Year 5: Multiplying by a two-digit number

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   23 = 20 + 3
   14 = 10 + 4
2. Draw the grid. Put the parts of 23 across the top and the parts of 14 down the side. Each box is one multiplication.

   ×	20	3
   10	?	?
   4	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   20 × 10: 2 × 1 = 2, then 2 zeros.  200
   3 × 10: 3 × 1 = 3, then 1 zero.  30
   20 × 4: 2 × 4 = 8, then 1 zero.  80
   3 × 4 = 12
4. Write the partial products in the grid. There is one answer for every box.

   ×	20	3
   10	200	30
   4	80	12

5. Add along each row. First add the boxes in each row.
   Row 10: 200 + 30 = 230
   Row 4: 80 + 12 = 92
6. Add the row totals. Now add the two rows together.
   230 + 92 = 322
7. Write the answer.
   23 × 14 = 322

Check. Estimate: 23 is about 20; 14 is about 10. 20 × 10 = 200. Our answer, 322, is a bit more than 200. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   48 = 40 + 8
   32 = 30 + 2
2. Draw the grid. Put the parts of 48 across the top and the parts of 32 down the side. Each box is one multiplication.

   ×	40	8
   30	?	?
   2	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   40 × 30: 4 × 3 = 12, then 2 zeros.  1,200
   8 × 30: 8 × 3 = 24, then 1 zero.  240
   40 × 2: 4 × 2 = 8, then 1 zero.  80
   8 × 2 = 16
4. Write the partial products in the grid. There is one answer for every box.

   ×	40	8
   30	1,200	240
   2	80	16

5. Add along each row. First add the boxes in each row.
   Row 30: 1,200 + 240 = 1,440
   Row 2: 80 + 16 = 96
6. Add the row totals. Now add the two rows together.
   1,440 + 96 = 1,536
7. Write the answer.
   48 × 32 = 1,536

Check. Estimate: 48 is about 50; 32 is about 30. 50 × 30 = 1,500. Our answer, 1,536, is close to 1,500. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   65 = 60 + 5
   27 = 20 + 7
2. Draw the grid. Put the parts of 65 across the top and the parts of 27 down the side. Each box is one multiplication.

   ×	60	5
   20	?	?
   7	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   60 × 20: 6 × 2 = 12, then 2 zeros.  1,200
   5 × 20: 5 × 2 = 10, then 1 zero.  100
   60 × 7: 6 × 7 = 42, then 1 zero.  420
   5 × 7 = 35
4. Write the partial products in the grid. There is one answer for every box.

   ×	60	5
   20	1,200	100
   7	420	35

5. Add along each row. First add the boxes in each row.
   Row 20: 1,200 + 100 = 1,300
   Row 7: 420 + 35 = 455
6. Add the row totals. Now add the two rows together.
   1,300 + 455 = 1,755
7. Write the answer.
   65 × 27 = 1,755

Check. Estimate: 65 is about 70; 27 is about 30. 70 × 30 = 2,100. Our answer, 1,755, is close to 2,100. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   79 = 70 + 9
   53 = 50 + 3
2. Draw the grid. Put the parts of 79 across the top and the parts of 53 down the side. Each box is one multiplication.

   ×	70	9
   50	?	?
   3	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   70 × 50: 7 × 5 = 35, then 2 zeros.  3,500
   9 × 50: 9 × 5 = 45, then 1 zero.  450
   70 × 3: 7 × 3 = 21, then 1 zero.  210
   9 × 3 = 27
4. Write the partial products in the grid. There is one answer for every box.

   ×	70	9
   50	3,500	450
   3	210	27

5. Add along each row. First add the boxes in each row.
   Row 50: 3,500 + 450 = 3,950
   Row 3: 210 + 27 = 237
6. Add the row totals. Now add the two rows together.
   3,950 + 237 = 4,187
7. Write the answer.
   79 × 53 = 4,187

Check. Estimate: 79 is about 80; 53 is about 50. 80 × 50 = 4,000. Our answer, 4,187, is close to 4,000. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   145 = 100 + 40 + 5
   23 = 20 + 3
2. Draw the grid. Put the parts of 145 across the top and the parts of 23 down the side. Each box is one multiplication.

   ×	100	40	5
   20	?	?	?
   3	?	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   100 × 20: 1 × 2 = 2, then 3 zeros.  2,000
   40 × 20: 4 × 2 = 8, then 2 zeros.  800
   5 × 20: 5 × 2 = 10, then 1 zero.  100
   100 × 3: 1 × 3 = 3, then 2 zeros.  300
   40 × 3: 4 × 3 = 12, then 1 zero.  120
   5 × 3 = 15
4. Write the partial products in the grid. There is one answer for every box.

