Year 5: Solving problems with multiplication
Part of the Year 5 long multiplication chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Solving problems with multiplication
Area. The area of a rectangle is its length multiplied by its width. A rectangle 8 cm long and 5 cm wide has 5 rows of 8 squares, so the area is 8 × 5 = 40 cm². If you know the area and one side, you can find the other side with the opposite (inverse) operation.
Squares and cubes. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. It is the area of a square with sides of 6. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64. It is the number of little cubes in a big cube that is 4 cubes long, wide and high. Do it in two steps: 4 × 4 = 16, then 16 × 4 = 64.
Factors and multiples. Factors help you multiply. To work out 15 × 14, write 14 as 2 × 7: 15 × 2 = 30, then 30 × 7 = 210. If a number is a multiple of 6, it is also a multiple of 2 and of 3. Mixed problems. Many problems need more than one step, and more than one operation. Draw a bar model, write down what each step finds, and remember that multiplication comes before addition and subtraction. And remember that the equals sign means “is the same value as”, so we can have calculations on both sides.
Key ideas
- Area of a rectangle = length × width. The unit is square units (cm², m²). A missing side is area ÷ the other side.
- Square number: n² = n × n. Cube number: n³ = n × n × n (do it in two steps).
- Use factors to make a multiplication easier: 15 × 14 = 15 × 2 × 7.
- In a multi-step problem, write down what each step finds. Check that you answer the question that was asked.
- The equals sign means “is the same value as”: both sides must balance.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area of a rectangle = length × width. Multiply the two sides.
- Draw the rectangle and label the sides. The area is the space inside. Rows of squares: 5 rows with 8 in each row.
- Write the rule for area. Area of a rectangle = length × width.8 × 5
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.8 = 8
- Multiply each part by 5. Use a times table fact, then put the zeros back.8 × 5 = 40
- Add the answers. Add them one at a time, starting with the biggest.40
- Write the answer with square units. Area is measured in square cm, written cm².40 cm²
Check. Estimate: 8 × 5 = 40. ✓
Watch out. Area is in square units (cm²). Perimeter is a length (cm), so do not mix them up.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area of a rectangle = length × width. Multiply the two sides.
- Draw the rectangle and label the sides. The area is the space inside. Rows of squares: 24 rows with 36 in each row.
- Write the rule for area. Area of a rectangle = length × width.36 × 24
- Split the multiplier. Multiplying by 24 means multiplying by 4 and by 20, then adding. Write the numbers in columns.24 = 20 + 4, so 36 × 24 = (36 × 4) + (36 × 20)
3 6 × 2 4 - Row 1: multiply by the 4 (the ones digit). This is short multiplication, starting with the ones. Work out 36 × 4.Ones: 6 × 4 = 24. Write 4 and carry 2.
Tens: 3 × 4 = 12, and 12 + 2 (carried) = 14. Write 4 and carry 1.
Hundreds: nothing left to multiply. Write the carried 1.
Row 1 = 144 - Row 2: write a 0 first, then multiply by the 2. The 2 is really 20, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 36 by 2.Write 0 in the ones column.
Ones: 6 × 2 = 12. Write 2 and carry 1.
Tens: 3 × 2 = 6, and 6 + 1 (carried) = 7. Write 7.
Row 2 = 72 with a 0 on the end = 720 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
3 6 × 2 4 1 4 4 36 × 4 + 7 2 0 36 × 20 8 6 4
144 + 720 = 864 - Write the answer with square units. Area is measured in square m, written m².864 m²
Check. Estimate: 36 is about 40; 24 is about 20. 40 × 20 = 800. Our answer, 864, is close to 800. ✓
Watch out. Area is in square units (m²). Perimeter is a length (m), so do not mix them up.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area = length × width. Divide the area by the side you know to find the missing side.
- Draw the rectangle. Mark the side we do not know with a ?.
- Write the rule for area. Area = length × width, so one side × the other side = the area.12 × ? = 168
- Use the opposite operation. Divide the area by the side you know.12 × 10 = 120, and 168 − 120 = 48
12 × 4 = 48
10 + 4 = 14, so 168 ÷ 12 = 14 - Write the answer. Include the unit.14 cm
Check. 12 × 14 = 168. ✓
Watch out. The missing side is a length, so the unit is cm (not cm²).
