Multiplying and dividing
By the end of this chapter you will be able to
- Count equal groups and add them to find the total
- Count the rows and columns of an array
- Write a sum using the signs ×, ÷ and =
- Use the 2, 5 and 10 times tables to multiply and divide
- Say whether a number is odd or even
- Use the order of numbers: × can be done in any order, ÷ cannot
You should already know
- Grouping and sharing with objects and pictures (Year 1, Chapter 9)
- Counting in 2s, 5s and 10s (Year 1, Chapter 9)
- Adding and subtracting numbers to 100 (Chapter 2)
Where this leads
- Fractions: halves and quarters of amounts (Chapter 4)
- Money and time, where equal groups help (Chapters 6 and 7)
National Curriculum for England, Key Stage 1 mathematics, Year 2. Content: Recall and use multiplication and division facts for the 2, 5 and 10 times tables, Recognise odd and even numbers, Write mathematical statements for multiplication and division using the signs x, divide and =, Know that multiplication can be done in any order and division cannot, Solve problems with equal groups, arrays and repeated addition.
Exercise P Do you remember?
Non-calculatorMethod. Count in 5s. Each jump is 5.
- Start at 5. Count in jumps of 5.5, 10, 15
- Count on to the gap. The next jump after 15 is 20.5, 10, 15, 20
- Write the answer. 20
Check. Count back in 5s from 20: 15, 10, 5. ✓
Watch out. Every jump is the same size, which is 5.
Method. Share the things one at a time into equal groups. Count how many are in one group.
- Share the things into equal groups. Put them into 2 equal groups, one at a time.6 shared into 2 groups
- Count what is in one group. Each group has 3 apples.3 in each group
- Write the answer. Each group gets 3.3
Check. Add 3, 2 times: 3 + 3 = 6. ✓
Watch out. Give one thing to each group in turn. Then count one group.
Method. Two equal numbers added together is a double.
- Find the first 7. Put 7 counters on the first ten frame.7
- Find the second 7. Put 7 counters on the second ten frame.7 + 7
- Write the answer. Seven and seven make fourteen.14
Check. Take 7 away from 14. You get 7. ✓
Watch out. Doubles are two equal groups.
Method. One more is the next number when counting on.
- Find 29. Put your finger on 29 on the number line.29
- Count on one. The next number after 29 is 30.29 + 1 = 30
- Write the answer. 30
Check. Count back one from 30: 29. ✓
Watch out. The tens digit changes when you go past 29.
Method. Work out both sums. If the answers match, the sums are the same.
- Work out 5 + 5. Five and five make ten.5 + 5 = 10
- Work out 4 + 6. Four and six make ten.4 + 6 = 10
- Compare. Both answers are 10, so the sums are the same.Yes. 5 + 5 = 4 + 6 = 10
Check. Both sums make ten. ✓
Watch out. Making ten first helps you add.
Equal groups and repeated addition
Some things come in equal groups. Each group has the same number of things.
Look at 3 groups. Each group has 2 balls. To find the total, add 2 for each group: 2 + 2 + 2 = 6.
Adding the same number again and again is called repeated addition.
Key ideas
- Equal groups have the same number of things in each group.
- Repeated addition adds the same number again and again.
- Add one number for each group.
- Counting in steps of the same size gives the same total.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 apples.2 in one group
- Add the same number once for each group. There are 4 groups, so add 2 4 times.2 + 2 + 2 + 2
- Write the total. The total is 8 apples.8
Check. Count in 2s: 2, 4, 6, 8. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 cubes.5 in one group
- Add the same number once for each group. There are 3 groups, so add 5 3 times.5 + 5 + 5
- Write the total. The total is 15 cubes.15
Check. Count in 5s: 5, 10, 15. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 10 eggs.10 in one group
- Add the same number once for each group. There are 2 groups, so add 10 2 times.10 + 10
- Write the total. The total is 20 eggs.20
Check. Count in 10s: 10, 20. ✓
Watch out. Count the groups first. Then add one number for each group.
Watch out
Watch out: 3 groups of 2 means add 2 three times: 2 + 2 + 2 = 6. Adding only two 2s gives 4, which is wrong.
Exam tip
Grown-up tip. Put 3 cups on the table with 2 pasta shapes in each cup. Ask: “How many cups? How many in each cup? How many altogether?” Then count in 2s to check.
Exercise 3A Equal groups and repeated addition
Non-calculatorKey words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 biscuits.2 in one group
- Add the same number once for each group. There are 4 groups, so add 2 4 times.2 + 2 + 2 + 2
- Write the total. The total is 8 biscuits.8
Check. Count in 2s: 2, 4, 6, 8. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 4 flowers.4 in one group
- Add the same number once for each group. There are 3 groups, so add 4 3 times.4 + 4 + 4
- Write the total. The total is 12 flowers.12
Check. Count in 4s: 4, 8, 12. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 6 stars.6 in one group
- Add the same number once for each group. There are 3 groups, so add 6 3 times.6 + 6 + 6
- Write the total. The total is 18 stars.18
Check. Count in 6s: 6, 12, 18. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 pears.2 in one group
- Add the same number once for each group. There are 5 groups, so add 2 5 times.2 + 2 + 2 + 2 + 2
- Write the total. The total is 10 pears.10
Check. Count in 2s: 2, 4, 6, 8, 10. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 4 marbles.4 in one group
- Add the same number once for each group. There are 4 groups, so add 4 4 times.4 + 4 + 4 + 4
- Write the total. The total is 16 marbles.16
Check. Count in 4s: 4, 8, 12, 16. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 4 cubes.4 in one group
- Add the same number once for each group. There are 2 groups, so add 4 2 times.4 + 4
- Write the total. The total is 8 cubes.8
Check. Count in 4s: 4, 8. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 3 sweets.3 in one group
- Add the same number once for each group. There are 4 groups, so add 3 4 times.3 + 3 + 3 + 3
- Write the total. The total is 12 sweets.12
Check. Count in 3s: 3, 6, 9, 12. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 6 crayons.6 in one group
- Add the same number once for each group. There are 2 groups, so add 6 2 times.6 + 6
- Write the total. The total is 12 crayons.12
Check. Count in 6s: 6, 12. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 3 stickers.3 in one group
- Add the same number once for each group. There are 5 groups, so add 3 5 times.3 + 3 + 3 + 3 + 3
- Write the total. The total is 15 stickers.15
Check. Count in 3s: 3, 6, 9, 12, 15. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Work out the total, then compare it with the claim.