   ×	100	40	5
   20	2,000	800	100
   3	300	120	15

5. Add along each row. First add the boxes in each row.
   Row 20: 2,000 + 800 + 100 = 2,900
   Row 3: 300 + 120 + 15 = 435
6. Add the row totals. Now add the two rows together.
   2,900 + 435 = 3,335
7. Write the answer.
   145 × 23 = 3,335

Check. Estimate: 145 is about 100; 23 is about 20. 100 × 20 = 2,000. Our answer, 3,335, is a bit more than 2,000. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   236 = 200 + 30 + 6
   41 = 40 + 1
2. Draw the grid. Put the parts of 236 across the top and the parts of 41 down the side. Each box is one multiplication.

   ×	200	30	6
   40	?	?	?
   1	?	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   200 × 40: 2 × 4 = 8, then 3 zeros.  8,000
   30 × 40: 3 × 4 = 12, then 2 zeros.  1,200
   6 × 40: 6 × 4 = 24, then 1 zero.  240
   200 × 1: 2 × 1 = 2, then 2 zeros.  200
   30 × 1: 3 × 1 = 3, then 1 zero.  30
   6 × 1 = 6
4. Write the partial products in the grid. There is one answer for every box.

   ×	200	30	6
   40	8,000	1,200	240
   1	200	30	6

5. Add along each row. First add the boxes in each row.
   Row 40: 8,000 + 1,200 + 240 = 9,440
   Row 1: 200 + 30 + 6 = 236
6. Add the row totals. Now add the two rows together.
   9,440 + 236 = 9,676
7. Write the answer.
   236 × 41 = 9,676

Check. Estimate: 236 is about 200; 41 is about 40. 200 × 40 = 8,000. Our answer, 9,676, is a bit more than 8,000. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts. A zero digit has no part, so it has no box.
   308 = 300 + 8
   52 = 50 + 2
2. Draw the grid. Put the parts of 308 across the top and the parts of 52 down the side. Each box is one multiplication.

   ×	300	8
   50	?	?
   2	?	?

3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   300 × 50: 3 × 5 = 15, then 3 zeros.  15,000
   8 × 50: 8 × 5 = 40, then 1 zero.  400
   300 × 2: 3 × 2 = 6, then 2 zeros.  600
   8 × 2 = 16
4. Write the partial products in the grid. There is one answer for every box.

   ×	300	8
   50	15,000	400
   2	600	16

5. Add along each row. First add the boxes in each row.
   Row 50: 15,000 + 400 = 15,400
   Row 2: 600 + 16 = 616
6. Add the row totals. Now add the two rows together.
   15,400 + 616 = 16,016
7. Write the answer.
   308 × 52 = 16,016

Check. Estimate: 308 is about 300; 52 is about 50. 300 × 50 = 15,000. Our answer, 16,016, is close to 15,000. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 26 means multiplying by 6 and by 20, then adding. Write the numbers in columns.
   26 = 20 + 6, so 34 × 26 = (34 × 6) + (34 × 20)
   

   		3	4
   ×		2	6

2. Row 1: multiply by the 6 (the ones digit). This is short multiplication, starting with the ones. Work out 34 × 6.
   Ones: 4 × 6 = 24. Write 4 and carry 2.
   Tens: 3 × 6 = 18, and 18 + 2 (carried) = 20. Write 0 and carry 2.
   Hundreds: nothing left to multiply. Write the carried 2.
   Row 1 = 204
3. Row 2: write a 0 first, then multiply by the 2. The 2 is really 20, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 34 by 2.
   Write 0 in the ones column.
   Ones: 4 × 2 = 8. Write 8.
   Tens: 3 × 2 = 6. Write 6.
   Row 2 = 68 with a 0 on the end = 680
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		3	4
   ×		2	6
   	2	0	4	34 × 6
   +	6	8	0	34 × 20
   	8	8	4

   
   204 + 680 = 884
5. Write the answer.
   34 × 26 = 884

Check. Estimate: 34 is about 30; 26 is about 30. 30 × 30 = 900. Our answer, 884, is close to 900. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 2, not by 20, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 43 means multiplying by 3 and by 40, then adding. Write the numbers in columns.
   43 = 40 + 3, so 57 × 43 = (57 × 3) + (57 × 40)
   

   			5	7
   ×			4	3

2. Row 1: multiply by the 3 (the ones digit). This is short multiplication, starting with the ones. Work out 57 × 3.
   Ones: 7 × 3 = 21. Write 1 and carry 2.
   Tens: 5 × 3 = 15, and 15 + 2 (carried) = 17. Write 7 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   Row 1 = 171
3. Row 2: write a 0 first, then multiply by the 4. The 4 is really 40, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 57 by 4.
   Write 0 in the ones column.
   Ones: 7 × 4 = 28. Write 8 and carry 2.
   Tens: 5 × 4 = 20, and 20 + 2 (carried) = 22. Write 2 and carry 2.
   Hundreds: nothing left to multiply. Write the carried 2.
   Row 2 = 228 with a 0 on the end = 2,280
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			5	7
   ×			4	3
   		1	7	1	57 × 3
   +	2	2	8	0	57 × 40
   	2	4	5	1

