Year 5: Combining operations, the equals sign and multi-step problems
Part of the Year 5 short division and remainders chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Combining operations, the equals sign and multi-step problems
When a calculation has more than one operation, the order matters. Look at 24 + 36 ÷ 4. If you add first you get 60 ÷ 4 = 15. But the correct answer is 33. The rule is: brackets first, then multiply and divide (from left to right), then add and subtract (from left to right). So 36 ÷ 4 = 9 comes first, and then 24 + 9 = 33. Brackets change the order. In (24 + 36) ÷ 4 the brackets say add first: 60 ÷ 4 = 15.
| Step | Do this | Example: 24 + 36 ÷ 4 |
|---|---|---|
| 1 | brackets | none |
| 2 | × and ÷, left to right | 36 ÷ 4 = 9, so 24 + 9 |
| 3 | + and −, left to right | 24 + 9 = 33 |
The equals sign (=) means “is the same value as”. It is not just a signal that the answer comes next. Both sides must balance, like a see-saw. In 56 ÷ 7 = 4 × [ ], the left side is 8, so the right side must be 8 too. What number multiplied by 4 makes 8? It is 2.
Multi-step problems. Read the story, draw a bar model, and work out one thing at a time. For example: a school buys 6 boxes of 24 pens and shares them equally between 8 classes. First find how many pens there are (6 × 24 = 144). Then share them (144 ÷ 8 = 18). Write down what each answer means.
Key ideas
- Order of operations: brackets first, then × and ÷ (left to right), then + and − (left to right).
- Do one step at a time and write down each new line.
- The equals sign means “is the same value as”. Work out the side you can, then find the missing number that balances the other side with an inverse operation.
- For multi-step problems: draw a bar model, find the missing amounts one at a time, and check you have answered the question.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.24 + 36 ÷ 4 = 24 + 9
- Add. Only + and − are left. Go from left to right.24 + 9 = 33
- Write the answer. 24 + 36 ÷ 4 = 33
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.(24 + 36) ÷ 4 = 60 ÷ 4
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.60 ÷ 4 = 15
- Write the answer. (24 + 36) ÷ 4 = 15
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the multiplication. × and ÷ come before + and −. If there are several, go from left to right.5 × 8 − 24 ÷ 6 = 40 − 24 ÷ 6
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.40 − 24 ÷ 6 = 40 − 4
- Subtract. Only + and − are left. Go from left to right.40 − 4 = 36
- Write the answer. 5 × 8 − 24 ÷ 6 = 36
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.100 − 72 ÷ 8 × 3 = 100 − 9 × 3
- Do the multiplication. × and ÷ come before + and −. If there are several, go from left to right.100 − 9 × 3 = 100 − 27
- Subtract. Only + and − are left. Go from left to right.100 − 27 = 73
- Write the answer. 100 − 72 ÷ 8 × 3 = 73
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.56 ÷ 7 = 8
- Look at the other side. It must also be worth 8. Picture a balance.
- Use the opposite operation. 4 times the missing number is 8. Undo the multiplication with a division.8 ÷ 4 = 2
- Write the missing number. Put it back in the box and read the whole sentence.2
Check. 4 × 2 = 8 and 56 ÷ 7 = 8. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.72 ÷ 8 = 9
- Look at the other side. It must also be worth 9. Picture a balance.
- Use the opposite operation. The missing number divided by 5 is 9. Undo the division with a multiplication.9 × 5 = 45
- Write the missing number. Put it back in the box and read the whole sentence.45
Check. 45 ÷ 5 = 9 and 72 ÷ 8 = 9. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.3 × 5 = 15
- Look at the other side. It must also be worth 15. Picture a balance.
- Use the opposite operation. 90 shared into groups of the missing size makes 15 groups, so the missing number is 90 divided by 15.90 ÷ 15 = 6
- Write the missing number. Put it back in the box and read the whole sentence.6
Check. 90 ÷ 6 = 15 and 3 × 5 = 15. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try each place where brackets could go. Work out each one using the order of operations. Only one will give the answer.