- Work out the total. 4 groups of 3: add 3 4 times.3 + 3 + 3 + 3 = 12
- Compare with the claim. The claim says 12. The total is 12.12 = 12
- Write the answer. The claim is right only if the two numbers match.Yes. 4 × 3 = 12
Check. Count in 3s: 3, 6, 9, 12. ✓
Watch out. Add the same number once for each group. Do not stop early.
Method. Work out the total, then compare it with the claim.
- Work out the total. 5 groups of 2: add 2 5 times.2 + 2 + 2 + 2 + 2 = 10
- Compare with the claim. The claim says 7. The total is 10.7 ≠ 10
- Write the answer. The claim is right only if the two numbers match.No. 5 × 2 = 10
Check. Count in 2s: 2, 4, 6, 8, 10. ✓
Watch out. Add the same number once for each group. Do not stop early.
Method. Work out the total of the 3 groups first. Then take one whole group away.
- Work out the total of 3 groups of 4. Each group has 4 pencils. There are 3 groups.4 + 4 + 4 = 12
- Take one group away. Ben gives away one group of 4 pencils.12 − 4 = 8
- Write the answer. Ben has 2 groups of 4 left.8
Check. 2 groups of 4 make 8: 4 + 4 = 8. ✓
Watch out. Take away a whole group, not one pencil.
Arrays
An array is a neat pattern of things in rows and columns. The rows go across. The columns go down. Each row has the same number of things.
To find the total, count the rows. Then count how many are in one row. Add that number once for each row.
Key ideas
- An array has rows across and columns down.
- Each row has the same number of things.
- Count the rows, and count the things in one row.
- Add one row for each row in the array.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 2 rows.2 rows
- Count the things in one row. Each row has 5 cubes.5 in each row
- Add the rows. Add 5 once for each row.5 + 5 = 10
Check. Count down the columns too. Both ways make 10. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 4 rows.4 rows
- Count the things in one row. Each row has 3 cakes.3 in each row
- Add the rows. Add 3 once for each row.3 + 3 + 3 + 3 = 12
Check. Count down the columns too. Both ways make 12. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 3 rows.3 rows
- Count the things in one row. Each row has 6 eggs.6 in each row
- Add the rows. Add 6 once for each row.6 + 6 + 6 = 18
Check. Count down the columns too. Both ways make 18. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Watch out
Watch out: count the rows and the things in one row. Do not count the columns and add them to the rows.
Exam tip
Grown-up tip. Lay out buttons in rows on the table. Ask: “How many rows? How many in each row? How many altogether?” Then turn the tray round and count the columns.
Exercise 3B Arrays
Non-calculatorMethod. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 2 rows.2 rows
- Count the things in one row. Each row has 4 biscuits.4 in each row
- Add the rows. Add 4 once for each row.4 + 4 = 8
Check. Count down the columns too. Both ways make 8. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 5 rows.5 rows
- Count the things in one row. Each row has 2 stars.2 in each row
- Add the rows. Add 2 once for each row.2 + 2 + 2 + 2 + 2 = 10
Check. Count down the columns too. Both ways make 10. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 3 rows.3 rows
- Count the things in one row. Each row has 3 balls.3 in each row
- Add the rows. Add 3 once for each row.3 + 3 + 3 = 9
Check. Count down the columns too. Both ways make 9. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 3 rows.3 rows
- Count the things in one row. Each row has 6 pears.6 in each row
- Add the rows. Add 6 once for each row.6 + 6 + 6 = 18
Check. Count down the columns too. Both ways make 18. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 2 rows.2 rows
- Count the things in one row. Each row has 3 bags.3 in each row
- Add the rows. Add 3 once for each row.3 + 3 = 6
Check. Count down the columns too. Both ways make 6. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 4 rows.4 rows
- Count the things in one row. Each row has 2 hats.2 in each row
- Add the rows. Add 2 once for each row.2 + 2 + 2 + 2 = 8
Check. Count down the columns too. Both ways make 8. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 3 rows.3 rows
- Count the things in one row. Each row has 5 books.5 in each row
- Add the rows. Add 5 once for each row.5 + 5 + 5 = 15
Check. Count down the columns too. Both ways make 15. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 6 rows.6 rows
- Count the things in one row. Each row has 2 shoes.2 in each row
- Add the rows. Add 2 once for each row.2 + 2 + 2 + 2 + 2 + 2 = 12
Check. Count down the columns too. Both ways make 12. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 4 rows.4 rows
- Count the things in one row. Each row has 5 books.5 in each row
- Add the rows. Add 5 once for each row.5 + 5 + 5 + 5 = 20
Check. Count down the columns too. Both ways make 20. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. The array has 3 rows. Share the 15 balls into 3 equal rows.
- Count the rows. There are 3 rows.3 rows
- Share the balls into the rows. Put 5 in each row: 5 + 5 + 5 = 15.5 + 5 + 5 = 15
- Write the answer. Each row has 5 balls.5
Check. 3 rows of 5 make 15. ✓
Watch out. Put one ball in each row at a time, so the rows stay equal.
