Multiplying and dividing
By the end of this chapter you will be able to
- Use arrays and the signs ×, ÷ and = to show equal groups
- Recall the 3 times table, and use the 2, 5 and 10 times tables to check facts
- Work out the 4 and 8 times tables by doubling
- Share and group, and write an answer with a remainder when some are left over
- Multiply a two-digit number by a one-digit number, using partition or the column method
- Solve correspondence, scaling and missing number problems
You should already know
- Counting in twos, fives and tens, and the 2, 5 and 10 times tables (Year 2)
- Repeated addition and equal groups (Year 2)
- Adding and subtracting numbers with up to three digits (Year 3, Chapter 2)
- Place value of a two-digit and a three-digit number (Year 3, Chapter 1)
Where this leads
- Fractions as parts of a whole, including unit fractions of a set of objects (Year 3, Chapter 4)
- Multiplying and dividing by 10 and 100 in Year 4
- Written short multiplication and division with larger numbers (Year 4)
National Curriculum for England, Key Stage 2 mathematics, Year 3. Content: Recall and use multiplication and division facts for the 3, 4 and 8 times tables, Write and calculate mathematical statements for multiplication and division using the tables known, including two-digit numbers times one-digit numbers, using mental and progressing to formal written methods, Solve problems, including missing number problems, involving multiplication and division, including scaling problems and correspondence problems in which n objects are connected to m objects.
Exercise P Do you remember?
Non-calculatorKey words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count in steps of 2, one step for each group.
- Start at 0. We will count in steps of 2.0
- Count in steps of 2, 5 times. Each step is one more group.2, 4, 6, 8, 10
- Write the answer. The 5th number in the count is the answer.10
Check. 10 ÷ 2 = 5. ✓
Watch out. Say the count out loud. Stop after 5 steps.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count in steps of 10, one step for each group.
- Start at 0. We will count in steps of 10.0
- Count in steps of 10, 4 times. Each step is one more group.10, 20, 30, 40
- Write the answer. The 4th number in the count is the answer.40
Check. 40 ÷ 10 = 4. ✓
Watch out. Say the count out loud. Stop after 4 steps.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 4 groups, and each group has 6.4 groups of 6
- Write it as a multiplication. 4 groups of 6 is 4 × 6.4 × 6
- Count in steps of 6. Each step adds one more group.6, 12, 18, 24
- Write the answer. 24
Check. 6 × 4 gives the same total, 24. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.10 ÷ 2
- Find the biggest multiple of 2 that is not more than 10. Try the 2 times table.2 × 5 = 10
- Check that nothing is left over. The division is exact.10 − 10 = 0
- Write the answer. The answer is the number in each group, plus any remainder.5
Check. 2 × 5 + 0 = 10. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Method. Add the tens, then add the ones, then put the two totals together.
- Add the tens. Tens are added with tens.40 + 30 = 70
- Add the ones. Ones are added with ones: 6 and 8 make 14.6 + 8 = 14
- Put the two totals together. Add the tens total and the ones total.70 + 14 = 84
- Write the answer. 84
Check. 84 − 38 = 46. ✓
Watch out. The 14 ones make 1 ten and 4 ones. Do not write 614.
Equal groups, arrays and the signs
Adding equal groups can take a long time. We can use multiplication instead. Three rows of four cones is four, four and four: 4 + 4 + 4 = 12.
We write this as 3 × 4 = 12. The sign × means "groups of" or "lots of". We read it as "three times four".
An array is a set of objects in rows and columns. This array has 3 equal rows, with 4 dots in each row. Turning an array round gives the same number of dots, so 3 × 4 = 4 × 3.
The sign ÷ means share or group. The sign = means that both sides have the same value.
Key ideas
- × means "groups of" or "lots of". 3 × 4 is 3 groups of 4.
- An array shows equal rows and columns of objects.
- Turning an array round does not change the total: 3 × 4 = 4 × 3.
- = means that both sides have the same value.
Key words. An array shows objects in equal rows and columns. The sign = means both sides have the same value.
Method. Count the rows and the dots in each row, then multiply.
- Count the rows. Each row is one equal group.4 rows
- Count the dots in one row. Every row has the same number of dots.5 dots in each row
- Add the equal groups. Repeated addition gives the total.5 + 5 + 5 + 5 = 20
- Write it as a multiplication. 4 groups of 5 is 4 × 5.4 × 5 = 20
- Write the answer. 20
Check. 20 ÷ 4 = 5, so each of the 4 rows has 5 dots. ✓
Watch out. Every row must have the same number of dots, or it is not an array.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 5 groups, and each group has 6.5 groups of 6
- Write it as a multiplication. 5 groups of 6 is 5 × 6.5 × 6
- Count in steps of 6. Each step adds one more group.6, 12, 18, 24, 30
- Write the answer. 30
Check. 6 × 5 gives the same total, 30. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Try each sign in the box. Only one sign makes the sentence true.
- Read the sentence. The box needs a sign that makes the sentence true.4 [ ] 3 = 12
- Try +. This is not 12.4 + 3 = 7
- Try −. This is not 12.4 − 3 = 1
- Try ×. This works.4 × 3 = 12
- Write the answer. Only one sign makes the sentence true.×
Check. The sentence 4 × 3 = 12 is true. ✓
Watch out. Check each sign before you choose. A sign can give a number close to the answer without being right.
Watch out
Watch out: 3 × 4 means 3 groups of 4, which is 12. It does not mean 3 + 4, which is 7.
Exam tip
Draw a quick array when you are not sure. Count the rows, then count the dots in one row.
Exercise 3A Equal groups, arrays and the signs
Non-calculatorKey words. An array shows objects in equal rows and columns. The sign = means both sides have the same value.
Method. Count the rows and the dots in each row, then multiply.
- Count the rows. Each row is one equal group.2 rows
- Count the dots in one row. Every row has the same number of dots.6 dots in each row
- Add the equal groups. Repeated addition gives the total.6 + 6 = 12
- Write it as a multiplication. 2 groups of 6 is 2 × 6.2 × 6 = 12
- Write the answer. 12
Check. 12 ÷ 2 = 6, so each of the 2 rows has 6 dots. ✓
Watch out. Every row must have the same number of dots, or it is not an array.
Key words. An array shows objects in equal rows and columns. The sign = means both sides have the same value.
Method. Count the rows and the dots in each row, then multiply.
- Count the rows. Each row is one equal group.5 rows
- Count the dots in one row. Every row has the same number of dots.2 dots in each row
- Add the equal groups. Repeated addition gives the total.2 + 2 + 2 + 2 + 2 = 10
- Write it as a multiplication. 5 groups of 2 is 5 × 2.5 × 2 = 10
- Write the answer. 10
Check. 10 ÷ 5 = 2, so each of the 5 rows has 2 dots. ✓
Watch out. Every row must have the same number of dots, or it is not an array.
Key words. An array shows objects in equal rows and columns. The sign = means both sides have the same value.
Method. Count the rows and the dots in each row, then multiply.
