M3M4 & M5M1 - Developing Area Formulas (1)

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).
M5M1. Students will extend their understanding of area of geometric plane figures.

In the past several posts, I discussed important ideas involved in helping students develop multiplication and division algorithms. With today's post, I want to start a new series on how to help students develop various area formulas (M3M1 & M5M1). In today's post, however, I want to focus my attention on the basic ideas about teaching and learning of area measurement.

I have discussed previously (December, 2008) some basic ideas of teaching measurement. As we teach measuring of any attribute (e.g., length, weight, area, angle, etc.), we must first help students understand the particular attribute we are trying to measure. Without understanding the attribute, measuring it will not make any sense. For that purpose, comparison activities are very useful. Typically, we start with direct comparisons, then move on to indirect comparisons. After students understand the concept, we can start thinking about quantifying the amount of the attribute, i.e., measuring the object. Many people suggest that we start with non-standards units first. One reason for this suggestion is that if we start with standard units, students will have to learn the idea of using units to quantify the attribute and the idea of standard units. Moreover, when we try to measure with standard units, we typically use measurement instruments, such as rulers and protractors. So, students will also have to learn how to use measurement instruments. If we start with non-standard units, students can focus on the notion of quantifying, or measuring, first. Once students understand how a particular attribute can be measured using a non-standard unit, they can use that understanding to both measuring with standard units and learning how to use measurement instruments. After all, the notion of "standard" units probably does not make sense unless you have some experiences with "non-standard" units.

So, what does this all mean when we are teaching the area measurement? Obviously, the first focus should be on helping students understand what area is about. To do so, it seems like we should have students engage in some comparison activities. So, for example, let's ask students to compare the following two rectangles (one is actually a square, but we know that squares are rectangles, don't we?).

Note that I am just showing you the drawing of rectangles, but you should give children cut out pieces to compare. Moreover, I included grid lines to show the dimensions of the rectangles, but you may not want to do so when you give these shapes to students.

Anyway, when students are given these two shapes and asked "Which is bigger?" you see generally two different ways students will compare these shapes. Some children will compare the shapes by overlapping them (on the left below). Others will put the shapes next to each other (on the right).

We can ask students if these two ways of comparing are actually comparing the same attribute (we may not want to use this particular word with 3rd graders). Students may not know, but they can understand that if the conclusions we get from these two ways are different, then, they couldn't be comparing the same attribute. So, have them try comparing these two shapes using both ways. For example, if we put the shapes next to each other and rotate one of them around the other shapes, you see that the two shapes are the same size:

When you overlap shapes, we see that the square is actually "bigger" than the rectangle:

Since the results are different, these two ways of comparison are indeed comparing two different attributes. We know that the first comparison was comparing the length around these shapes, or perimeter, while the second comparison is about area, i.e., the amount of space inside the shapes.

I am not saying that children will understand the difference between the perimeter and the area by doing this one activity. However, it is important for children to have a number of comparison activities to compare the length around and comparing the amount of space inside. When two same objects give different results like the one above several times, students might develop a better sense of the difference between these two attributes.

Once students understand what the area as an attribute is about, we can now move into the discussion of measuring it. So, in the next post, we will discuss M3M1, in which students think about the area of rectangles and squares.

M4N3 - Developing multiplication algorithms (7)

M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

So far, we have discussed the following:
(1) extending the multiplication table to 10x10
(2) multiplying multiples of 10 and 100 by 1-digit numbers
(3) multiplying 2- and 3-digit number by 1-digit numbers
(4) multiplying by multiples of 10.

Now, we are ready to tackle multiplication of 2- and 3-digit numbers by 2-digit numbers. Before we get started, I wanted to say that, to me, teaching of an algorithm means helping students make their own strategies into written procedures instead of imposing a specific algorithm upon students. Of course, that doesn't mean "anything goes." Rather, teachers must think carefully about how to influence students' thinking naturally. Moreover, it may be possible for teachers to sequence students' experiences in such a way that the algorithm students develop "naturally" is something very similar to, or exactly the same as, the conventional algorithm. For that purpose, the area model of multiplication can play a very important role. Therefore, the use of the model along with base-10 blocks before reaching this point is an integral part of the process. So, how do we help students expand their written methods into multiplication of 2- and 3-digit numbers by 2-digit numbers?

Let's think about 12x23 first. How can students use what they have learned so far to think about ways to calculate this problem? There are at least three possible ways. At the most abstract level, students might be able to think of 12x23 as 12x20+12x3 - i.e., 23 groups of 12 can be split into 20 groups of 12 and 3 groups of 12. Then, each of 12x20 and 12x3 are already discussed. If students can think about this way, they can record the process using the vertical notation,
or .

The notation on the right is basically the standard algorithm for multiplication.

Another possibility is for students to go back to the area representation of multiplication. 12x23 means that we are making a rectangle with the dimension of 12 units by 23 units. The product is represented by the area of this rectangle. So, if you construct this rectangle using base-10 blocks, and using the fewest number of blocks (i.e., use large blocks whenever possible), you can make a rectangle like this one:

By examining the arrangement, we see that there are 1 by 2 rectangles made of flats (200), 2 by 2 rectangles of longs (40), 1 by 3 rectangles made of longs (30) and 2 by 3 rectangles of units (6). So, the product is 200+40+30+6=276. After students have become comfortable with the area model representation with base-10 blocks, you may want to encourage students to move toward drawing instead of using actual base-10 blocks. Sometimes you can make this transition simply by giving students multiplication problems with larger factors. Students will realize that actually making rectangles using base-10 blocks is too tedious.