Method. Work out the total, then compare it with the claim.
- Work out the total. 2 groups of 5: add 5 2 times.5 + 5 = 10
- Compare with the claim. The claim says 10. The total is 10.10 = 10
- Write the answer. The claim is right only if the two numbers match.Yes. 2 × 5 = 10
Check. Count in 5s: 5, 10. ✓
Watch out. Add the same number once for each group. Do not stop early.
Method. Count each array. Compare the totals.
- Count 4 rows of 3. Each row has 3 things, and there are 4 rows.3 + 3 + 3 + 3 = 12
- Count 3 rows of 4. Each row has 4 things, and there are 3 rows.4 + 4 + 4 = 12
- Write the answer. Both arrays have the same number of things.No. Both arrays have 12 things.
Check. Turn one array round and count it again. You get 12 too. ✓
Watch out. Turning an array round does not change how many things there are.
Method. Try to share the 12 counters into 5 equal rows.
- Try 5 rows. Each row needs the same number of counters.12 counters, 5 rows
- Share the counters into the rows. Put 2 in each of the 5 rows. That uses 10 counters.2 + 2 + 2 + 2 + 2 = 10
- Write the answer. There are 2 counters left over, so the rows are not equal.No. 2 are left over.
Check. 10 counters in the rows and 2 left over make 12. ✓
Watch out. If some things are left over, the rows cannot all be the same.
The signs ×, ÷ and =
We can write equal groups as a sum with signs. The sign × means groups of. The sign ÷ means share into equal groups. The sign = means is the same as.
Look at 3 groups of 2 apples. Three groups of 2 is written 3 × 2 = 6.
Sharing 6 apples into 3 equal groups gives 2 in each group. This is written 6 ÷ 3 = 2.
Key ideas
- × means groups of, or lots of.
- ÷ means share into equal groups.
- = means the same as.
- Read a sum from left to right.
Method. Write the groups as a sum with the sign ×, then work out the total.
- Read the groups. There are 4 groups of 2.4 groups of 2
- Write it with the sign ×. The sign × means groups of.4 × 2
- Work out the total. Add 2, 4 times.4 × 2 = 8
Check. 2 + 2 + 2 + 2 = 8. ✓
Watch out. The first number says how many groups. The second says how many in each group.
Method. Write the sharing as a sum with the sign ÷.
- Share the things equally. Put 10 things into 2 equal groups.10 shared into 2 groups
- Count what is in one group. Each group has 5.5 in each group
- Write it with the sign ÷. The sign ÷ means share into equal groups.10 ÷ 2 = 5
Check. Add 5, 2 times: 5 + 5 = 10. ✓
Watch out. The number being shared comes first. The number of groups comes second.
Method. Work out what the sum means, then choose the sign.
- Read the sum. 3 and 4 have a sign missing.3 ? 4 = 12
- Think what the sign means. 3 groups of 4 make 12.3 × 4 = 12
- Write the sign. The sign × means groups of.×
Check. 3 × 4 = 12. ✓
Watch out. Groups of is ×. Share into groups is ÷.
Watch out
Watch out: 6 ÷ 3 is not 3 ÷ 6. The number being shared always comes first.
Exam tip
Grown-up tip. Put out 3 plates with 2 buttons on each. Ask: “How many buttons altogether?” Then write the sum together: 3 × 2 = 6.
Exercise 3C The signs ×, ÷ and =
Non-calculatorMethod. Work out what the sum means, then choose the sign.
- Read the sum. 2 and 6 have a sign missing.2 ? 6 = 12
- Think what the sign means. 2 groups of 6 make 12.2 × 6 = 12
- Write the sign. The sign × means groups of.×
Check. 2 × 6 = 12. ✓
Watch out. Groups of is ×. Share into groups is ÷.
Method. Work out what the sum means, then choose the sign.
- Read the sum. 12 and 3 have a sign missing.12 ? 3 = 4
- Think what the sign means. Share 12 into 3 equal groups: 4 in each group.12 ÷ 3 = 4
- Write the sign. The sign ÷ means share into equal groups.÷
Check. 12 ÷ 3 = 4. ✓
Watch out. Groups of is ×. Share into groups is ÷.
Method. Write the groups as a sum with the sign ×, then work out the total.
- Read the groups. There are 5 groups of 2.5 groups of 2
- Write it with the sign ×. The sign × means groups of.5 × 2
- Work out the total. Add 2, 5 times.5 × 2 = 10
Check. 2 + 2 + 2 + 2 + 2 = 10. ✓
Watch out. The first number says how many groups. The second says how many in each group.
Method. Write the sharing as a sum with the sign ÷.
- Share the things equally. Put 8 things into 4 equal groups.8 shared into 4 groups
- Count what is in one group. Each group has 2.2 in each group
- Write it with the sign ÷. The sign ÷ means share into equal groups.8 ÷ 4 = 2
Check. Add 2, 4 times: 2 + 2 + 2 + 2 = 8. ✓
Watch out. The number being shared comes first. The number of groups comes second.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 7 cups.7 in one group
- Add the same number once for each group. There are 3 groups, so add 7 3 times.7 + 7 + 7
- Write the total. The total is 21 cups.21
Check. Count in 7s: 7, 14, 21. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 2 socks in each group.2 in each group
- Count the groups. Keep making groups until none are left. There are 7 groups.7 groups
- Write the answer. 14 ÷ 2 means how many groups of 2.7
Check. Count in 2s: 2, 4, 6, 8, 10, 12, 14. ✓
Watch out. Count the groups, not the things in each group.
Method. Write the groups as a sum with the sign ×, then work out the total.