- Count the rows. Each row is one equal group.3 rows
- Count the dots in one row. Every row has the same number of dots.7 dots in each row
- Add the equal groups. Repeated addition gives the total.7 + 7 + 7 = 21
- Write it as a multiplication. 3 groups of 7 is 3 × 7.3 × 7 = 21
- Write the answer. 21
Check. 21 ÷ 3 = 7, so each of the 3 rows has 7 dots. ✓
Watch out. Every row must have the same number of dots, or it is not an array.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 3 groups, and each group has 5.3 groups of 5
- Write it as a multiplication. 3 groups of 5 is 3 × 5.3 × 5
- Count in steps of 5. Each step adds one more group.5, 10, 15
- Write the answer. 15
Check. 5 × 3 gives the same total, 15. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 6 groups, and each group has 4.6 groups of 4
- Write it as a multiplication. 6 groups of 4 is 6 × 4.6 × 4
- Count in steps of 4. Each step adds one more group.4, 8, 12, 16, 20, 24
- Write the answer. 24
Check. 4 × 6 gives the same total, 24. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 4 groups, and each group has 10.4 groups of 10
- Write it as a multiplication. 4 groups of 10 is 4 × 10.4 × 10
- Count in steps of 10. Each step adds one more group.10, 20, 30, 40
- Write the answer. 40
Check. 10 × 4 gives the same total, 40. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 6 groups, and each group has 2.6 groups of 2
- Write it as a multiplication. 6 groups of 2 is 6 × 2.6 × 2
- Count in steps of 2. Each step adds one more group.2, 4, 6, 8, 10, 12
- Write the answer. 12
Check. 2 × 6 gives the same total, 12. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 5 groups, and each group has 3.5 groups of 3
- Write it as a multiplication. 5 groups of 3 is 5 × 3.5 × 3
- Count in steps of 3. Each step adds one more group.3, 6, 9, 12, 15
- Write the answer. 15
Check. 3 × 5 gives the same total, 15. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Try each sign in the box. Only one sign makes the sentence true.
- Read the sentence. The box needs a sign that makes the sentence true.3 [ ] 5 = 15
- Try +. This is not 15.3 + 5 = 8
- Try −. This is not 15.3 − 5 = -2
- Try ×. This works.3 × 5 = 15
- Write the answer. Only one sign makes the sentence true.×
Check. The sentence 3 × 5 = 15 is true. ✓
Watch out. Check each sign before you choose. A sign can give a number close to the answer without being right.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Try each sign in the box. Only one sign makes the sentence true.
- Read the sentence. The box needs a sign that makes the sentence true.12 [ ] 3 = 4
- Try +. This is not 4.12 + 3 = 15
- Try −. This is not 4.12 − 3 = 9
- Try ×. This is not 4.12 × 3 = 36
- Try ÷. This works.12 ÷ 3 = 4
- Write the answer. Only one sign makes the sentence true.÷
Check. The sentence 12 ÷ 3 = 4 is true. ✓
Watch out. Check each sign before you choose. A sign can give a number close to the answer without being right.
Key words. An array shows objects in equal rows and columns. The sign = means both sides have the same value.
Method. Turn the array round and count the dots again.
- Look at the array. 3 rows, and 4 dots in each row.3 × 4 = 12
- Turn the array round. Now there are 4 rows, and 3 dots in each row.4 rows of 3
- Write the multiplication. The number of dots has not changed.4 × 3 = 12
- Write the answer. 4 × 3 = 12
Check. Both arrays have 12 dots. ✓
Watch out. Turning the array round gives a different multiplication with the same answer.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count the equal groups, write them as a multiplication, then work out the total.
- Count the equal groups. There are 6 groups, and each group has 3.6 groups of 3
- Write it as a multiplication. 6 groups of 3 is 6 × 3.6 × 3
- Count in steps of 3. Each step adds one more group.3, 6, 9, 12, 15, 18
- Write the answer. 18
Check. 3 × 6 gives the same total, 18. ✓
Watch out. Repeated addition and multiplication give the same answer. Multiplication is quicker for many groups.
Key words. An array shows objects in equal rows and columns. The sign = means both sides have the same value.
Method. Turn the array round and count the dots again.
- Count the dots in 3 rows of 4. Each row has 4 dots.3 × 4 = 12
- Turn the array round to 4 rows of 3. The same dots are now in 4 rows.4 × 3 = 12
- Compare the two answers. The number of dots is the same, so the answers are the same.12 = 12
- Decide. Sam is not right. The order of the numbers does not change the answer.No, both are 12
Check. 3 × 4 = 4 × 3 = 12. ✓
Watch out. Changing the order of the numbers does not change the product.
The 3 times table and the 2, 5 and 10 facts
The 3 times table gives the answer to 3 × a number. Count in threes to find the answer: 3, 6, 9, 12, 15, and so on.
You do not have to count every time. Use a fact you know, and add one more group of 3. For example, 3 × 6 = 18, so 3 × 7 = 18 + 3 = 21.
The 2, 5 and 10 times tables are useful checks. Multiples of 2 are even numbers. Multiples of 5 end in 0 or 5. Multiples of 10 end in 0.
Key ideas
- 3 × 4 means 3 groups of 4, or 4 groups of 3.
- Count in threes, or use a fact you know and add one more group of 3.
- Turn the numbers round: 5 × 3 is the same as 3 × 5.
- Use the 2, 5 and 10 tables to check: 10 × 3 = 30 and 2 × 9 = 18.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 6. We know 6 groups of 3.3 × 6 = 18
- Add one more group of 3. One more group makes 7 groups.18 + 3 = 21
- Write the answer. 21
Check. 21 ÷ 3 = 7. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Turn the numbers round, then count in steps of 3.
- Turn the numbers round. Changing the order does not change the answer.5 × 3 = 3 × 5
- Count in steps of 3, 5 times. Each step is one group of 3.3, 6, 9, 12, 15
- Write the answer. 15
Check. 15 ÷ 5 = 3. ✓
Watch out. The order of the numbers can change, but the answer stays the same.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count in steps of 3, one step for each group.
- Start at 0. We will count in steps of 3.0
- Count in steps of 3, 9 times. Each step is one more group.3, 6, 9, 12, 15, 18, 21, 24, 27
- Write the answer. The 9th number in the count is the answer.27
Check. 27 ÷ 3 = 9. ✓
Watch out. Say the count out loud. Stop after 9 steps.
Watch out
Watch out: 3 × 4 is 12. It is not 3 + 4, which is 7. When you multiply, count groups; do not add the numbers.
Exam tip
Check a multiplication by using a second fact. If 3 × 6 is 18, then 3 × 7 must be 3 more, which is 21.
Exercise 3B The 3 times table and the 2, 5 and 10 facts
Non-calculatorKey words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Count in steps of 3, one step for each group.
