Year 4: Equal groups, arrays and swapping the order
Part of the Year 4 times tables chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Equal groups, arrays and swapping the order
Multiplication is a quick way to add equal groups. Imagine 3 bags with 4 apples in each bag. You could add 4 + 4 + 4 = 12. It is quicker to write 3 × 4 = 12. We read it as “3 groups of 4”, “3 lots of 4” or “3 times 4”. The first number is how many groups. The second number is how many are in each group.
An array puts the objects in neat rows (across) and columns (down). This array has 3 rows with 4 dots in each row. It shows 3 × 4 = 12. Now turn the array on its side. There are 4 rows with 3 dots in each row. That is 4 × 3 = 12. It is the same number of dots! So 3 × 4 = 4 × 3. This is called commutativity: you can multiply in either order and get the same answer. It means that you only need to learn half of the facts.
You can also count in steps on a number line. Each hop is the same size. Six hops of 4 take you to 24, so 6 × 4 = 24. Counting in steps is called skip counting.
Two special numbers. Multiplying by 1 means 1 group, or groups of 1. 1 group of 7 is 7, and 7 groups of 1 is 7. So 7 × 1 = 7. Multiplying by 1 does not change a number. Multiplying by 0 means no groups, or groups with nothing in them. 5 plates with 0 biscuits on each plate is 0 biscuits. So 5 × 0 = 0 and 0 × 5 = 0. Anything times 0 is 0.
Key words. A product is the answer to a multiplication. The numbers you multiply are called factors. A multiple of a number is in its times table. The inverse of multiplying is dividing, and you will meet it in Section 3.5.
Key ideas
- Multiplication means adding equal groups: 3 × 4 = 4 + 4 + 4 = 12.
- An array has rows and columns. r rows with c dots in each row is r × c.
- Multiplication can be done in either order: 3 × 4 = 4 × 3. (This is called commutativity.)
- Any number × 1 is that same number.
- Any number × 0 is 0.
Key words. Multiplication is a quick way to add equal groups. A group is a set of things together.
Method. Count the groups and how many are in each group. All the groups are equal, so you can add them up or multiply.
- Count the groups. Each plate is one group.There are 4 plates.
- Count how many are in each group. The groups are equal, so count in one group only.Each plate has 6 biscuits.
- Write it as an addition. Equal groups can be added.6 + 6 + 6 + 6 = 24
- Write it as a multiplication. 4 groups of 6 is 4 × 6.4 × 6 = 24
Check. Count in 6s: 6, 12, 18, 24. The last number is 24. ✓
Watch out. The first number says how many groups. The second number says how many in each group.
Key words. An array is dots or objects in rows and columns. Rows go across and columns go down.
Method. Count the rows and the columns. An array with r rows and c dots in each row has r × c dots.
- Count the rows. Rows go across.There are 5 rows.
- Count the columns. Columns go down. Each row has the same number of dots.There are 7 columns.
- Write the multiplication. 5 rows of 7 is 5 × 7. Count in 7s, 5 times.7, 14, 21, 28, 35
5 × 7 = 35 - Turn the array on its side. Now there are 7 rows of 5. The total is the same.7 × 5 = 35
Check. Count in 5s instead: 5, 10, 15, 20, 25, 30, 35. ✓
Watch out. The total is the same both ways round, but the sentence is different. 3 rows of 4 is 3 × 4.
Key words. Commutative means you can swap the order and the answer stays the same.
Method. Multiplication can be done in either order, so 9 × 4 has the same answer as 4 × 9.
- Write down the fact Mei knows. 4 × 9 = 36
- Swap the numbers round. 4 groups of 9 and 9 groups of 4 have the same total. You can see this by turning an array on its side.9 × 4 = 4 × 9
- Write the answer. 9 × 4 = 36
Check. Count in 9s four times: 9, 18, 27, 36. ✓
Watch out. Swapping the order works for multiplication. It does not work for subtraction or division.
Method. Multiplying by 1 leaves the number the same.
- Swap the numbers round. Order does not matter in multiplication, so 7 × 1 = 1 × 7. Use the 1 times table trick.7 × 1 = 1 × 7
- Say what the multiplication means. 1 group of 7 is just 7.1 group of 7 = 7
- Write the answer. 7 × 1 = 7
Check. 7 groups of 1, or 1 groups of 7: both are 7. ✓
Watch out. Multiplying by 1 does not change the number. It does not make it 0 or 1.