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A square number is a number multiplied by itself. 7² = 7 × 7.
- Write the square as a multiplication. 7² means 7 multiplied by itself: two 7s.7² = 7 × 7
- Picture it as a square of dots. There are 7 rows of 7.
- Work it out. Use your times table facts.7 × 7 = 49
Check. 7 rows of 7: 49. And 49 is a square number. ✓
Watch out. The small 2 means “multiplied by itself”. 7² is not 7 × 2 = 14.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A square number is a number multiplied by itself. 15² = 15 × 15.
- Write the square as a multiplication. 15² means 15 multiplied by itself: two 15s.15² = 15 × 15
- Split the multiplier. Multiplying by 15 means multiplying by 5 and by 10, then adding. Write the numbers in columns.15 = 10 + 5, so 15 × 15 = (15 × 5) + (15 × 10)
1 5 × 1 5 - Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 15 × 5.Ones: 5 × 5 = 25. Write 5 and carry 2.
Tens: 1 × 5 = 5, and 5 + 2 (carried) = 7. Write 7.
Row 1 = 75 - Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 15 by 1.Write 0 in the ones column.
Ones: 5 × 1 = 5. Write 5.
Tens: 1 × 1 = 1. Write 1.
Row 2 = 15 with a 0 on the end = 150 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
1 5 × 1 5 7 5 15 × 5 + 1 5 0 15 × 10 2 2 5
75 + 150 = 225
Check. 15 rows of 15: 225. And 225 is a square number. ✓
Watch out. The small 2 means “multiplied by itself”. 15² is not 15 × 2 = 30.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A cube number is a number multiplied by itself twice. 4³ = 4 × 4 × 4. Do it in two steps.
- Write the cube as a multiplication. 4³ means three 4s multiplied together.4³ = 4 × 4 × 4
- Do the first two numbers. Work in two steps. First multiply the first two numbers.4 × 4 = 16
- Picture it as a cube. 4 layers, each with 4 × 4 = 16 little cubes.
- Multiply the answer by 4 again. There is one more 4 to multiply by.16 × 4
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.16 = 10 + 6
- Multiply each part by 4. Use a times table fact, then put the zeros back.10 × 4 = 40
6 × 4 = 24 - Add the answers. Add them one at a time, starting with the biggest.40 + 24 = 64
- Write the answer. 4³ = 64
Check. 64 ÷ 4 = 16 and 16 ÷ 4 = 4. ✓
Watch out. The small 3 does not mean “times 3”. 4³ is 4 × 4 × 4, not 4 × 3.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A cube number is a number multiplied by itself twice. 6³ = 6 × 6 × 6. Do it in two steps.
- Write the cube as a multiplication. 6³ means three 6s multiplied together.6³ = 6 × 6 × 6
- Do the first two numbers. Work in two steps. First multiply the first two numbers.6 × 6 = 36
- Multiply the answer by 6 again. There is one more 6 to multiply by.36 × 6
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.36 = 30 + 6
- Multiply each part by 6. Use a times table fact, then put the zeros back.30 × 6 = 180
6 × 6 = 36 - Add the answers. Add them one at a time, starting with the biggest.180 + 36 = 216
- Write the answer. 6³ = 216
Check. 216 ÷ 6 = 36 and 36 ÷ 6 = 6. ✓
Watch out. The small 3 does not mean “times 3”. 6³ is 6 × 6 × 6, not 6 × 3.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.
Method. Draw a bar model, choose the operation, then work it out.
- Draw a bar model. A table shows how the amounts match up.
