Chapter 7 of 8

All 8 chapters
  1. 1Equal groups, readunderstood
  2. 2The 2, 5 and 10 times tables, readunderstood
  3. 3Sharing and grouping, readunderstood
  4. 4The 3, 4 and 8 times tables, readunderstood
  5. 5The 6, 7 and 9 times tables, readunderstood
  6. 6Times and divide facts together, readunderstood
  7. 7Remainders, readunderstood
  8. 8Times and divide with bigger numbers, readunderstood

M2.3.7 Chapter 7

Remainders

I have 17 stones. I make groups of 5. What about the extra stones?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Share 17 into groups of 5 and say what is left over
  • Decide whether to round up or down in a story

Some numbers do not share out evenly. A few things are left over.

Take 17 stones. Make groups of 5. You can make 3 full groups. That uses 15 stones.

The 2 stones left over are the remainder. We write 17 ÷ 5 = 3, remainder 2.

Draw it first

Share 17 stones into groups of 5

  1. Seventeen stones. Step 1 shows them loose, labelled 17 stones. Step 2 puts 15 of them in 3 groups of 5 and labels the groups 3 groups of 5. Step 3 labels the 2 stones left over remainder 2.17 stones

    1 Put out 17 stones.

  2. Seventeen stones. Step 1 shows them loose, labelled 17 stones. Step 2 puts 15 of them in 3 groups of 5 and labels the groups 3 groups of 5. Step 3 labels the 2 stones left over remainder 2.3 groups of 5

    2 Make groups of 5. You can make 3 full groups. Two stones are in no group.

  3. Seventeen stones. Step 1 shows them loose, labelled 17 stones. Step 2 puts 15 of them in 3 groups of 5 and labels the groups 3 groups of 5. Step 3 labels the 2 stones left over remainder 2.3 groups of 5remainder 2

    3 The 2 stones left over are the remainder.

You can always check. Groups times size, plus the remainder, gives the start. 3 × 5 + 2 = 17.

The remainder is always smaller than the group size. Here 2 is smaller than 5. If it were 5, you could make one more group.

Two pictures of 20 stones. The top picture shows 3 groups of 5 and 5 stones apart, labelled 3 groups of 5, 5 left over. The bottom picture shows 4 groups of 5, labelled 5 left over makes a 4th group.3 groups of 5, 5 left over5 left over makes a 4th group
A remainder of 5 is not a remainder. Those 5 stones make one more group.

Worked example

Work out 23 ÷ 4.

  1. Count in 4s: 4, 8, 12, 16, 20. The next one, 24, is too big.
  2. So 5 groups of 4 fit. 5 × 4 = 20.
  3. What is left? 23 − 20 = 3.

Answer: 5 groups, remainder 3

In a story, you must decide what to do with the remainder. Look at two stories.

Story one. 17 children must ride in cars. A car holds 5. How many cars?

17 ÷ 5 = 3, remainder 2. Two children still need a seat. They need another car. So 4 cars. This is to round up. You go up to the next whole number.

Seventeen dots for children. Three groups of 5 are labelled 3 full cars. A fourth group of 2 dots is labelled car 4.3 full carscar 4
The 2 children left over need a car. Round up to 4 cars.

Story two. 17 eggs go in boxes. A box holds 5. How many full boxes?

Again 17 ÷ 5 = 3, remainder 2. Two eggs do not fill a box. So 3 full boxes. This is to round down. You drop the remainder.

Seventeen stones stand for the eggs. Three rings of 5 stones, one ring for each box, are labelled 3 full boxes. Two stones outside the rings are labelled left over.3 full boxesleft over
The 2 eggs left over do not fill a box. Round down to 3 boxes.

See it three ways

With things

Put out 17 stones. Take away groups of 5 until you cannot take another.

As a picture

Three groups of 5 stones. Two more stones sit apart from the groups, labelled remainder 2.remainder 2

In symbols

17 ÷ 5 = 3, remainder 2. Check: 3 × 5 + 2 = 17.

Try it

Share 14 stones two ways

You need: 14 stones, beans or sticks.

  1. Put out 14 stones.
  2. Make groups of 4. Count the full groups. Count the stones left over.
  3. Put them all back. Now make groups of 7.
  4. Count the full groups. Count the stones left over.

What you will see: Groups of 4 leave 2 stones. Groups of 7 leave none. The remainder is 0 when a number shares out exactly.

If you cannot try it, this is what happens.
Two pictures of 14 stones. The first has 3 groups of 4 and 2 stones apart, labelled 3 groups of 4, remainder 2. The second has 2 groups of 7 and nothing apart, labelled 2 groups of 7, remainder 0.3 groups of 4, remainder 22 groups of 7, remainder 0

A wrong idea many people have

The remainder is always dropped. 17 children in cars of 5 need 3 cars.

Why it feels right: The sum 17 ÷ 5 gives 3. So 3 looks like the answer.

The test that breaks it: Put 17 stones in 3 groups of 5. Two stones are in no group. In the story, two children would have no seat.

What is true: Read what the story asks. If the rest still need a place, round up. If only full groups count, round down.

Key words

remainder
what is left over when things do not share out evenly.
round up
to go up to the next whole number.
round down
to drop the remainder and keep the whole number.

Back to your guess

The answer is B: There are 3 groups of 5, and 2 stones left over.