- Read the groups. There are 3 groups of 4.3 groups of 4
- Write it with the sign ×. The sign × means groups of.3 × 4
- Work out the total. Add 4, 3 times.3 × 4 = 12
Check. 4 + 4 + 4 = 12. ✓
Watch out. The first number says how many groups. The second says how many in each group.
Method. Write the sharing as a sum with the sign ÷.
- Share the things equally. Put 12 things into 4 equal groups.12 shared into 4 groups
- Count what is in one group. Each group has 3.3 in each group
- Write it with the sign ÷. The sign ÷ means share into equal groups.12 ÷ 4 = 3
Check. Add 3, 4 times: 3 + 3 + 3 + 3 = 12. ✓
Watch out. The number being shared comes first. The number of groups comes second.
Method. Write the sharing as a sum with the sign ÷.
- Share the things equally. Put 12 things into 3 equal groups.12 shared into 3 groups
- Count what is in one group. Each group has 4.4 in each group
- Write it with the sign ÷. The sign ÷ means share into equal groups.12 ÷ 3 = 4
Check. Add 4, 3 times: 4 + 4 + 4 = 12. ✓
Watch out. The number being shared comes first. The number of groups comes second.
Method. Work out the total, then compare it with the claim.
- Work out the total. 3 groups of 5: add 5 3 times.5 + 5 + 5 = 15
- Compare with the claim. The claim says 8. The total is 15.8 ≠ 15
- Write the answer. The claim is right only if the two numbers match.No. 3 × 5 = 15
Check. Count in 5s: 5, 10, 15. ✓
Watch out. Add the same number once for each group. Do not stop early.
Method. Share 3 into 3 equal groups. Then compare with Jo.
- Share 3 into 3 groups. Each group gets 1.3 ÷ 3 = 1
- Compare with Jo. Jo says the answer is 9. But 3 ÷ 3 is 1.9 ≠ 1
- Write the answer. Jo is wrong.No. 3 ÷ 3 = 1, not 9.
Check. Three groups of 1 make 3. ✓
Watch out. Check each side of the sign separately.
The 2, 5 and 10 times tables
The 2 times table counts in 2s: 2, 4, 6, 8, 10, 12. The 5 times table counts in 5s: 5, 10, 15, 20, 25, 30. The 10 times table counts in 10s: 10, 20, 30, 40, 50.
Learn these facts. Each multiplication fact also gives a division fact. If 4 × 5 = 20, then 20 ÷ 5 = 4.
Key ideas
- Learn the 2, 5 and 10 times tables.
- Count in 2s, 5s or 10s to check an answer.
- Every multiplication fact gives a division fact.
- 6 × 10 = 60, because 6 tens are 60.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 socks.2 in one group
- Add the same number once for each group. There are 4 groups, so add 2 4 times.2 + 2 + 2 + 2
- Write the total. The total is 8 socks.8
Check. Count in 2s: 2, 4, 6, 8. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 stars.5 in one group
- Add the same number once for each group. There are 3 groups, so add 5 3 times.5 + 5 + 5
- Write the total. The total is 15 stars.15
Check. Count in 5s: 5, 10, 15. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 10 things in each group.10 in each group
- Count the groups. Keep making groups until none are left. There are 4 groups.4 groups
- Write the answer. 40 ÷ 10 means how many groups of 10.4
Check. Count in 10s: 10, 20, 30, 40. ✓
Watch out. Count the groups, not the things in each group.
Watch out
Watch out: 5 × 4 is 20, not 9. Count in 5s: 5, 10, 15, 20. That is four jumps.
Exam tip
Grown-up tip. Count in 2s with socks, in 5s with fingers on two hands, and in 10s with bundles of ten. Ask: “What is 4 fives?” and “How many fives make 20?”
Exercise 3D The 2, 5 and 10 times tables
Non-calculatorKey words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 socks.2 in one group
- Add the same number once for each group. There are 5 groups, so add 2 5 times.2 + 2 + 2 + 2 + 2
- Write the total. The total is 10 socks.10
Check. Count in 2s: 2, 4, 6, 8, 10. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 socks.2 in one group
- Add the same number once for each group. There are 7 groups, so add 2 7 times.2 + 2 + 2 + 2 + 2 + 2 + 2
- Write the total. The total is 14 socks.14
Check. Count in 2s: 2, 4, 6, 8, 10, 12, 14. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 stars.5 in one group
- Add the same number once for each group. There are 4 groups, so add 5 4 times.5 + 5 + 5 + 5
- Write the total. The total is 20 stars.20
Check. Count in 5s: 5, 10, 15, 20. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 stars.5 in one group
- Add the same number once for each group. There are 9 groups, so add 5 9 times.5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5
- Write the total. The total is 45 stars.45
Check. Count in 5s: 5, 10, 15, 20, 25, 30, 35, 40, 45. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 10 things.10 in one group
- Add the same number once for each group. There are 6 groups, so add 10 6 times.10 + 10 + 10 + 10 + 10 + 10
- Write the total. The total is 60 things.60
Check. Count in 10s: 10, 20, 30, 40, 50, 60. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 2 socks in each group.2 in each group
- Count the groups. Keep making groups until none are left. There are 6 groups.6 groups
- Write the answer. 12 ÷ 2 means how many groups of 2.6
Check. Count in 2s: 2, 4, 6, 8, 10, 12. ✓
Watch out. Count the groups, not the things in each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 5 stars in each group.5 in each group
- Count the groups. Keep making groups until none are left. There are 5 groups.5 groups
- Write the answer. 25 ÷ 5 means how many groups of 5.5
Check. Count in 5s: 5, 10, 15, 20, 25. ✓
Watch out. Count the groups, not the things in each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 10 things in each group.10 in each group
- Count the groups. Keep making groups until none are left. There are 7 groups.7 groups
- Write the answer. 70 ÷ 10 means how many groups of 10.7
Check. Count in 10s: 10, 20, 30, 40, 50, 60, 70. ✓
Watch out. Count the groups, not the things in each group.