When students become comfortable with drawing rectangles, they might realize that it is still rather tedious. This is when you may be able to suggest if they could use an adaptation of a notation that we used when we were multiplying 2- or 3-digit number by 1-digit number. Some students may be able to start at this point, without going all the way back to using base-10 blocks. That judgment must be made by teachers, using their knowledge of students. Anyway, the notation might look like this for 12x23:

Again, after students have become fluent with this notation, you might want to bring their attention to the four products (in the example here, 200, 40, 30, and 6). Noticing that these are the products of the two tens digits, the tens digit and the ones digit (in both direction) and the two ones digits. So, you can introduce a new notation that records the same information as this diagram does:

You can then negotiate with your students a consistent order in which you calculate these four products (typically called "partial products) so that we can make sure that we have accounted for all of them. If you really want students to understand the conventional multiplication algorithm, you will start with the ones digit of the multiplier (the bottom number) and multiply the ones and then the tens digits of the multiplicand (the top number). You will then multiply the tens digit of the multiplier with the ones and then the tens digits of the multiplicand. So, this problem would look like this:

If you combine the first two partial products and the last two partial products, you will have:

Note that the example we used, 12x23, did not involve any re-grouping. In a way, this is the most "basic" situation. As students move from one notation to another, you may want to consider moving back to a basic situation. Once students become comfortable with the notation (area model, symbolic notation, or whatever), then you want to look at other situations such as those involving re-grouping and a 0 in the factor/product.

When extending the multiplicand to 3-digit numbers, for example, 587x34, you may want to go back to the diagram notation - it will be rather difficult to actually model these multiplication with base-10 blocks. From the diagram, you can move to the notation that will explicitly record all partial products, then eventually to the conventional algorithm.


As usual, you do want to pay close attention to the numbers (factors) you use. Some students have difficulty with 0's - either in the factors or in the product/partial products, so you want to pay particular attention to those situations.

M3N3d - Developing multiplication algorithms (6)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
d.Understand the effect on the product when multiplying by multiples of 10.

This standard talks about multiplying by multiples of 10, for example 37x30. This situation is different from multiplying multiples of 10, 100, etc. (which we have discussed in a previous post) because we now have 30 groups of 37. Now, if we study this idea after students have already developed a paper-and-pencil algorithm, these problems can be considered as a special case where there will be a 0 in the product. So, procedurally, there are different ways to deal with these problems. Some will carry out the calculation exactly in the same manner as they do with other multipliers:

After students get used to this calculation, they might try to combine the steps to make it more efficient:


From this perspective, this multiplication isn't much different from something like 35x18. The important idea is that we have to write a 0 in the ones place as a place holder.

However, M3N3d states that students must understand "the effect on the product when multiplying by multiples of 10." Moreover, according to the GPS, students do not study how to multiply by 2-digit number until Grade 4 (next post). So, it seems rather odd to talk about multiplying by multiples of 10, which are 2-digit number, at this point. If students' don't know how to multiply by 2-digit number, then we can't focus on the procedural aspect discussed above. Rather, we want students to understand what is going on when we multiply by multiples of 10. Although we cannot use the idea of 10 as a unit in the same way as we did when we were multiplying multiples of 10, we can still use the idea of 10 as a unit when the multipliers are multiples of 10. For example, you can think of 37x30 as 37x3x10. Alternately, you can think of 37x30 as 37x10x3. Either way, multiplying a 2- or 3-digit number by 3 is something students have already learned. What students may not have studied is multiplying 2- (or 3-) digit number by 10. So, that seems to be the primary focus of this standard.

As we explore multiplying 2- and 3-digit numbers by 10, we may again want to go back to the area model of multiplication. For example, if students are to model 17x10 using base-10 blocks, they might at first construct something like this by simply extending what they have done previously:

At this point, some might notice that we can actually use a flat on the left side since there are 10 longs. Moreover, on the right side, since there are 10 rows of units, we can replace each column by a long, resulting in an arrangement like this:

Students can also record the process more abstractly like this, too:

They can also consider cases like 40x10 by extending their thinking of 40 as four 10's. If you have 10 groups of four 10's, you can think of that as 4 groups of ten 10's as well, or 100x4.

From these exploration, students may notice that when you multiply 2- and 3-digit numbers by 10, the product will contain the same set of numerals in the same order but every numeral is moved one place to the left - and there is a 0 in the ones place as a place holder.

Although we may be able to consider multiplying by multiples of 10 as a special case of multiplying by 2-digit numbers, students still need to learn the effect of multiplying by 10 before they can explore multiplying by 2-digit numbers. Moreover, once you study the effect of multiplying by 10, extending it to multiplication by multiples of 10 may be useful to help students deepen their understanding of multiplication operation. Although the formal study of properties of multiplication is done in Grade 4, Grade 3 students can use the associative property to reason about the effect of multiplying by multiples of 10. I believe that's the point behind this standard, not just the procedural fluency.