M3N3c - Developing multiplication algorithms (5)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
In the previous post, I discussed how students can develop a paper-and-pencil algorithm for multiplying 2-digit numbers by 1-digit numbers. Let's consider how we can help students extend the procedure to multiplication of 3-digit numbers by 1-digit number.

How can we multiply 312 x 3? How can students use what they have learned so far to calculate this? One possibility is to think of 312 as 300+12. Then, we can multiply 300x3 and 12x3. Both of these are already learned ideas. If students have already understood how to multiply a 2-digit number by a 1-digit number using a paper-and-pencil method, they can then combine their learning and record this multiplication something like this:

When extending the multiplicand from 2-digit to 3-digit, therefore, there isn't really any new concept involved. Even the idea of looking at 312x3 as 300x3+12x3 is really the same idea as looking at 12x3 as 10x3+2x3, i.e., the distributive property of multiplication, which will be formally studied in Grade 4.

One important thing to think about when we study multiplying 3-digit numbers by 1-digit numbers is different situations where re-grouping must take place, or when there is a 0 (or more) in either the multiplicand or the product. The example we just saw, 312x3, does not involve re-grouping and there is no 0 in the multiplicand nor the product. So, in a way, it is a "general" case of multiplying 3-digit numbers by 1-digit numbers. But, here are some of other cases:
Re-grouping is involved
• 227x3
• 227x5
• 162x3
• etc.

0 is involved
• 406x7
• 365x4
• 527x4
• etc.
I encourage you to think about other cases. As teachers, we must also think about how we want to deal with them. We can carefully sequence those cases and have students think about how they can adapt the written procedure they developed those situations. As you do, it will be helpful if you explicitly ask students what is different about each case compared to the most general one that we start with.

As we look at those special cases, it is important that students understand what is actually happening when we are multiplying 3-digit numbers by 1-digit numbers. For that, it might be useful to go back to the notation system that we used when we developed when we were multiplying 2-digit numbers by 1-digit numbers. For example, let's think about 427x4. Since we can think of 427x4 as 400x4+27x4, and we can use a pictorial notation like this:

Or, we can use more symbolic notation like this (with the previous agreement that we start recording with the partial product of the ones digits first):

We can combine some of the steps involved in this notation and develop a notation like this:

No matter how you approach this topic, what we cannot do is to start with the standard algorithm, which is the most sophisticated way of recording the processes. Help students extend what they have previously learned, which may be the standard algorithm for multiplying 2-digit numbers by 1-digit number by thinking about the structure of numbers and the meaning of operations. If necessary, go back to the intermediate notations that were used while developing the algorithm for multiplying 2-digit number by 1-digit numbers. By experiencing this extension, students can then think about how they can extend the algorithm for multiplying 3-digit numbers by 1-digit numbers to multiplying 4-digit (or even longer) numbers by 1-digit numbers. They have not only the experiences of multiplying two numbers but also the experience of "extending" their procedure from one case to another. So they can ask themselves not just "How did I multiply 2- or 3-digit numbers by 1-digit numbers?" but also "How did I extend the algorithm for multiplying 2-digit numbers to 3-digit numbers?" Therefore, when teaching multiplication of 3-digit numbers by 1-digit numbers, what is important is not the procedure but the idea of how to extend the previously learned procedure (2-digit multiplicands) to a new situation (3-digit multiplicands).

M3N3c - Developing multiplication algorithms (4)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
This is the fourth in a series of posts in which I am discussing the development of multiplication algorithms. Up to this point, students were calculating mentally. The focus has been more on consolidating students' understanding of our number system and the meaning of multiplication by using those understanding to figure out multiplication beyond the basic facts. Today's standard is the first step toward developing paper-and-pencil algorithms. As I begin my post, let me emphasize that teaching of an algorithm for any operation should focus on helping students develop the algorithm on their own. In other words, we need to move away from the show-and-tell approach where teachers show students how to multiply using the multiplication algorithm and then have them practice over and over. Practice is important, but students should first develop the algorithm themselves. Of course, that does NOT mean that we just leave students on their own. Rather, teachers must plan carefully to guide students' thinking.

One useful idea in developing a multiplication algorithm is the area model of multiplication. In Grade 3, students learn about area of rectangles and squares. When students cover a rectangle with unit squares, they notice that they are arranged rows and columns of equal sizes. Because all rows (or columns) are equal, we can use multiplication to efficiently determine the area. This idea can be used to model multiplication where the two factors are represented by the two dimensions of a rectangle and the product is represented by the area. So, for example, 4x6=24 can be modeled as shown below.

Notice that since you can turn the rectangles around without changing the area, this is also a useful model to show why the commutative property of multiplication is true. It is also useful to model the distributive property of multiplication.

When you model multiplication problems like 14x7 using base-10 blocks, you can certainly try to make 7 groups of 14 (1 long and 4 units). However, we want to encourage students to organize the model more systematically using the area model. The area model representation will make it much easier to determine the product by observation if the same type of blocks are grouped together.

Eventually, we want to help students move beyond modeling with actual base-10 blocks. One useful approach to do so is to have students draw what they would have done with base-10 blocks. Thus, drawing the picture like the one above. Grid papers can be very helpful in that process. However, as they become comfortable with drawing pictures, they realize that drawing can be rather tedious. Given our goal is to determine the product, what we want to know is how many longs and how many units we have. Thus, we can model the multiplication explicitly showing only the information we need. Here is an example for 14x7.

