Chapter 3 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

M3.2.3 Chapter 3

Long multiplication

How do I work out 346 × 52 on paper?

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 a 3 digit number by a 2 digit number
  • Explain why the second row starts with a zero

A shop has 52 boxes. Each box holds 346 pencils. How many pencils is that? You know how to multiply by one digit. Now the multiplier has two digits.

52 means 50 + 2. So multiply 346 by 2, then multiply 346 by 50, then add. First look at it as a grid.

Draw it first

The grid for 346 × 52

  1. A grid method table. The columns are 300, 40 and 6. The rows are 50 and 2. The top row holds 15 000, 2 000 and 300. The bottom row holds 600, 80 and 12. Under the grid are the words total 17 992 and not to scale.300406502not to scale

    1 Split both numbers by place. 346 is 300, 40 and 6. 52 is 50 and 2.

  2. A grid method table. The columns are 300, 40 and 6. The rows are 50 and 2. The top row holds 15 000, 2 000 and 300. The bottom row holds 600, 80 and 12. Under the grid are the words total 17 992 and not to scale.30040650215 0002 0003006008012not to scale

    2 Multiply each pair. There are six boxes, such as 50 × 300 = 15 000.

  3. A grid method table. The columns are 300, 40 and 6. The rows are 50 and 2. The top row holds 15 000, 2 000 and 300. The bottom row holds 600, 80 and 12. Under the grid are the words total 17 992 and not to scale.30040650215 0002 0003006008012total 17 992not to scale

    3 Add all six boxes. The total is 17 992.

Long multiplication does the same job with two rows. Each row is a partial product. It is part of the answer, not the whole answer.

Draw it first

Long multiplication, row by row

  1. The calculation 346 times 52 written in columns. Row 1 is 692. Row 2 is 17 300, with a ringed 0 in the ones place. The total under the second line is 17 992.64325×

    1 Write 346 above 52, with each digit in its own column. Draw a line.

  2. The calculation 346 times 52 written in columns. Row 1 is 692. Row 2 is 17 300, with a ringed 0 in the ones place. The total under the second line is 17 992.64325×296

    2 Row 1: multiply 346 by the 2 ones. 346 × 2 = 692.

  3. The calculation 346 times 52 written in columns. Row 1 is 692. Row 2 is 17 300, with a ringed 0 in the ones place. The total under the second line is 17 992.64325×29600371+

    3 Row 2: write a 0 first. Then multiply 346 by the 5 tens. 346 × 5 = 1 730, so the row reads 17 300.

  4. The calculation 346 times 52 written in columns. Row 1 is 692. Row 2 is 17 300, with a ringed 0 in the ones place. The total under the second line is 17 992.64325×29600371+29971

    4 Add the rows. 692 + 17 300 = 17 992.

Why does row 2 start with a 0? The 5 is not 5 ones. It is 5 tens. So row 2 is 346 × 50. The 0 is a placeholder. It keeps the ones place empty, so each digit lands in the right place.

Worked example

Work out 346 × 52.

  1. Row 1: multiply 346 by the 2 ones. 346 × 2 = 692.
  2. Row 2: write a 0 in the ones place first. This row is for tens.
  3. Multiply 346 by the 5 tens. 346 × 5 = 1 730. Write 1 730 in front of the 0, so the row reads 17 300.
  4. Add the two rows. 692 + 17 300 = 17 992.

Answer: 17 992

Worked example

Work out 128 × 37.

  1. Row 1: 128 × 7 = 896.
  2. Row 2: start with a 0 in the ones place.
  3. Then 128 × 3 = 384, so row 2 reads 3 840.
  4. Add the rows. 896 + 3 840 = 4 736.

Answer: 4 736

Worked example

Work out 408 × 63.

  1. Row 1: 3 × 408 = 1 224.
  2. Row 2 is for the 6 tens, so it starts with a 0 in the ones place.
  3. Then 6 × 408 = 2 448.
  4. Put the 0 after it. Row 2 reads 24 480.
  5. Add the rows. 1 224 + 24 480 = 25 704.

Answer: 25 704

See it three ways

With things

Make 23 rows of stones with 14 stones in each row. Think of it as 20 rows and 3 rows.

As a picture

A grid method table. The columns are 10 and 4. The rows are 20 and 3. The boxes hold 200 and 80 in the top row, and 30 and 12 in the bottom row. Under the grid are the words total 322 and not to scale.104203200803012total 322not to scale

In symbols

23 × 14 = 20 × 14 + 3 × 14 = 280 + 42 = 322.

Try it

Cut a rectangle in two

You need: squared paper, or a page where you draw 13 squares by 12 squares, a pencil.

  1. Draw a rectangle 13 squares wide and 12 squares tall.
  2. Draw a line to cut off the top 10 rows. The bottom 2 rows are left.
  3. Count the squares in each block by multiplying. Add the two counts.