Bunches 1 15 Roses 12 ? - Choose the operation. There are 15 equal bunches of 12 roses. 15 = 3 × 5, so multiply by 3 and then by 5.12 × 15
- Split 15 into factors. A factor pair gives two easier multiplications that make the same answer.15 = 3 × 5
- Multiply by 3. Use the 3 times table and partition if you need to.12 × 3 = 36
- Multiply by 5. Use the 5 times table and partition if you need to.36 × 5 = 30 × 5 + 6 × 5 = 150 + 30 = 180
- Write the answer. Include the unit.180 roses
Check. Estimate: 12 is about 10; 15 is about 20. 10 × 20 = 200. Our answer, 180, is close to 200. ✓
Watch out. Multiply by each factor, one after the other. Do not add the factors: 15 is 3 × 5, not 3 + 5.
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 long bar is all the pencils. The pencils that are not broken are the total less 15.
- Work out how many pencils there are. 6 boxes with 24 pencils in each. Multiply.24 × 6 = 144
- Take away the broken pencils. 15 pencils are broken, so subtract them.144 − 15 = 129
- Write the answer. Include the unit.129 pencils
Check. 129 + 15 = 144. ✓
Watch out. There are two steps. Do not stop after the multiplication.
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. There are two different groups of coaches, so we work out each one.
- Work out the seats in the 12 big coaches. Each holds 53 passengers. Use long multiplication.12 × 53: 53 × 2 = 106 and 53 × 10 = 530
106 + 530 = 636 - Work out the seats in the 9 small coaches. Each holds 49 passengers. Use compensation: 49 is 1 less than 50.9 × 49 = 9 × 50 − 9 = 450 − 9 = 441
- Add the two totals. We want all the seats.636 + 441 = 1077
- Write the answer. Include the unit.1077 seats
Check. Estimate: 12 × 50 = 600 and 9 × 50 = 450, so about 1,050. ✓
Watch out. Check you have used the right number of seats for each size of coach.
Watch out
6² is not 6 × 2. The small 2 means “multiplied by itself”: 6² = 6 × 6 = 36. Also, area is in square units (cm²), but a missing side is a length (cm). In a multi-step problem, check at the end that your answer is the thing the question asked for.Exercise 4G Solving problems with multiplication
Non-calculatorKey words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area of a rectangle = length × width. Multiply the two sides.
- Draw the rectangle and label the sides. The area is the space inside. Rows of squares: 6 rows with 9 in each row.
- Write the rule for area. Area of a rectangle = length × width.9 × 6
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.9 = 9
- Multiply each part by 6. Use a times table fact, then put the zeros back.9 × 6 = 54
- Add the answers. Add them one at a time, starting with the biggest.54
- Write the answer with square units. Area is measured in square cm, written cm².54 cm²
Check. Estimate: 9 × 6 = 54. ✓
Watch out. Area is in square units (cm²). Perimeter is a length (cm), so do not mix them up.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area of a rectangle = length × width. Multiply the two sides.
- Draw the rectangle and label the sides. The area is the space inside. Rows of squares: 12 rows with 15 in each row.
- Write the rule for area. Area of a rectangle = length × width.15 × 12
- Split the multiplier. Multiplying by 12 means multiplying by 2 and by 10, then adding. Write the numbers in columns.12 = 10 + 2, so 15 × 12 = (15 × 2) + (15 × 10)
1 5 × 1 2 - Row 1: multiply by the 2 (the ones digit). This is short multiplication, starting with the ones. Work out 15 × 2.Ones: 5 × 2 = 10. Write 0 and carry 1.
Tens: 1 × 2 = 2, and 2 + 1 (carried) = 3. Write 3.
Row 1 = 30 - Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 15 by 1.Write 0 in the ones column.
Ones: 5 × 1 = 5. Write 5.
Tens: 1 × 1 = 1. Write 1.
Row 2 = 15 with a 0 on the end = 150 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
1 5 × 1 2 3 0 15 × 2 + 1 5 0 15 × 10 1 8 0
30 + 150 = 180 - Write the answer with square units. Area is measured in square m, written m².180 m²
Check. Estimate: 15 is about 20; 12 is about 10. 20 × 10 = 200. Our answer, 180, is close to 200. ✓
Watch out. Area is in square units (m²). Perimeter is a length (m), so do not mix them up.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area of a rectangle = length × width. Multiply the two sides.
- Draw the rectangle and label the sides. The area is the space inside. Rows of squares: 18 rows with 24 in each row.
