Chapter 1 of 8

All 8 chapters
  1. 1Multiplying and dividing in your head, readunderstood
  2. 2Short multiplication, readunderstood
  3. 3Long multiplication, readunderstood
  4. 4Short division, readunderstood
  5. 5Long division, readunderstood
  6. 6Order of operations, readunderstood
  7. 7Estimate, then check, readunderstood
  8. 8Choosing what to do, readunderstood

M3.2.1 Chapter 1

Multiplying and dividing in your head

How can I work out 25 × 16 without writing it down?

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 and divide by 10, 100 and 1 000
  • Use doubling and halving to work out 25 × 16
  • Multiply multiples of ten, like 30 × 40

Kofi plants 16 rows of seeds. Each row has 25 seeds. How many seeds is that? You can find out in your head, with three short tricks.

The first trick is about place value. When you multiply by 10, every digit moves one place to the left. A 0 fills the empty ones place.

Draw it first

Move the digits to multiply by 10 and 100

  1. A place value chart with the columns thousands, hundreds, tens and ones, shown three times. First it shows 35. Then it shows 35 × 10 = 350. Last it shows 35 × 100 = 3 500. Each time the digits move to the left.35thousandshundredstensones35

    1 Write 35 in the chart. The 3 is in the tens place and the 5 is in the ones place.

  2. A place value chart with the columns thousands, hundreds, tens and ones, shown three times. First it shows 35. Then it shows 35 × 10 = 350. Last it shows 35 × 100 = 3 500. Each time the digits move to the left.35 × 10 = 350thousandshundredstensones350

    2 Multiply by 10. Both digits move one place to the left, and a 0 fills the ones place. The answer is 350.

  3. A place value chart with the columns thousands, hundreds, tens and ones, shown three times. First it shows 35. Then it shows 35 × 10 = 350. Last it shows 35 × 100 = 3 500. Each time the digits move to the left.35 × 100 = 3 500thousandshundredstensones3500

    3 Multiply by 100. Both digits move two places to the left, and two zeros fill the gap. The answer is 3 500.

To divide by 10, 100 or 1 000, move the digits the other way, to the right. 4 500 ÷ 100 = 45, because each digit moves two places to the right.

Two place value charts with the columns thousands, hundreds, tens and ones. The top chart shows 4 500. The bottom chart shows 4 500 ÷ 100 = 45. The digits have moved two places to the right.4 500thousandshundredstensones45004 500 ÷ 100 = 45thousandshundredstensones45
Dividing by 100 moves each digit two places to the right.

Worked example

Work out 4 500 ÷ 100.

  1. Dividing by 100 moves every digit two places to the right.
  2. The 4 moves from thousands to tens. The 5 moves from hundreds to ones.
  3. The two zeros move off the end and are not needed.
  4. So 4 500 ÷ 100 = 45.

Answer: 45

The second trick is doubling and halving. To double a number is to make it twice as big. To halve it is to cut it into two equal parts.

A table with two columns, first and second. The rows are 25 and 16, then 50 and 8, then 100 and 4, then 200 and 2. In every row the two numbers multiply to 400.firstsecond251650810042002
Each row doubles the first number and halves the second. Every row multiplies to 400.

Worked example

Work out 25 × 16 by doubling and halving.

  1. Write the calculation as 25 × 16.
  2. Halve 16 to make 8, and double 25 to make 50.
  3. Do it again. Halve 8 to make 4, and double 50 to make 100.
  4. Now the calculation is 100 × 4. You can work it out at once: 100 × 4 = 400.
  5. So 25 × 16 = 400.

Answer: 400

The third trick is for multiples of ten, such as 30 and 40. Leave out the zeros at the end of each number. Multiply the other digits. Then put those zeros back.

See it three ways

With things

A farmer plants 30 rows of 40 trees. Think of 30 as 3 tens and 40 as 4 tens.

As a picture

A grid method table with one cell. The column is 40 and the row is 30. The cell holds the product 1 200. Under it the picture says total 1 200 and not to scale.40301 200total 1 200not to scale

In symbols

3 × 4 = 12, so 30 × 40 = 1 200. Two zeros go back after the 12.

Worked example

Work out 30 × 40.

  1. 30 × 40 means 3 tens × 4 tens. First work out 3 × 4 = 12.
  2. Each number had one zero, so the answer needs two zeros after the 12.
  3. Put them back. 12 with two zeros after it is 1 200.
  4. Check another way. 30 × 4 = 120.
  5. Then multiply by 10. 120 × 10 = 1 200.