- Start at 0. We will count in steps of 3.0
- Count in steps of 3, 2 times. Each step is one more group.3, 6
- Write the answer. The 2th number in the count is the answer.6
Check. 6 ÷ 3 = 2. ✓
Watch out. Say the count out loud. Stop after 2 steps.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 4. We know 4 groups of 3.3 × 4 = 12
- Add one more group of 3. One more group makes 5 groups.12 + 3 = 15
- Write the answer. 15
Check. 15 ÷ 3 = 5. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 7. We know 7 groups of 3.3 × 7 = 21
- Add one more group of 3. One more group makes 8 groups.21 + 3 = 24
- Write the answer. 24
Check. 24 ÷ 3 = 8. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 10. We know 10 groups of 3.3 × 10 = 30
- Add one more group of 3. One more group makes 11 groups.30 + 3 = 33
- Write the answer. 33
Check. 33 ÷ 3 = 11. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 11. We know 11 groups of 3.3 × 11 = 33
- Add one more group of 3. One more group makes 12 groups.33 + 3 = 36
- Write the answer. 36
Check. 36 ÷ 3 = 12. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 2. We know 2 groups of 3.3 × 2 = 6
- Add one more group of 3. One more group makes 3 groups.6 + 3 = 9
- Write the answer. 9
Check. 9 ÷ 3 = 3. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Turn the numbers round, then count in steps of 3.
- Turn the numbers round. Changing the order does not change the answer.4 × 3 = 3 × 4
- Count in steps of 3, 4 times. Each step is one group of 3.3, 6, 9, 12
- Write the answer. 12
Check. 12 ÷ 4 = 3. ✓
Watch out. The order of the numbers can change, but the answer stays the same.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 5. We know 5 groups of 3.3 × 5 = 15
- Add one more group of 3. One more group makes 6 groups.15 + 3 = 18
- Write the answer. 18
Check. 18 ÷ 3 = 6. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 10.
- Use a fact you know: 10 × 3. We know 3 groups of 10.10 × 3 = 30
- Add one more group of 10. One more group makes 4 groups.30 + 10 = 40
- Write the answer. 40
Check. 40 ÷ 10 = 4. ✓
Watch out. Adding one more group of 10 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 5.
- Use a fact you know: 5 × 6. We know 6 groups of 5.5 × 6 = 30
- Add one more group of 5. One more group makes 7 groups.30 + 5 = 35
- Write the answer. 35
Check. 35 ÷ 5 = 7. ✓
Watch out. Adding one more group of 5 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 2.
- Use a fact you know: 2 × 8. We know 8 groups of 2.2 × 8 = 16
- Add one more group of 2. One more group makes 9 groups.16 + 2 = 18
- Write the answer. 18
Check. 18 ÷ 2 = 9. ✓
Watch out. Adding one more group of 2 is quicker than starting again.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 8. We know 8 groups of 3.3 × 8 = 24
- Add one more group of 3. One more group makes 9 groups.24 + 3 = 27
- Write the answer. 27
Check. 27 ÷ 3 = 9. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.3 × [ ] = 21
- Use the inverse: divide the total by the number you know. Division undoes multiplication.21 ÷ 3 = 7
- Check by multiplying. The check must give the total.3 × 7 = 21 ✓
- Write the answer. 7
Check. 3 × 7 = 21. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.[ ] × 3 = 30
- Use the inverse: divide the total by the number you know. Division undoes multiplication.30 ÷ 3 = 10
- Check by multiplying. The check must give the total.3 × 10 = 30 ✓
- Write the answer. 10
Check. 3 × 10 = 30. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for 3 × 5, then add one more group of 3.
- Work out 3 × 5. We know this fact from the table.3 × 5 = 15
- Add one more group of 3. Mia added only 1, but one more group is 3.15 + 3 = 18
- Compare. The answer is 18, not 16.18 ≠ 16
- Decide. Mia is not right.No, 3 × 6 = 18
Check. 3 × 6 = 18. ✓
Watch out. One more group of 3 adds 3, not 1.
The 4 and 8 times tables
The 4 times table is double the 2 times table. To find 4 × 6, first find 2 × 6 = 12. Then double it: 4 × 6 = 24.
The 8 times table is double the 4 times table. To find 8 × 6, double 4 × 6 = 24 to get 48.
This bar model shows 3 equal groups of 4.
Key ideas
- To find 4 × a number, double the 2 times table fact.
- To find 8 × a number, double the 4 times table fact.
- Double 12 is 24, and double 24 is 48.
- Use the tables to check each other: 8 × 3 = 24 and 4 × 6 = 24.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 6. We know this fact.2 × 6 = 12
- Double it to get 4 × 6. Doubling means adding the number to itself.12 + 12 = 24
- Write the answer. 24
Check. 24 ÷ 4 = 6. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 5. We know this fact.2 × 5 = 10
- Double it to get 4 × 5. Doubling means adding the number to itself.10 + 10 = 20
- Double again to get 8 × 5. Doubling the 4 times table gives the 8 times table.20 + 20 = 40
- Write the answer. 40
Check. 40 ÷ 8 = 5. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 7. We know this fact.2 × 7 = 14
- Double it to get 4 × 7. Doubling means adding the number to itself.14 + 14 = 28
- Double again to get 8 × 7. Doubling the 4 times table gives the 8 times table.28 + 28 = 56
- Write the answer. 56
Check. 56 ÷ 8 = 7. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Watch out
Watch out: 8 × 4 is not 8 + 4. It means 8 groups of 4, which is 32. Adding 8 + 4 gives only 12.
Exam tip
Write each double on paper, so you can check each step. Every double is exactly twice the number before it.
Exercise 3C The 4 and 8 times tables
Non-calculatorKey words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 3. We know this fact.2 × 3 = 6
- Double it to get 4 × 3. Doubling means adding the number to itself.6 + 6 = 12
- Write the answer. 12
Check. 12 ÷ 4 = 3. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 5. We know this fact.2 × 5 = 10
- Double it to get 4 × 5. Doubling means adding the number to itself.10 + 10 = 20
- Write the answer. 20
Check. 20 ÷ 4 = 5. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 9. We know this fact.2 × 9 = 18
- Double it to get 4 × 9. Doubling means adding the number to itself.18 + 18 = 36
- Write the answer. 36
Check. 36 ÷ 4 = 9. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 12. We know this fact.2 × 12 = 24
- Double it to get 4 × 12. Doubling means adding the number to itself.24 + 24 = 48
- Write the answer. 48
Check. 48 ÷ 4 = 12. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 2. We know this fact.2 × 2 = 4
- Double it to get 4 × 2. Doubling means adding the number to itself.4 + 4 = 8
- Double again to get 8 × 2. Doubling the 4 times table gives the 8 times table.8 + 8 = 16
- Write the answer. 16
Check. 16 ÷ 8 = 2. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 6. We know this fact.2 × 6 = 12
- Double it to get 4 × 6. Doubling means adding the number to itself.12 + 12 = 24
- Double again to get 8 × 6. Doubling the 4 times table gives the 8 times table.24 + 24 = 48
- Write the answer. 48
Check. 48 ÷ 8 = 6. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 9. We know this fact.2 × 9 = 18
- Double it to get 4 × 9. Doubling means adding the number to itself.18 + 18 = 36
- Double again to get 8 × 9. Doubling the 4 times table gives the 8 times table.36 + 36 = 72
- Write the answer. 72
Check. 72 ÷ 8 = 9. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 3. We know this fact.2 × 3 = 6
- Double it to get 4 × 3. Doubling means adding the number to itself.6 + 6 = 12
- Double again to get 8 × 3. Doubling the 4 times table gives the 8 times table.12 + 12 = 24
- Write the answer. 24
Check. 24 ÷ 8 = 3. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 4. We know this fact.2 × 4 = 8
- Double it to get 4 × 4. Doubling means adding the number to itself.8 + 8 = 16
- Double again to get 8 × 4. Doubling the 4 times table gives the 8 times table.16 + 16 = 32
- Write the answer. 32
Check. 32 ÷ 8 = 4. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 11. We know this fact.2 × 11 = 22
- Double it to get 4 × 11. Doubling means adding the number to itself.22 + 22 = 44
- Double again to get 8 × 11. Doubling the 4 times table gives the 8 times table.44 + 44 = 88
- Write the answer. 88
Check. 88 ÷ 8 = 11. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 7. We know this fact.2 × 7 = 14
- Double it to get 4 × 7. Doubling means adding the number to itself.14 + 14 = 28
- Write the answer. 28
Check. 28 ÷ 4 = 7. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 11. We know this fact.2 × 11 = 22
- Double it to get 4 × 11. Doubling means adding the number to itself.22 + 22 = 44
- Write the answer. 44
Check. 44 ÷ 4 = 11. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Work out 4 × 8 by doubling, then compare it with 36.