- Write the rule for area. Area of a rectangle = length × width.24 × 18
- Split the multiplier. Multiplying by 18 means multiplying by 8 and by 10, then adding. Write the numbers in columns.18 = 10 + 8, so 24 × 18 = (24 × 8) + (24 × 10)
2 4 × 1 8 - Row 1: multiply by the 8 (the ones digit). This is short multiplication, starting with the ones. Work out 24 × 8.Ones: 4 × 8 = 32. Write 2 and carry 3.
Tens: 2 × 8 = 16, and 16 + 3 (carried) = 19. Write 9 and carry 1.
Hundreds: nothing left to multiply. Write the carried 1.
Row 1 = 192 - Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 24 by 1.Write 0 in the ones column.
Ones: 4 × 1 = 4. Write 4.
Tens: 2 × 1 = 2. Write 2.
Row 2 = 24 with a 0 on the end = 240 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
2 4 × 1 8 1 9 2 24 × 8 + 2 4 0 24 × 10 4 3 2
192 + 240 = 432 - Write the answer with square units. Area is measured in square cm, written cm².432 cm²
Check. Estimate: 24 is about 20; 18 is about 20. 20 × 20 = 400. Our answer, 432, is close to 400. ✓
Watch out. Area is in square units (cm²). Perimeter is a length (cm), so do not mix them up.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area = length × width. Divide the area by the side you know to find the missing side.
- Draw the rectangle. Mark the side we do not know with a ?.
- Write the rule for area. Area = length × width, so one side × the other side = the area.8 × ? = 96
- Use the opposite operation. Divide the area by the side you know.8 × 10 = 80, and 96 − 80 = 16
8 × 2 = 16
10 + 2 = 12, so 96 ÷ 8 = 12 - Write the answer. Include the unit.12 cm
Check. 8 × 12 = 96. ✓
Watch out. The missing side is a length, so the unit is cm (not cm²).
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
Method. Area = length × width. Divide the area by the side you know to find the missing side.
- Draw the rectangle. Mark the side we do not know with a ?.
- Write the rule for area. Area = length × width, so one side × the other side = the area.14 × ? = 252
- Use the opposite operation. Divide the area by the side you know.14 × 10 = 140, and 252 − 140 = 112
14 × 8 = 112
10 + 8 = 18, so 252 ÷ 14 = 18 - Write the answer. Include the unit.18 cm
Check. 14 × 18 = 252. ✓
Watch out. The missing side is a length, so the unit is cm (not cm²).
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A square number is a number multiplied by itself. 9² = 9 × 9.
- Write the square as a multiplication. 9² means 9 multiplied by itself: two 9s.9² = 9 × 9
- Work it out. Use your times table facts.9 × 9 = 81
Check. 9 rows of 9: 81. And 81 is a square number. ✓
Watch out. The small 2 means “multiplied by itself”. 9² is not 9 × 2 = 18.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A square number is a number multiplied by itself. 12² = 12 × 12.
- Write the square as a multiplication. 12² means 12 multiplied by itself: two 12s.12² = 12 × 12
- Work it out. Use your times table facts.12 × 12 = 144
Check. 12 rows of 12: 144. And 144 is a square number. ✓
Watch out. The small 2 means “multiplied by itself”. 12² is not 12 × 2 = 24.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A square number is a number multiplied by itself. 15² = 15 × 15.
- Write the square as a multiplication. 15² means 15 multiplied by itself: two 15s.15² = 15 × 15
- Split the multiplier. Multiplying by 15 means multiplying by 5 and by 10, then adding. Write the numbers in columns.15 = 10 + 5, so 15 × 15 = (15 × 5) + (15 × 10)
1 5 × 1 5 - Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 15 × 5.Ones: 5 × 5 = 25. Write 5 and carry 2.
Tens: 1 × 5 = 5, and 5 + 2 (carried) = 7. Write 7.
Row 1 = 75 - Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 15 by 1.Write 0 in the ones column.
Ones: 5 × 1 = 5. Write 5.
Tens: 1 × 1 = 1. Write 1.
