18 - 3 \times {(5 - 7)}^{2} \div 6$ without a calculator.

Show the working

Brackets first: $5 - 7 = - 2$ .
$18 - 3 \times {(- 2)}^{2} \div 6$
Indices next. The bracket stays with the square, so ${(- 2)}^{2} = (- 2) \times (- 2) = 4$ .
$18 - 3 \times 4 \div 6$
Multiplication and division have the same rank, so work left to right: $3 \times 4 = 12$ , then $12 \div 6 = 2$ .
$18 - 2$
Subtraction last.
$18 - 2 = 16$
Notice the two wrong paths. Taking ${(- 2)}^{2}$ as $- 4$ gives $18 - (- 2) = 20$ . Doing $18 - 3$ first gives $15 \times 4 \div 6 = 10$ . One line per operation stops both.

Answer: $16$

Example 2. Work out (a) $2 \frac{2}{3} - 1 \frac{3}{4}$ (b) $3 \frac{1}{3} \div 1 \frac{1}{4}$ , giving your answer to (b) as a mixed number.

Show the working

(a) Change both mixed numbers to improper fractions: $2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}$ and $1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$ .
The LCM of 3 and 4 is 12. Write both fractions over 12, then subtract the numerators.
$\frac{8}{3} - \frac{7}{4} = \frac{32}{12} - \frac{21}{12} = \frac{11}{12}$
Check the size with decimals: about $2.7 - 1.8 = 0.9$ , and $\frac{11}{12}$ is about $0.92$ . The answer is sensible.
(b) Change to improper fractions: $3 \frac{1}{3} = \frac{10}{3}$ and $1 \frac{1}{4} = \frac{5}{4}$ .
Dividing by $\frac{5}{4}$ is the same as multiplying by its reciprocal, $\frac{4}{5}$ . Turn over the second fraction only.
$\frac{10}{3} \div \frac{5}{4} = \frac{10}{3} \times \frac{4}{5}$
Cancel: 10 and 5 share a factor of 5.
$\frac{2}{3} \times \frac{4}{1} = \frac{8}{3}$
Change back to a mixed number, because the question asks for one: $8 \div 3 = 2$ remainder $2$ .
$\frac{8}{3} = 2 \frac{2}{3}$

Answer: (a) $\frac{11}{12}$ (b) $2 \frac{2}{3}$

Example 3. At 06:00 the temperature on a hill was $- 8$ °C. By 13:00 it had risen by 13 degrees. After 13:00 the temperature fell by $1.5$ degrees every hour for 6 hours. (a) Work out the temperature at 13:00. (b) Work out the temperature at 19:00. (c) How many degrees warmer was it at 19:00 than at 06:00?

Show the working

(a) A rise means add.
$- 8 + 13 = 5$
(b) Work out the total fall first: 6 hours at $1.5$ degrees each.
$6 \times 1.5 = 9$
A fall means subtract. Start from the 13:00 value, not from the 06:00 value.
$5 - 9 = - 4$
(c) 'How many degrees warmer' is a difference: higher minus lower. $- 4$ is higher than $- 8$ .
$- 4 - (- 8) = - 4 + 8 = 4$
Check on a number line: from $- 8$ up to $- 4$ is 4 steps.

Answer: (a) $5$ °C (b) $- 4$ °C (c) $4$ degrees warmer

Example 4. (a) Work out $0.7 \times 0.08$ without a calculator. (b) Work out $8.4 \div 0.07$ without a calculator. (c) Use your calculator to work out $\frac{4.7 + 2.38}{1.6 \times 0.25}$ .

Show the working

(a) Multiply without the decimal points: $7 \times 8 = 56$ . The question has $1 + 2 = 3$ decimal places, so the answer has 3.
$0.7 \times 0.08 = 0.056$
Size check: multiplying $0.08$ by $0.7$ , which is less than 1, must give less than $0.08$ . It does.
(b) Multiply both numbers by 100, so that you divide by a whole number.
$8.4 \div 0.07 = 840 \div 7 = 120$
Size check: dividing $8.4$ by $0.07$ , which is less than 1, must give an answer bigger than $8.4$ . It does.
(c) The fraction bar groups the top and the bottom. Work out each one and write it down.
$4.7 + 2.38 = 7.08 1.6 \times 0.25 = 0.4$
Now divide.
$\frac{7.08}{0.4} = 17.7$
To do it in one line, type $(4.7 + 2.38) \div (1.6 \times 0.25)$ . Without the brackets the calculator works out $4.7 + 2.38 \div 1.6 \times 0.25 = 5.071875$ , which is wrong.

Answer: (a) $0.056$ (b) $120$ (c) $17.7$

Where people go wrong

Mistake: Working strictly from left to right, e.g. $5 + 3 \times 4 = 8 \times 4 = 32$ .

Instead: Multiply before you add: $5 + 3 \times 4 = 5 + 12 = 17$ .