Method. Zero groups, or groups with nothing in them, always make 0 in total.
- Swap the numbers round. Order does not matter in multiplication, so 9 × 0 = 0 × 9. Use the 0 times table trick.9 × 0 = 0 × 9
- Say what the multiplication means. 0 × 9 means 0 groups of 9. There are no groups, so there is nothing.0 groups of 9 = 0
- Write the answer. Anything multiplied by 0 is 0.9 × 0 = 0
Check. Adding groups: 0 + 0 + ... = 0. ✓
Watch out. Do not write the other number. Multiplying by 0 always gives 0, not the number you started with.
Method. Each hop is 4. Add 4 to the number before each box. The last number on the line is 6 × 4.
- Find the size of each hop. The labels at the start show it.0 to 4: each hop is 4.
- Fill the boxes. Add 4 to the number before the box.12 + 4 = 16
20 is the next number, and 20 + 4 = 24 - Say what 6 × 4 is. There are 6 hops, so the last number is 6 × 4.6 × 4 = 24
Check. Count in 4s: 4, 8, 12, 16, 20, 24. Six numbers. ✓
Watch out. There are 6 hops but 7 numbers on the line, because the line starts at 0.
Watch out
Be careful with 0 and 1. 7 × 1 is 7 (not 1) and 7 × 0 is 0 (not 7). Also, do not mix up rows and columns: 3 rows of 4 is 3 × 4, and 4 rows of 3 is 4 × 3. The answer is the same, but they are different pictures.Exercise 3A Groups, arrays, swapping and multiplying by 0 and 1
Non-calculatorKey words. Multiplication is a quick way to add equal groups. A group is a set of things together.
Method. Count the groups and how many are in each group. All the groups are equal, so you can add them up or multiply.
- Count the groups. Each box is one group.There are 3 boxes.
- Count how many are in each group. The groups are equal, so count in one group only.Each box has 5 dots.
- Write it as an addition. Equal groups can be added.5 + 5 + 5 = 15
- Write it as a multiplication. 3 groups of 5 is 3 × 5.3 × 5 = 15
Check. Count in 5s: 5, 10, 15. The last number is 15. ✓
Watch out. The first number says how many groups. The second number says how many in each group.
Key words. Multiplication is a quick way to add equal groups. A group is a set of things together.
Method. Count the groups and how many are in each group. All the groups are equal, so you can add them up or multiply.
- Count the groups. Each box is one group.There are 6 boxes.
- Count how many are in each group. The groups are equal, so count in one group only.Each box has 2 dots.
- Write it as an addition. Equal groups can be added.2 + 2 + 2 + 2 + 2 + 2 = 12
- Write it as a multiplication. 6 groups of 2 is 6 × 2.6 × 2 = 12
Check. Count in 2s: 2, 4, 6, 8, 10, 12. The last number is 12. ✓
Watch out. The first number says how many groups. The second number says how many in each group.
Key words. Multiplication is a quick way to add equal groups. A group is a set of things together.
Method. Count the groups and how many are in each group. All the groups are equal, so you can add them up or multiply.
- Count the groups. Each box is one group.There are 4 boxes.
- Count how many are in each group. The groups are equal, so count in one group only.Each box has 4 dots.
- Write it as an addition. Equal groups can be added.4 + 4 + 4 + 4 = 16
- Write it as a multiplication. 4 groups of 4 is 4 × 4.4 × 4 = 16
Check. Count in 4s: 4, 8, 12, 16. The last number is 16. ✓
Watch out. The first number says how many groups. The second number says how many in each group.
Key words. Multiplication is a quick way to add equal groups. A group is a set of things together.
Method. Count the groups and how many are in each group. All the groups are equal, so you can add them up or multiply.
- Count the groups. Each box is one group.There are 5 boxes.
- Count how many are in each group. The groups are equal, so count in one group only.Each box has 3 dots.
- Write it as an addition. Equal groups can be added.3 + 3 + 3 + 3 + 3 = 15
- Write it as a multiplication. 5 groups of 3 is 5 × 3.5 × 3 = 15
Check. Count in 3s: 3, 6, 9, 12, 15. The last number is 15. ✓
Watch out. The first number says how many groups. The second number says how many in each group.