Row 2 = 15 with a 0 on the end = 150 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
1 5 × 1 5 7 5 15 × 5 + 1 5 0 15 × 10 2 2 5
75 + 150 = 225
Check. 15 rows of 15: 225. And 225 is a square number. ✓
Watch out. The small 2 means “multiplied by itself”. 15² is not 15 × 2 = 30.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A square number is a number multiplied by itself. 20² = 20 × 20.
- Write the square as a multiplication. 20² means 20 multiplied by itself: two 20s.20² = 20 × 20
- Split 20 into 2 × 10. Multiplying by 20 is the same as multiplying by 2 and then by 10.20 = 2 × 10
- Multiply by 2. Use short multiplication.Ones: 0 × 2 = 0. Write 0.
Tens: 2 × 2 = 4. Write 4.
20 × 2 = 40 - Multiply by 10. Move every digit one place to the left and write a 0 place holder.40 × 10 = 400
Check. 20 rows of 20: 400. And 400 is a square number. ✓
Watch out. The small 2 means “multiplied by itself”. 20² is not 20 × 2 = 40.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A cube number is a number multiplied by itself twice. 3³ = 3 × 3 × 3. Do it in two steps.
- Write the cube as a multiplication. 3³ means three 3s multiplied together.3³ = 3 × 3 × 3
- Do the first two numbers. Work in two steps. First multiply the first two numbers.3 × 3 = 9
- Multiply the answer by 3 again. There is one more 3 to multiply by.9 × 3
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.9 = 9
- Multiply each part by 3. Use a times table fact, then put the zeros back.9 × 3 = 27
- Add the answers. Add them one at a time, starting with the biggest.27
- Write the answer. 3³ = 27
Check. 27 ÷ 3 = 9 and 9 ÷ 3 = 3. ✓
Watch out. The small 3 does not mean “times 3”. 3³ is 3 × 3 × 3, not 3 × 3.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A cube number is a number multiplied by itself twice. 5³ = 5 × 5 × 5. Do it in two steps.
- Write the cube as a multiplication. 5³ means three 5s multiplied together.5³ = 5 × 5 × 5
- Do the first two numbers. Work in two steps. First multiply the first two numbers.5 × 5 = 25
- Multiply the answer by 5 again. There is one more 5 to multiply by.25 × 5
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.25 = 20 + 5
- Multiply each part by 5. Use a times table fact, then put the zeros back.20 × 5 = 100
5 × 5 = 25 - Add the answers. Add them one at a time, starting with the biggest.100 + 25 = 125
- Write the answer. 5³ = 125
Check. 125 ÷ 5 = 25 and 25 ÷ 5 = 5. ✓
Watch out. The small 3 does not mean “times 3”. 5³ is 5 × 5 × 5, not 5 × 3.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A cube number is a number multiplied by itself twice. 8³ = 8 × 8 × 8. Do it in two steps.
- Write the cube as a multiplication. 8³ means three 8s multiplied together.8³ = 8 × 8 × 8
- Do the first two numbers. Work in two steps. First multiply the first two numbers.8 × 8 = 64
- Multiply the answer by 8 again. There is one more 8 to multiply by.64 × 8
- Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.64 = 60 + 4
- Multiply each part by 8. Use a times table fact, then put the zeros back.60 × 8 = 480
4 × 8 = 32 - Add the answers. Add them one at a time, starting with the biggest.480 + 32 = 512
- Write the answer. 8³ = 512
Check. 512 ÷ 8 = 64 and 64 ÷ 8 = 8. ✓
Watch out. The small 3 does not mean “times 3”. 8³ is 8 × 8 × 8, not 8 × 3.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
Method. A cube number is a number multiplied by itself twice. 10³ = 10 × 10 × 10. Do it in two steps.
- Write the cube as a multiplication. 10³ means three 10s multiplied together.10³ = 10 × 10 × 10
- Do the first two numbers. Work in two steps. First multiply the first two numbers.10 × 10 = 100
- Multiply the answer by 10 again. There is one more 10 to multiply by.100 × 10
- Say what multiplying by 10 does. Each digit becomes 10 times bigger, so it moves 1 place to the left. The digits move. The decimal point does not.1 hundred becomes 1 thousand
- Move every digit 1 place to the left. Use a place-value chart and follow the arrows.
