The four operations.
Use the right operation in the right order, with integers, fractions and decimals.
Use brackets and operation order · calculate with negatives, fractions and decimals · model practical situations
The order changes the answer
A calculation is an instruction with a structure. Without agreed rules, 2 + 3 × 4 could produce either 14 or 20. The order of operations tells every reader to interpret the same written calculation in the same way.
Brackets mark a group to complete first. Powers and roots come next, then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar also groups its whole numerator and denominator, so treat each part as a bracketed calculation.
Brackets first, then powers and roots, then multiplication and division, then addition and subtraction. At the same priority, work from left to right.
Compare 18 ÷ 3 × 2 and 18 ÷ (3 × 2).
Without brackets, work left to right: 18 ÷ 3 × 2 = 6 × 2 = 12.
With brackets, calculate inside first: 18 ÷ (3 × 2) = 18 ÷ 6 = 3.
Calculate 7 + 3 × (8 − 5)².
Pause and explainIn 18 ÷ 3 × 2, do you multiply 3 by 2 first?
No. Division and multiplication have equal priority. Work left to right: 18 ÷ 3 = 6, then 6 × 2 = 12.
Negative numbers: follow the operation
Use temperature to give a negative number meaning. Starting at −3°C and warming by 5°C takes you through zero to 2°C. Addition describes a change, not always an increase in the size of the written digits.
Subtracting a negative reverses the direction of that change. Multiplication signs follow a consistent pattern as well: one negative factor makes a negative product; two negative factors make a positive product. Separate the sign decision from the ordinary numerical calculation.
Add and subtract
Adding a positive moves right; adding a negative moves left.
−3 + 8 = 5
−3 + (−8) = −11
4 − (−6) = 4 + 6 = 10
Subtracting a number means adding its opposite.
Multiply and divide
Same signs give a positive result. Different signs give a negative result.
(−4) × (−3) = 12
(−4) × 3 = −12
18 ÷ (−6) = −3
These sign rules do not replace the rules for addition.
The temperature rises from −7°C to 5°C. Find the change.
It is a rise of 12°C. A later fall of 9°C gives 5 − 9 = −4°C.
Pause and explainWhy is 4 − (−2) equal to 6?
Removing a decrease of 2 has the same effect as increasing by 2. On a number line, move two steps right from 4.
Fractions and mixed numbers
You can add three quarters and one quarter because the pieces have the same size. You cannot add thirds and quarters simply by counting their pieces together: a third is not the same-sized part as a quarter.
A common denominator renames both fractions using equal-sized parts. For multiplication, take a fraction of another fraction. For division, ask how many of the divisor fit inside the first amount; this leads to multiplying by its reciprocal. Convert mixed numbers before multiplying or dividing.
Use a common denominator
23 + 14 = 812 + 312 = 1112.
56 − 14 = 1012 − 312 = 712.
Multiply across; cancel common factors
34 × 109 = 3036 = 56. You can cancel 3 with 9, and 10 with 4, before multiplying.
Multiply by the reciprocal of the divisor
35 ÷ 910 = 35 × 109 = 23. Only invert the fraction you divide by.
Calculate 213 − 34 ÷ 112.
- Convert mixed numbers: 73 − 34 ÷ 32.
- Divide before subtracting: 34 × 23 = 12.
- Subtract: 73 − 12 = 146 − 36 = 116 = 156.
Pause and explainWhy is 12 + 13 not 25?
The original pieces have different sizes. Renaming them as 36 and 26 gives 56.
Decimals and practical calculations
Decimals still use ordinary place value. In addition and subtraction, hundredths must line up with hundredths, so align decimal points. In multiplication and division, think about the size of the answer as well as the written digits.
For division such as 3.6 ÷ 0.4, multiply both numbers by 10 to obtain 36 ÷ 4. The quotient stays unchanged because both quantities were scaled equally. In practical problems, write the unit so you know what the result measures.
For addition and subtraction, align decimal points. For multiplication, calculate with integers then restore the total number of decimal places. For division, multiply both numbers by the same power of 10 to make the divisor an integer.
Work out 0.24 × 1.5.
24 × 15 = 360. The factors have 3 decimal places in total, so the answer is 0.360 = 0.36.
Work out 4.68 ÷ 0.12.
Multiply both numbers by 100:
Check: 39 × 0.12 = 4.68.
Three notebooks cost $2.75 each. Find the change from $10.
The brackets in 3 × (10 − 2.75) would describe a different situation.
Pause and explainShould 0.3 × 0.2 be larger or smaller than 0.3?
Smaller: multiplying by 0.2 means taking one fifth of 0.3. The result is 0.06.
Your turn. Show the order.
Before calculating, mark any brackets and identify the first operation. For fractions, decide whether the operation needs a common denominator or a reciprocal; these are different methods for different jobs.
Use a rough estimate to predict the sign and size of each answer. If an exact calculation disagrees wildly with that prediction, check signs, decimal places and the order of operations before moving on.
Calculate 20 − 12 ÷ 3 × 2.
Reveal worked answer
Work out (−5)² − 4 × (−3).
Reveal worked answer
Calculate 134 + 23 × 38.
Reveal worked answer
Calculate 6.3 ÷ 0.09 and 2.04 − 0.785.