- Work out the calculation with no brackets. Use the order of operations to see what we get now.30 + 6 ÷ 3 = 32, but we need 12
- Try each place for the brackets. Put the brackets round two numbers that are next to each other, and work out the answer.
With brackets Value = 12? (30 + 6) ÷ 3 12 yes 30 + (6 ÷ 3) 32 no - Pick the one that works. Only one gives the target.(30 + 6) ÷ 3 = 12
Check. (30 + 6) ÷ 3 = 12. ✓
Watch out. Brackets change which calculation is done first.
Key words. A bar model is a drawing that shows the amounts in a problem as bars, so you can see which calculation to do.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. 6 boxes of 24 pens, then the total is shared.
- Find the number of pens. 6 boxes with 24 pens in each, so multiply.6 × 24 = 144
- Share the pens. The pens are shared equally between 8 classes, so divide by 8.144 ÷ 8 = 18
- Write the answer. Include the unit.18 pens
Check. 18 × 8 = 144, and 144 = 6 × 24. ✓
Watch out. Do the multiplication first to find the total, then divide.
Key words. A bar model is a drawing that shows the amounts in a problem as bars, so you can see which calculation to do.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. The £100 is split into the cost of the books and the money that is left over.
- Find the cost of the books. 4 books at £12 each, so multiply.4 × 12 = 48
- Find how much is left. Take the cost of the books away from £100.100 − 48 = 52
- Share what is left. The money left is shared equally between 4 friends, so divide by 4.52 ÷ 4 = 13
- Write the answer. Include the unit.£13
Check. 13 × 4 = 52 and 52 + 48 = 100. ✓
Watch out. Read the question again: you want the amount for each friend, not the amount left.
Watch out
Do not work from left to right when there are mixed operations. 24 + 36 ÷ 4 is 33, not 15. The division comes first. And the equals sign is not an instruction to “work it out”: in 56 ÷ 7 = 4 × [ ] both sides must have the same value (8), so the missing number is 2, not 8.Exercise 5G Combining operations, the equals sign and multi-step problems
Non-calculatorKey words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.18 + 24 ÷ 6 = 18 + 4
- Add. Only + and − are left. Go from left to right.18 + 4 = 22
- Write the answer. 18 + 24 ÷ 6 = 22
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.40 − 36 ÷ 9 = 40 − 4
- Subtract. Only + and − are left. Go from left to right.40 − 4 = 36
- Write the answer. 40 − 36 ÷ 9 = 36
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the multiplication. × and ÷ come before + and −. If there are several, go from left to right.7 × 4 + 56 ÷ 8 = 28 + 56 ÷ 8
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.28 + 56 ÷ 8 = 28 + 7
- Add. Only + and − are left. Go from left to right.28 + 7 = 35
- Write the answer. 7 × 4 + 56 ÷ 8 = 35
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.60 ÷ 5 − 3 × 2 = 12 − 3 × 2
- Do the multiplication. × and ÷ come before + and −. If there are several, go from left to right.12 − 3 × 2 = 12 − 6
- Subtract. Only + and − are left. Go from left to right.12 − 6 = 6
- Write the answer. 60 ÷ 5 − 3 × 2 = 6
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.(18 + 24) ÷ 6 = 42 ÷ 6
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.42 ÷ 6 = 7
- Write the answer. (18 + 24) ÷ 6 = 7
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.(40 − 36) × 9 = 4 × 9
- Do the multiplication. × and ÷ come before + and −. If there are several, go from left to right.4 × 9 = 36
- Write the answer. (40 − 36) × 9 = 36
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.7 × (4 + 56 ÷ 8) = 7 × (4 + 7)
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.7 × (4 + 7) = 7 × 11
- Do the multiplication. × and ÷ come before + and −. If there are several, go from left to right.7 × 11 = 77
- Write the answer. 7 × (4 + 56 ÷ 8) = 77
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.60 ÷ (5 + 7) = 60 ÷ 12
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.60 ÷ 12 = 5
- Write the answer. 60 ÷ (5 + 7) = 5
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.20 − 8 ÷ 4 = 20 − 2
- Subtract. Only + and − are left. Go from left to right.20 − 2 = 18
- Write the answer. 20 − 8 ÷ 4 = 18
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Use the order of operations: brackets first, then × and ÷, then + and −. Do one step at a time.
