Chapter 8 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

M2.3.8 Chapter 8

Times and divide with bigger numbers

What happens to 35 when I times it by 10?

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

  • Multiply by 10 and 100 and see how the digits move
  • Multiply numbers like 23 × 4 on paper
  • Divide numbers like 84 ÷ 4 on paper

Big numbers use the same ideas as small ones. First, times 10.

The place value of a digit is what it is worth. It depends on where the digit stands.

35 is 3 tens and 5 ones. Ten times as much is 3 hundreds and 5 tens.

Draw it first

Times 10 moves the digits left

  1. A place value chart with the headings hundreds, tens and ones holds 35. The label × 10 sits under it. A second chart with the same headings holds 350. The last label reads 35 × 10 = 350.hundredstensones35

    1 Write 35 on a chart. The 3 is in tens. The 5 is in ones.

  2. A place value chart with the headings hundreds, tens and ones holds 35. The label × 10 sits under it. A second chart with the same headings holds 350. The last label reads 35 × 10 = 350.hundredstensones35× 10hundredstensones350

    2 Times 10. Each digit moves one place left. The 3 is in hundreds. The 5 is in tens.

  3. A place value chart with the headings hundreds, tens and ones holds 35. The label × 10 sits under it. A second chart with the same headings holds 350. The last label reads 35 × 10 = 350.hundredstensones35× 10hundredstensones35035 × 10 = 350

    3 The ones place is empty, so it holds a 0. 35 × 10 = 350.

Times 100 moves every digit two places left. 35 × 100 = 3500.

A place value chart with the headings thousands, hundreds, tens and ones holds 35. The label × 100 sits under it. A second chart with the same headings holds 3500, with the 3 in thousands and the 5 in hundreds.thousandshundredstensones35× 100thousandshundredstensones3500
Times 100 moves each digit two places left. The two empty places hold 0.

Now a number such as 23 × 4. Split 23 into 20 and 3. To split a number into parts like this is to partition it.

Times each part by 4. Then add. We draw a box for each part.

Draw it first

Work out 23 × 4 in boxes

  1. Two boxes in a row. Step 1 writes 23 = 20 + 3 on top, then 20 over the wide box, 3 over the narrow box, and 4 beside both. Step 2 writes 80 in the wide box and 12 in the narrow box. Step 3 writes 80 + 12 = 92 and 23 × 4 = 92.23 = 20 + 32034

    1 Split 23 into 20 and 3. Draw a box for each. Write 4 beside them.

  2. Two boxes in a row. Step 1 writes 23 = 20 + 3 on top, then 20 over the wide box, 3 over the narrow box, and 4 beside both. Step 2 writes 80 in the wide box and 12 in the narrow box. Step 3 writes 80 + 12 = 92 and 23 × 4 = 92.23 = 20 + 320348012

    2 Times each part by 4. 20 × 4 = 80 and 3 × 4 = 12.

  3. Two boxes in a row. Step 1 writes 23 = 20 + 3 on top, then 20 over the wide box, 3 over the narrow box, and 4 beside both. Step 2 writes 80 in the wide box and 12 in the narrow box. Step 3 writes 80 + 12 = 92 and 23 × 4 = 92.23 = 20 + 32034801280 + 12 = 9223 × 4 = 92

    3 Add the two boxes. 80 + 12 = 92. So 23 × 4 = 92.

Worked example

Work out 36 × 5.

  1. Split 36 into 30 and 6.
  2. 30 × 5 = 150.
  3. 6 × 5 = 30.
  4. Add. 150 + 30 = 180.

Answer: 36 × 5 = 180

Dividing works the same way. Split the big number into parts that share out well.

Take 84 ÷ 4. Split 84 into 80 and 4. Both parts divide by 4.