Answer: 1 200

Try it

Double one, halve the other

You need: 24 small stones, beans or buttons.

  1. Lay the stones in 4 rows of 6. Count them.
  2. Move row 2 to the end of row 1. Move row 4 to the end of row 3. Now you have 2 rows of 12.
  3. Count again, without moving a stone.

What you will see: There are 24 both times. You halved the number of rows and doubled the length of each row. So 4 × 6 = 2 × 12.

If you cannot try it, this is what happens.
Two rectangles of tiles with 24 tiles each. The first has 4 rows of 6 and is labelled 4 rows of 6. The second has 2 rows of 12 and is labelled 2 rows of 12.4 rows of 62 rows of 12

A wrong idea many people have

To keep the answer the same, double both numbers.

Why it feels right: Doubling and halving are both fair changes, so doubling twice feels fair too.

The test that breaks it: Try it on 25 × 16 = 400. Double both and you get 50 × 32. That is 1 600, which is four times 400.

What is true: Double one number and halve the other. If you double both, the answer becomes four times as big.

Key words

doubling
making a number twice as big.
halving
cutting a number into two equal parts.
multiples of ten
numbers you get by multiplying a whole number by 10, such as 30, 40 and 700.

Back to your guess

The answer is B: Double 25 and halve 16, again and again. The answer is 400.

Doubling one number and halving the other keeps the answer the same. 25 × 16 = 50 × 8 = 100 × 4 = 400. The first idea adds only 6, but you need 6 more lots of 25.

Check yourself

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

  1. Work out 67 × 100.

    See the answer

    6 700

    Multiplying by 100 moves each digit two places to the left. 67 becomes 6 700.

  2. Work out 8 000 ÷ 100.

    See the answer

    80

    Dividing by 100 moves each digit two places to the right. 8 000 becomes 80.

  3. Which calculation has the same answer as 25 × 16?

    Tap your answer.

    1. Yes, that is right. Double 25 to make 50 and halve 16 to make 8. 50 × 8 = 400, the same as 25 × 16.

    2. Not this one. You doubled both numbers. That makes the answer four times bigger. 50 × 32 = 1 600, not 400. Go to M3.2.1. Try another.

    3. Not this one. You halved 16 but did not double 25. That halves the answer. 25 × 8 = 200, not 400. Go to M3.2.1. Try another.

    4. Not this one. 25 is odd, so it does not halve into a whole number. 12 × 32 = 384, not 400. Go to M3.2.1. Try another.

    See the answer

    50 × 8

    Double 25 to make 50 and halve 16 to make 8. 50 × 8 = 400, the same as 25 × 16.

    • If you chose B: You doubled both numbers. That makes the answer four times bigger. 50 × 32 = 1 600, not 400. Go to M3.2.1.
    • If you chose C: You halved 16 but did not double 25. That halves the answer. 25 × 8 = 200, not 400. Go to M3.2.1.
    • If you chose D: 25 is odd, so it does not halve into a whole number. 12 × 32 = 384, not 400. Go to M3.2.1.

  4. How do the digits of 62 move when you multiply 62 by 1 000?

    Tap your answer.

    1. Not this one. Multiplying is not adding. Multiplying by 1 000 makes each digit worth 1 000 times as much. Go to M3.2.1. Try another.

    2. Not this one. That is multiplying by 10. Each zero in 1 000 moves the digits one more place, and 1 000 has three zeros. Go to M3.2.1. Try another.

    3. Yes, that is right. 1 000 has three zeros, so each digit moves three places to the left. 62 becomes 62 000.

    4. Not this one. Moving to the right makes a number smaller. That is dividing. Multiplying moves the digits to the left. Go to M3.2.1. Try another.

    See the answer

    Three places to the left, so the answer is 62 000.

    1 000 has three zeros, so each digit moves three places to the left. 62 becomes 62 000.

    • If you chose A: Multiplying is not adding. Multiplying by 1 000 makes each digit worth 1 000 times as much. Go to M3.2.1.
    • If you chose B: That is multiplying by 10. Each zero in 1 000 moves the digits one more place, and 1 000 has three zeros. Go to M3.2.1.
    • If you chose D: Moving to the right makes a number smaller. That is dividing. Multiplying moves the digits to the left. Go to M3.2.1.