- Work out 2 × 8. We know this fact from the table.2 × 8 = 16
- Double to get 4 × 8. Double the answer.16 + 16 = 32
- Compare with 36. Ruby used the answer to 4 × 9 instead.32 ≠ 36
- Decide. Ruby is not right.No, 4 × 8 = 32
Check. 4 × 9 = 36, which is one more group of 4 than 4 × 8. ✓
Watch out. 4 × 9 is 4 more than 4 × 8, so the two answers are different.
Dividing: sharing and grouping
Division can mean sharing or grouping. Sharing asks: how many does each person get? Grouping asks: how many groups can we make?
Sharing 12 stickers between 3 friends gives 4 each, because 3 × 4 = 12. We write 12 ÷ 3 = 4.
Sometimes the numbers do not share exactly. The number left over is the remainder. For 14 ÷ 3, the biggest multiple of 3 that is not more than 14 is 3 × 4 = 12. So 14 ÷ 3 = 4 remainder 2, because 14 − 12 = 2.
The remainder is always smaller than the number you divide by.
Key ideas
- Sharing asks: how many does each person get?
- Grouping asks: how many groups can we make?
- Use a multiplication fact to help: 3 × 4 = 12, so 12 ÷ 3 = 4.
- A remainder is left over. It is always smaller than the number you divide by.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.12 ÷ 3
- Find the biggest multiple of 3 that is not more than 12. Try the 3 times table.3 × 4 = 12
- Check that nothing is left over. The division is exact.12 − 12 = 0
- Write the answer. The answer is the number in each group, plus any remainder.4
Check. 3 × 4 + 0 = 12. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 4 can be made from 20.20 ÷ 4
- Find the biggest multiple of 4 that is not more than 20. Try the 4 times table.4 × 5 = 20
- Check that nothing is left over. The division is exact.20 − 20 = 0
- Write the answer. The answer is the number in each group, plus any remainder.5
Check. 4 × 5 + 0 = 20. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.17 ÷ 3
- Find the biggest multiple of 3 that is not more than 17. Try the 3 times table.3 × 5 = 15
- Find what is left over. 2 is left over, and 2 is smaller than 3.17 − 15 = 2
- Write the answer. The answer is the number in each group, plus any remainder.5 remainder 2
Check. 3 × 5 + 2 = 17. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Watch out
Watch out: 14 ÷ 3 is 4 remainder 2, not 4 remainder 3. Check it: 3 × 4 = 12, and 12 + 2 = 14. If the remainder is as big as the number you divide by, you can make one more group.
Exam tip
When some are left over, write the answer as "4 remainder 2". The remainder is a whole number, not a fraction.
Exercise 3D Dividing: sharing and grouping
Non-calculatorKey words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.15 ÷ 3
- Find the biggest multiple of 3 that is not more than 15. Try the 3 times table.3 × 5 = 15
- Check that nothing is left over. The division is exact.15 − 15 = 0
- Write the answer. The answer is the number in each group, plus any remainder.5
Check. 3 × 5 + 0 = 15. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 5 can be made from 35.35 ÷ 5
- Find the biggest multiple of 5 that is not more than 35. Try the 5 times table.5 × 7 = 35
- Check that nothing is left over. The division is exact.35 − 35 = 0
- Write the answer. The answer is the number in each group, plus any remainder.7
Check. 5 × 7 + 0 = 35. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.18 ÷ 2
- Find the biggest multiple of 2 that is not more than 18. Try the 2 times table.2 × 9 = 18
- Check that nothing is left over. The division is exact.18 − 18 = 0
- Write the answer. The answer is the number in each group, plus any remainder.9
Check. 2 × 9 + 0 = 18. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.21 ÷ 3
- Find the biggest multiple of 3 that is not more than 21. Try the 3 times table.3 × 7 = 21
- Check that nothing is left over. The division is exact.21 − 21 = 0
- Write the answer. The answer is the number in each group, plus any remainder.7
Check. 3 × 7 + 0 = 21. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.24 ÷ 8
- Find the biggest multiple of 8 that is not more than 24. Try the 8 times table.8 × 3 = 24
- Check that nothing is left over. The division is exact.24 − 24 = 0
- Write the answer. The answer is the number in each group, plus any remainder.3
Check. 8 × 3 + 0 = 24. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 4 can be made from 32.32 ÷ 4
- Find the biggest multiple of 4 that is not more than 32. Try the 4 times table.4 × 8 = 32
- Check that nothing is left over. The division is exact.32 − 32 = 0
- Write the answer. The answer is the number in each group, plus any remainder.8
Check. 4 × 8 + 0 = 32. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.14 ÷ 3
- Find the biggest multiple of 3 that is not more than 14. Try the 3 times table.3 × 4 = 12
- Find what is left over. 2 is left over, and 2 is smaller than 3.14 − 12 = 2
- Write the answer. The answer is the number in each group, plus any remainder.4 remainder 2
Check. 3 × 4 + 2 = 14. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 4 can be made from 22.22 ÷ 4
- Find the biggest multiple of 4 that is not more than 22. Try the 4 times table.4 × 5 = 20
- Find what is left over. 2 is left over, and 2 is smaller than 4.22 − 20 = 2
- Write the answer. The answer is the number in each group, plus any remainder.5 remainder 2
Check. 4 × 5 + 2 = 22. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 8 can be made from 26.26 ÷ 8
- Find the biggest multiple of 8 that is not more than 26. Try the 8 times table.8 × 3 = 24
- Find what is left over. 2 is left over, and 2 is smaller than 8.26 − 24 = 2
- Write the answer. The answer is the number in each group, plus any remainder.3 remainder 2
Check. 8 × 3 + 2 = 26. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 5 can be made from 25.25 ÷ 5
- Find the biggest multiple of 5 that is not more than 25. Try the 5 times table.5 × 5 = 25
- Check that nothing is left over. The division is exact.25 − 25 = 0
- Write the answer. The answer is the number in each group, plus any remainder.5
Check. 5 × 5 + 0 = 25. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 4 can be made from 27.27 ÷ 4
- Find the biggest multiple of 4 that is not more than 27. Try the 4 times table.4 × 6 = 24
- Find what is left over. 3 is left over, and 3 is smaller than 4.27 − 24 = 3
- Write the answer. The answer is the number in each group, plus any remainder.6 remainder 3
Check. 4 × 6 + 3 = 27. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 10 can be made from 40.40 ÷ 10
- Find the biggest multiple of 10 that is not more than 40. Try the 10 times table.10 × 4 = 40
- Check that nothing is left over. The division is exact.40 − 40 = 0
- Write the answer. The answer is the number in each group, plus any remainder.4
Check. 10 × 4 + 0 = 40. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of 3 that is not more than 29, then find what is left over.