- Work out the brackets first. Brackets always come first. Inside them, × and ÷ come before + and −.(20 − 8) ÷ 4 = 12 ÷ 4
- Do the division. × and ÷ come before + and −. If there are several, go from left to right.12 ÷ 4 = 3
- Write the answer. (20 − 8) ÷ 4 = 3
Check. Work through the steps again, and check the order: brackets, then × and ÷, then + and −.
Watch out. Do not just work from left to right. Brackets first, then × and ÷, then + and −.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.48 ÷ 6 = 8
- Look at the other side. It must also be worth 8. Picture a balance.
- Use the opposite operation. 2 times the missing number is 8. Undo the multiplication with a division.8 ÷ 2 = 4
- Write the missing number. Put it back in the box and read the whole sentence.4
Check. 2 × 4 = 8 and 48 ÷ 6 = 8. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.72 ÷ 9 = 8
- Look at the other side. It must also be worth 8. Picture a balance.
- Use the opposite operation. The missing number divided by 4 is 8. Undo the division with a multiplication.8 × 4 = 32
- Write the missing number. Put it back in the box and read the whole sentence.32
Check. 32 ÷ 4 = 8 and 72 ÷ 9 = 8. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.5 × 6 = 30
- Look at the other side. It must also be worth 30. Picture a balance.
- Use the opposite operation. 60 shared into groups of the missing size makes 30 groups, so the missing number is 60 divided by 30.60 ÷ 30 = 2
- Write the missing number. Put it back in the box and read the whole sentence.2
Check. 60 ÷ 2 = 30 and 5 × 6 = 30. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.36 ÷ 4 = 9
- Look at the other side. It must also be worth 9. Picture a balance.
- Use the opposite operation. 5 add the missing number makes 9. Undo the addition with a subtraction.9 − 5 = 4
- Write the missing number. Put it back in the box and read the whole sentence.4
Check. 5 + 4 = 9 and 36 ÷ 4 = 9. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.5 × 5 = 25
- Look at the other side. It must also be worth 25. Picture a balance.
- Use the opposite operation. 60 take away the missing number leaves 25, so the missing number is what is taken away: 60 − 25.60 − 25 = 35
- Write the missing number. Put it back in the box and read the whole sentence.35
Check. 60 − 35 = 25 and 5 × 5 = 25. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. The equals sign means both sides have the same value. Work out the side you can, then find the number that makes the other side match.
- Work out the side with no missing number. The two sides of an equals sign must have the same value, so first find the value of the side we can work out.8 × 3 = 24
- Look at the other side. It must also be worth 24. Picture a balance.
- Use the opposite operation. The missing number take away 7 leaves 24. Undo the subtraction with an addition.24 + 7 = 31
- Write the missing number. Put it back in the box and read the whole sentence.31
Check. 31 − 7 = 24 and 8 × 3 = 24. Both sides are equal. ✓
Watch out. The equals sign does not mean “the answer comes next”. It means both sides balance.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. Work out the value of each side. The statement is true only if both sides have the same value.
- Work out the left side. Use the order of operations.48 ÷ 6 = 8
- Work out the right side. Do the same for the other side.4 × 2 = 8
- Compare. The equals sign says both sides have the same value.8 = 8: the statement is true
Check. Read the sentence back with the two values.
Watch out. The equals sign means “is the same value as”, not “the answer is”.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. Work out the value of each side. The statement is true only if both sides have the same value.
- Work out the left side. Use the order of operations.36 ÷ 4 + 2 = 9 + 2
9 + 2 = 11 - Work out the right side. Do the same for the other side.6 × 2 = 12
- Compare. The equals sign says both sides have the same value.11 ≠ 12: the statement is false
Check. Read the sentence back with the two values.