- Fill the empty places with zeros. The ones column is empty. A zero in it is a place holder: it keeps every digit in its right column.100 × 10 = 1,000
- Write the answer. 10³ = 1,000
Check. 1,000 ÷ 10 = 100 and 100 ÷ 10 = 10. ✓
Watch out. The small 3 does not mean “times 3”. 10³ is 10 × 10 × 10, not 10 × 3.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.
Method. 14 is 2 × 7. Multiply 15 by each factor in turn.
- Split 14 into factors. A factor pair gives two easier multiplications that make the same answer.14 = 2 × 7
- Multiply by 2. Double it.15 × 2 = 10 × 2 + 5 × 2 = 20 + 10 = 30
- Multiply by 7. Use the 7 times table and partition if you need to.30 × 7 = 210
- Write the answer. 15 × 14 = 210
Check. Estimate: 15 is about 20; 14 is about 10. 20 × 10 = 200. Our answer, 210, is close to 200. ✓
Watch out. Multiply by each factor, one after the other. Do not add the factors: 14 is 2 × 7, not 2 + 7.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.
Method. 18 is 2 × 9. Multiply 35 by each factor in turn.
- Split 18 into factors. A factor pair gives two easier multiplications that make the same answer.18 = 2 × 9
- Multiply by 2. Double it.35 × 2 = 30 × 2 + 5 × 2 = 60 + 10 = 70
- Multiply by 9. Use the 9 times table and partition if you need to.70 × 9 = 630
- Write the answer. 35 × 18 = 630
Check. Estimate: 35 is about 40; 18 is about 20. 40 × 20 = 800. Our answer, 630, is a bit less than 800. ✓
Watch out. Multiply by each factor, one after the other. Do not add the factors: 18 is 2 × 9, not 2 + 9.
Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.
Method. 15 is 3 × 5. Multiply 24 by each factor in turn.
- Split 15 into factors. A factor pair gives two easier multiplications that make the same answer.15 = 3 × 5
- Multiply by 3. Use the 3 times table and partition if you need to.24 × 3 = 20 × 3 + 4 × 3 = 60 + 12 = 72
- Multiply by 5. Use the 5 times table and partition if you need to.72 × 5 = 70 × 5 + 2 × 5 = 350 + 10 = 360
- Write the answer. 24 × 15 = 360
Check. Estimate: 24 is about 20; 15 is about 20. 20 × 20 = 400. Our answer, 360, is close to 400. ✓
Watch out. Multiply by each factor, one after the other. Do not add the factors: 15 is 3 × 5, not 3 + 5.
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. It does not mean “the answer comes next”.
Method. The two sides of an equals sign must have the same value. Work out the side you can, then find the missing number that makes the other side match.
- Work out the side with no missing number. The equals sign means both sides have the same value, so we need this value first.8 × 5 = 40
- Say what the other side must be. The other side has to balance, so it must also be worth the same.15 + ? = 40
- Ask the question the missing number answers. Use the inverse (opposite) operation or a fact you know.What do I add to 15 to make 40?
40 − 15 = 25 - Write the missing number and read the sentence back. Both sides must be the same.25 (15 + 25 = 40)
Check. Put 25 in the box: 8 × 5 = 40 and 15 + 25 = 40. Both sides are equal. ✓
Watch out. The equals sign does not mean “now write the answer”. It means “is the same as”, so there can be a calculation on both sides.
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. It does not mean “the answer comes next”.
Method. The two sides of an equals sign must have the same value. Work out the side you can, then find the missing number that makes the other side match.
- Work out the side with no missing number. The equals sign means both sides have the same value, so we need this value first.9 × 4 = 36
- Say what the other side must be. The other side has to balance, so it must also be worth the same.? − 12 = 36
- Ask the question the missing number answers. Use the inverse (opposite) operation or a fact you know.What number, take away 12, leaves 36?