Method. Count in 4s. Count how many jumps you need.
- Count in the known number. Count in 4s until you reach 28.4, 8, 12, 16, 20, 24, 28
- Count the jumps. It takes 7 jumps to reach 28.7 jumps
- Write the missing number. The missing number is the number of jumps.7
Check. 4 × 7 = 28. ✓
Watch out. The missing number is how many times you say the number, not the total.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 socks.2 in one group
- Add the same number once for each group. There are 3 groups, so add 2 3 times.2 + 2 + 2
- Write the total. The total is 6 socks.6
Check. Count in 2s: 2, 4, 6. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 stars.5 in one group
- Add the same number once for each group. There are 6 groups, so add 5 6 times.5 + 5 + 5 + 5 + 5 + 5
- Write the total. The total is 30 stars.30
Check. Count in 5s: 5, 10, 15, 20, 25, 30. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 10 pencils.10 in one group
- Add the same number once for each group. There are 5 groups, so add 10 5 times.10 + 10 + 10 + 10 + 10
- Write the total. The total is 50 pencils.50
Check. Count in 10s: 10, 20, 30, 40, 50. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 10 things in each group.10 in each group
- Count the groups. Keep making groups until none are left. There are 4 groups.4 groups
- Write the answer. 40 ÷ 10 means how many groups of 10.4
Check. Count in 10s: 10, 20, 30, 40. ✓
Watch out. Count the groups, not the things in each group.
Method. Count in 5s on the number line. Look at each number.
- Count in 5s. Start at 5 and jump on by 5 each time.5, 10, 15, 20, 25, 30
- Look at the last digit. Every number in the 5s ends in 5 or 0.5, 0, 5, 0, 5, 0
- Write the answer. Every jump lands on a 5 or a 0.Yes. Every number in the 5s ends in 0 or 5.
Check. The 10s are in the 5s too, so the rule holds. ✓
Watch out. Count on the number line to check.
Odd and even numbers
Some numbers can be made into pairs with none left over. These are even numbers. Other numbers have one left over. These are odd numbers.
4 shoes make 2 pairs with none left over, so 4 is even. 5 shoes make 2 pairs with 1 left over, so 5 is odd.
Key ideas
- Even numbers make pairs with none left over.
- Odd numbers have one left over.
- Even numbers end in 0, 2, 4, 6 or 8.
- Odd numbers end in 1, 3, 5, 7 or 9.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 6 things into pairs.3 pairs
- Look for one left over. There is none left over.none left over
- Write the answer. A number with none left over is even.even
Check. 3 pairs make 6. 6 is even. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 7 things into pairs.3 pairs
- Look for one left over. 3 pairs and 1 left over.1 left over
- Write the answer. A number with one left over is odd.odd
Check. 3 pairs make 6. 7 is odd. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 14 things into pairs.7 pairs
- Look for one left over. There is none left over.none left over
- Write the answer. A number with none left over is even.even
Check. 7 pairs make 14. 14 is even. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Watch out
Watch out: only the last digit tells you if a number is odd or even. A number ending in 5 is odd.
Exam tip
Grown-up tip. Use socks or gloves. Ask: “Can we make pairs with none left over?” Odd and even is all about pairs.
Exercise 3E Odd and even numbers
Non-calculatorKey words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 9 things into pairs.4 pairs
- Look for one left over. 4 pairs and 1 left over.1 left over
- Write the answer. A number with one left over is odd.odd
Check. 4 pairs make 8. 9 is odd. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 10 things into pairs.5 pairs
- Look for one left over. There is none left over.none left over
- Write the answer. A number with none left over is even.even
Check. 5 pairs make 10. 10 is even. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 13 things into pairs.6 pairs
- Look for one left over. 6 pairs and 1 left over.1 left over
- Write the answer. A number with one left over is odd.odd
Check. 6 pairs make 12. 13 is odd. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 18 things into pairs.9 pairs
- Look for one left over. There is none left over.none left over
- Write the answer. A number with none left over is even.even
Check. 9 pairs make 18. 18 is even. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Method. Count the balls in each group. Make pairs. Look for one left over.
- Look at 7 balls. 7 balls make 3 pairs and 1 left over.7 is odd
- Look at 12 balls. 12 balls make 6 pairs with none left over.12 is even
- Write the answer. The even number is 12.12
Check. 6 pairs make 12, with none left over. ✓
Watch out. Look for the group that makes pairs with none left over.
Method. Look for one left over when you make pairs.
- Look at 8. 8 makes 4 pairs, with none left over. 8 is even.8 is even
- Look at 11. 11 makes 5 pairs and 1 left over. 11 is odd.11 is odd
- Look at 16 and 21. 16 makes 8 pairs. 21 makes 10 pairs and 1 left over.16 is even, 21 is odd
- Write the answer. The odd numbers are 11 and 21.11, 21
Check. 11 and 21 each have one left over when they make pairs. ✓
Watch out. The odd numbers all end in 1, 3, 5, 7 or 9.
Method. Make pairs of the children. Check for any child left out.
- Make pairs. Put the 12 children into pairs.6 pairs
- Look for any left over. No child is left out.none left over
- Write the answer. With none left over, all the children can make pairs.Yes. 12 is even.
Check. 6 pairs make 12 children. ✓
Watch out. Even numbers make pairs with none left over.
Method. Make as many pairs as you can. Count what is left over.
- Make pairs. 15 cubes make 7 pairs.7 pairs use 14 cubes
- Count what is left. Take 14 away from 15. One cube is left.15 − 14 = 1
- Write the answer. One cube is left over, so 15 is odd.1
Check. 7 pairs make 14, and 14 + 1 = 15. ✓
Watch out. The number left over tells you if the number is odd.