- Work out 3 × 9. Try the 3 times table.3 × 9 = 27
- Find what is left over. 29 is 2 more than 27.29 − 27 = 2
- Compare. The remainder is 2, which is not 0. So 29 is not exactly 9 threes.2 ≠ 0
- Decide. Ben is not right. There are 2 left over.No, it is 9 remainder 2
Check. 3 × 9 + 2 = 29. ✓
Watch out. Use the multiplication fact first, then check what is left over.
Multiplying a two-digit number by a one-digit number
A two-digit number has tens and ones. In 23 there are 2 tens and 3 ones, so 23 = 20 + 3.
Method 1: partition. Split 23 into 20 and 3. Multiply each part by 4, then add: 4 × 20 = 80 and 4 × 3 = 12. So 4 × 23 = 80 + 12 = 92.
Method 2: the column method. Start with the ones. Multiply the ones and write the ones digit. If the answer is ten or more, carry the tens to the next column. Then multiply the tens and add the carried number.
Both methods give the same answer. You can also estimate first: 4 × 23 is about 4 × 20 = 80, so an answer of 92 is sensible.
Key ideas
- Partition: split the two-digit number into tens and ones, then multiply each part.
- Column method: multiply the ones first, then the tens.
- If a product is ten or more, write the ones digit and carry the tens to the next column.
- Estimate first: 4 × 23 is about 4 × 20 = 80, so the answer must be close to 80.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Split the two-digit number into tens and ones, multiply each part, then add.
- Split the number into tens and ones. Multiplying the parts is easier than multiplying the whole number.23 = 20 + 3
- Multiply the tens. Each ten is worth ten ones.4 × 20 = 80
- Multiply the ones. Each one is worth one.4 × 3 = 12
- Add the two parts. Put the two answers together.80 + 12 = 92
- Write the answer. 92
Check. 92 ÷ 4 = 23. ✓
Watch out. Do not forget to multiply the ones. Only the tens and ones add to the answer.
| Hundreds | Tens | Ones | |
| 3 | 7 | ||
| × | 3 | ||
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.37 × 3
- Multiply the ones first. Write 1 and carry 2 to the tens.7 × 3 = 21
- Multiply the tens. Write 1 and carry 1 to the hundreds.3 × 3 = 9, + 2 carried = 11
- Write the carried number in the next column. There is nothing left to multiply, so the carry is written.write 1
- Write the answer. Read the digits from the bottom line.111
Check. 111 ÷ 3 = 37. ✓
Watch out. Add the carried number before you write the next digit.
| Tens | Ones | |
| 1 | 5 | |
| × | 6 | |
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.15 × 6
- Multiply the ones first. Write 0 and carry 3 to the tens.5 × 6 = 30
- Multiply the tens. Write 9. Nothing is carried.1 × 6 = 6, + 3 carried = 9
- Write the answer. Read the digits from the bottom line.90
Check. 90 ÷ 6 = 15. ✓
Watch out. Add the carried number before you write the next digit.
Watch out
Watch out: 4 × 23 is not 80 + 3 = 83. The ones must be multiplied too: 4 × 3 = 12. Also, 3 × 37 is 111, not 19. When you multiply the tens, add the carried 2 first: 3 × 3 = 9, and 9 + 2 = 11.
Exam tip
Estimate before you start, so you know whether your answer is sensible. Check each carry with the table facts you know.
Exercise 3E Multiplying a two-digit number by a one-digit number
Non-calculatorKey words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Split the two-digit number into tens and ones, multiply each part, then add.
- Split the number into tens and ones. Multiplying the parts is easier than multiplying the whole number.14 = 10 + 4
- Multiply the tens. Each ten is worth ten ones.2 × 10 = 20
- Multiply the ones. Each one is worth one.2 × 4 = 8
- Add the two parts. Put the two answers together.20 + 8 = 28
- Write the answer. 28
Check. 28 ÷ 2 = 14. ✓
Watch out. Do not forget to multiply the ones. Only the tens and ones add to the answer.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Split the two-digit number into tens and ones, multiply each part, then add.
- Split the number into tens and ones. Multiplying the parts is easier than multiplying the whole number.12 = 10 + 2
- Multiply the tens. Each ten is worth ten ones.5 × 10 = 50
- Multiply the ones. Each one is worth one.5 × 2 = 10
- Add the two parts. Put the two answers together.50 + 10 = 60
- Write the answer. 60
Check. 60 ÷ 5 = 12. ✓
Watch out. Do not forget to multiply the ones. Only the tens and ones add to the answer.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Split the two-digit number into tens and ones, multiply each part, then add.