36 + 12 = 48 - Write the missing number and read the sentence back. Both sides must be the same.48 (48 − 12 = 36)
Check. Put 48 in the box: 9 × 4 = 36 and 48 − 12 = 36. Both sides are equal. ✓
Watch out. The equals sign does not mean “now write the answer”. It means “is the same as”, so there can be a calculation on both sides.
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. It does not mean “the answer comes next”.
Method. The two sides of an equals sign must have the same value. Work out the side you can, then find the missing number that makes the other side match.
- Work out the side with no missing number. The equals sign means both sides have the same value, so we need this value first.6 × 7 = 42
- Say what the other side must be. The other side has to balance, so it must also be worth the same.50 − ? = 42
- Ask the question the missing number answers. Use the inverse (opposite) operation or a fact you know.50 take away what leaves 42?
50 − 42 = 8 - Write the missing number and read the sentence back. Both sides must be the same.8 (50 − 8 = 42)
Check. Put 8 in the box: 6 × 7 = 42 and 50 − 8 = 42. Both sides are equal. ✓
Watch out. The equals sign does not mean “now write the answer”. It means “is the same as”, so there can be a calculation on both sides.
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. It does not mean “the answer comes next”.
Method. Work out the value of the side with no missing number. The other side must have the same value.
- Work out the left side. The equals sign means both sides are worth the same. Find the value of the side with no missing number.4 × 9 = 36
- Say what the right side must be. It has to balance the left side.6 × ? = 36
- Ask: how many 6s make 36? Count in steps, or use the inverse operation.36 ÷ 6 = 6
- Read the sentence back. Both sides must be equal.4 × 9 = 6 × 6 (both are 36)
Check. 6 × 6 = 36 and 4 × 9 = 36. ✓
Watch out. If you double one number you must halve the other to keep the same answer. The two sides need the same VALUE, not the same numbers.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width.
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 area is the first thing to find.
- Work out the area of the garden. Area = length × width.14 × 9 = 126 m²
- Work out the cost of the seed. Seed costs £2 for every m², so multiply the area by 2.126 × £2 = £252
- Write the answer. Include the unit.£252
Check. 252 ÷ 2 = 126, and 126 = 14 × 9. ✓
Watch out. Area first, then cost.
Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.
Method. Choose the operation, then work it out.
- Choose the operation. There are 15 rows with 15 tiles in each row. That is 15 × 15 = 15².15 × 15
- Split the multiplier. Multiplying by 15 means multiplying by 5 and by 10, then adding. Write the numbers in columns.15 = 10 + 5, so 15 × 15 = (15 × 5) + (15 × 10)
1 5 × 1 5 - Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 15 × 5.Ones: 5 × 5 = 25. Write 5 and carry 2.
Tens: 1 × 5 = 5, and 5 + 2 (carried) = 7. Write 7.
Row 1 = 75 - Row 2: write a 0 first, then multiply by the 1. The 1 is really 10, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 15 by 1.Write 0 in the ones column.
Ones: 5 × 1 = 5. Write 5.
Tens: 1 × 1 = 1. Write 1.
Row 2 = 15 with a 0 on the end = 150 - Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.
1 5 × 1 5 7 5 15 × 5 + 1 5 0 15 × 10 2 2 5
75 + 150 = 225 - Write the answer. Include the unit.225 tiles
Check. Estimate: 15 is about 20; 15 is about 20. 20 × 20 = 400. Our answer, 225, is a bit less than 400. ✓
Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 1, not by 10, and the answer is ten times too small.
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.
- Count the rolls on the big trays. 6 trays with 24 rolls on each.6 × 24 = 144
- Count the rolls on the small trays. 4 trays with 18 rolls on each.4 × 18 = 72
- Add them. We want all the rolls.144 + 72 = 216
- Write the answer. Include the unit.216 rolls
Check. Estimate: 6 × 25 = 150 and 4 × 20 = 80, so about 230. ✓
Watch out. There are two sizes of tray, so there are two multiplications.
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 empty seats are all the seats less the 135 sold.