Method. Share the sweets into 2 equal groups. Check that none are left over.
- Share the sweets. Each friend gets 10 sweets.20 ÷ 2 = 10
- Check for any left over. There are none left over.10 + 10 = 20
- Write the answer. The sweets can be shared equally.Yes. 10 + 10 = 20.
Check. Each friend has 10, and 10 + 10 = 20. ✓
Watch out. An even number can always be shared into 2 equal groups.
Method. Test an even number. Look at its last digit.
- Test an even number. Take 4. Make pairs: 2 pairs with none left over.4 = 2 pairs
- Look at the last digit. 4 ends in 4, not 0.4 is even
- Write the answer. Even numbers can end in 0, 2, 4, 6 or 8.No. 4 is even, but it ends in 4.
Check. Make 4 into 2 pairs. None are left over. ✓
Watch out. Even numbers do not all end in 0.
Any order, and division is different
Multiplication can be done in any order. Look at 2 rows of 5 stars. Turn the array round: 5 rows of 2 stars. Both arrays have 10 stars.
So 2 × 5 = 10, and 5 × 2 = 10.
Division is different. Sharing 10 into 2 equal groups gives 5 in each group: 10 ÷ 2 = 5. But 2 ÷ 10 is not the same. The order matters.
Key ideas
- Multiplication can be done in any order.
- Turning an array round keeps the same total.
- Division cannot be done in any order.
- Check a multiplication by counting the array the other way.
Method. Work out both ways. If the answers match, the order does not matter.
- Work out the first way. 5 groups of 2 is 10.5 × 2 = 10
- Swap the numbers and work out the other way. 2 groups of 5 is also 10.2 × 5 = 10
- Compare the answers. The two answers are the same, so the order does not matter.Yes. 5 × 2 = 2 × 5
Check. Both ways make 10. Count the array the other way to check. ✓
Watch out. Multiplication works in any order. Division does not.
Method. Division cannot be done in any order. Try both ways and compare.
- Work out the first way. Share 10 into 2 equal groups. Each group has 5.10 ÷ 2 = 5
- Try the other way. Now try to share 2 into 10 equal groups. There are not enough things.2 ÷ 10 is not a whole number
- Compare the answers. The answers are not the same, so the order matters for division.No. 10 ÷ 2 is not 2 ÷ 10
Check. 10 ÷ 2 = 5, but 2 ÷ 10 is not 5. ✓
Watch out. Division needs the number being shared first. Swapping the numbers changes the answer.
Method. Swapping the two numbers does not change the total when you multiply.
- Look at the fact you know. 6 groups of 3 make 18.6 × 3 = 18
- Swap the numbers. Now there are 3 groups of 6.3 × 6
- Work out the answer. Swapping the numbers does not change the total.18
Check. 3 groups of 6 add up to 18: 6 + 6 + 6 = 18. ✓
Watch out. Multiplication can be done in any order. Check by counting the other way.
Watch out
Watch out: 10 ÷ 2 is 5, but 2 ÷ 10 is not 5. You cannot swap the numbers in a division.
Exam tip
Grown-up tip. Use 2 rows of 5 buttons. Turn the tray round and ask: “Is it still 10?” Then share 10 buttons between 2 plates and ask: “Can we do it the other way round?”
Exercise 3F Any order, and division is different
Non-calculatorMethod. Swapping the two numbers does not change the total when you multiply.
- Look at the fact you know. 2 groups of 6 make 12.2 × 6 = 12
- Swap the numbers. Now there are 6 groups of 2.6 × 2
- Work out the answer. Swapping the numbers does not change the total.12
Check. 6 groups of 2 add up to 12: 2 + 2 + 2 + 2 + 2 + 2 = 12. ✓
Watch out. Multiplication can be done in any order. Check by counting the other way.
Method. Swapping the two numbers does not change the total when you multiply.
- Look at the fact you know. 5 groups of 4 make 20.5 × 4 = 20
- Swap the numbers. Now there are 4 groups of 5.4 × 5
- Work out the answer. Swapping the numbers does not change the total.20
Check. 4 groups of 5 add up to 20: 5 + 5 + 5 + 5 = 20. ✓
Watch out. Multiplication can be done in any order. Check by counting the other way.
Method. Swapping the two numbers does not change the total when you multiply.
- Look at the fact you know. 10 groups of 3 make 30.10 × 3 = 30
- Swap the numbers. Now there are 3 groups of 10.3 × 10
- Work out the answer. Swapping the numbers does not change the total.30
Check. 3 groups of 10 add up to 30: 10 + 10 + 10 = 30. ✓
Watch out. Multiplication can be done in any order. Check by counting the other way.
Method. Work out both ways. If the answers match, the order does not matter.
- Work out the first way. 4 groups of 5 is 20.4 × 5 = 20
- Swap the numbers and work out the other way. 5 groups of 4 is also 20.5 × 4 = 20
- Compare the answers. The two answers are the same, so the order does not matter.Yes. 4 × 5 = 5 × 4
Check. Both ways make 20. Count the array the other way to check. ✓
Watch out. Multiplication works in any order. Division does not.
Method. Division cannot be done in any order. Try both ways and compare.
- Work out the first way. Share 8 into 2 equal groups. Each group has 4.8 ÷ 2 = 4
- Try the other way. Now try to share 2 into 8 equal groups. There are not enough things.2 ÷ 8 is not a whole number
- Compare the answers. The answers are not the same, so the order matters for division.No. 8 ÷ 2 is not 2 ÷ 8
Check. 8 ÷ 2 = 4, but 2 ÷ 8 is not 4. ✓
Watch out. Division needs the number being shared first. Swapping the numbers changes the answer.