Watch out. The equals sign means “is the same value as”, not “the answer is”.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. Work out the value of each side. The statement is true only if both sides have the same value.
- Work out the left side. Use the order of operations.5 × 8 = 40
- Work out the right side. Do the same for the other side.80 ÷ 2 = 40
- Compare. The equals sign says both sides have the same value.40 = 40: the statement is true
Check. Read the sentence back with the two values.
Watch out. The equals sign means “is the same value as”, not “the answer is”.
Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. A missing number is the number that makes both sides balance.
Method. Work out the value of each side. The statement is true only if both sides have the same value.
- Work out the left side. Use the order of operations.72 ÷ 9 = 8
- Work out the right side. Do the same for the other side.2 × 3 + 3 = 6 + 3
6 + 3 = 9 - Compare. The equals sign says both sides have the same value.8 ≠ 9: the statement is false
Check. Read the sentence back with the two values.
Watch out. The equals sign means “is the same value as”, not “the answer is”.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try each place where brackets could go. Work out each one using the order of operations. Only one will give the answer.
- Work out the calculation with no brackets. Use the order of operations to see what we get now.8 + 4 × 3 = 20, but we need 36
- Try each place for the brackets. Put the brackets round two numbers that are next to each other, and work out the answer.
With brackets Value = 36? (8 + 4) × 3 36 yes 8 + (4 × 3) 20 no - Pick the one that works. Only one gives the target.(8 + 4) × 3 = 36
Check. (8 + 4) × 3 = 36. ✓
Watch out. Brackets change which calculation is done first.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try each place where brackets could go. Work out each one using the order of operations. Only one will give the answer.
- Work out the calculation with no brackets. Use the order of operations to see what we get now.48 ÷ 6 − 2 = 6, but we need 12
- Try each place for the brackets. Put the brackets round two numbers that are next to each other, and work out the answer.
With brackets Value = 12? (48 ÷ 6) − 2 6 no 48 ÷ (6 − 2) 12 yes - Pick the one that works. Only one gives the target.48 ÷ (6 − 2) = 12
Check. 48 ÷ (6 − 2) = 12. ✓
Watch out. Brackets change which calculation is done first.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try every sign in every gap, in order. Use the order of operations to work out each one. Only one gives the answer.
- List the choices. There are 4 signs for each gap, so we try them all in an organised way.2 gaps, 4 signs each: 16 calculations
- Work each one out. Use the order of operations. A division must come out exactly. We want the answer 14.
36 + 4 + 5 = 45 36 + 4 − 5 = 35 36 + 4 × 5 = 56 36 + 4 ÷ 5 = no 36 − 4 + 5 = 37 36 − 4 − 5 = 27 36 − 4 × 5 = 16 36 − 4 ÷ 5 = no 36 × 4 + 5 = 149 36 × 4 − 5 = 139 36 × 4 × 5 = 720 36 × 4 ÷ 5 = no 36 ÷ 4 + 5 = 14 ✓ 36 ÷ 4 − 5 = 4 36 ÷ 4 × 5 = 45 36 ÷ 4 ÷ 5 = no - Pick the one that works. Only one of them gives the target.36 ÷ 4 + 5 = 14
Check. 36 ÷ 4 + 5 = 14. ✓
Watch out. Remember: × and ÷ are done before + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try every sign in every gap, in order. Use the order of operations to work out each one. Only one gives the answer.
- List the choices. There are 4 signs for each gap, so we try them all in an organised way.2 gaps, 4 signs each: 16 calculations
- Work each one out. Use the order of operations. A division must come out exactly. We want the answer 8.
40 + 8 + 3 = 51 40 + 8 − 3 = 45 40 + 8 × 3 = 64 40 + 8 ÷ 3 = no 40 − 8 + 3 = 35 40 − 8 − 3 = 29 40 − 8 × 3 = 16 40 − 8 ÷ 3 = no 40 × 8 + 3 = 323 40 × 8 − 3 = 317 40 × 8 × 3 = 960 40 × 8 ÷ 3 = no 40 ÷ 8 + 3 = 8 ✓ 40 ÷ 8 − 3 = 2 40 ÷ 8 × 3 = 15 40 ÷ 8 ÷ 3 = no - Pick the one that works. Only one of them gives the target.40 ÷ 8 + 3 = 8
Check. 40 ÷ 8 + 3 = 8. ✓
Watch out. Remember: × and ÷ are done before + and −.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try every sign in every gap, in order. Use the order of operations to work out each one. Only one gives the answer.
- List the choices. There are 4 signs for each gap, so we try them all in an organised way.2 gaps, 4 signs each: 16 calculations
- Work each one out. Use the order of operations. A division must come out exactly. We want the answer 12.
24 + 6 + 2 = 32 24 + 6 − 2 = 28 24 + 6 × 2 = 36 24 + 6 ÷ 2 = 27 24 − 6 + 2 = 20 24 − 6 − 2 = 16 24 − 6 × 2 = 12 ✓ 24 − 6 ÷ 2 = 21 24 × 6 + 2 = 146 24 × 6 − 2 = 142 24 × 6 × 2 = 288 24 × 6 ÷ 2 = 72 24 ÷ 6 + 2 = 6 24 ÷ 6 − 2 = 2 24 ÷ 6 × 2 = 8 24 ÷ 6 ÷ 2 = 2 - Pick the one that works. Only one of them gives the target.24 − 6 × 2 = 12
Check. 24 − 6 × 2 = 12. ✓
Watch out. Remember: × and ÷ are done before + and −.
Key words. A bar model is a drawing that shows the amounts in a problem as bars, so you can see which calculation to do.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. A bar model shows what we know and what we need to find out.
- Find the number of tins. 5 shelves of 36 tins, so multiply.5 × 36 = 180
- Put the tins into boxes. Boxes of 9 tins, so divide the total by 9.180 ÷ 9 = 20
- Write the answer. Include the unit.20 boxes
Check. Work backwards with the opposite operations to check.
Watch out. Do one step at a time and say what each answer means.
Key words. A bar model is a drawing that shows the amounts in a problem as bars, so you can see which calculation to do.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. A bar model shows what we know and what we need to find out.
- Find how many stickers are left. She gives 70 away, so subtract.250 − 70 = 180
- Share the rest. The stickers are shared equally between 6 friends, so divide by 6.180 ÷ 6 = 30
- Write the answer. Include the unit.30 stickers
Check. Work backwards with the opposite operations to check.
Watch out. Do one step at a time and say what each answer means.
Key words. A bar model is a drawing that shows the amounts in a problem as bars, so you can see which calculation to do.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. A bar model shows what we know and what we need to find out.
- Find the number of chairs. 8 rows of 45, so multiply.8 × 45 = 360
- Take away the chairs removed. 30 chairs are taken away.360 − 30 = 330
- Share into stacks. 5 equal stacks, so divide by 5.330 ÷ 5 = 66
- Write the answer. Include the unit.66 chairs
Check. Work backwards with the opposite operations to check.
Watch out. Do one step at a time and say what each answer means.
Key words. Multiplying and dividing are inverse (opposite) operations: one undoes the other. A fact family is a set of multiplication and division facts that use the same three numbers.
Method. The area of a rectangle is length × width. To find a missing side, divide the area by the side you know.
- Draw the rectangle. The area is 96 cm² and the length is 8 cm.
- Choose the operation. Area = length × width. Multiplying is undone by dividing, so width = area ÷ length.96 ÷ 8
- Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.
8 9 6 - Divide the tens. Always start on the left, with the biggest place value.9 ÷ 8 = 1, remainder 1
Write 1 above the 9.
Exchange the remainder: 1 ten becomes 10 ones. Carry 1 to the next digit. - Divide the ones. Move to the next digit on the right. Add the 1 that was carried.The carried 1 makes 10 ones. With the 6 ones there are 10 + 6 = 16 ones.