  5. The chart shows 4 500. What is 4 500 ÷ 10?

    A place value chart with the columns thousands, hundreds, tens and ones, showing 4 500.thousandshundredstensones4500
    Tap your answer.

    1. Not this one. Dividing by 10 always changes the number. Each digit moves one place to the right. Go to M3.2.1. Try another.

    2. Not this one. That is dividing by 100. Dividing by 10 moves the digits only one place to the right. Go to M3.2.1. Try another.

    3. Not this one. The digits moved to the left. That is multiplying by 10. Dividing moves them to the right. Go to M3.2.1. Try another.

    4. Yes, that is right. Dividing by 10 moves each digit one place to the right. 4 500 becomes 450.

    See the answer

    450

    Dividing by 10 moves each digit one place to the right. 4 500 becomes 450.

    • If you chose A: Dividing by 10 always changes the number. Each digit moves one place to the right. Go to M3.2.1.
    • If you chose B: That is dividing by 100. Dividing by 10 moves the digits only one place to the right. Go to M3.2.1.
    • If you chose C: The digits moved to the left. That is multiplying by 10. Dividing moves them to the right. Go to M3.2.1.

  6. Maya works out 35 × 12. She doubles 35 to make 70 and halves 12 to make 6. What is 70 × 6?

    Tap your answer.

    1. Not this one. You may have doubled both numbers, so 70 × 12. Halve 12 when you double 35. 70 × 6 uses the halved number. Go to M3.2.1. Try another.

    2. Yes, that is right. 70 × 6 = 420. This is the same as 35 × 12, because doubling one number and halving the other keeps the answer.

    3. Not this one. You may have used 35 × 6. Both changes are needed: double 35 and halve 12, then multiply 70 by 6. Go to M3.2.1. Try another.

    4. Not this one. You added 70 and 6. The calculation is a multiplication, so work out 70 × 6. Go to M3.2.1. Try another.

    See the answer

    420

    70 × 6 = 420. This is the same as 35 × 12, because doubling one number and halving the other keeps the answer.

    • If you chose A: You may have doubled both numbers, so 70 × 12. Halve 12 when you double 35. 70 × 6 uses the halved number. Go to M3.2.1.
    • If you chose C: You may have used 35 × 6. Both changes are needed: double 35 and halve 12, then multiply 70 by 6. Go to M3.2.1.
    • If you chose D: You added 70 and 6. The calculation is a multiplication, so work out 70 × 6. Go to M3.2.1.

  7. The grid shows the pieces 50 and 60. What goes in the box?

    A grid method table with the column 50 and the row 60, and an empty box. Under it is the label not to scale.5060not to scale
    Tap your answer.

    1. Yes, that is right. Work out 5 × 6 = 30, then put back two zeros. 50 × 60 = 3 000.

    2. Not this one. You worked out 5 × 6 = 30, then put back only one zero. Each number has a zero, so put back two. Go to M3.2.1. Try another.

    3. Not this one. You put back three zeros. 50 and 60 have one zero each, so two zeros go back after 30. Go to M3.2.1. Try another.

    4. Not this one. You added 50 and 60. The box holds the product, so multiply. Go to M3.2.1. Try another.

    See the answer

    3 000

    Work out 5 × 6 = 30, then put back two zeros. 50 × 60 = 3 000.

    • If you chose B: You worked out 5 × 6 = 30, then put back only one zero. Each number has a zero, so put back two. Go to M3.2.1.
    • If you chose C: You put back three zeros. 50 and 60 have one zero each, so two zeros go back after 30. Go to M3.2.1.
    • If you chose D: You added 50 and 60. The box holds the product, so multiply. Go to M3.2.1.

  8. Work out 60 × 70.

    See the answer

    4 200

    First work out 6 × 7 = 42. Then put back the two zeros. 60 × 70 = 4 200.

  9. Use doubling and halving to work out 35 × 14.

    See the answer

    490

    Double 35 to make 70. Halve 14 to make 7. Then 70 × 7 = 490.

Keep it

Copy this into your exercise book.

To multiply by 10, 100 or 1 000, move every digit left. To divide, move them right. Double one number and halve the other. For 30 × 40, work out 3 × 4, then put back two zeros.

Draw: Draw a place value chart with 35 in it, and under it 350. Draw an arrow from each digit to its new place.

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 (792 KB). It opens on any phone.