Method. Swapping the two numbers does not change the total when you multiply.
- Look at the fact you know. 3 groups of 5 make 15.3 × 5 = 15
- Swap the numbers. Now there are 5 groups of 3.5 × 3
- Work out the answer. Swapping the numbers does not change the total.15
Check. 5 groups of 3 add up to 15: 3 + 3 + 3 + 3 + 3 = 15. ✓
Watch out. Multiplication can be done in any order. Check by counting the other way.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 7 shoes.7 in one group
- Add the same number once for each group. There are 2 groups, so add 7 2 times.7 + 7
- Write the total. The total is 14 shoes.14
Check. Count in 7s: 7, 14. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 shoes.2 in one group
- Add the same number once for each group. There are 7 groups, so add 2 7 times.2 + 2 + 2 + 2 + 2 + 2 + 2
- Write the total. The total is 14 shoes.14
Check. Count in 2s: 2, 4, 6, 8, 10, 12, 14. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Count each array. Compare the totals.
- Count Mo’s sweets. 4 rows of 2 sweets make 8 sweets.2 + 2 + 2 + 2 = 8
- Count Ruby’s sweets. 2 rows of 4 sweets make 8 sweets.4 + 4 = 8
- Compare. Both have the same number of sweets.Same. Both have 8.
Check. Turning an array round keeps the total the same. ✓
Watch out. Count the sweets, not the rows.
Method. Try to make 4 rows of 3 with the same 12 cups.
- Count the cups. Dev has 12 cups in total.12 cups
- Make 4 rows of 3. Put 3 cups in each of 4 rows.3 + 3 + 3 + 3 = 12
- Write the answer. The same 12 cups fit into 4 rows of 3.Yes. 3 + 3 + 3 + 3 = 12.
Check. 4 rows of 3 make 12, the same as 3 rows of 4. ✓
Watch out. The same number of things can make arrays of different shapes.
Method. Division cannot be done in any order. Try both ways and compare.
- Work out the first way. Share 12 into 3 equal groups. Each group has 4.12 ÷ 3 = 4
- Try the other way. Now try to share 3 into 12 equal groups. There are not enough things.3 ÷ 12 is not a whole number
- Compare the answers. The answers are not the same, so the order matters for division.No. 12 ÷ 3 is not 3 ÷ 12
Check. 12 ÷ 3 = 4, but 3 ÷ 12 is not 4. ✓
Watch out. Division needs the number being shared first. Swapping the numbers changes the answer.
Method. Check each sum on its own. Then find the one that is wrong.
- Check 24 ÷ 6. Share 24 into 6 equal groups. Each group has 4.24 ÷ 6 = 4
- Check 6 ÷ 24. Share 6 into 24 equal groups. There are not enough.6 ÷ 24 is not 4
- Write the answer. Sam’s wrong statement is the one about 6 ÷ 24.6 ÷ 24 = 4
Check. 4 × 6 = 24, but 6 ÷ 24 is not a whole number of groups. ✓
Watch out. Division is not in any order. Check each statement the right way round.
Chapter summary
| Skill | Method |
|---|---|
| Equal groups and repeated addition | Count the groups. Add the same number once for each group. |
| Arrays | Count the rows. Count the things in one row. Add one row for each row. |
| Signs | × means groups of. ÷ means share into equal groups. = means the same as. |
| 2, 5 and 10 times tables | Count in 2s, 5s or 10s. Each multiplication fact gives a division fact. |
| Odd and even | Even numbers make pairs with none left over. Odd numbers have one left over. |
| Order | Multiplication can be done in any order. Division cannot. |
Can you…?
- I can add equal groups to find the total
- I can count the rows and columns of an array
- I can write a sum using ×, ÷ and =
- I know the 2, 5 and 10 times tables, and their division facts
- I can say whether a number is odd or even
- I know multiplication can be done in any order, but division cannot
Key terms
- equal groups
- groups that each have the same number of things
- repeated addition
- adding the same number again and again
- array
- things arranged in rows and columns
- row
- a line of things going across
- column
- a line of things going down
- multiply
- make equal groups; the sign is ×
- divide
- share into equal groups, or count how many groups fit; the sign is ÷
- odd number
- a number with one left over when you make pairs
- even number
- a number that makes pairs with none left over
Exercise 3Z Check-up
Non-calculatorKey words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 7 marbles.7 in one group
- Add the same number once for each group. There are 2 groups, so add 7 2 times.7 + 7
- Write the total. The total is 14 marbles.14
Check. Count in 7s: 7, 14. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Count the rows. Then add the number in one row, once for each row.
- Count the rows. There are 4 rows.4 rows
- Count the things in one row. Each row has 2 hats.2 in each row
- Add the rows. Add 2 once for each row.2 + 2 + 2 + 2 = 8
Check. Count down the columns too. Both ways make 8. ✓
Watch out. Rows go across. Columns go down. Count each row once, and do not skip any.
Method. Write the groups as a sum with the sign ×, then work out the total.
- Read the groups. There are 5 groups of 3.5 groups of 3
- Write it with the sign ×. The sign × means groups of.5 × 3
- Work out the total. Add 3, 5 times.5 × 3 = 15
Check. 3 + 3 + 3 + 3 + 3 = 15. ✓
Watch out. The first number says how many groups. The second says how many in each group.
Method. Write the sharing as a sum with the sign ÷.