Key words. An array is dots or objects in rows and columns. Rows go across and columns go down.
Method. Count the rows and the columns. An array with r rows and c dots in each row has r × c dots.
- Count the rows. Rows go across.There are 2 rows.
- Count the columns. Columns go down. Each row has the same number of dots.There are 9 columns.
- Write the multiplication. 2 rows of 9 is 2 × 9. Count in 9s, 2 times.9, 18
2 × 9 = 18 - Turn the array on its side. Now there are 9 rows of 2. The total is the same.9 × 2 = 18
Check. Count in 2s instead: 2, 4, 6, 8, 10, 12, 14, 16, 18. ✓
Watch out. The total is the same both ways round, but the sentence is different. 3 rows of 4 is 3 × 4.
Key words. An array is dots or objects in rows and columns. Rows go across and columns go down.
Method. Count the rows and the columns. An array with r rows and c dots in each row has r × c dots.
- Count the rows. Rows go across.There are 3 rows.
- Count the columns. Columns go down. Each row has the same number of dots.There are 7 columns.
- Write the multiplication. 3 rows of 7 is 3 × 7. Count in 7s, 3 times.7, 14, 21
3 × 7 = 21 - Turn the array on its side. Now there are 7 rows of 3. The total is the same.7 × 3 = 21
Check. Count in 3s instead: 3, 6, 9, 12, 15, 18, 21. ✓
Watch out. The total is the same both ways round, but the sentence is different. 3 rows of 4 is 3 × 4.
Key words. An array is dots or objects in rows and columns. Rows go across and columns go down.
Method. Count the rows and the columns. An array with r rows and c dots in each row has r × c dots.
- Count the rows. Rows go across.There are 5 rows.
- Count the columns. Columns go down. Each row has the same number of dots.There are 6 columns.
- Write the multiplication. 5 rows of 6 is 5 × 6. Count in 6s, 5 times.6, 12, 18, 24, 30
5 × 6 = 30 - Turn the array on its side. Now there are 6 rows of 5. The total is the same.6 × 5 = 30
Check. Count in 5s instead: 5, 10, 15, 20, 25, 30. ✓
Watch out. The total is the same both ways round, but the sentence is different. 3 rows of 4 is 3 × 4.
Key words. An array is dots or objects in rows and columns. Rows go across and columns go down.
Method. Count the rows and the columns. An array with r rows and c dots in each row has r × c dots.
- Count the rows. Rows go across.There are 4 rows.
- Count the columns. Columns go down. Each row has the same number of dots.There are 8 columns.
- Write the multiplication. 4 rows of 8 is 4 × 8. Count in 8s, 4 times.8, 16, 24, 32
4 × 8 = 32 - Turn the array on its side. Now there are 8 rows of 4. The total is the same.8 × 4 = 32
Check. Count in 4s instead: 4, 8, 12, 16, 20, 24, 28, 32. ✓
Watch out. The total is the same both ways round, but the sentence is different. 3 rows of 4 is 3 × 4.
(a) 7 + 7 + 7 + 7 (b) 5 + 5 + 5 + 5 + 5 + 5 (c) 9 + 9 + 9 (d) 10 + 10 + 10 + 10 + 10
Method. Adding the same number again and again is a multiplication. Count how many times the number is added.
- Count how many numbers are being added. The number 7 appears each time.7 appears 4 times.
- Write it as a multiplication. 4 lots of 7 is 4 × 7.4 × 7
- Work out the answer. Use a times table fact.7 + 7 = 14
double 10 = 20, double 4 = 8, so 20 + 8 = 28
4 × 7 = 28
Check. Add them up: 7 + 7 + 7 + 7 = 28. ✓
Watch out. Count the numbers carefully. 4 lots of 7 is 4 × 7, not 7 × 4 (though the total is the same).
Method. Adding the same number again and again is a multiplication. Count how many times the number is added.
- Count how many numbers are being added. The number 5 appears each time.5 appears 6 times.
- Write it as a multiplication. 6 lots of 5 is 6 × 5.6 × 5
- Work out the answer. Use a times table fact.10 × 6 = 60
half of 60 = 30
6 × 5 = 30
Check. Add them up: 5 + 5 + 5 + 5 + 5 + 5 = 30. ✓
Watch out. Count the numbers carefully. 6 lots of 5 is 6 × 5, not 5 × 6 (though the total is the same).
