Multiplication How do we develop this concept with our students?

Basic Representations

Array 3 X 5 or 3 5 or three by five 3 5 X = 15

Area Model 3 X 5 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  Discover  Investigate  Understand  Communicate

Identity Property n 1 = n

Zero Property n0=0  This can cause conceptual challenges for students.  This property can not be demonstrated as an array or with the area model. Try it!  Show n0 in sets. Place it in context.  5 hops of 0 on a number line  Water 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

Associative Property  The Grouping Property of Multiplication  Changing 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 times 2 is 24. There are 24 bags of chips. (34) = = 24

Associative Property of Multiplication 4 sets of 2 is 8. 8 times 3 is 24. There are 24 bags of chips. (42) = = 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 = 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 ten = 3 tens 3 2 ten = 6 tens

3 4 hundreds Explore Number Strings 3 4 = tens = 12 tens= hundreds = 12 hundreds=1200

Finally, look for patterns in number form. N 10N = 3 3 1ten = 3 tens= = hundred = 3 hundreds= = tens = 6 tens = = 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 X2030X8 2X208X2