- Split the number into tens and ones. Multiplying the parts is easier than multiplying the whole number.26 = 20 + 6
- Multiply the tens. Each ten is worth ten ones.4 × 20 = 80
- Multiply the ones. Each one is worth one.4 × 6 = 24
- Add the two parts. Put the two answers together.80 + 24 = 104
- Write the answer. 104
Check. 104 ÷ 4 = 26. ✓
Watch out. Do not forget to multiply the ones. Only the tens and ones add to the answer.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.49 × 3
- Multiply the ones first. Write 7 and carry 2 to the tens.9 × 3 = 27
- Multiply the tens. Write 4 and carry 1 to the hundreds.4 × 3 = 12, + 2 carried = 14
- Write the carried number in the next column. There is nothing left to multiply, so the carry is written.write 1
- Write the answer. Read the digits from the bottom line.147
Check. 147 ÷ 3 = 49. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.13 × 8
- Multiply the ones first. Write 4 and carry 2 to the tens.3 × 8 = 24
- Multiply the tens. Write 0 and carry 1 to the hundreds.1 × 8 = 8, + 2 carried = 10
- Write the carried number in the next column. There is nothing left to multiply, so the carry is written.write 1
- Write the answer. Read the digits from the bottom line.104
Check. 104 ÷ 8 = 13. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.27 × 6
- Multiply the ones first. Write 2 and carry 4 to the tens.7 × 6 = 42
- Multiply the tens. Write 6 and carry 1 to the hundreds.2 × 6 = 12, + 4 carried = 16
- Write the carried number in the next column. There is nothing left to multiply, so the carry is written.write 1
- Write the answer. Read the digits from the bottom line.162
Check. 162 ÷ 6 = 27. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.46 × 2
- Multiply the ones first. Write 2 and carry 1 to the tens.6 × 2 = 12
- Multiply the tens. Write 9. Nothing is carried.4 × 2 = 8, + 1 carried = 9
- Write the answer. Read the digits from the bottom line.92
Check. 92 ÷ 2 = 46. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.21 × 3
- Multiply the ones first. Write 3. Nothing is carried.1 × 3 = 3
- Multiply the tens. Write 6. Nothing is carried.2 × 3 = 6
- Write the answer. Read the digits from the bottom line.63
Check. 63 ÷ 3 = 21. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.18 × 4
- Multiply the ones first. Write 2 and carry 3 to the tens.8 × 4 = 32
- Multiply the tens. Write 7. Nothing is carried.1 × 4 = 4, + 3 carried = 7
- Write the answer. Read the digits from the bottom line.72
Check. 72 ÷ 4 = 18. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.16 × 5
- Multiply the ones first. Write 0 and carry 3 to the tens.6 × 5 = 30
- Multiply the tens. Write 8. Nothing is carried.1 × 5 = 5, + 3 carried = 8
- Write the answer. Read the digits from the bottom line.80
Check. 80 ÷ 5 = 16. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.12 × 7
- Multiply the ones first. Write 4 and carry 1 to the tens.2 × 7 = 14
- Multiply the tens. Write 8. Nothing is carried.1 × 7 = 7, + 1 carried = 8
- Write the answer. Read the digits from the bottom line.84
Check. 84 ÷ 7 = 12. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.15 × 4
- Multiply the ones first. Write 0 and carry 2 to the tens.5 × 4 = 20
- Multiply the tens. Write 6. Nothing is carried.1 × 4 = 4, + 2 carried = 6
- Write the answer. Read the digits from the bottom line.60
Check. 60 ÷ 4 = 15. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Work out 4 × 20 and 4 × 5, then add the two answers.
- Work out 4 × 20. Multiply the tens part.4 × 20 = 80
- Work out 4 × 5. Multiply the ones part.4 × 5 = 20
- Add the two parts. Put the two answers together.80 + 20 = 100
- Write the answer. 100
Check. 100 ÷ 4 = 25. ✓
Watch out. Partition the 25 into 20 and 5. Multiply each part by 4, then add.
Correspondence, scaling and missing numbers
Some problems connect two groups of things. Each item in one group can go with each item in the other group. These are correspondence problems.
Make a grid or an array to count the pairs. Each dot below is one outfit. There are 3 shirts, and each one goes with 4 pairs of shorts: 3 × 4 = 12 outfits.
Scaling problems compare two amounts. "Three times as many" means multiply by 3.
Missing numbers can be found with the inverse. In [ ] × 8 = 40, the box is 40 ÷ 8 = 5.
Key ideas
- Correspondence: multiply the number in one group by the number in the other group.
- Adding gives the wrong answer for pairs. 3 + 4 = 7, but there are 12 outfits.
- "Three times as many" means multiply by 3.
- Missing numbers: use the inverse. [ ] × 4 = 20 means 20 ÷ 4.
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 5 sandwiches and 2 drinks.5 sandwiches and 2 drinks
- Pair each item with each item in the other group. Each of the 5 sandwiches can go with each of the 2 drinks.5 × 2 = 10
- Write the answer. The answer is the number of different pairs.10
Check. Count one row at a time: 2 + 2 + 2 + 2 + 2 = 10. ✓
Watch out. Adding gives 7, which is not the number of pairs (10).
Key words. Times as many means multiply. "Three times as many" means 3 groups of the first amount.
Method. 3 times as many means multiply by 3.
- Find the starting amount. 5 is the amount we start with.5
- Multiply by 3. 3 times as many means 3 groups of 5.5 × 3 = 15
- Write the answer. 15
Check. 15 ÷ 3 = 5. ✓
Watch out. Times as many means multiply. Do not add.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.[ ] × 8 = 40
- Use the inverse: divide the total by the number you know. Division undoes multiplication.40 ÷ 8 = 5
- Check by multiplying. The check must give the total.8 × 5 = 40 ✓
- Write the answer. 5
Check. 8 × 5 = 40. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Watch out
Watch out: 3 shirts and 4 pairs of shorts make 12 outfits, not 7. You must pair each shirt with each pair of shorts.
Exam tip
For a missing number, write the sum or product in full first. Then find the box with division, and check it by multiplying.
Exercise 3F Correspondence, scaling and missing numbers
Non-calculatorKey words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 5 hats and 3 scarves.5 hats and 3 scarves
- Pair each item with each item in the other group. Each of the 5 hats can go with each of the 3 scarves.5 × 3 = 15
- Write the answer. The answer is the number of different pairs.15
Check. Count one row at a time: 3 + 3 + 3 + 3 + 3 = 15. ✓
Watch out. Adding gives 8, which is not the number of pairs (15).
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 4 coats and 2 hats.4 coats and 2 hats
- Pair each item with each item in the other group. Each of the 4 coats can go with each of the 2 hats.4 × 2 = 8
- Write the answer. The answer is the number of different pairs.8
Check. Count one row at a time: 2 + 2 + 2 + 2 = 8. ✓
Watch out. Adding gives 6, which is not the number of pairs (8).
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 7 pencils and 2 erasers.7 pencils and 2 erasers
- Pair each item with each item in the other group. Each of the 7 pencils can go with each of the 2 erasers.7 × 2 = 14
- Write the answer. The answer is the number of different pairs.14
Check. Count one row at a time: 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14. ✓
Watch out. Adding gives 9, which is not the number of pairs (14).
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 3 sheets of paper and 5 colours.3 sheets of paper and 5 colours
- Pair each item with each item in the other group. Each of the 3 sheets of paper can go with each of the 5 colours.3 × 5 = 15
- Write the answer. The answer is the number of different pairs.15
Check. Count one row at a time: 5 + 5 + 5 = 15. ✓
Watch out. Adding gives 8, which is not the number of pairs (15).
Key words. Times as many means multiply. "Three times as many" means 3 groups of the first amount.
Method. 3 times as many means multiply by 3.
- Find the starting amount. 6 is the amount we start with.6
- Multiply by 3. 3 times as many means 3 groups of 6.6 × 3 = 18
- Write the answer. 18
Check. 18 ÷ 3 = 6. ✓
Watch out. Times as many means multiply. Do not add.
Key words. Times as many means multiply. "Three times as many" means 3 groups of the first amount.
Method. 4 times as many means multiply by 4.
- Find the starting amount. 8 is the amount we start with.8
- Multiply by 4. 4 times as many means 4 groups of 8.8 × 4 = 32
- Write the answer. 32
Check. 32 ÷ 4 = 8. ✓
Watch out. Times as many means multiply. Do not add.