Left to right is the rule only for operations of the same rank. Multiplication and division come before addition and subtraction, wherever they are written.

Mistake: Reading BIDMAS as 'always divide before you multiply', e.g. $12 \div 4 \times 3 = 12 \div 12 = 1$ .

Instead: Division and multiplication have the same rank. Work left to right: $12 \div 4 \times 3 = 3 \times 3 = 9$ .

The letters are only a memory aid. D and M share one level, and A and S share the next one.

Mistake: Writing $- 3^{2} = 9$ , or ${(- 2)}^{2} = - 4$ .

Instead: $- 3^{2} = - (3^{2}) = - 9$ , but ${(- 3)}^{2} = 9$ and ${(- 2)}^{2} = 4$ .

An index applies only to what it is attached to. Without a bracket, the power is on the 3 alone, and the minus sign is applied after the squaring. With a bracket, the negative number is multiplied by itself, and two negatives give a positive.

Mistake: Adding fractions by adding the tops and adding the bottoms, e.g. $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$ .

Instead: Use a common denominator: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .

The denominator names the size of the pieces, and you can only add pieces of the same size. A quick check shows the error: $\frac{2}{5}$ is less than $\frac{1}{2}$ , but adding something positive to $\frac{1}{2}$ cannot make it smaller.

Mistake: Treating 'subtract a negative' as 'subtract', e.g. $- 4 - (- 9) = - 13$ .

Instead: Two minus signs side by side become a plus: $- 4 - (- 9) = - 4 + 9 = 5$ .

Subtraction finds the gap. From $- 9$ up to $- 4$ is 5 steps, so $- 4$ is 5 more than $- 9$ .

Check yourself

Work out (a) $5 + 3 \times 4 - 2$ (b) $24 \div 6 \times 2 - 3$ .
Answer
(a) $5 + 12 - 2 = 15$ (b) $4 \times 2 - 3 = 8 - 3 = 5$
Work out (a) $- 4 - (- 9)$ (b) $(- 6) \times (- 3) \div (- 2)$ .
Answer
(a) $- 4 + 9 = 5$ (b) $18 \div (- 2) = - 9$
Write down the value of ${(- 3)}^{2}$ and the value of $- 3^{2}$ .
Answer
${(- 3)}^{2} = 9$ and $- 3^{2} = - 9$ .
Work out $\frac{3}{4} + \frac{5}{6}$ . Give your answer as a mixed number.
Answer
$\frac{9}{12} + \frac{10}{12} = \frac{19}{12} = 1 \frac{7}{12}$
Work out $2 \frac{1}{2} \div \frac{5}{8}$ .
Answer
$\frac{5}{2} \times \frac{8}{5} = \frac{40}{10} = 4$
Without a calculator, work out (a) $2.4 \times 0.3$ (b) $1.56 \div 0.4$ .
Answer
(a) $24 \times 3 = 72$ , then 2 decimal places: $0.72$ (b) $15.6 \div 4 = 3.9$
Write in two pairs of brackets to make this true: $8 - 2 \times 3 + 1 = 24$ .
Answer
$(8 - 2) \times (3 + 1) = 6 \times 4 = 24$
A freezer is at $- 18$ °C. The kitchen is at $21$ °C. Find the difference between the two temperatures.
Answer
$21 - (- 18) = 21 + 18 = 39$ degrees.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Work out $\frac{2}{3} + \frac{4}{5} \times \frac{5}{8}$ . Give your answer as a mixed number. 2 marks No calculator

Mark scheme

M1 4/5 × 5/8 worked out first, i.e. 20/40 or 1/2 seen
A1 1 1/6 cao

(b) Show that $3 \frac{1}{2} - 1 \frac{5}{6} = 1 \frac{2}{3}$ . 3 marks No calculator

Mark scheme

M1 7/2 and 11/6 seen, or another correct method such as 3 − 1 = 2 with 1/2 − 5/6 seen
M1 21/6 − 11/6 oe, both fractions correct over one common denominator (or 3/6 − 5/6 oe in the whole-number method)
A1 10/6 or 5/3 seen (or 2 − 2/6 = 1 4/6 in the whole-number method) and shown equal to 1 2/3, with no errors in the working

(c) Work out $2 \frac{1}{4} \div 1 \frac{4}{5}$ . Give your answer as a mixed number in its simplest form. 3 marks No calculator

Mark scheme

B1 9/4 and 9/5 seen
M1 9/4 × 5/9 oe, i.e. multiplying by the reciprocal of their second fraction
A1 1 1/4 cao

Question 2.