   
   171 + 2,280 = 2,451
5. Write the answer.
   57 × 43 = 2,451

Check. Estimate: 57 is about 60; 43 is about 40. 60 × 40 = 2,400. Our answer, 2,451, is close to 2,400. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 4, not by 40, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 75 means multiplying by 5 and by 70, then adding. Write the numbers in columns.
   75 = 70 + 5, so 68 × 75 = (68 × 5) + (68 × 70)
   

   			6	8
   ×			7	5

2. Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 68 × 5.
   Ones: 8 × 5 = 40. Write 0 and carry 4.
   Tens: 6 × 5 = 30, and 30 + 4 (carried) = 34. Write 4 and carry 3.
   Hundreds: nothing left to multiply. Write the carried 3.
   Row 1 = 340
3. Row 2: write a 0 first, then multiply by the 7. The 7 is really 70, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 68 by 7.
   Write 0 in the ones column.
   Ones: 8 × 7 = 56. Write 6 and carry 5.
   Tens: 6 × 7 = 42, and 42 + 5 (carried) = 47. Write 7 and carry 4.
   Hundreds: nothing left to multiply. Write the carried 4.
   Row 2 = 476 with a 0 on the end = 4,760
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			6	8
   ×			7	5
   		3	4	0	68 × 5
   +	4	7	6	0	68 × 70
   	5	1	0	0

   
   340 + 4,760 = 5,100
5. Write the answer.
   68 × 75 = 5,100

Check. Estimate: 68 is about 70; 75 is about 80. 70 × 80 = 5,600. Our answer, 5,100, is close to 5,600. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 7, not by 70, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 38 means multiplying by 8 and by 30, then adding. Write the numbers in columns.
   38 = 30 + 8, so 92 × 38 = (92 × 8) + (92 × 30)
   

   			9	2
   ×			3	8

2. Row 1: multiply by the 8 (the ones digit). This is short multiplication, starting with the ones. Work out 92 × 8.
   Ones: 2 × 8 = 16. Write 6 and carry 1.
   Tens: 9 × 8 = 72, and 72 + 1 (carried) = 73. Write 3 and carry 7.
   Hundreds: nothing left to multiply. Write the carried 7.
   Row 1 = 736
3. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 92 by 3.
   Write 0 in the ones column.
   Ones: 2 × 3 = 6. Write 6.
   Tens: 9 × 3 = 27. Write 7 and carry 2.
   Hundreds: nothing left to multiply. Write the carried 2.
   Row 2 = 276 with a 0 on the end = 2,760
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			9	2
   ×			3	8
   		7	3	6	92 × 8
   +	2	7	6	0	92 × 30
   	3	4	9	6

   
   736 + 2,760 = 3,496
5. Write the answer.
   92 × 38 = 3,496

Check. Estimate: 92 is about 90; 38 is about 40. 90 × 40 = 3,600. Our answer, 3,496, is close to 3,600. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 35 means multiplying by 5 and by 30, then adding. Write the numbers in columns.
   35 = 30 + 5, so 246 × 35 = (246 × 5) + (246 × 30)
   

   		2	4	6
   ×			3	5

2. Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 246 × 5.
   Ones: 6 × 5 = 30. Write 0 and carry 3.
   Tens: 4 × 5 = 20, and 20 + 3 (carried) = 23. Write 3 and carry 2.
   Hundreds: 2 × 5 = 10, and 10 + 2 (carried) = 12. Write 2 and carry 1.
   Thousands: nothing left to multiply. Write the carried 1.
   Row 1 = 1,230
3. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 246 by 3.
   Write 0 in the ones column.
   Ones: 6 × 3 = 18. Write 8 and carry 1.
   Tens: 4 × 3 = 12, and 12 + 1 (carried) = 13. Write 3 and carry 1.
   Hundreds: 2 × 3 = 6, and 6 + 1 (carried) = 7. Write 7.
   Row 2 = 738 with a 0 on the end = 7,380
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		2	4	6
   ×			3	5
   	1	2	3	0	246 × 5
   +	7	3	8	0	246 × 30
   	8	6	1	0

   
   1,230 + 7,380 = 8,610
5. Write the answer.
   246 × 35 = 8,610

Check. Estimate: 246 is about 200; 35 is about 40. 200 × 40 = 8,000. Our answer, 8,610, is close to 8,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 28 means multiplying by 8 and by 20, then adding. Write the numbers in columns.
   28 = 20 + 8, so 417 × 28 = (417 × 8) + (417 × 20)
   

   			4	1	7
   ×				2	8

2. Row 1: multiply by the 8 (the ones digit). This is short multiplication, starting with the ones. Work out 417 × 8.
   Ones: 7 × 8 = 56. Write 6 and carry 5.
   Tens: 1 × 8 = 8, and 8 + 5 (carried) = 13. Write 3 and carry 1.
   Hundreds: 4 × 8 = 32, and 32 + 1 (carried) = 33. Write 3 and carry 3.
   Thousands: nothing left to multiply. Write the carried 3.
   Row 1 = 3,336
3. Row 2: write a 0 first, then multiply by the 2. The 2 is really 20, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 417 by 2.
   Write 0 in the ones column.
   Ones: 7 × 2 = 14. Write 4 and carry 1.
   Tens: 1 × 2 = 2, and 2 + 1 (carried) = 3. Write 3.
   Hundreds: 4 × 2 = 8. Write 8.
   Row 2 = 834 with a 0 on the end = 8,340
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			4	1	7
   ×				2	8
   		3	3	3	6	417 × 8
   +		8	3	4	0	417 × 20
   	1	1	6	7	6

