Multiplication

StandardsThird Grade Common Core3.3 NBT Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Third Grade CA StandardsNS 2.4 Solve simple problems involving multiplication of multi-digit numbers by one-digit numbers (3,671 x 3 = __).

Fourth Grade Common Core4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.4.5.1 NBT - Solve problems involving multiplication of multi-digit numbers by two-digit numbers.

Fourth Grade CA StandardsNS 3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multi digit number by a two-digit number and for dividing a multi digit number by a one-digit number; use relationships between them to simplify computations and to check results.NS 3.3 Solve problems involving multiplication of multi digit numbers by two-digit numbers.

Multiplication• Students should use methods they understand and can explain while

multiplying.– Visual representations such as area models that students draw and connect

to equations are useful for this purpose.

• By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see written algorithms as abbreviations or summaries of their reasoning about quantities.

• In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.

Area models enhance the understanding of geometry and multiplication. This is an area model of 7x5 using built using base ten blocks. The red base ten blocks represent the factors as if we were making a square with a length of 7 and a width of 5. The product is represented by the blue base ten blocks.

Area Models

Build an area model of 10 x 5 using base ten blocks.

10 x 5

Build an area model of 10x10 using base ten blocks.

10 x 10

Build an area model of 12x15 using base ten blocks.

10 x 10

2 x 10

10 x 5

2 x 5

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two

two-digit numbers, using strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

10 + 5

10+

2

10 x 10 = 100 10x5 = 50

10x2= 20 2x5=10

12 15

x

100 + 50 + 20 + 10 = 180

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two

two-digit numbers, using strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

30 + 1

40+

2

1,200 40

60 2

42 31

x

1,200 + 60 + 40 + 2 = 1,302

Connecting the area model to the algorithm

200 + 40 + 6

7

246 7 x

2 4 6 7

x

4

2 4 6 7 2

x

42

2 8

2 4 6 7 2

x

4

8

2

41,

422801,400

2 4 6 7 2

x

4

8

2

41,+ 2 4 0

2271,

1

Two-Step Solution1) Multiply all the numbers and

record the answers.2) Add the numbers from the top

and record the final answer.

3,000 + 200 + 40 + 67

3, 246 7

x

3, 2 4 6 7

x

4

2

422801,40021,000

3, 2 4 6 7 2

x

4

8

2

3, 2 4 6 7 2

x

4

8

2

4

3, 2 4 6 7 2

x

4

8

2

4

11

1,2+ 1, 2 4 0

1

2272,2

Connecting the area model to the algorithmTwo-Step Solution1) Multiply all the numbers and

record the answers.2) Add the numbers from the top

and record the final answer.

2,000 + 300 + 50 + 86

2, 358 6

x

2, 3 5 8 6

x

4

8

483001,80012,000

2, 3 5 8 6 8

x

4

0

3

2, 3 5 8 6 8

x

4

0

3

8

2, 3 5 8 6 8

x

4

0

3

8

11

2,1+ 1, 3 4 0

1

8414,1

Connecting the area model to the algorithmTwo-Step Solution1) Multiply all the numbers and

record the answers.2) Add the numbers from the top

and record the final answer.

2, 3 5 8 6

x

431

1

+ 1, 3 4 0 1 2, 8 0 8

1 4, 1 4 8

2, 3 5 8 6

x

4

84

3

1

2

4,1

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Two-Step Solution1) Multiply all the numbers and

record the answers.2) Add the numbers from the top

and record the final answer.

One-Step SolutionMultiply each number and add the

number from the top before recording a final answer answer.

We are multiplying by a one-digit whole number. Mathematicians prefer to complete this problem in one step.

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two

two-digit numbers, using strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

20 + 3

30+

6

600 90

120 18

36 23

x

600 + 120 + 90 + 18

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

36 23

x

Step one: 3 x 6 = 18Step two: 3 x 30 = 90Step three: 20 x 6 = 120Step four: 20 x 30 = 600Step five: 828

18 (3 x 6) 90 (3 x 30) 120 (20 x 6)

600 (20 x 30)

+ ____ 828

36 x 23 = (30 + 6) x (20 + 3)36 x 23 = (30 + 6) x 20 + (30 + 6) x 336 x 23 = (30 x 20) + (6 x 20) + (30 x 3) + (6 x 3)

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. llustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Step one: 3 x 6 = 18Step two: 3 x 30 = 90Step three: 20 x 6 = 120Step four: 20 x 30 = 600Step five: 828

8

1 36 23

x 36 23

x

8

1

9

36 23

x

8

1

92

1

36 23

x

8

1

92 0

1

608

1

92 0

1

6

817

1

+ 011

828

+

36 23

x

Notice how step five was two addition problems.The next slide will show us how to do step five with only one addition problem.

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two

two-digit numbers, using strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Step one: 3 x 6 = 18Step two: 3 x 30 = 90Step three: 20 x 6 = 120Step four: 20 x 30 = 600Step five: 828

8

1 36 23

x 36 23

x

8

1

0

36 23

x

8

1

02

1

36 23

x

8

1

02 0

1

708

1

02 0

1

7

828

+

36 23

x

1 1 1 1

This is the preferred method, and this is how we will do all future multiplication problems.

4.5 NBT– Multiply a whole number of up to four digits by a one-digit whole number, and multiply two

two-digit numbers, using strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Step one: 6 x 8 = 48Step two: 6 x 40 = 240

Step three: 70 x 8 = 560Step four: 70 x 40 = 2,800

Step five: 3,648

8

4 48 76

x 48 76

x

8

4

8

48 76

x

8

4

86

5

48 76

x

8

4

86 0

5

302 2 2

3

6 x 40 is 240. Then I add the 40 from step one and I have 280. This is 8 tens and 2 hundreds. Notice how 288 is the total of steps one and two below. Why is that?

70 x 40 is 2800. Then I add the 500 from step three and I have 3,300. This is 3 hundreds and 3 thousands. Notice how 3,360 is the total of

steps three and four below. Why is that?

8

4

86 0

5

3

846

+

48 76

x

23

1

3,

What is the best method for solving this problem? Why?

368,423 x 7,286