What you will see: The top block has 10 × 13 = 130 squares. The bottom block has 2 × 13 = 26. Together they make 156, so 12 × 13 = 156.

If you cannot try it, this is what happens.
Two blocks of squares, one above the other, each 13 squares wide. The top block has 10 rows and is labelled 10 rows of 13 = 130. The bottom block has 2 rows and is labelled 2 rows of 13 = 26.10 rows of 13 = 1302 rows of 13 = 26

A wrong idea many people have

The 0 on the second row is only for neatness. You can leave it out.

Why it feels right: The row is already lined up under the first row, so the 0 looks like an extra.

The test that breaks it: Leave the 0 out of 346 × 52. Row 2 becomes 1 730, and 692 + 1 730 = 2 422. But 350 × 50 is about 17 500, so 2 422 is far too small.

What is true: The 5 is worth 50, so the row is 346 × 50. The 0 holds the ones place and moves the other digits to the left.

Key words

partial product
one row of a long multiplication, a part of the full answer.
placeholder
a 0 that holds a place empty, so the other digits stay in the right places.

Back to your guess

The answer is C: Multiply 346 by the 2, then by the 50, and add the two rows. The answer is 17 992.

The 5 in 52 is worth 50, so one row is 346 × 50 = 17 300. The other row is 346 × 2 = 692. The two rows add to 17 992. The first idea treats the 5 as 5 ones. The third forgets the 2.

Check yourself

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

  1. Work out 24 × 13 on paper.

    See the answer

    312

    Row 1: 24 × 3 = 72. Row 2: 24 × 10 = 240. Add the rows. 72 + 240 = 312.

  2. A library has 38 shelves. Each shelf holds 245 books. How many books is that?

    See the answer

    9 310

    Row 1: 245 × 8 = 1 960. Row 2: 245 × 30 = 7 350. Add the rows. 1 960 + 7 350 = 9 310.

  3. What is 245 × 36?

    Tap your answer.

    1. Not this one. This is only row 2, 245 × 30. The 6 ones still need their own row, 245 × 6 = 1 470. Go to M3.2.3. Try another.

    2. Not this one. You may have left out the 0 in row 2. 245 × 3 = 735, but the 3 is worth 30, so row 2 is 7 350, not 735. Go to M3.2.3. Try another.

    3. Not this one. This is only row 1, 245 × 6. The 3 tens still need their own row, 245 × 30 = 7 350. Go to M3.2.3. Try another.

    4. Yes, that is right. Row 1: 245 × 6 = 1 470. Row 2: 245 × 30 = 7 350. Add the rows. 1 470 + 7 350 = 8 820.

    See the answer

    8 820

    Row 1: 245 × 6 = 1 470. Row 2: 245 × 30 = 7 350. Add the rows. 1 470 + 7 350 = 8 820.

    • If you chose A: This is only row 2, 245 × 30. The 6 ones still need their own row, 245 × 6 = 1 470. Go to M3.2.3.
    • If you chose B: You may have left out the 0 in row 2. 245 × 3 = 735, but the 3 is worth 30, so row 2 is 7 350, not 735. Go to M3.2.3.
    • If you chose C: This is only row 1, 245 × 6. The 3 tens still need their own row, 245 × 30 = 7 350. Go to M3.2.3.

  4. Why does the second row of 346 × 52 start with a 0?

    The calculation 346 times 52 written in columns, with row 1 and row 2 shown. Row 2 ends in a ringed 0.64325×29600371+
    Tap your answer.

    1. Yes, that is right. The 5 in 52 means 5 tens. So row 2 is 346 × 50, and the zero holds the ones place.

    2. Not this one. The zero does a job. Without it, the digits of row 2 sit one place too far right. Each is worth ten times too little. Go to M3.2.3. Try another.

    3. Not this one. Not every number times 5 ends in a zero. 347 × 5 = 1 735. The zero is there because the 5 means 5 tens. Go to M3.2.3. Try another.

    4. Not this one. Row 1 is 346 × 2 = 692, which does not end in a zero. The zero belongs to row 2, the row for the tens. Go to M3.2.3. Try another.

    See the answer

    The digit is worth tens. The row is ten times as big, and the zero holds the ones place.

    The 5 in 52 means 5 tens. So row 2 is 346 × 50, and the zero holds the ones place.

    • If you chose B: The zero does a job. Without it, the digits of row 2 sit one place too far right. Each is worth ten times too little. Go to M3.2.3.
    • If you chose C: Not every number times 5 ends in a zero. 347 × 5 = 1 735. The zero is there because the 5 means 5 tens. Go to M3.2.3.
    • If you chose D: Row 1 is 346 × 2 = 692, which does not end in a zero. The zero belongs to row 2, the row for the tens. Go to M3.2.3.

  5. What are the two rows of 215 × 43?

    The calculation 215 times 43 written in columns, with no rows worked out yet.51234×
    Tap your answer.