16 ÷ 8: 2 × 8 = 16. Write 2 above the 6.
Nothing is left over, so there is no remainder. - Read the answer. Read the digits above the line from left to right.
1 2 8 9 16
96 ÷ 8 = 12 - Write the answer. Include the unit.12 cm
Check. 8 × 12 = 96. ✓
Watch out. The answer is a length in cm, not an area in cm².
Key words. A rate compares two amounts, such as £12 for 4 pens or 60 km each hour. A unit rate is the amount for just one (one pen, one hour, one litre). The word “per” means “for each” and tells you to divide.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. Each box is one minute. 7 boxes make 1,260 pages.
- Find the pages printed in one minute. The same number is printed each minute, so divide by 7.1,260 ÷ 7 = 180
- Find the pages printed in 10 minutes. 10 minutes of 180 pages, so multiply by 10.180 × 10 = 1,800
- Write the answer. Include the unit.1,800 pages
Check. 180 × 7 = 1,260. ✓
Watch out. Find the amount for one minute first.
Key words. The remainder is what is left over after you divide. The quotient is the answer to a division. Whether you round the answer up, round it down or use the remainder depends on the question.
Method. Break the problem into steps. Work out one thing at a time and write down what each answer means.
- Draw a bar model. 200 beads make 33 bracelets of 6 beads and 2 beads are left over.
- Find the number of bracelets. Each bracelet uses 6 beads, so divide 200 by 6 and ignore the remainder (only whole bracelets count).
0 3 3 r 2 6 2 20 20
200 ÷ 6 = 33 r 2, so 33 whole bracelets - Find the money. Each bracelet is sold for £3, so multiply by 3.33 × 3 = 99
- Write the answer. Include the unit.£99
Check. 33 × 6 = 198 and 198 + 2 = 200, and 33 × 3 = 99. ✓
Watch out. Round down: the 2 extra beads cannot make a whole bracelet.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.Divide first: 36 ÷ 4 = 9. Then add: 24 + 9 = 33.
- Compare with the answer in the question. Zara wrote 15. She did 24 + 36 = 60 and then 60 ÷ 4 = 15.
- Find the mistake. Zara worked from left to right. But division comes before addition, so 36 ÷ 4 must be done first.
- Write the correct answer. The correct answer is 33.
Check. If she wanted to add first, the calculation needs brackets: (24 + 36) ÷ 4 = 15.
Watch out. Without brackets, × and ÷ are done before + and −.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.2 + 3 + 4 = 9 and 2 + (3 + 4) = 9 (brackets did not change it)
10 − 6 ÷ 2 = 7 and (10 − 6) ÷ 2 = 2 (brackets changed it) - Look for one that does not work. One example that breaks the rule is called a counter-example.(10 − 6) ÷ 2 = 2 but 10 − 6 ÷ 2 = 7.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.Sometimes the brackets are around the part that would be done first anyway, so nothing changes. If they change the order, the answer can change.
- Decide. Sometimes
Check. We found one example each way. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.12 ÷ 6 = 2 and 12 ÷ 2 ÷ 3 = 6 ÷ 3 = 2
60 ÷ 6 = 10 and 60 ÷ 2 ÷ 3 = 30 ÷ 3 = 10
252 ÷ 6 = 42 and 252 ÷ 2 ÷ 3 = 126 ÷ 3 = 42 - Explain why. A reason shows what is happening for every number, not just the ones we tried.6 = 2 × 3, so dividing by 2 and then by 3 splits the number into 2 equal parts and then each part into 3. That is 6 equal parts altogether.
- Decide. Always
Check. All three examples matched. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Work out the answer to every one. The odd one out has a different answer from the others.
- Work out each one. Use place value or a known fact.
Calculation Answer 3 × 4 12 48 ÷ 4 12 (20 + 4) ÷ 2 12 20 − 4 ÷ 2 18 - Find the odd one out. Three have the same answer, 12. One is different.20 − 4 ÷ 2 = 18
- Write the answer. The odd one out is 20 − 4 ÷ 2.