   
   3,336 + 8,340 = 11,676
5. Write the answer.
   417 × 28 = 11,676

Check. Estimate: 417 is about 400; 28 is about 30. 400 × 30 = 12,000. Our answer, 11,676, is close to 12,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 2, not by 20, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 64 means multiplying by 4 and by 60, then adding. Write the numbers in columns.
   64 = 60 + 4, so 583 × 64 = (583 × 4) + (583 × 60)
   

   			5	8	3
   ×				6	4

2. Row 1: multiply by the 4 (the ones digit). This is short multiplication, starting with the ones. Work out 583 × 4.
   Ones: 3 × 4 = 12. Write 2 and carry 1.
   Tens: 8 × 4 = 32, and 32 + 1 (carried) = 33. Write 3 and carry 3.
   Hundreds: 5 × 4 = 20, and 20 + 3 (carried) = 23. Write 3 and carry 2.
   Thousands: nothing left to multiply. Write the carried 2.
   Row 1 = 2,332
3. Row 2: write a 0 first, then multiply by the 6. The 6 is really 60, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 583 by 6.
   Write 0 in the ones column.
   Ones: 3 × 6 = 18. Write 8 and carry 1.
   Tens: 8 × 6 = 48, and 48 + 1 (carried) = 49. Write 9 and carry 4.
   Hundreds: 5 × 6 = 30, and 30 + 4 (carried) = 34. Write 4 and carry 3.
   Thousands: nothing left to multiply. Write the carried 3.
   Row 2 = 3,498 with a 0 on the end = 34,980
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			5	8	3
   ×				6	4
   		2	3	3	2	583 × 4
   +	3	4	9	8	0	583 × 60
   	3	7	3	1	2

   
   2,332 + 34,980 = 37,312
5. Write the answer.
   583 × 64 = 37,312

Check. Estimate: 583 is about 600; 64 is about 60. 600 × 60 = 36,000. Our answer, 37,312, is close to 36,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 6, not by 60, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 47 means multiplying by 7 and by 40, then adding. Write the numbers in columns.
   47 = 40 + 7, so 729 × 47 = (729 × 7) + (729 × 40)
   

   			7	2	9
   ×				4	7

2. Row 1: multiply by the 7 (the ones digit). This is short multiplication, starting with the ones. Work out 729 × 7.
   Ones: 9 × 7 = 63. Write 3 and carry 6.
   Tens: 2 × 7 = 14, and 14 + 6 (carried) = 20. Write 0 and carry 2.
   Hundreds: 7 × 7 = 49, and 49 + 2 (carried) = 51. Write 1 and carry 5.
   Thousands: nothing left to multiply. Write the carried 5.
   Row 1 = 5,103
3. Row 2: write a 0 first, then multiply by the 4. The 4 is really 40, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 729 by 4.
   Write 0 in the ones column.
   Ones: 9 × 4 = 36. Write 6 and carry 3.
   Tens: 2 × 4 = 8, and 8 + 3 (carried) = 11. Write 1 and carry 1.
   Hundreds: 7 × 4 = 28, and 28 + 1 (carried) = 29. Write 9 and carry 2.
   Thousands: nothing left to multiply. Write the carried 2.
   Row 2 = 2,916 with a 0 on the end = 29,160
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			7	2	9
   ×				4	7
   		5	1	0	3	729 × 7
   +	2	9	1	6	0	729 × 40
   	3	4	2	6	3

   
   5,103 + 29,160 = 34,263
5. Write the answer.
   729 × 47 = 34,263

Check. Estimate: 729 is about 700; 47 is about 50. 700 × 50 = 35,000. Our answer, 34,263, is close to 35,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 4, not by 40, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 26 means multiplying by 6 and by 20, then adding. Write the numbers in columns.
   26 = 20 + 6, so 1,534 × 26 = (1,534 × 6) + (1,534 × 20)
   

   		1	5	3	4
   ×				2	6

2. Row 1: multiply by the 6 (the ones digit). This is short multiplication, starting with the ones. Work out 1,534 × 6.
   Ones: 4 × 6 = 24. Write 4 and carry 2.
   Tens: 3 × 6 = 18, and 18 + 2 (carried) = 20. Write 0 and carry 2.
   Hundreds: 5 × 6 = 30, and 30 + 2 (carried) = 32. Write 2 and carry 3.
   Thousands: 1 × 6 = 6, and 6 + 3 (carried) = 9. Write 9.
   Row 1 = 9,204
3. Row 2: write a 0 first, then multiply by the 2. The 2 is really 20, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 1,534 by 2.
   Write 0 in the ones column.
   Ones: 4 × 2 = 8. Write 8.
   Tens: 3 × 2 = 6. Write 6.
   Hundreds: 5 × 2 = 10. Write 0 and carry 1.
   Thousands: 1 × 2 = 2, and 2 + 1 (carried) = 3. Write 3.
   Row 2 = 3,068 with a 0 on the end = 30,680
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		1	5	3	4
   ×				2	6
   		9	2	0	4	1,534 × 6
   +	3	0	6	8	0	1,534 × 20
   	3	9	8	8	4

