Sunday, September 6, 2009

M3N3/M4N3 -- developing multiplication algorithms

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

In the previous 2 posts, I discussed division algorithms. So, in the next few posts, I would like to discuss multiplication algorithms. Today, as the first entry on multiplication algorithm, I want to discuss an overview of teaching and learning multiplication algorithms.

Students are introduced to multiplication in Grade 2. The GPS (M2N3) states that students should construct the multiplication table and correctly multiply 1-digit numbers. What is not quite clear is where multiplication involving 0 as a factor (either the multiplicand or the multiplier) should be discussed. Many US textbooks introduce multiplication with 0 and 1 as factors fairly early on in their discussion of multiplication. In contrast, in the Japanese textbooks, multiplication with 1 as the multiplicand, i.e., 1x1, 1x2, 1x3,..., are discussed AFTER students study the 9's facts. [Note: in the Japanese notation, the first number is the multiplicand, i.e., the number in a group.] They do not discuss 0 as a factor until the 3rd grade. They do this because the emphasis in Grade 2 is developing the meaning of multiplication first. For children, considering 1, or even 0, item as a "group" may be strange. From the equal group perspective of multiplication, therefore, 0 and 1 as the multiplicand are special cases. Therefore, they start with more general cases first (2's through 9's), then discuss the special cases (1's and 0's). Textbooks often treat 0's and 1's early because getting the answers is easy. However, if our focus is on the meaning of multiplication, that may not be a wise choice.

Anyway, after students study 1x1 through 9x9 in Grade 2 (and possibly 0's), students are expected to learn to multiply larger numbers in Grades 3 and 4 (M3N3 and M4N3). So, by the end of the 4th grade, we want students to be able to calculate problems like 512 x 43. Using the conventional algorithm, we can calculate this problem as shown below:

With this algorithm, we can calculate this problem by performing 6 basic multiplication and 5 basic addition. In fact, with our base-10 numeration system, once we learn the basic addition and multiplication facts, we can perform the basic 4 operations with any size numbers. Although this is not explicitly spelled out in the GPS, we would like students to understand this merit of our number system as a result of learning the computational algorithms.

Of course, this is by the end of Grade 4, and we have to think about how to help students go from knowing only the 1-digit multiplication facts to that point. So, how should we organize our instruction? What are some important mile markers in this endeavor?

Here are some important understandings students need.
* We can think of 512x43 as 512x40+512x3.
* 512x3 can be thought of as 500x3+10x3+2x3.
* 512x40 can be thought of as 512x10x4.

The first idea involves the use of the distributive property. Although the formal study of the properties of operations is in Grade 4, students use the distributive property as they construct the multiplication table. For example, they might have thought of 7x6 as7x5+7. Or, they thought of 8x7 as 8x5+8x2. So, this is not a completely new idea. However, multiplying 512x3 certainly is. So, they need to learn how to multiply 2- and 3-digit numbers by 1-digit number. The third idea uses the associative property of multiplication. Students my have used it to find something like 7x4 as 7x2x2. So, the use of property itself may not be new, but 512x10 certainly is. Students must learn how to multiply numbers by 10 before they think about multiplying a number by multiples of 10.

So, from this example, we can see 5 important mile markers of multiplication instruction in Grades 3 and 4.
a. Expand the basic multiplication up to 10x10. [M3N3b]
b. Understand how to multiply multiples of 10 and 100 by 1-digit number (e.g., 30x8, 400x6, etc.).
c. Understand how to multiply 2- and 3-digit numbers (but not multiples of 10 and 100) by 1-digit numbers. [M3N3c]
d. Understand how to multiply by multiples of 10. [M3N3d]
e. Understand how to multiply be general 2-digit numbers [M4N3]

As you can see, all but one mile marker is explicitly noted by the GPS. Starting next entry, I will discuss these 5 mile markers in more details.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.