Key words. Times as many means multiply. "Three times as many" means 3 groups of the first amount.
Method. 2 times as many means multiply by 2.
- Find the starting amount. 9 is the amount we start with.9
- Multiply by 2. 2 times as many means 2 groups of 9.9 × 2 = 18
- Write the answer. 18
Check. 18 ÷ 2 = 9. ✓
Watch out. Times as many means multiply. Do not add.
Key words. Times as many means multiply. "Three times as many" means 3 groups of the first amount.
Method. 5 times as many means multiply by 5.
- Find the starting amount. 4 is the amount we start with.4
- Multiply by 5. 5 times as many means 5 groups of 4.4 × 5 = 20
- Write the answer. 20
Check. 20 ÷ 5 = 4. ✓
Watch out. Times as many means multiply. Do not add.
Key words. Times as many means multiply. "Three times as many" means 3 groups of the first amount.
Method. 10 times as many means multiply by 10.
- Find the starting amount. 3 is the amount we start with.3
- Multiply by 10. 10 times as many means 10 groups of 3.3 × 10 = 30
- Write the answer. 30
Check. 30 ÷ 10 = 3. ✓
Watch out. Times as many means multiply. Do not add.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.[ ] × 3 = 27
- Use the inverse: divide the total by the number you know. Division undoes multiplication.27 ÷ 3 = 9
- Check by multiplying. The check must give the total.3 × 9 = 27 ✓
- Write the answer. 9
Check. 3 × 9 = 27. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.6 × [ ] = 42
- Use the inverse: divide the total by the number you know. Division undoes multiplication.42 ÷ 6 = 7
- Check by multiplying. The check must give the total.6 × 7 = 42 ✓
- Write the answer. 7
Check. 6 × 7 = 42. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.[ ] × 4 = 36
- Use the inverse: divide the total by the number you know. Division undoes multiplication.36 ÷ 4 = 9
- Check by multiplying. The check must give the total.4 × 9 = 36 ✓
- Write the answer. 9
Check. 4 × 9 = 36. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.8 × [ ] = 48
- Use the inverse: divide the total by the number you know. Division undoes multiplication.48 ÷ 8 = 6
- Check by multiplying. The check must give the total.8 × 6 = 48 ✓
- Write the answer. 6
Check. 8 × 6 = 48. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Work out the apples and the pears separately, then add them.
- Work out the apples. There are 4 boxes with 3 apples in each.4 × 3 = 12
- Work out the pears. There are 4 boxes with 2 pears in each.4 × 2 = 8
- Add the two totals. Add the apples and the pears together.12 + 8 = 20
- Write the answer. 20
Check. 20 − 8 = 12. ✓
Watch out. Count the fruit in each box, and make sure you add the apples and the pears.
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Count the pairs, not the items. Compare the answer with the sum.
- Count the pairs. Each shirt can go with each pair of shorts.3 × 4 = 12
- Compare with the sum. Adding gives the number of items, not the number of outfits.3 + 4 = 7
- Decide. Ravi is not right. There are 12 outfits.No, there are 12 outfits
Check. Count one row at a time: 4 + 4 + 4 = 12. ✓
Watch out. When every item pairs with every other item, multiply.
Chapter summary
| Skill | Method |
|---|---|
| Equal groups and arrays | Count the rows and the dots in each row, then multiply. The sign × means groups of. |
| The 2, 3, 4, 5, 8 and 10 times tables | Count in steps, or use a fact you know and add one more group. Double the 2 times table for the 4 times table, and double the 4 times table for the 8 times table. |
| Sharing and grouping | Find the biggest multiple of the number you divide by that is not more than the total. What is left over is the remainder. |
| Two-digit times one-digit | Partition the number into tens and ones, multiply each part, then add. Or use the column method and carry the tens to the next column. |
| Correspondence problems | Each item in one group pairs with each item in the other group, so multiply the two numbers. |
| Scaling problems | "Three times as many" means multiply by 3. |
| Missing numbers | Use the inverse operation. Divide the total by the number you know, then check by multiplying. |
Can you…?
- I can use arrays and the signs ×, ÷ and = to show equal groups.
- I can recall the 3, 4 and 8 times tables, and use the 2, 5 and 10 tables to check.
- I can work out the 4 and 8 times tables by doubling.
- I can share and group, and write an answer with a remainder when some are left over.
- I can multiply a two-digit number by a one-digit number, using partition or the column method.
- I can solve correspondence, scaling and missing number problems.
Key terms
- Multiply
- To find the total of equal groups. The sign is ×.
- Product
- The answer to a multiplication.
- Divide
- To share equally, or to make equal groups. The sign is ÷.
- Array
- Objects arranged in equal rows and columns.
- Remainder
- The number left over when a number does not divide exactly.
- Partition
- To split a number into parts, such as tens and ones, to make a calculation easier.
- Carry
- To move a ten (or a hundred) to the next column on the left in a column method.
- Correspondence
- Each item in one group is paired with each item in another group.
- Inverse
- The operation that undoes another. Division undoes multiplication.
- Times as many
- Multiply an amount to find how many times as many there are.
Exercise 3Z Check-up
Non-calculatorKey words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 4 times table.
- Start with the 2 times table: 2 × 7. We know this fact.2 × 7 = 14
- Double it to get 4 × 7. Doubling means adding the number to itself.14 + 14 = 28
- Write the answer. 28
Check. 28 ÷ 4 = 7. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 10. We know this fact.2 × 10 = 20
- Double it to get 4 × 10. Doubling means adding the number to itself.20 + 20 = 40
- Double again to get 8 × 10. Doubling the 4 times table gives the 8 times table.40 + 40 = 80
- Write the answer. 80
Check. 80 ÷ 8 = 10. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 3.
- Use a fact you know: 3 × 9. We know 9 groups of 3.3 × 9 = 27
- Add one more group of 3. One more group makes 10 groups.27 + 3 = 30
- Write the answer. 30
Check. 30 ÷ 3 = 10. ✓
Watch out. Adding one more group of 3 is quicker than starting again.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 4 can be made from 28.28 ÷ 4
- Find the biggest multiple of 4 that is not more than 28. Try the 4 times table.4 × 7 = 28
- Check that nothing is left over. The division is exact.28 − 28 = 0
- Write the answer. The answer is the number in each group, plus any remainder.7
Check. 4 × 7 + 0 = 28. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.34 × 3
- Multiply the ones first. Write 2 and carry 1 to the tens.4 × 3 = 12
- Multiply the tens. Write 0 and carry 1 to the hundreds.3 × 3 = 9, + 1 carried = 10
- Write the carried number in the next column. There is nothing left to multiply, so the carry is written.write 1
- Write the answer. Read the digits from the bottom line.102
Check. 102 ÷ 3 = 34. ✓
Watch out. Add the carried number before you write the next digit.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.[ ] × 5 = 35
- Use the inverse: divide the total by the number you know. Division undoes multiplication.35 ÷ 5 = 7
- Check by multiplying. The check must give the total.5 × 7 = 35 ✓
- Write the answer. 7
Check. 5 × 7 = 35. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.17 ÷ 2
- Find the biggest multiple of 2 that is not more than 17. Try the 2 times table.2 × 8 = 16
- Find what is left over. 1 is left over, and 1 is smaller than 2.17 − 16 = 1
- Write the answer. The answer is the number in each group, plus any remainder.8 remainder 1
Check. 2 × 8 + 1 = 17. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 6 can be made from 29.29 ÷ 6
- Find the biggest multiple of 6 that is not more than 29. Try the 6 times table.6 × 4 = 24
- Find what is left over. 5 is left over, and 5 is smaller than 6.29 − 24 = 5
- Write the answer. The answer is the number in each group, plus any remainder.4 remainder 5
Check. 6 × 4 + 5 = 29. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 6 caps and 4 jackets.6 caps and 4 jackets
- Pair each item with each item in the other group. Each of the 6 caps can go with each of the 4 jackets.6 × 4 = 24
- Write the answer. The answer is the number of different pairs.24
Check. Count one row at a time: 4 + 4 + 4 + 4 + 4 + 4 = 24. ✓
Watch out. Adding gives 10, which is not the number of pairs (24).