   
   9,204 + 30,680 = 39,884
5. Write the answer.
   1,534 × 26 = 39,884

Check. Estimate: 1,534 is about 2,000; 26 is about 30. 2,000 × 30 = 60,000. Our answer, 39,884, is a bit less than 60,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 2, not by 20, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 34 means multiplying by 4 and by 30, then adding. Write the numbers in columns.
   34 = 30 + 4, so 2,718 × 34 = (2,718 × 4) + (2,718 × 30)
   

   		2	7	1	8
   ×				3	4

2. Row 1: multiply by the 4 (the ones digit). This is short multiplication, starting with the ones. Work out 2,718 × 4.
   Ones: 8 × 4 = 32. Write 2 and carry 3.
   Tens: 1 × 4 = 4, and 4 + 3 (carried) = 7. Write 7.
   Hundreds: 7 × 4 = 28. Write 8 and carry 2.
   Thousands: 2 × 4 = 8, and 8 + 2 (carried) = 10. Write 0 and carry 1.
   Ten thousands: nothing left to multiply. Write the carried 1.
   Row 1 = 10,872
3. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 2,718 by 3.
   Write 0 in the ones column.
   Ones: 8 × 3 = 24. Write 4 and carry 2.
   Tens: 1 × 3 = 3, and 3 + 2 (carried) = 5. Write 5.
   Hundreds: 7 × 3 = 21. Write 1 and carry 2.
   Thousands: 2 × 3 = 6, and 6 + 2 (carried) = 8. Write 8.
   Row 2 = 8,154 with a 0 on the end = 81,540
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		2	7	1	8
   ×				3	4
   	1	0	8	7	2	2,718 × 4
   +	8	1	5	4	0	2,718 × 30
   	9	2	4	1	2

   
   10,872 + 81,540 = 92,412
5. Write the answer.
   2,718 × 34 = 92,412

Check. Estimate: 2,718 is about 3,000; 34 is about 30. 3,000 × 30 = 90,000. Our answer, 92,412, is close to 90,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 57 means multiplying by 7 and by 50, then adding. Write the numbers in columns.
   57 = 50 + 7, so 4,096 × 57 = (4,096 × 7) + (4,096 × 50)
   

   			4	0	9	6
   ×					5	7

2. Row 1: multiply by the 7 (the ones digit). This is short multiplication, starting with the ones. Work out 4,096 × 7.
   Ones: 6 × 7 = 42. Write 2 and carry 4.
   Tens: 9 × 7 = 63, and 63 + 4 (carried) = 67. Write 7 and carry 6.
   Hundreds: 0 × 7 = 0, and 0 + 6 (carried) = 6. Write 6.
   Thousands: 4 × 7 = 28. Write 8 and carry 2.
   Ten thousands: nothing left to multiply. Write the carried 2.
   Row 1 = 28,672
3. Row 2: write a 0 first, then multiply by the 5. The 5 is really 50, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 4,096 by 5.
   Write 0 in the ones column.
   Ones: 6 × 5 = 30. Write 0 and carry 3.
   Tens: 9 × 5 = 45, and 45 + 3 (carried) = 48. Write 8 and carry 4.
   Hundreds: 0 × 5 = 0, and 0 + 4 (carried) = 4. Write 4.
   Thousands: 4 × 5 = 20. Write 0 and carry 2.
   Ten thousands: nothing left to multiply. Write the carried 2.
   Row 2 = 20,480 with a 0 on the end = 204,800
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			4	0	9	6
   ×					5	7
   		2	8	6	7	2	4,096 × 7
   +	2	0	4	8	0	0	4,096 × 50
   	2	3	3	4	7	2

   
   28,672 + 204,800 = 233,472
5. Write the answer.
   4,096 × 57 = 233,472

Check. Estimate: 4,096 is about 4,000; 57 is about 60. 4,000 × 60 = 240,000. Our answer, 233,472, is close to 240,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 5, not by 50, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. 30 is 3 tens. Multiply by 3, then by 10.

1. Split 30 into 3 × 10. Multiplying by 30 is the same as multiplying by 3 and then by 10.
   30 = 3 × 10
2. Multiply by 3. Use short multiplication.
   Ones: 6 × 3 = 18. Write 8 and carry 1.
   Tens: 4 × 3 = 12, and 12 + 1 (carried) = 13. Write 3 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   46 × 3 = 138
3. Multiply by 10. Move every digit one place to the left and write a 0 place holder.
   138 × 10 = 1,380
4. Write the answer.
   46 × 30 = 1,380

Check. Estimate: 46 is about 50. 50 × 30 = 1,500. Our answer, 1,380, is close to 1,500. ✓

Watch out. The 0 in 30 matters. Do not forget to multiply by 10 at the end.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. 40 is 4 tens. Multiply by 4, then by 10.