Method. Adding the same number again and again is a multiplication. Count how many times the number is added.
- Count how many numbers are being added. The number 9 appears each time.9 appears 3 times.
- Write it as a multiplication. 3 lots of 9 is 3 × 9.3 × 9
- Work out the answer. Use a times table fact.10 × 3 = 30
30 − 3 = 27
3 × 9 = 27
Check. Add them up: 9 + 9 + 9 = 27. ✓
Watch out. Count the numbers carefully. 3 lots of 9 is 3 × 9, not 9 × 3 (though the total is the same).
Method. Adding the same number again and again is a multiplication. Count how many times the number is added.
- Count how many numbers are being added. The number 10 appears each time.10 appears 5 times.
- Write it as a multiplication. 5 lots of 10 is 5 × 10.5 × 10
- Work out the answer. Use a times table fact.5 tens
5 → 50
5 × 10 = 50
Check. Add them up: 10 + 10 + 10 + 10 + 10 = 50. ✓
Watch out. Count the numbers carefully. 5 lots of 10 is 5 × 10, not 10 × 5 (though the total is the same).
Method. Find the step between the numbers, then add it each time to fill the gaps.
- Find the step. Look at two numbers that are next to each other. How much bigger is the second?3 to 6: 6 − 3 = 3. We are counting in 3s.
- Fill each gap. Add 3 to the number before the gap.6 + 3 = 9
12 + 3 = 15 - Say the whole sequence. Read it from the start to check it sounds right.3, 6, 9, 12, 15, 18
Check. Every number is a multiple of 3: 3, 6, 9, 12, 15, 18. ✓
Watch out. Check each jump is the same size (3), including the jump into and out of each gap.
Method. Find the step between the numbers, then add it each time to fill the gaps.
- Find the step. Look at two numbers that are next to each other. How much bigger is the second?20 to 24: 24 − 20 = 4. We are counting in 4s.
- Fill each gap. Add 4 to the number before the gap.4 + 4 = 8
12 + 4 = 16 - Say the whole sequence. Read it from the start to check it sounds right.4, 8, 12, 16, 20, 24
Check. Every number is a multiple of 4: 4, 8, 12, 16, 20, 24. ✓
Watch out. Check each jump is the same size (4), including the jump into and out of each gap.
Method. Find the step between the numbers, then add it each time to fill the gaps.
- Find the step. Look at two numbers that are next to each other. How much bigger is the second?5 to 10: 10 − 5 = 5. We are counting in 5s.
- Fill each gap. Add 5 to the number before the gap.15 + 5 = 20
25 + 5 = 30
35 + 5 = 40 - Say the whole sequence. Read it from the start to check it sounds right.5, 10, 15, 20, 25, 30, 35, 40
Check. Every number is a multiple of 5: 5, 10, 15, 20, 25, 30, 35, 40. ✓
Watch out. Check each jump is the same size (5), including the jump into and out of each gap.
Method. Find the step between the numbers, then add it each time to fill the gaps.
- Find the step. Look at two numbers that are next to each other. How much bigger is the second?2 to 4: 4 − 2 = 2. We are counting in 2s.
- Fill each gap. Add 2 to the number before the gap.4 + 2 = 6
8 + 2 = 10
12 + 2 = 14 - Say the whole sequence. Read it from the start to check it sounds right.2, 4, 6, 8, 10, 12, 14, 16
Check. Every number is a multiple of 2: 2, 4, 6, 8, 10, 12, 14, 16. ✓
Watch out. Check each jump is the same size (2), including the jump into and out of each gap.
Key words. Commutative means that the order does not change the answer. Multiplication is commutative.
Method. Multiplication can be done in either order. The same two numbers give the same answer.
- Look at the numbers you can see. Both sides must use 6 and 4.6 × 4 = 4 × ?
- Use the idea that order does not matter. 6 × 4 = 4 × 6.6 × 4 = 24 and 4 × 6 = 24
- Write the missing number. It is the number that is not already on that side.The missing number is 6.
Check. Work out both sides: they are both 24. ✓
Watch out. This only works for multiplication. 8 − 3 is not the same as 3 − 8.
Key words. Commutative means that the order does not change the answer. Multiplication is commutative.