Three groups of 5 use 15 stones. Two are left over. That is the remainder. A story then tells you what to do with it.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Put 20 stones in groups of 6. How many stones are left over?

    Tap your answer.

    1. Yes, that is right. 3 groups of 6 use 18 stones. 20 − 18 = 2 left over.

    2. Not this one. That is the number of full groups. 3 groups of 6 use 18 stones. Now take 18 from 20. Try another.

    3. Not this one. You took away one group of 6. Keep making groups until no full group is left. Try another.

    See the answer

    2

    3 groups of 6 use 18 stones. 20 − 18 = 2 left over.

    • If you chose B: That is the number of full groups. 3 groups of 6 use 18 stones. Now take 18 from 20.
    • If you chose C: You took away one group of 6. Keep making groups until no full group is left.

  2. Work out 29 ÷ 5. Say the groups and the remainder.

    See the answer

    5 groups, remainder 4

    5 × 5 = 25. 29 − 25 = 4. The remainder is smaller than 5, so it is right.

  3. Lucía takes 23 children across a river. A boat holds 6. How many boats?

    Tap your answer.

    1. Yes, that is right. 23 ÷ 6 = 3, remainder 5. Five children still need a boat. Round up to 4.

    2. Not this one. You dropped the remainder. 5 children would have no boat. Round up. Try another.

    3. Not this one. Four boats hold 24 children. That is enough for 23. Try another.

    See the answer

    4 boats

    23 ÷ 6 = 3, remainder 5. Five children still need a boat. Round up to 4.

    • If you chose B: You dropped the remainder. 5 children would have no boat. Round up.
    • If you chose C: Four boats hold 24 children. That is enough for 23.

  4. Joseph has 23 sticks. A bundle needs 6 sticks. How many full bundles can he make?

    See the answer

    3 bundles

    23 ÷ 6 = 3, remainder 5. Five sticks cannot make a full bundle. Round down to 3.

  5. From an earlier chapter

    Share 24 beans fairly between 4 friends. How many beans does each friend get?

    See the answer

    6 beans each

    24 ÷ 4 = 6, because 4 × 6 = 24. Nothing is left over.

  6. Stones go in groups of 5. Which number can be a remainder?

    Tap your answer.

    1. Not this one. With 5 left over you could make one more group. A remainder is smaller than the group. Try another.

    2. Yes, that is right. The remainder is always smaller than the group size. Only 3 is smaller than 5.

    3. Not this one. With 7 left over you could make one more group of 5. A remainder is smaller than 5. Try another.

    See the answer

    3

    The remainder is always smaller than the group size. Only 3 is smaller than 5.

    • If you chose A: With 5 left over you could make one more group. A remainder is smaller than the group.
    • If you chose C: With 7 left over you could make one more group of 5. A remainder is smaller than 5.

  7. Sipho has 31 beans. A pot needs 7 beans. How many full pots can he fill?

    Tap your answer.

    1. Not this one. You rounded up. A 5th pot would hold only 3 beans. It is not full. Try another.

    2. Not this one. The 3 is the remainder, the beans left over. Count the full pots instead. Try another.

    3. Yes, that is right. 4 × 7 = 28. 31 − 28 = 3 beans are left. Only full pots count, so round down to 4.

    See the answer

    4 pots

    4 × 7 = 28. 31 − 28 = 3 beans are left. Only full pots count, so round down to 4.

    • If you chose A: You rounded up. A 5th pot would hold only 3 beans. It is not full.
    • If you chose B: The 3 is the remainder, the beans left over. Count the full pots instead.

  8. A boat holds 4 people. How many boats do these people need?

    Dots for people waiting to cross a river, in two rows. The label is people waiting.people waiting
    Tap your answer.

    1. Not this one. You dropped the remainder. 3 people would be left on the bank. Round up. Try another.

    2. Not this one. The 4 is how many one boat holds. Count how many boats everyone needs. Try another.

    3. Yes, that is right. 11 ÷ 4 = 2, remainder 3. Three people still need a boat. Round up to 3.

    See the answer

    3 boats

    11 ÷ 4 = 2, remainder 3. Three people still need a boat. Round up to 3.

    • If you chose A: You dropped the remainder. 3 people would be left on the bank. Round up.
    • If you chose B: The 4 is how many one boat holds. Count how many boats everyone needs.

  9. Which number makes groups of 4 with nothing left over?

    Tap your answer.

    1. Not this one. 14 is even, but 3 groups of 4 use 12. Two are left over. Try another.

    2. Not this one. 34 ends in 4. It is not in the 4 times table. 8 groups of 4 use 32. Two are left over. Try another.

    3. Yes, that is right. 5 × 4 = 20. So 5 groups of 4 use all 20. None are left over.

    See the answer

    20

    5 × 4 = 20. So 5 groups of 4 use all 20. None are left over.

    • If you chose A: 14 is even, but 3 groups of 4 use 12. Two are left over.
    • If you chose B: 34 ends in 4. It is not in the 4 times table. 8 groups of 4 use 32. Two are left over.

Keep it

Copy this into your exercise book.

The remainder is what is left over. Groups times size, plus remainder, gives the start. 17 ÷ 5 = 3, remainder 2. In a story, round up if the rest need a place.

Draw: 17 stones. Ring 3 groups of 5. Show the 2 left over. Write 3, remainder 2.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (905 KB). It opens on any phone.