Draw it first

Work out 84 ÷ 4 with blocks

  1. Base ten blocks. Step 1 shows 84 as 8 tens and 4 ones, labelled 84. Step 2 shares them into 4 equal groups of 2 tens and 1 one, each labelled 21, and the line 4 equal groups. Step 3 writes 80 ÷ 4 = 20, 4 ÷ 4 = 1 and 84 ÷ 4 = 20 + 1 = 21.84

    1 Show 84 as 8 tens and 4 ones.

  2. Base ten blocks. Step 1 shows 84 as 8 tens and 4 ones, labelled 84. Step 2 shares them into 4 equal groups of 2 tens and 1 one, each labelled 21, and the line 4 equal groups. Step 3 writes 80 ÷ 4 = 20, 4 ÷ 4 = 1 and 84 ÷ 4 = 20 + 1 = 21.212121214 equal groups

    2 Share them into 4 equal groups. Each group gets 2 tens and 1 one.

  3. Base ten blocks. Step 1 shows 84 as 8 tens and 4 ones, labelled 84. Step 2 shares them into 4 equal groups of 2 tens and 1 one, each labelled 21, and the line 4 equal groups. Step 3 writes 80 ÷ 4 = 20, 4 ÷ 4 = 1 and 84 ÷ 4 = 20 + 1 = 21.212121214 equal groups80 ÷ 4 = 204 ÷ 4 = 184 ÷ 4 = 20 + 1 = 21

    3 Each group is 21. 80 ÷ 4 = 20 and 4 ÷ 4 = 1. So 84 ÷ 4 = 21.

Worked example

Work out 78 ÷ 3.

  1. Split 78 into 60 and 18. Both divide by 3.
  2. 60 ÷ 3 = 20.
  3. 18 ÷ 3 = 6.
  4. Add. 20 + 6 = 26.

Answer: 78 ÷ 3 = 26

See it three ways

With things

Make 4 piles. Each pile has 2 bundles of ten sticks and 3 single sticks. Count the tens, then the ones.

As a picture

Four groups of base ten blocks, each labelled 23. Each group has 2 tens and 3 ones.23232323

In symbols

23 × 4 = 20 × 4 + 3 × 4. 20 × 4 = 80 and 3 × 4 = 12. 80 + 12 = 92.

Try it

Times 10 on a paper chart

You need: a pencil, a sheet of paper, or an old page.

  1. Draw three columns. Write hundreds, tens and ones over them.
  2. Write 27 in the chart. The 2 is in tens. The 7 is in ones.
  3. Now times 10. Move each digit one column to the left.
  4. Write 0 in the empty ones column. Read the number.

What you will see: You read 270. Check it. Add 27 ten times. You also get 270.

If you cannot try it, this is what happens.
A place value chart with the headings hundreds, tens and ones holds 27. The label × 10 sits under it. A second chart with the same headings holds 270.hundredstensones27× 10hundredstensones270

A wrong idea many people have

23 × 4 = 812, because 2 × 4 = 8 and 3 × 4 = 12.

Why it feels right: You times each digit by 4. It seems fair to write the answers side by side.

The test that breaks it: Add 23 four times. 23 + 23 = 46. 46 + 46 = 92. The answer is 92. It is far less than 812.

What is true: The 2 is 2 tens, not 2 ones. 2 tens × 4 = 80. Then add 3 × 4 = 12. 80 + 12 = 92.

Key words

place value
what a digit is worth, because of the place it stands in.
partition
to split a number into parts, such as 20 and 3. The parts use tables you know.

Back to your guess

The answer is B: Each digit moves one place left. It becomes 350.

Times 10 moves every digit one place to the left. The empty ones place holds a 0. So 35 × 10 = 350.

Check yourself

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

  1. What is 46 × 10?

    Tap your answer.

    1. Yes, that is right. 4 tens and 6 ones become 4 hundreds and 6 tens. The ones place holds 0.

    2. Not this one. You added 10. Times 10 moves every digit one place left. Try another.

    3. Not this one. You wrote 10 after the 46. Move the digits left and put 0 in the ones place. Try another.

    See the answer

    460

    4 tens and 6 ones become 4 hundreds and 6 tens. The ones place holds 0.

    • If you chose B: You added 10. Times 10 moves every digit one place left.
    • If you chose C: You wrote 10 after the 46. Move the digits left and put 0 in the ones place.

  2. Mei has 6 bags. Each bag holds 100 beans. How many beans is that?

    See the answer

    600 beans

    Times 100 moves the 6 two places left, from ones to hundreds. 6 × 100 = 600.

  3. Work out 32 × 3. Split 32 into 30 and 2.

    See the answer

    96

    30 × 3 = 90. 2 × 3 = 6. 90 + 6 = 96.