Check. The other three all equal 12. ✓
Watch out. Work out all of them. Do not decide from just the first two.
Key words. The order of operations says which part of a calculation to do first: brackets first, then × and ÷ (left to right), then + and − (left to right).
Method. Try every sign in every gap, in order. Use the order of operations to work out each one. Only one gives the answer.
- List the choices. There are 4 signs for each gap, so we try them all in an organised way.3 gaps, 4 signs each: 64 calculations
- Work each one out. Use the order of operations. A division must come out exactly. We want the answer 14.
24 + 6 + 2 + 5 = 37 24 + 6 + 2 − 5 = 27 24 + 6 + 2 × 5 = 40 24 + 6 + 2 ÷ 5 = no 24 + 6 − 2 + 5 = 33 24 + 6 − 2 − 5 = 23 24 + 6 − 2 × 5 = 20 24 + 6 − 2 ÷ 5 = no 24 + 6 × 2 + 5 = 41 24 + 6 × 2 − 5 = 31 24 + 6 × 2 × 5 = 84 24 + 6 × 2 ÷ 5 = no 24 + 6 ÷ 2 + 5 = 32 24 + 6 ÷ 2 − 5 = 22 24 + 6 ÷ 2 × 5 = 39 24 + 6 ÷ 2 ÷ 5 = no 24 − 6 + 2 + 5 = 25 24 − 6 + 2 − 5 = 15 24 − 6 + 2 × 5 = 28 24 − 6 + 2 ÷ 5 = no 24 − 6 − 2 + 5 = 21 24 − 6 − 2 − 5 = 11 24 − 6 − 2 × 5 = 8 24 − 6 − 2 ÷ 5 = no 24 − 6 × 2 + 5 = 17 24 − 6 × 2 − 5 = 7 24 − 6 × 2 × 5 = no 24 − 6 × 2 ÷ 5 = no 24 − 6 ÷ 2 + 5 = 26 24 − 6 ÷ 2 − 5 = 16 24 − 6 ÷ 2 × 5 = 9 24 − 6 ÷ 2 ÷ 5 = no 24 × 6 + 2 + 5 = 151 24 × 6 + 2 − 5 = 141 24 × 6 + 2 × 5 = 154 24 × 6 + 2 ÷ 5 = no 24 × 6 − 2 + 5 = 147 24 × 6 − 2 − 5 = 137 24 × 6 − 2 × 5 = 134 24 × 6 − 2 ÷ 5 = no 24 × 6 × 2 + 5 = 293 24 × 6 × 2 − 5 = 283 24 × 6 × 2 × 5 = 1,440 24 × 6 × 2 ÷ 5 = no 24 × 6 ÷ 2 + 5 = 77 24 × 6 ÷ 2 − 5 = 67 24 × 6 ÷ 2 × 5 = 360 24 × 6 ÷ 2 ÷ 5 = no 24 ÷ 6 + 2 + 5 = 11 24 ÷ 6 + 2 − 5 = 1 24 ÷ 6 + 2 × 5 = 14 ✓ 24 ÷ 6 + 2 ÷ 5 = no 24 ÷ 6 − 2 + 5 = 7 24 ÷ 6 − 2 − 5 = no 24 ÷ 6 − 2 × 5 = no 24 ÷ 6 − 2 ÷ 5 = no 24 ÷ 6 × 2 + 5 = 13 24 ÷ 6 × 2 − 5 = 3 24 ÷ 6 × 2 × 5 = 40 24 ÷ 6 × 2 ÷ 5 = no 24 ÷ 6 ÷ 2 + 5 = 7 24 ÷ 6 ÷ 2 − 5 = no 24 ÷ 6 ÷ 2 × 5 = 10 24 ÷ 6 ÷ 2 ÷ 5 = no - Pick the one that works. Only one of them gives the target.24 ÷ 6 + 2 × 5 = 14
Check. 24 ÷ 6 + 2 × 5 = 14. ✓
Watch out. Remember: × and ÷ are done before + and −.