1. Split 40 into 4 × 10. Multiplying by 40 is the same as multiplying by 4 and then by 10.
   40 = 4 × 10
2. Multiply by 4. Use short multiplication.
   Ones: 5 × 4 = 20. Write 0 and carry 2.
   Tens: 2 × 4 = 8, and 8 + 2 (carried) = 10. Write 0 and carry 1.
   Hundreds: 1 × 4 = 4, and 4 + 1 (carried) = 5. Write 5.
   125 × 4 = 500
3. Multiply by 10. Move every digit one place to the left and write a 0 place holder.
   500 × 10 = 5,000
4. Write the answer.
   125 × 40 = 5,000

Check. Estimate: 125 is about 100. 100 × 40 = 4,000. Our answer, 5,000, is a bit more than 4,000. ✓

Watch out. The 0 in 40 matters. Do not forget to multiply by 10 at the end.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. 60 is 6 tens. Multiply by 6, then by 10.

1. Split 60 into 6 × 10. Multiplying by 60 is the same as multiplying by 6 and then by 10.
   60 = 6 × 10
2. Multiply by 6. Use short multiplication.
   Ones: 8 × 6 = 48. Write 8 and carry 4.
   Tens: 0 × 6 = 0, and 0 + 4 (carried) = 4. Write 4.
   Hundreds: 3 × 6 = 18. Write 8 and carry 1.
   Thousands: nothing left to multiply. Write the carried 1.
   308 × 6 = 1,848
3. Multiply by 10. Move every digit one place to the left and write a 0 place holder.
   1,848 × 10 = 18,480
4. Write the answer.
   308 × 60 = 18,480

Check. Estimate: 308 is about 300. 300 × 60 = 18,000. Our answer, 18,480, is close to 18,000. ✓

Watch out. The 0 in 60 matters. Do not forget to multiply by 10 at the end.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. 20 is 2 tens. Multiply by 2, then by 10.

1. Split 20 into 2 × 10. Multiplying by 20 is the same as multiplying by 2 and then by 10.
   20 = 2 × 10
2. Multiply by 2. Use short multiplication.
   Ones: 0 × 2 = 0. Write 0.
   Tens: 0 × 2 = 0. Write 0.
   Hundreds: 5 × 2 = 10. Write 0 and carry 1.
   Thousands: 1 × 2 = 2, and 2 + 1 (carried) = 3. Write 3.
   1,500 × 2 = 3,000
3. Multiply by 10. Move every digit one place to the left and write a 0 place holder.
   3,000 × 10 = 30,000
4. Write the answer.
   1,500 × 20 = 30,000

Check. Estimate: 1,500 is about 2,000. 2,000 × 20 = 40,000. Our answer, 30,000, is a bit less than 40,000. ✓

Watch out. The 0 in 20 matters. Do not forget to multiply by 10 at the end.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Row 1 is 426 × 5. Multiply by the ones digit with short multiplication.

1. Multiply 426 by 5. Short multiplication: start at the right and carry when you need to.
   Ones: 6 × 5 = 30. Write 0 and carry 3.
   Tens: 2 × 5 = 10, and 10 + 3 (carried) = 13. Write 3 and carry 1.
   Hundreds: 4 × 5 = 20, and 20 + 1 (carried) = 21. Write 1 and carry 2.
   Thousands: nothing left to multiply. Write the carried 2.
2. Write the row. This is the first row.
   Row = 2,130

Check. 2,130 is about 400 × 5 = 2,000. ✓

Watch out. Add each carry after multiplying that column.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Row 2 is 426 × 30. Write a 0 place holder, then multiply by the tens digit.

1. Write the place holder. This row is for the tens digit, so it is worth tens. Write 0 in the ones column first.
   Write 0 in the ones column.
2. Multiply 426 by 3. Short multiplication: start at the right and carry when you need to.
   Ones: 6 × 3 = 18. Write 8 and carry 1.
   Tens: 2 × 3 = 6, and 6 + 1 (carried) = 7. Write 7.
   Hundreds: 4 × 3 = 12. Write 2 and carry 1.
   Thousands: nothing left to multiply. Write the carried 1.
3. Write the row. Put the digits in front of the 0.
   Row = 12,780

Check. 12,780 is about 400 × 30 = 12,000. ✓

Watch out. The second row always ends in a 0 place holder.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Multiply by the ones digit on one row. Write a 0 place holder, then multiply by the tens digit on the next row. Add the two rows.