Once students become comfortable with modeling multiplying 2-digit number by 1-digit number this way, we can ask if they can think of a way to represent this model using a vertical notation like we did with addition and subtraction. Here are two possibilities:

Students can see that 70+28 and 28+70 are the same. Thus, we can write it either way. At this point, it is ok to suggest that we agree to write the product of the ones digit first. Again, after students practice this notation, they might notice that the ones digit for the partial product of the tens digit on the multiplicand and the multiplier is always 0. Therefore, the ones' digit of the product is always the ones digit of the partial product of the ones digit of the multiplicand and the multiplier, 8 in the example above. Then, we have to add the tens digits of the partial products to find the product. This process can be combined if you use a notation like this:

This may be a slightly different notation than some of us are used to, where the tens digit of the partial product above the tens digit of the multiplicand. That notation sometimes causes students to add the re-grouped digit and the tens digit of the multiplicand before multiplying by the multiplier - that is students end up doing (2+1)x7 instead of 1x7+2. Writing the re-grouped digit below the horizontal bar (the equal sign) might minimize that error.

In the next post, I will discuss how this procedure may be extended to multiplying 3-digit numbers

M3N3 Developing multiplication algorithms (3)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.

This is the third in a series of posts in which I discuss the development of multiplication algorithms in Grades 3 and 4. If you haven't read the first two, I encourage you to do so, either before or after reading this post.

One idea that is important as children continue to develop multiplication algorithms yet not explicitly mentioned in the GPS is the idea of multiplying multiples of 10 and 100, such as 40x7 and 600x3. Note, multiplying by multiples of 10 and 100 is a Grade 4 standard.

So, how can students make sense of multiplying multiples of 10 and 100? So, let's think about 40x7. Again, just a reminder that I am following the Japanese convention of writing the multiplicand (number in a group) first. Therefore, this problem is asking us to find the total amount when there are 40 groups in each and there are 4 groups.

An important idea here is the understanding of 10 and 100 (and 1000) as unit. When students first learned simple addition and subtraction of 2-digit numbers such as 30+40 and 70-20 in Grade 1, they used the idea of 10 as a unit. Since 30 and 40 are made up of 3 and 4 tens, putting those two numbers together meant there are seven 10's, or 70. In a similar way, we can think of 40x7 using 10 as a unit. Since there are 7 groups of 40, or 7 groups of four 10's, we see that there are 7x4=28 tens altogether. Therefore, the product is 280. Similarly, you can think of 600x3 as 3 groups of six 100's, or 3x6=18 hundreds, i.e., 1800.

The idea of 10 and 100 (and 1000) as units was the focus of M2N1(b):
Understand the relative magnitudes of numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones.
Although the second part of the statement emphasizes looking at 3-digit numbers as composed of hundreds, tens, and ones, it is also important for children to understand numbers like 280 as twenty-eight 10's. This notion of relative magnitude ("relative size" in M3N1(b)) plays an important role in students' mathematics learning in the future. Therefore, it is important to help them deepen and consolidate their understanding as we teach multiplication of multiples of tens and hundreds.

M3N3(b) - Developing multiplication algorithms (2)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. b. Know the multiplication facts with understanding and fluency to 10 x 10.
The new idea here is multiplication with 10 as a factor, either the multiplicand or the multiplier. Let's first look at the cases where 10 is used as the multiplicand, i.e., 10x1, 10x2, 10x3, .... These can be interpreted as one 10, two 10's, three 10's, ... respectively. In Grade 1, when students studied the numbers up to 100, this is something they should have encountered. Thus, they can use that particular prior knowledge to figure out what these facts will be, i.e., 10x1=10, 10x2=20, 10x3=30, ....

What about 10 as the multiplier, i.e., 1x10, 2x10, 3x10, .... These means, respectively, ten 1's, ten 2's, ten 3's, .... For adults, these are obvious and we might think they should be obvious to children, too. However, although young students can answer the problem on the left very quickly, many of the same students have much more difficult time with the problem on the right.

So, how can students think about problems like 3x10? Hopefully, when they were constructing their multiplication table, they have used the idea that when the multiplier increases by 1, the product increases by the multiplicand. For example, the answer for 3x6 should be 3 (the multiplicand) more than 3x5. This idea, then, can be used to think about 3x10. The answer to 3x10 should be 3 more than 3x9, which is a part of the basic fact they have learned in Grade 2. This idea is really a particular case of the distributive property, which students will formally study in Grade 4. However, the distributive property plays an important role as students think about how to multiply by larger numbers. Therefore, it may be useful if this idea is discussed explicitly in classrooms.

Some of us grew up memorizing the multiplication table up to 12x12. Even today, some teachers/schools/districts still make their students consider the multiplication table up to 12x12. Although it may have some usefulness in everyday situations to know the multiplication facts of 11's and 12's, there is really no particular mathematical reason for expanding the multiplication table to 12x12. Once students develop an algorithm for multiplying by 2-digit numbers, they can calculate anything beyond 10x10 using the algorithm. On the other hand, students can also use the property of multiplication and think of 11's and 12's as simply 10's and 1's or 10's and 2's. Thus, 7x12 can be thought of as 7x10+7x2 and 12x8 can be thought of as 10x8+2x8. Perhaps it is much more important for students to develop that form of flexible thinking than simply memorizing those facts.

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.