- Work out how many seats there are. 25 rows of 32 seats. 25 is a quarter of 100, so multiply by 100 and halve twice.32 × 100 = 3,200, 3,200 ÷ 2 = 1,600, 1,600 ÷ 2 = 800
- Take away the seats that are sold. 135 tickets have been sold, so subtract.800 − 135 = 665
- Write the answer. Include the unit.665 seats
Check. 665 + 135 = 800. ✓
Watch out. There are two steps: multiply, then subtract.
Key words. A square number is a number multiplied by itself: 6² = 6 × 6 = 36. A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64.
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.6² means 6 multiplied by itself.
6 × 6 = 36 - Compare with the answer in the question. Priya wrote 12, which is 6 × 2.
- Find the mistake. Priya multiplied 6 by 2. The small 2 means “squared”, so multiply 6 by itself.
- Write the correct answer. 6² = 36
Check. 6 rows of 6 dots make 36 dots. ✓
Watch out. The small 2 is a power, not a multiplier.
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.4 = 2 × 2 (even).
9 = 3 × 3 (odd). - Look for one that does not work. One example that breaks the rule is called a counter-example.9 is a square number and it is odd.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.A square number is even when the number being squared is even, and odd when it is odd.
- Decide. Sometimes
Check. We found one even and one odd square number. ✓
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.Rectangle 8 by 5: area 40, which is in the 8 times table and the 5 times table.
Rectangle 6 by 4: area 24, in the 6 and 4 times tables. - Explain why. A reason shows what is happening for every number, not just the ones we tried.The area is length × width. So it is a multiple of the length and a multiple of the width.
- Decide. Always
Check. Both areas were multiples of both sides. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.
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 6² 36 4 × 9 36 3 × 13 39 2 × 18 36 - Find the odd one out. Three have the same answer, 36. One is different.3 × 13 = 39
- Write the answer. The odd one out is 3 × 13.
Check. The other three all equal 36. ✓
Watch out. Work out all of them. Do not decide from just the first two.
Key words. A factor pair is two whole numbers that multiply to make a number. The product is the answer to a multiplication. The sum is the answer to an addition.
Method. List every pair of whole numbers that multiply to make the product (the factor pairs). Then find the pair that also has the right sum.
- List the factor pairs. These are the pairs of whole numbers with a product of 72. Start with 1 and work up.72 = 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12, 8 × 9
- Add each pair. We need the pair with a sum of 17.
Pair Sum Sum is 17? 1 × 72 = 72 1 + 72 = 73 no 2 × 36 = 72 2 + 36 = 38 no 3 × 24 = 72 3 + 24 = 27 no 4 × 18 = 72 4 + 18 = 22 no 6 × 12 = 72 6 + 12 = 18 no 8 × 9 = 72 8 + 9 = 17 yes - Write the answer. Only one pair works.The numbers are 8 and 9.
Check. 8 × 9 = 72 and 8 + 9 = 17. ✓
Watch out. Check BOTH clues, not just one.
Key words. Area is the amount of flat space inside a shape. It is measured in square units, such as cm². The area of a rectangle is its length multiplied by its width. The perimeter is the distance all the way around the outside of a shape.
Method. The area is length × width, so the sides are a factor pair of the area. List the pairs, then find the perimeter of each.
- List the factor pairs of the area. Length × width = 60.60 = 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10
- Work out the perimeter for each pair. The perimeter is the distance all the way round: 2 × (length + width).
Sides Area Perimeter Perimeter is 32 cm? 1 cm by 60 cm 1 × 60 = 60 2 × (1 + 60) = 122 cm no 2 cm by 30 cm 2 × 30 = 60 2 × (2 + 30) = 64 cm no 3 cm by 20 cm 3 × 20 = 60 2 × (3 + 20) = 46 cm no 4 cm by 15 cm 4 × 15 = 60 2 × (4 + 15) = 38 cm no 5 cm by 12 cm 5 × 12 = 60 2 × (5 + 12) = 34 cm no 6 cm by 10 cm 6 × 10 = 60 2 × (6 + 10) = 32 cm yes - Write the answer. Only one rectangle fits both clues.The sides are 6 cm and 10 cm.
Check. 6 × 10 = 60 and 2 × (6 + 10) = 32. ✓
Watch out. Use both clues. The area clue gives the pairs; the perimeter clue picks one.