1. Split the multiplier. Multiplying by 35 means multiplying by 5 and by 30, then adding. Write the numbers in columns.
   35 = 30 + 5, so 426 × 35 = (426 × 5) + (426 × 30)
   

   			4	2	6
   ×				3	5

2. Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 426 × 5.
   Ones: 6 × 5 = 30. Write 0 and carry 3.
   Tens: 2 × 5 = 10, and 10 + 3 (carried) = 13. Write 3 and carry 1.
   Hundreds: 4 × 5 = 20, and 20 + 1 (carried) = 21. Write 1 and carry 2.
   Thousands: nothing left to multiply. Write the carried 2.
   Row 1 = 2,130
3. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 426 by 3.
   Write 0 in the ones column.
   Ones: 6 × 3 = 18. Write 8 and carry 1.
   Tens: 2 × 3 = 6, and 6 + 1 (carried) = 7. Write 7.
   Hundreds: 4 × 3 = 12. Write 2 and carry 1.
   Thousands: nothing left to multiply. Write the carried 1.
   Row 2 = 1,278 with a 0 on the end = 12,780
4. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			4	2	6
   ×				3	5
   		2	1	3	0	426 × 5
   +	1	2	7	8	0	426 × 30
   	1	4	9	1	0

   
   2,130 + 12,780 = 14,910
5. Write the answer.
   426 × 35 = 14,910

Check. Estimate: 426 is about 400; 35 is about 40. 400 × 40 = 16,000. Our answer, 14,910, is close to 16,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Each box in the grid is the column number multiplied by the row number. Find the numbers for this box, then multiply.

1. Find the numbers for the box. Look along the top of the grid and down the side.
   Top: 40.   Side: 30.
2. Multiply them. Multiply the digits, then count the zeros of both numbers and put them back.
   40 × 30: 4 × 3 = 12, then 2 zeros.  1,200

Check. 1,200 ÷ 30 = 40. ✓

Watch out. Do not forget the zeros: 30 × 40 is 1,200, not 12.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Each box in the grid is the column number multiplied by the row number. Find the numbers for this box, then multiply.

1. Find the numbers for the box. Look along the top of the grid and down the side.
   Top: 7.   Side: 6.
2. Multiply them. Multiply the digits, then count the zeros of both numbers and put them back.
   7 × 6 = 42

Check. 42 ÷ 6 = 7. ✓

Watch out. Do not forget the zeros: 30 × 40 is 1,200, not 12.

Key words. To partition a number is to split it into parts, such as 47 = 40 + 7. A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each one.

Method. Split both numbers into place-value parts. Multiply every part by every part (one box each), then add all the boxes.

1. Partition both numbers. Split each number into its place-value parts.
   47 = 40 + 7
   36 = 30 + 6
2. Draw the grid. Put the parts of 47 across the top and the parts of 36 down the side. Each box is one multiplication.
3. Multiply to fill every box. Multiply the digits, then count the zeros of BOTH numbers and put them back.
   40 × 30: 4 × 3 = 12, then 2 zeros.  1,200
   7 × 30: 7 × 3 = 21, then 1 zero.  210
   40 × 6: 4 × 6 = 24, then 1 zero.  240
   7 × 6 = 42
4. Write the partial products in the grid. There is one answer for every box.
5. Add along each row. First add the boxes in each row.
   Row 30: 1,200 + 210 = 1,410
   Row 6: 240 + 42 = 282
6. Add the row totals. Now add the two rows together.
   1,410 + 282 = 1,692
7. Write the answer.
   47 × 36 = 1,692

Check. Estimate: 47 is about 50; 36 is about 40. 50 × 40 = 2,000. Our answer, 1,692, is close to 2,000. ✓

Watch out. Count the boxes: (number of parts in the top number) × (number of parts in the side number). Missing one box gives a wrong answer. And keep the zeros.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Choose the operation, then work it out.

1. Choose the operation. There are 28 equal groups of 53, so multiply.
   53 × 28
2. Split the multiplier. Multiplying by 28 means multiplying by 8 and by 20, then adding. Write the numbers in columns.
   28 = 20 + 8, so 53 × 28 = (53 × 8) + (53 × 20)
   

   			5	3
   ×			2	8

3. Row 1: multiply by the 8 (the ones digit). This is short multiplication, starting with the ones. Work out 53 × 8.
   Ones: 3 × 8 = 24. Write 4 and carry 2.
   Tens: 5 × 8 = 40, and 40 + 2 (carried) = 42. Write 2 and carry 4.
   Hundreds: nothing left to multiply. Write the carried 4.
   Row 1 = 424
4. Row 2: write a 0 first, then multiply by the 2. The 2 is really 20, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 53 by 2.
   Write 0 in the ones column.
   Ones: 3 × 2 = 6. Write 6.
   Tens: 5 × 2 = 10. Write 0 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   Row 2 = 106 with a 0 on the end = 1,060
5. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   			5	3
   ×			2	8
   		4	2	4	53 × 8
   +	1	0	6	0	53 × 20
   	1	4	8	4

   
   424 + 1,060 = 1,484
6. Write the answer. Include the unit.
   1,484 passengers

Check. Estimate: 53 is about 50; 28 is about 30. 50 × 30 = 1,500. Our answer, 1,484, is close to 1,500. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 2, not by 20, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Choose the operation, then work it out.