Method. Multiplication can be done in either order. The same two numbers give the same answer.
- Look at the numbers you can see. Both sides must use 7 and 9.7 × ? = 9 × 7
- Use the idea that order does not matter. 7 × 9 = 9 × 7.7 × 9 = 63 and 9 × 7 = 63
- Write the missing number. It is the number that is not already on that side.The missing number is 9.
Check. Work out both sides: they are both 63. ✓
Watch out. This only works for multiplication. 8 − 3 is not the same as 3 − 8.
Key words. Commutative means that the order does not change the answer. Multiplication is commutative.
Method. Multiplication can be done in either order. The same two numbers give the same answer.
- Look at the numbers you can see. Both sides must use 8 and 3.? × 8 = 8 × 3
- Use the idea that order does not matter. 8 × 3 = 3 × 8.8 × 3 = 24 and 3 × 8 = 24
- Write the missing number. It is the number that is not already on that side.The missing number is 3.
Check. Work out both sides: they are both 24. ✓
Watch out. This only works for multiplication. 8 − 3 is not the same as 3 − 8.
Key words. Commutative means that the order does not change the answer. Multiplication is commutative.
Method. Multiplication can be done in either order. The same two numbers give the same answer.
- Look at the numbers you can see. Both sides must use 5 and 12.5 × 12 = ? × 5
- Use the idea that order does not matter. 5 × 12 = 12 × 5.5 × 12 = 60 and 12 × 5 = 60
- Write the missing number. It is the number that is not already on that side.The missing number is 12.
Check. Work out both sides: they are both 60. ✓
Watch out. This only works for multiplication. 8 − 3 is not the same as 3 − 8.
Method. Multiplying by 1 leaves the number the same.
- Swap the numbers round. Order does not matter in multiplication, so 7 × 1 = 1 × 7. Use the 1 times table trick.7 × 1 = 1 × 7
- Say what the multiplication means. 1 group of 7 is just 7.1 group of 7 = 7
- Write the answer. 7 × 1 = 7
Check. 7 groups of 1, or 1 groups of 7: both are 7. ✓
Watch out. Multiplying by 1 does not change the number. It does not make it 0 or 1.
Method. Zero groups, or groups with nothing in them, always make 0 in total.
- Say what the multiplication means. 0 × 9 means 0 groups of 9. There are no groups, so there is nothing.0 groups of 9 = 0
- Write the answer. Anything multiplied by 0 is 0.0 × 9 = 0
Check. Adding groups: 0 + 0 + ... = 0. ✓
Watch out. Do not write the other number. Multiplying by 0 always gives 0, not the number you started with.
Method. Multiplying by 1 leaves the number the same.
- Say what the multiplication means. 1 group of 12 is just 12.1 group of 12 = 12
- Write the answer. 1 × 12 = 12
Check. 1 groups of 12, or 12 groups of 1: both are 12. ✓
Watch out. Multiplying by 1 does not change the number. It does not make it 0 or 1.
Method. Zero groups, or groups with nothing in them, always make 0 in total.
- Swap the numbers round. Order does not matter in multiplication, so 12 × 0 = 0 × 12. Use the 0 times table trick.12 × 0 = 0 × 12
- Say what the multiplication means. 0 × 12 means 0 groups of 12. There are no groups, so there is nothing.0 groups of 12 = 0
- Write the answer. Anything multiplied by 0 is 0.12 × 0 = 0
Check. Adding groups: 0 + 0 + ... = 0. ✓
Watch out. Do not write the other number. Multiplying by 0 always gives 0, not the number you started with.
Method. Multiplying by 1 leaves the number the same.
- Say what the multiplication means. 1 group of 1 is just 1.1 group of 1 = 1
- Write the answer. 1 × 1 = 1
Check. 1 groups of 1, or 1 groups of 1: both are 1. ✓
Watch out. Multiplying by 1 does not change the number. It does not make it 0 or 1.
Method. Zero groups, or groups with nothing in them, always make 0 in total.
- Swap the numbers round. Order does not matter in multiplication, so 8 × 0 = 0 × 8. Use the 0 times table trick.8 × 0 = 0 × 8
- Say what the multiplication means. 0 × 8 means 0 groups of 8. There are no groups, so there is nothing.0 groups of 8 = 0
- Write the answer. Anything multiplied by 0 is 0.8 × 0 = 0
Check. Adding groups: 0 + 0 + ... = 0. ✓
Watch out. Do not write the other number. Multiplying by 0 always gives 0, not the number you started with.