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.16 × 3
- Multiply the ones first. Write 8 and carry 1 to the tens.6 × 3 = 18
- Multiply the tens. Write 4. Nothing is carried.1 × 3 = 3, + 1 carried = 4
- Write the answer. Read the digits from the bottom line.48
Check. 48 ÷ 3 = 16. ✓
Watch out. Add the carried number before you write the next digit.
Total 20 marks. Allow about 20 minutes.
Exercise 3X Practice test questions
Non-calculatorKey words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the fact for one group less, then add one more group of 5.
- Use a fact you know: 5 × 8. We know 8 groups of 5.5 × 8 = 40
- Add one more group of 5. One more group makes 9 groups.40 + 5 = 45
- Write the answer. 45
Check. 45 ÷ 5 = 9. ✓
Watch out. Adding one more group of 5 is quicker than starting again.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. We ask how many groups of 4 can be made from 36.36 ÷ 4
- Find the biggest multiple of 4 that is not more than 36. Try the 4 times table.4 × 9 = 36
- Check that nothing is left over. The division is exact.36 − 36 = 0
- Write the answer. The answer is the number in each group, plus any remainder.9
Check. 4 × 9 + 0 = 36. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.14 × 6
- Multiply the ones first. Write 4 and carry 2 to the tens.4 × 6 = 24
- Multiply the tens. Write 8. Nothing is carried.1 × 6 = 6, + 2 carried = 8
- Write the answer. Read the digits from the bottom line.84
Check. 84 ÷ 6 = 14. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.38 × 2
- Multiply the ones first. Write 6 and carry 1 to the tens.8 × 2 = 16
- Multiply the tens. Write 7. Nothing is carried.3 × 2 = 6, + 1 carried = 7
- Write the answer. Read the digits from the bottom line.76
Check. 76 ÷ 2 = 38. ✓
Watch out. Add the carried number before you write the next digit.
Key words. A multiplication is equal groups. The sign × means "groups of" or "lots of". In 3 × 4, the 3 is the number of groups and the 4 is the size of each group. The answer is the product.
Method. Use the 2 times table, then double it to make the 8 times table.
- Start with the 2 times table: 2 × 7. We know this fact.2 × 7 = 14
- Double it to get 4 × 7. Doubling means adding the number to itself.14 + 14 = 28
- Double again to get 8 × 7. Doubling the 4 times table gives the 8 times table.28 + 28 = 56
- Write the answer. 56
Check. 56 ÷ 8 = 7. ✓
Watch out. Check the doubling. Each double must be exactly twice the number before it.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Use the inverse: divide the total by the number you know.
- Look at the calculation. The box is one of the two numbers being multiplied.[ ] × 4 = 44
- Use the inverse: divide the total by the number you know. Division undoes multiplication.44 ÷ 4 = 11
- Check by multiplying. The check must give the total.4 × 11 = 44 ✓
- Write the answer. 11
Check. 4 × 11 = 44. ✓
Watch out. Divide the total, not the other way round. Then multiply back to check.
Key words. The sign ÷ means share or group. A remainder is the number left over when a number does not divide exactly.
Method. Find the biggest multiple of the divisor that fits, then find what is left over.
- Write the division. Sharing means each person gets the same amount.19 ÷ 4
- Find the biggest multiple of 4 that is not more than 19. Try the 4 times table.4 × 4 = 16
- Find what is left over. 3 is left over, and 3 is smaller than 4.19 − 16 = 3
- Write the answer. The answer is the number in each group, plus any remainder.4 remainder 3
Check. 4 × 4 + 3 = 19. ✓
Watch out. The remainder must be smaller than the number you divide by. If it is not, you can make one more group.
Key words. A missing number is shown by a box, [ ]. Use the inverse operation to find it: division undoes multiplication.
Method. Find the total on the shelves, then take away the tins sold.
- Find the total on the shelves. 5 shelves with 14 tins on each.5 × 14 = 70
- Take away the tins sold. The number left is the total minus the number sold.70 − 27 = 43
- Check. Add the tins sold to the tins left.43 + 27 = 70 ✓
- Write the answer. 43
Check. 43 + 27 = 70. ✓
Watch out. The question asks for the tins left, not the total on the shelves.
Key words. A correspondence problem pairs each item in one group with each item in another group. The number of pairs is found by multiplying.
Method. Multiply the two numbers, because each item pairs with each item in the other group.
- Name the two groups. We have 4 T-shirts and 6 caps.4 T-shirts and 6 caps
- Pair each item with each item in the other group. Each of the 4 T-shirts can go with each of the 6 caps.4 × 6 = 24
- Write the answer. The answer is the number of different pairs.24
Check. Count one row at a time: 6 + 6 + 6 + 6 = 24. ✓
Watch out. Adding gives 10, which is not the number of pairs (24).
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.24 × 3
- Multiply the ones first. Write 2 and carry 1 to the tens.4 × 3 = 12
- Multiply the tens. Write 7. Nothing is carried.2 × 3 = 6, + 1 carried = 7
- Write the answer. Read the digits from the bottom line.72
Check. 72 ÷ 3 = 24. ✓
Watch out. Add the carried number before you write the next digit.
Key words. In a column layout, each digit is multiplied in its own column. A carry is a ten or a hundred written small above the next column to the left.
Method. Multiply the ones first. Write the ones digit, and carry any tens to the next column.
- Write the numbers in columns. Line up the ones under the ones, and the tens under the tens.72 × 2
- Multiply the ones first. Write 4. Nothing is carried.2 × 2 = 4
- Multiply the tens. Write 4 and carry 1 to the hundreds.7 × 2 = 14
- Write the carried number in the next column. There is nothing left to multiply, so the carry is written.write 1
- Write the answer. Read the digits from the bottom line.144
Check. 144 ÷ 2 = 72. ✓
Watch out. Add the carried number before you write the next digit.