Arithmetic operations and the order of calculation
Calculate with signed numbers, fractions and decimals while keeping the meaning of each operation clear.
Core and Extended.
Before you begin
- Convert mixed numbers to improper fractions.
- Locate positive and negative numbers on a number line.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 4 − (−3)?
What you will learn
- Calculate with negative numbers, fractions and decimals.
- Apply brackets and the order of operations.
- Check a calculation using an inverse or estimate.
Signs describe a change
Adding a negative amount is the same numerical change as subtracting its positive magnitude: 5 + (−8) = −3. Subtracting a negative reverses that change: 5 − (−8) = 13. Write the two signs separately before combining them. A number-line move can check the result, especially for temperatures or account balances.
For multiplication or division, equal signs give a positive result and different signs give a negative result. Thus (−6) × (−4) = 24 and (−24) ÷ 6 = −4. This is a rule for products and quotients; it does not mean the sum of two negatives is positive. For example, −6 + (−4) = −10.
Fraction operations use equal-sized pieces
Addition and subtraction need a common denominator: 3/4 − 2/3 = 9/12 − 8/12 = 1/12. Multiplication does not: multiply numerators and denominators, cancelling common factors when useful. To divide by a non-zero fraction, multiply by its reciprocal. Convert a mixed number before using these rules.
For decimals, align place values when adding or subtracting. Multiplication can be checked using whole-number products and the decimal scale. Dividing both dividend and divisor by the same non-zero value preserves a quotient; for instance, 1.26 ÷ 0.3 = 12.6 ÷ 3 = 4.2. An estimate helps catch a misplaced decimal.
Follow the structure of the expression
Evaluate brackets first, then powers and roots, then multiplication and division, then addition and subtraction. Multiplication and division share a priority and run from left to right. So 24 ÷ 6 × 2 = 4 × 2 = 8; it is not 24 ÷ 12. Addition and subtraction also share a priority.
A fraction bar groups its whole numerator and denominator. The expression (8 + 4)/(5 − 2) is 12/3 = 4, whereas 8 + 4/5 − 2 has a different value. Write brackets when typing a fraction into a calculator, and keep working visible even when the final arithmetic is short.
Check the scale and the context
Before an exact calculation, decide whether the answer should be positive, negative, greater than one or less than one. Dividing a positive amount by a fraction smaller than one makes the numerical quotient larger. This makes sense if the quotient counts small portions. An answer with the wrong sign or scale needs investigation.
Afterwards, reverse the last operation where possible. If 1.26 ÷ 0.3 = 4.2, check that 4.2 × 0.3 = 1.26. In a word problem, attach units to the result and explain any remainder or rounding. Counting full containers is different from measuring an amount that allows a fractional answer.
Pause and explain
What is 18 ÷ 3 × 2?
Worked example
Calculate −3 + 2(5 − 8)² ÷ 6.
Show the worked solution
- Inside the brackets, 5 − 8 = −3; squaring gives 9.
- Multiply and divide from left to right: 2 × 9 ÷ 6 = 3.
- Then add: −3 + 3 = 0.
Answer 0
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate 5/6 − 1/4 + 1/3.
Give me a hint
Use twelfths so every term counts equal-sized pieces.
Compare my reasoning
- Write 10/12 − 3/12 + 4/12.
- Combine the numerators: (10 − 3 + 4)/12 = 11/12.
- The result is slightly below one, consistent with the original quantities.
11/12
Look for these in your work
- Used one common denominator.
- Kept the subtraction sign attached to its term.
Calculate (−1.2) × 0.5 + 3/4 ÷ 1/2.
Give me a hint
Do each product or quotient before adding.
Compare my reasoning
- (−1.2) × 0.5 = −0.6.
- 3/4 ÷ 1/2 = 3/4 × 2 = 1.5.
- Add: −0.6 + 1.5 = 0.9.
0.9
Look for these in your work
- Inverted the divisor in the fraction calculation.
- Combined the signed results correctly.
An account starts at −$18. A $45 payment arrives, then three $12 purchases are made. Find the final balance and explain its sign.
Give me a hint
Treat each purchase as a negative change.
Compare my reasoning
- After the payment the balance is −18 + 45 = 27.
- The purchases total 3 × 12 = 36, so the balance becomes 27 − 36 = −9.
- The negative balance means $9 is owed, rather than $9 available to spend.
−$9; $9 is owed.
Look for these in your work
- Modelled all three purchases.
- Interpreted the negative answer in context.
Common mistakes
- Applying a multiplication sign rule to addition.
- Doing multiplication before every division instead of working left to right.
- Adding fraction numerators and denominators separately.
What can you explain now?
Evaluate 20 − 3 × (2 + 4).
Compare with the explanation
2
The bracket gives 6, multiplication gives 18, then subtract from 20.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Without notes tomorrow, explain why 12 ÷ 3 × 2 is 8 and why 12 ÷ (3 × 2) is 2. Create a fraction calculation that equals one.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources