Multiplication

How do we develop this concept with our students?

Basic Representations

Array3 X 5 or 3 • 5 or three by five

3

5X555

3 + 3 + 3 + 3 + 3 = 15

Area Model

3

X5

You will want to move away from drawing an array quickly. The area model is a more efficient way to draw a representation of an array. Filling in each square is time consuming.

Sets

The Properties

DiscoverInvestigateUnderstandCommunicate

Identity Property n • 1 = n

Zero Property n•0=0This can cause conceptual challenges for

students.This property can not be demonstrated as an

array or with the area model. Try it!Show n•0 in sets. Place it in context.

5 hops of 0 on a number lineWater has 0 grams of fat. How many grams of fat

does 5 glasses of water have?

Commutative Property

• The Order Property of Multiplication• Changing the order of the factors does not

change the product.• a • b = b • a• The product of a times b will have the same

value as b times a.

Commutative Property of Multiplication Representations

Commutative Property of Multiplication Representations

Associative Property

The Grouping Property of MultiplicationChanging the way you group the factors does

not change the product.(a • b) • c = a • (b • c)The product of a times b then multiplied by c

will have the same value as b times c and then multiplied by a.

Associative Property of Multiplication

3 sets of 4 is 12. 12 times 2 is 24. There are 24 bags of chips.(3•4) = 12 12 • 2 = 24

Associative Property of Multiplication

4 sets of 2 is 8. 8 times 3 is 24. There are 24 bags of chips.(4•2) = 8 8 • 3 = 24

Multiplying by a power of 10

• This pattern is essential to understand as students move to multiplying larger values.

• Students can discover why I have a certain number of zeros in my product when I multiply by a power of 10.

• Discover what happens to the decimal point.

You DO NOT want to tell them, “Just count the zeros and add them

to your product.”• Look at 5 • 40• 5 • 4 = 20 The zero is already there. Many

students do not see the difference between adding one zero and a factor that has a zero in the ones place.

• Thinking 5 times 4 tens is 20 tens is much more precise.

Build It First. Many, many times.

3 • 1

3 • 1 ten = 3 tens

3 • 2 ten = 6 tens

3 • 4 hundreds

Explore Number Strings3 • 4 = 12 3 • 4 tens = 12 tens= 120

3 • 4 hundreds = 12 hundreds=1200

Finally, look for patterns in number form.

N • 10 N • 100

3 • 1 = 3 3 • 1 = 3

3 • 1ten = 3 tens= 30 3 • 10 = 30

3 • 1 hundred = 3 hundreds= 300

3 • 2 = 6 3 • 2 = 6

3 • 2 tens = 6 tens = 60 3 • 2 0= 60

3 • 2 hundreds = 6 hundreds = 600

Now you can discover the math generalization or rule.

Put it all together. Represent 8 X 7

1.Build it with color tiles.

2.Represent it with pictures.

3.Represent it with numbers

4.Solve it with an algorithm.

32 X 5 Build It

Let’s solve 32X28 with partial products.

30

2

20 8

30X20 30X8

2X20 8X2