- Share the things equally. Put 12 things into 2 equal groups.12 shared into 2 groups
- Count what is in one group. Each group has 6.6 in each group
- Write it with the sign ÷. The sign ÷ means share into equal groups.12 ÷ 2 = 6
Check. Add 6, 2 times: 6 + 6 = 12. ✓
Watch out. The number being shared comes first. The number of groups comes second.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 socks.2 in one group
- Add the same number once for each group. There are 9 groups, so add 2 9 times.2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
- Write the total. The total is 18 socks.18
Check. Count in 2s: 2, 4, 6, 8, 10, 12, 14, 16, 18. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 stars.5 in one group
- Add the same number once for each group. There are 7 groups, so add 5 7 times.5 + 5 + 5 + 5 + 5 + 5 + 5
- Write the total. The total is 35 stars.35
Check. Count in 5s: 5, 10, 15, 20, 25, 30, 35. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 10 things.10 in one group
- Add the same number once for each group. There are 8 groups, so add 10 8 times.10 + 10 + 10 + 10 + 10 + 10 + 10 + 10
- Write the total. The total is 80 things.80
Check. Count in 10s: 10, 20, 30, 40, 50, 60, 70, 80. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 17 things into pairs.8 pairs
- Look for one left over. 8 pairs and 1 left over.1 left over
- Write the answer. A number with one left over is odd.odd
Check. 8 pairs make 16. 17 is odd. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Method. Work out both ways. If the answers match, the order does not matter.
- Work out the first way. 4 groups of 6 is 24.4 × 6 = 24
- Swap the numbers and work out the other way. 6 groups of 4 is also 24.6 × 4 = 24
- Compare the answers. The two answers are the same, so the order does not matter.Yes. 4 × 6 = 6 × 4
Check. Both ways make 24. Count the array the other way to check. ✓
Watch out. Multiplication works in any order. Division does not.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 5 sweets.5 in one group
- Add the same number once for each group. There are 6 groups, so add 5 6 times.5 + 5 + 5 + 5 + 5 + 5
- Write the total. The total is 30 sweets.30
Check. Count in 5s: 5, 10, 15, 20, 25, 30. ✓
Watch out. Count the groups first. Then add one number for each group.
Total 14 marks. Allow about 14 minutes.
Exercise 3X End-of-chapter quiz
Non-calculatorKey words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 2 shoes.2 in one group
- Add the same number once for each group. There are 6 groups, so add 2 6 times.2 + 2 + 2 + 2 + 2 + 2
- Write the total. The total is 12 shoes.12
Check. Count in 2s: 2, 4, 6, 8, 10, 12. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 10 things.10 in one group
- Add the same number once for each group. There are 9 groups, so add 10 9 times.10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10
- Write the total. The total is 90 things.90
Check. Count in 10s: 10, 20, 30, 40, 50, 60, 70, 80, 90. ✓
Watch out. Count the groups first. Then add one number for each group.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 8 stars.8 in one group
- Add the same number once for each group. There are 5 groups, so add 8 5 times.8 + 8 + 8 + 8 + 8
- Write the total. The total is 40 stars.40
Check. Count in 8s: 8, 16, 24, 32, 40. ✓
Watch out. Count the groups first. Then add one number for each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 2 socks in each group.2 in each group
- Count the groups. Keep making groups until none are left. There are 8 groups.8 groups
- Write the answer. 16 ÷ 2 means how many groups of 2.8
Check. Count in 2s: 2, 4, 6, 8, 10, 12, 14, 16. ✓
Watch out. Count the groups, not the things in each group.
Method. Make groups of the same size, then count how many groups there are.
- Make groups of the same size. Put 5 stars in each group.5 in each group
- Count the groups. Keep making groups until none are left. There are 3 groups.3 groups
- Write the answer. 15 ÷ 5 means how many groups of 5.3
Check. Count in 5s: 5, 10, 15. ✓
Watch out. Count the groups, not the things in each group.
Key words. An even number makes pairs. An odd number has one left over.
Method. Put the things into pairs. Look for one left over.
- Make pairs. Put the 21 things into pairs.10 pairs
- Look for one left over. 10 pairs and 1 left over.1 left over
- Write the answer. A number with one left over is odd.odd
Check. 10 pairs make 20. 21 is odd. ✓
Watch out. Odd numbers have one left over. Even numbers have none left over.
Method. Count in 5s. Count how many jumps you need.
- Count in the known number. Count in 5s until you reach 35.5, 10, 15, 20, 25, 30, 35
- Count the jumps. It takes 7 jumps to reach 35.7 jumps
- Write the missing number. The missing number is the number of jumps.7
Check. 5 × 7 = 35. ✓
Watch out. The missing number is how many times you say the number, not the total.
Method. Work out what the sum means, then choose the sign.
- Read the sum. 24 and 4 have a sign missing.24 ? 4 = 6
- Think what the sign means. Share 24 into 4 equal groups: 6 in each group.24 ÷ 4 = 6
- Write the sign. The sign ÷ means share into equal groups.÷
Check. 24 ÷ 4 = 6. ✓
Watch out. Groups of is ×. Share into groups is ÷.
Method. Work out both ways. If the answers match, the order does not matter.
- Work out the first way. 5 groups of 3 is 15.5 × 3 = 15
- Swap the numbers and work out the other way. 3 groups of 5 is also 15.3 × 5 = 15
- Compare the answers. The two answers are the same, so the order does not matter.Yes. 5 × 3 = 3 × 5
Check. Both ways make 15. Count the array the other way to check. ✓
Watch out. Multiplication works in any order. Division does not.
Key words. Equal groups have the same number of things in each group.
Method. Add the same number once for each group. Count the groups first.
- Look at one group. Each group has 4 books.4 in one group
- Add the same number once for each group. There are 6 groups, so add 4 6 times.4 + 4 + 4 + 4 + 4 + 4
- Write the total. The total is 24 books.24
Check. Count in 4s: 4, 8, 12, 16, 20, 24. ✓
Watch out. Count the groups first. Then add one number for each group.