1. Choose the operation. There are 25 equal groups of 144, so multiply.
   144 × 25
2. Split the multiplier. Multiplying by 25 means multiplying by 5 and by 20, then adding. Write the numbers in columns.
   25 = 20 + 5, so 144 × 25 = (144 × 5) + (144 × 20)
   

   		1	4	4
   ×			2	5

3. Row 1: multiply by the 5 (the ones digit). This is short multiplication, starting with the ones. Work out 144 × 5.
   Ones: 4 × 5 = 20. Write 0 and carry 2.
   Tens: 4 × 5 = 20, and 20 + 2 (carried) = 22. Write 2 and carry 2.
   Hundreds: 1 × 5 = 5, and 5 + 2 (carried) = 7. Write 7.
   Row 1 = 720
4. Row 2: write a 0 first, then multiply by the 2. The 2 is really 20, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 144 by 2.
   Write 0 in the ones column.
   Ones: 4 × 2 = 8. Write 8.
   Tens: 4 × 2 = 8. Write 8.
   Hundreds: 1 × 2 = 2. Write 2.
   Row 2 = 288 with a 0 on the end = 2,880
5. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		1	4	4
   ×			2	5
   		7	2	0	144 × 5
   +	2	8	8	0	144 × 20
   	3	6	0	0

   
   720 + 2,880 = 3,600
6. Write the answer. Include the unit.
   3,600 pencils

Check. Estimate: 144 is about 100; 25 is about 30. 100 × 30 = 3,000. Our answer, 3,600, is close to 3,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 2, not by 20, and the answer is ten times too small.

Key words. Long multiplication is a written method for multiplying by a number with two or more digits. You multiply by each digit of the second number on its own row, then add the rows. A place holder zero goes at the end of the tens row, because that row is worth tens.

Method. Choose the operation, then work it out.

1. Choose the operation. There are 36 equal groups of 235, so multiply.
   235 × 36
2. Split the multiplier. Multiplying by 36 means multiplying by 6 and by 30, then adding. Write the numbers in columns.
   36 = 30 + 6, so 235 × 36 = (235 × 6) + (235 × 30)
   

   		2	3	5
   ×			3	6

3. Row 1: multiply by the 6 (the ones digit). This is short multiplication, starting with the ones. Work out 235 × 6.
   Ones: 5 × 6 = 30. Write 0 and carry 3.
   Tens: 3 × 6 = 18, and 18 + 3 (carried) = 21. Write 1 and carry 2.
   Hundreds: 2 × 6 = 12, and 12 + 2 (carried) = 14. Write 4 and carry 1.
   Thousands: nothing left to multiply. Write the carried 1.
   Row 1 = 1,410
4. Row 2: write a 0 first, then multiply by the 3. The 3 is really 30, so this row is worth tens. The 0 in the ones column is a place holder. Now multiply 235 by 3.
   Write 0 in the ones column.
   Ones: 5 × 3 = 15. Write 5 and carry 1.
   Tens: 3 × 3 = 9, and 9 + 1 (carried) = 10. Write 0 and carry 1.
   Hundreds: 2 × 3 = 6, and 6 + 1 (carried) = 7. Write 7.
   Row 2 = 705 with a 0 on the end = 7,050
5. Add the two rows. The sum of the rows is the answer, because 26 = 20 + 6.

   		2	3	5
   ×			3	6
   	1	4	1	0	235 × 6
   +	7	0	5	0	235 × 30
   	8	4	6	0

   
   1,410 + 7,050 = 8,460
6. Write the answer. Include the unit.
   8,460 books

Check. Estimate: 235 is about 200; 36 is about 40. 200 × 40 = 8,000. Our answer, 8,460, is close to 8,000. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it you are multiplying by 3, not by 30, and the answer is ten times too small.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

1. Try some numbers. Test the statement with some examples.
   Try 347 × 26: second row 6,940 (ends in 0).
   Try 58 × 43: second row 2,320 (ends in 0).
2. Explain why. A reason shows what is happening for every number, not just the ones we tried.
   The second row is the number multiplied by the tens digit, written with a place holder. It is a multiple of 10, so it ends in 0.
3. Decide.
   Always

Check. Both second rows end in 0. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

1. Try some numbers. Test the statement with some examples.
   12 × 11 = 132 (3 digits).
   34 × 26 = 884 (3 digits).
   45 × 32 = 1,440 (4 digits).
2. Look for one that does not work. One example that breaks the rule is called a counter-example.
   The first two have 3 digits, so it is not always 4 digits.
3. Explain why. A reason shows what is happening for every number, not just the ones we tried.
   The smallest 2-digit × 2-digit is 10 × 10 = 100 (3 digits). The biggest is 99 × 99 = 9,801 (4 digits). So the answer has 3 or 4 digits.
4. Decide.
   Sometimes

Check. We found both kinds of answer. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.