Method. Multiplying by 2 is the same as doubling. Double the number.
- Swap the numbers round. Order does not matter in multiplication, so 3 × 2 = 2 × 3. Use the 2 times table trick.3 × 2 = 2 × 3
- Double the number. Doubling means adding the number to itself.3 + 3 = 6
- Write the answer. 3 × 2 = 6
Check. Check by dividing: 6 ÷ 2 = 3. ✓
Watch out. Doubling is not adding 2. To double 7, add 7 + 7, not 7 + 2.
Method. Five is half of ten. Multiply by 10 first, then halve the answer.
- Multiply by 10. This is easy: move the digits one place to the left.10 × 4 = 40
- Halve the answer. 5 is half of 10, so 5 lots of a number is half of 10 lots.half of 40 = 20
- Write the answer. Answers in the 5 times table end in 0 or 5.5 × 4 = 20
Check. Check by dividing: 20 ÷ 4 = 5. ✓
Watch out. Halve the answer to 10 × the number. Do not halve the number before you multiply by 10 and then forget to multiply.
Method. To multiply by 10, every digit moves one place to the left. The ones become tens.
- Think about tens. 10 × 6 means 6 lots of 10, which is 6 tens.6 tens
- Move the digits one place to the left. A tens column needs a 0 in the ones place.6 → 60
- Write the answer. 10 × 6 = 60
Check. Check by dividing: 60 ÷ 6 = 10. ✓
Watch out. The 10 times table has a 0 in the ones place every time. Put the 0 on the end, but think “tens” so you understand why.
Method. Multiplying by 2 is the same as doubling. Double the number.
- Double the number. Doubling means adding the number to itself.9 + 9 = 18
- Write the answer. 2 × 9 = 18
Check. The digits of 18 add up to 9: 1 + 8 = 9. ✓
Watch out. Doubling is not adding 2. To double 7, add 7 + 7, not 7 + 2.
Method. Five is half of ten. Multiply by 10 first, then halve the answer.
- Multiply by 10. This is easy: move the digits one place to the left.10 × 7 = 70
- Halve the answer. 5 is half of 10, so 5 lots of a number is half of 10 lots.half of 60 = 30, half of 10 = 5, so 30 + 5 = 35
- Write the answer. Answers in the 5 times table end in 0 or 5.5 × 7 = 35
Check. Check by dividing: 35 ÷ 5 = 7. ✓
Watch out. Halve the answer to 10 × the number. Do not halve the number before you multiply by 10 and then forget to multiply.
Method. To multiply by 10, every digit moves one place to the left. The ones become tens.
- Think about tens. 10 × 10 means 10 lots of 10, which is 10 tens.10 tens
- Move the digits one place to the left. A tens column needs a 0 in the ones place.10 → 100
- Write the answer. 10 × 10 = 100
Check. Check by dividing: 100 ÷ 10 = 10. ✓
Watch out. The 10 times table has a 0 in the ones place every time. Put the 0 on the end, but think “tens” so you understand why.
Method. Five is half of ten. Multiply by 10 first, then halve the answer.
- Swap the numbers round. Order does not matter in multiplication, so 4 × 5 = 5 × 4. Use the 5 times table trick.4 × 5 = 5 × 4
- Multiply by 10. This is easy: move the digits one place to the left.10 × 4 = 40
- Halve the answer. 5 is half of 10, so 5 lots of a number is half of 10 lots.half of 40 = 20
- Write the answer. Answers in the 5 times table end in 0 or 5.4 × 5 = 20
Check. Check by dividing: 20 ÷ 4 = 5. ✓
Watch out. Halve the answer to 10 × the number. Do not halve the number before you multiply by 10 and then forget to multiply.
Method. Multiplying by 2 is the same as doubling. Double the number.
- Double the number. Doubling means adding the number to itself.8 + 8 = 16
- Write the answer. 2 × 8 = 16
Check. Check by dividing: 16 ÷ 2 = 8. ✓
Watch out. Doubling is not adding 2. To double 7, add 7 + 7, not 7 + 2.
Method. There are 5 equal groups, so multiply the number in one group by the number of groups.