  4. Work out 68 ÷ 4. Split 68 into 40 and 28.

    See the answer

    17

    40 ÷ 4 = 10. 28 ÷ 4 = 7. 10 + 7 = 17.

  5. From an earlier chapter

    Use 8 × 7 = 56 to answer 56 ÷ 7.

    See the answer

    8

    Ask: 7 times what makes 56? 8 × 7 = 56. So 56 ÷ 7 = 8.

  6. The chart shows 4. What is 4 × 100?

    A place value chart with the headings hundreds, tens and ones holds 4 in the ones place.hundredstensones4
    Tap your answer.

    1. Not this one. That is times 10, one place left. Times 100 moves the 4 two places left. Try another.

    2. Not this one. You wrote 100 after the 4. Move the 4 two places left. Fill the empty places with 0. Try another.

    3. Yes, that is right. Times 100 moves each digit two places left. The 4 moves from ones to hundreds. 4 × 100 = 400.

    See the answer

    400

    Times 100 moves each digit two places left. The 4 moves from ones to hundreds. 4 × 100 = 400.

    • If you chose A: That is times 10, one place left. Times 100 moves the 4 two places left.
    • If you chose B: You wrote 100 after the 4. Move the 4 two places left. Fill the empty places with 0.

  7. Tariq works out 34 × 3. Which answer is right?

    Tap your answer.

    1. Not this one. You wrote 9 and 12 side by side. The 3 is 3 tens: 30 × 3 = 90. Then add 4 × 3 = 12. Try another.

    2. Not this one. You forgot that the 3 is 3 tens. 30 × 3 = 90, not 9. Then add 4 × 3 = 12. Try another.

    3. Yes, that is right. Split 34 into 30 and 4. 30 × 3 = 90. 4 × 3 = 12. 90 + 12 = 102.

    See the answer

    102

    Split 34 into 30 and 4. 30 × 3 = 90. 4 × 3 = 12. 90 + 12 = 102.

    • If you chose A: You wrote 9 and 12 side by side. The 3 is 3 tens: 30 × 3 = 90. Then add 4 × 3 = 12.
    • If you chose B: You forgot that the 3 is 3 tens. 30 × 3 = 90, not 9. Then add 4 × 3 = 12.

  8. Work out 96 ÷ 3. Split 96 into 90 and 6. What is the answer?

    Tap your answer.

    1. Not this one. You wrote 30 and 2 side by side. Add them: 30 + 2. Try another.

    2. Yes, that is right. 90 ÷ 3 = 30. 6 ÷ 3 = 2. 30 + 2 = 32.

    3. Not this one. That is only 90 ÷ 3. Divide the 6 by 3 as well, and add. Try another.

    See the answer

    32

    90 ÷ 3 = 30. 6 ÷ 3 = 2. 30 + 2 = 32.

    • If you chose A: You wrote 30 and 2 side by side. Add them: 30 + 2.
    • If you chose C: That is only 90 ÷ 3. Divide the 6 by 3 as well, and add.

  9. The groups of blocks are all the same. How many are there in all?

    Three groups of base ten blocks, each labelled one group. Each group has 1 ten and 4 ones.one groupone groupone group
    Tap your answer.

    1. Not this one. You wrote 3 and 12 side by side. 3 × 10 = 30 and 3 × 4 = 12. Add them. Try another.

    2. Yes, that is right. Each group is 1 ten and 4 ones, which is 14. 3 × 10 = 30 and 3 × 4 = 12. 30 + 12 = 42.

    3. Not this one. You added 3 and 14. Times means 3 groups of 14. Go to M2.3.1. Try another.

    See the answer

    42

    Each group is 1 ten and 4 ones, which is 14. 3 × 10 = 30 and 3 × 4 = 12. 30 + 12 = 42.

    • If you chose A: You wrote 3 and 12 side by side. 3 × 10 = 30 and 3 × 4 = 12. Add them.
    • If you chose C: You added 3 and 14. Times means 3 groups of 14. Go to M2.3.1.

Keep it

Copy this into your exercise book.

Times 10 moves each digit one place left. Split to multiply: 23 × 4 = 80 + 12 = 92. Split to divide: 84 ÷ 4 = 20 + 1 = 21.

Draw: A box grid. Write 20 and 3 across the top and 4 at the side. Write 80 and 12 inside.

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.