    1. Not this one. Row 2 is 215 × 40, and 215 × 4 = 860 is only the start. The 4 is worth 40, so row 2 is 8 600. Go to M3.2.3. Try another.

    2. Yes, that is right. Row 1 is 215 × 3 = 645. Row 2 is 215 × 40 = 8 600.

    3. Not this one. You may have started with the 4. Row 1 is for the ones digit, 3. Row 2 is for the 4 tens. Go to M3.2.3. Try another.

    4. Not this one. You put back one zero too many. The 4 is 4 tens, so row 2 is 215 × 40 = 8 600. Go to M3.2.3. Try another.

    See the answer

    645 and 8 600

    Row 1 is 215 × 3 = 645. Row 2 is 215 × 40 = 8 600.

    • If you chose A: Row 2 is 215 × 40, and 215 × 4 = 860 is only the start. The 4 is worth 40, so row 2 is 8 600. Go to M3.2.3.
    • If you chose C: You may have started with the 4. Row 1 is for the ones digit, 3. Row 2 is for the 4 tens. Go to M3.2.3.
    • If you chose D: You put back one zero too many. The 4 is 4 tens, so row 2 is 215 × 40 = 8 600. Go to M3.2.3.

  6. Joseph works out 128 × 37. He writes 896 and 384, and adds them to get 1 280. What went wrong?

    Tap your answer.

    1. Not this one. Check with an estimate. 130 × 40 is about 5 200, so 1 280 is far too small. Go to M3.2.3. Try another.

    2. Not this one. 896 + 384 = 1 280 is right. The slip is in row 2, which needs its zero. Go to M3.2.3. Try another.

    3. Not this one. Row 1 is for the ones digit, 7, so row 1 is 128 × 7 = 896. That row is right. Go to M3.2.3. Try another.

    4. Yes, that is right. The 3 in 37 means 3 tens, so row 2 is 128 × 30 = 3 840. Then 896 + 3 840 = 4 736.

    See the answer

    The second row should be 3 840, not 384. He left out the zero.

    The 3 in 37 means 3 tens, so row 2 is 128 × 30 = 3 840. Then 896 + 3 840 = 4 736.

    • If you chose A: Check with an estimate. 130 × 40 is about 5 200, so 1 280 is far too small. Go to M3.2.3.
    • If you chose B: 896 + 384 = 1 280 is right. The slip is in row 2, which needs its zero. Go to M3.2.3.
    • If you chose C: Row 1 is for the ones digit, 7, so row 1 is 128 × 7 = 896. That row is right. Go to M3.2.3.

  7. The grid shows 346 × 52. Which boxes add up to row 2, the row for the 50?

    A grid method table. The columns are 300, 40 and 6. The rows are 50 and 2. The top row holds 15 000, 2 000 and 300. The bottom row holds 600, 80 and 12. Under the grid are the words not to scale.30040650215 0002 0003006008012not to scale
    Tap your answer.

    1. Yes, that is right. The row for the 50 holds 50 × 300 = 15 000, 50 × 40 = 2 000 and 50 × 6 = 300. They add to 17 300, which is row 2.

    2. Not this one. These are the boxes for the 2, the bottom row. They add up to 692, which is row 1. Go to M3.2.3. Try another.

    3. Not this one. These are the boxes in the first column. Row 2 is the whole row for the 50, across all three columns. Go to M3.2.3. Try another.

    4. Not this one. These are the pieces of 346, the heads of the columns. The boxes below them hold the products. Go to M3.2.3. Try another.

    See the answer

    15 000, 2 000 and 300

    The row for the 50 holds 50 × 300 = 15 000, 50 × 40 = 2 000 and 50 × 6 = 300. They add to 17 300, which is row 2.

    • If you chose B: These are the boxes for the 2, the bottom row. They add up to 692, which is row 1. Go to M3.2.3.
    • If you chose C: These are the boxes in the first column. Row 2 is the whole row for the 50, across all three columns. Go to M3.2.3.
    • If you chose D: These are the pieces of 346, the heads of the columns. The boxes below them hold the products. Go to M3.2.3.

  8. Work out 307 × 48.

    See the answer

    14 736

    Row 1: 307 × 8 = 2 456. Row 2: 307 × 40 = 12 280. Add the rows. 2 456 + 12 280 = 14 736.

  9. From an earlier chapter

    Use the grid method to work out 4 × 1 263.

    See the answer

    5 052

    Split 1 263 into 1 000, 200, 60 and 3. The boxes are 4 000, 800, 240 and 12. 4 000 + 800 + 240 + 12 = 5 052.

Keep it

Copy this into your exercise book.

Multiply by the ones. Then write a 0 and multiply by the tens. Add the two rows. The 0 is there because the second row is worth tens.

Draw: Draw 346 × 52 in columns with the two rows. Ring the 0 at the end of row 2 and write "row for the tens" beside it.

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

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