- Draw a bar model. 5 equal bars show 5 equal boxes. Each bar is 6. We want the total.
- Choose the operation. Equal groups: multiply the number of groups by the amount in each group.5 × 6
- Work it out. Use a times table trick.10 × 6 = 60
half of 60 = 30
5 × 6 = 30 - Write the answer. Include the unit.30 eggs
Check. Divide back: 30 ÷ 5 = 6. ✓
Watch out. Check that you multiplied. Equal groups mean the total is bigger than the size of one group.
Method. There are 4 equal groups, so multiply the number in one group by the number of groups.
- Draw a bar model. 4 equal bars show 4 equal spiders. Each bar is 8. We want the total.
- Choose the operation. Equal groups: multiply the number of groups by the amount in each group.4 × 8
- Work it out. Use a times table trick.8 + 8 = 16
double 10 = 20, double 6 = 12, so 20 + 12 = 32
4 × 8 = 32 - Write the answer. Include the unit.32 legs
Check. Divide back: 32 ÷ 4 = 8. ✓
Watch out. Check that you multiplied. Equal groups mean the total is bigger than the size of one group.
Method. Look at the array. Equal rows are equal groups, so we multiply, we do not add the two numbers.
- Look at what Zainab did. She added the number of rows and the number in each row.3 + 8 = 11
- Say why that is wrong. 3 rows of 8 means 3 groups of 8. We need 8 + 8 + 8, or 3 × 8.8 + 8 + 8 = 24
- Multiply to find the number of chairs. Count in 8s: 8, 16, 24.3 × 8 = 24
Check. Turn the array round: 8 rows of 3 is 8 × 3 = 24. ✓
Watch out. “Rows and columns” means equal groups. Multiply the two numbers, do not add them.
Method. Try some numbers. If the rule works for every number you try, it is probably always true.
- Try some examples. 1 group of a number is just that number.1 × 4 = 4 1 × 12 = 12 9 × 1 = 9 100 × 1 = 100
- Explain why. Multiplying by 1 means one group, and one group of any number is just that number.It does not matter which number you start with.
- Decide. Always
Check. Check with an array: 1 row of 6 dots has 6 dots. ✓
Watch out. “Always” means it works every time. One example is not enough to prove it, so give a reason too.
Method. Work out each one. The odd one out has a different answer from the others.
- Work out A and B. Both use the numbers 4 and 6. Order does not matter.4 × 6 = 24 and 6 × 4 = 24
- Work out C. This one is an addition, not a multiplication.4 + 6 = 10
- Work out D. 2 groups of 12.2 × 12 = 24
- Choose the odd one out. A, B and D all make 24. C makes 10.C (4 + 6)
Check. Count in 4s six times: 4, 8, 12, 16, 20, 24. ✓
Watch out. Check the sign (+ or ×) as well as the numbers.
Method. Think about what 9 × 0 means: 9 groups, and each group has 0 in it.
- Say what the multiplication means. 9 × 0 is 9 groups of 0.0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0
- Add the groups. Adding 0 nine times does not make anything.0 + 0 + ... + 0 = 0
- Answer Amira. There are nine groups, but nothing in them.No. 9 × 0 = 0
Check. Swap the order: 0 × 9 = 0 groups of 9 = 0. ✓
Watch out. Anything multiplied by 0 is 0. The number of groups does not matter if each group is empty.
Method. Look for pairs of numbers that multiply to make 12. Each pair makes an array. Turning the array gives a second array.
- Try 1 row. 12 dots in one long row.1 × 12 = 12
- Try 2 rows. Divide 12 into 2 equal rows.2 × 6 = 12
- Try 3 rows. 3 × 4 = 12
- Try 4, 5, 6... rows. 4 rows gives 4 × 3, which is the 3-row array turned round. 5 does not fit (5 × 2 = 10 and 5 × 3 = 15). 6 rows gives 6 × 2.4 × 3 = 12 6 × 2 = 12 12 × 1 = 12
- Count the arrays. The pictures show 3 different shapes. None of them is a square, so each shape can be turned to make a second array.3 shapes × 2 = 6 arrays
Check. Each fact has its partner: 1 × 12 and 12 × 1, 2 × 6 and 6 × 2, 3 × 4 and 4 × 3. ✓
Watch out. Check that every row has the same number of dots. No gaps and no leftover dots.