(a) Work out $20 - 12 \div 4 \times 2$ . 1 mark No calculator

Mark scheme

B1 14 cao

(b) Write one pair of brackets in this calculation to make it true. $20 - 12 \div 4 \times 2 = 4$ 1 mark No calculator

Mark scheme

B1 (20 − 12) ÷ 4 × 2 = 4 only

(c) Work out ${(- 5)}^{2} - 3 \times (- 4)$ . 2 marks No calculator

Mark scheme

M1 25 and −12 seen, or 25 − (−12) or 25 + 12 seen
A1 37 cao

Question 3. The temperatures at noon in five towns were: town A $- 7$ °C, town B $3$ °C, town C $- 12$ °C, town D $- 1$ °C, town E $8$ °C.

(a) Work out the difference between the highest and the lowest of these temperatures. 2 marks No calculator

Mark scheme

M1 8 − (−12) or 8 + 12 seen, or a count on a number line from −12 to 8
A1 20 cao

(b) After noon, the temperature in town E fell by $2.5$ degrees every hour for 6 hours. Work out the temperature in town E at the end of the 6 hours. 2 marks No calculator

Mark scheme

M1 6 × 2.5 = 15 seen, or 8 − 6 × 2.5 oe
A1 −7 cao

Question 4.

(a) Calculate $\frac{7.2 + 3 \times 1.6}{{(4.3 - 1.8)}^{2}}$ . 2 marks

Mark scheme

M1 12 or 6.25 seen
A1 1.92 cao

(b) A ribbon is $4.8$ m long. Ana cuts all of it into pieces that are each $0.15$ m long. Work out the number of pieces she cuts. 2 marks No calculator

Mark scheme

M1 4.8 ÷ 0.15 written as 480 ÷ 15 oe
A1 32 cao

How this is examined

Adding, subtracting or dividing mixed numbers with every step on the page: A 2- or 3-mark Work out or Show (that) with no calculator allowed, asking for the answer as a mixed number or as a fraction in its simplest form Both mixed numbers written as improper fractions. For $+$ or $-$ , both fractions written over one common denominator. For $\div$ , the second fraction turned over and multiplied. Then the result simplified and, if asked, written as a mixed number. In a Show (that), the given answer earns nothing by itself: the improper fractions, the common-denominator line and the simplified result must all be written.
Working out a calculation with negatives and powers in the right order: A 1- or 2-mark Work out with no calculator that mixes brackets, a square and negative numbers, such as ${(- a)}^{2} - b \times (- c)$ The values written in BIDMAS order, so that the square and the signed product can each be seen and given the method mark, then the final number. ${(- a)}^{2}$ is positive, and subtracting a negative product becomes adding.
Finding a temperature change or a difference that crosses zero: A short context with a list or table of temperatures, asking for a difference between two of them or for a temperature after a rise or a repeated fall, 1 or 2 marks each The calculation written with its signs, e.g. higher $-$ lower $= 9 - (- 14)$ , then a positive difference in degrees. For a repeated change, the total change (number of hours $\times$ change per hour) worked out first, then added to or subtracted from the starting value once.
Using a calculator on a fraction-bar calculation without losing the brackets: A 2-mark Calculate or Work out with a sum or a power on the top or the bottom of a fraction The value of the top and the value of the bottom each written down, which can earn the method mark, then the final answer. Typed in one line, the top and the bottom each need their own brackets. Give an exact answer in full; give a non-exact answer to 3 significant figures.
Multiplying and dividing decimals without a calculator: A 1- or 2-mark Work out, or a short context such as cutting a length into equal pieces For $\times$ : the whole-number product, then the correct total number of decimal places. For $\div$ : both numbers multiplied by the same power of 10 so the divisor is whole, e.g. $3.6 \div 0.12 = 360 \div 12$ , written down, then the answer.
Writing brackets into a calculation to make it true: A 1-mark question that gives a calculation and its answer and asks for one or two pairs of brackets Brackets placed so that the calculation, worked out in BIDMAS order, gives exactly the stated answer, and only as many pairs as the question asks for. Work it out once more with the brackets in place to check.

Words to know

Integer: A whole number, positive, zero or negative: $\dots, - 2, - 1, 0, 1, 2, \dots$
The four operations: Addition, subtraction, multiplication and division.
Order of operations (BIDMAS): Brackets, then indices, then $\times$ and $\div$ from left to right, then $+$ and $-$ from left to right.
Numerator: The top number of a fraction. It counts how many pieces there are.
Denominator: The bottom number of a fraction. It names the size of the pieces.
Improper fraction: A fraction whose numerator is bigger than or equal to its denominator, e.g. $\frac{19}{12}$ .
Mixed number: A whole number and a fraction written together, e.g. $1 \frac{7}{12}$ , which means $1 + \frac{7}{12}$ .
Common denominator: A denominator shared by two or more fractions, so that they can be added or subtracted. The LCM of the denominators is the smallest one.
Reciprocal: 1 divided by a number. For a fraction, turn it upside down: the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$ .
Difference: The gap between two values: the higher one minus the lower one, given as a positive number.
Decimal place: A position for a digit after the decimal point. $0.024$ has 3 decimal places.

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