M5N4(d) - Modeling Multiplication & Division of Fractions

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.
d. Model the multiplication and division of common fractions.

In the last three posts, I discussed multiplication and division of decimal numbers that do not depend on the knowledge of multiplication and division of fractions. That was necessary because in the GPS decimal multiplication and division are discussed prior to fraction multiplication and division. In this post, I would like to discuss multiplication and division of fractions. I have previously discussed this topic (November, 2007). In the post, I briefly discussed how the area model may be used to represent multiplication of fractions, as well as the double number line representation that can be used for both multiplication and division. So, in today's post, I want to focus on how to model division of fractions.

As students are introduced to division operation in Grade 3, they are expected to understand that "division m many equal parts of a given size or amount may be taken away from the who as in repeated subtraction, and the second is determining the size of the parts when the whole is separated into a given number of equal parts as in a sharing model" (M3N4b). We discussed how these interpretations must be extended as the number of "groups" become decimal numbers - whether as the divisor in a fair sharing problem or as the quotient in a measurement division problem.

The situation is basically the same with fraction multiplication and division. If the divisor is a whole number, we can use the fair sharing interpretation. When the divisor becomes a fraction, we must use either the extended meaning of fair sharing, that is, an operation that determines the per-one quantity, or the measurement interpretation. The measurement interpretation is much more easily modeled using manipulatives such as pattern blocks. We can model 3/4 divided by 1/6 this way: First, let's represent a whole using two hexagon pieces together. Then, 3 trapezoids will represent 3/4 and a blue rhombus will represent 1/6. The division question is asking how many blue pieces will fit in the 3 red trapezoids together. You can easily show that 4 blue pieces will fit completely inside the 3 red trapezoids. The remaining section is 1/2 of a blue rhombus. Therefore, the quotient is 4 1/2.

How else can we model division of fractions? In particular, how can we model fraction division that may also reflect the inverse relationship between multiplication and division? In my previous post on this standard, I mentioned how the area model of multiplication may be used to represent multiplication of fractions. In this model, the two dimensions of a rectangle represent the two factors and the product is represented by the area of the rectangle (in relationship to the unit rectangle). Thus, the figure below represents 1/3 x 2/3:

So, is there a way to represent division using the area model? For example, how can we model 3/4÷2/5? [I encourage you to think about how you may be able to represent this division using pattern blocks. You may find it a bit cumbersome.]

Since division is the inverse operation of multiplication, 3/4 must be the area of the rectangle, and the divisior, 3/4, is one of the two dimensions. Thus, we are trying to determine the other dimension of the rectangle so that the area will be 3/4. So, how can we model this? I'm sure that there are different ways, but here is one possibility.

Let's start by first representing 3/4:

Of course, this fraction has the dimension of 1 unit (vertically) by 3/4 units (horizontally). What we want is a rectangle that has the same area as the yellow rectangle but has 2/5 as one of the dimensions. If we say that the vertical dimension is 2/5 units, then we are looking for the horizontal dimension, as shown in the figure below:

So, how can we find the horizontal dimension? First, let's first draw in the segments showing the fifths in the yellow rectangle.

Now, we can see that another set of 2/5 by 3/4 rectangle (shown in green below) can be shifted to fit inside the rectangle whose vertical dimension is 2/5.

Now, two of the remaining 3 small rectangles (shown in blue) can be shifted.

Finally, the remaining small rectangle has to be split into two equal parts (shown in red).

So, how long is the horizontal dimension, which will be the quotient? Each of the small rectangle has the horizontal dimension of 1/4 unit. Clearly, we have 7 1/4-units. Finally, the red segment is a half of the small rectangle, or a half of 1/4. Thus, we have 7/4 and 1/8, or 15/8 altogether.

The figure below shows 1 2/3 ÷ 3/4.



Now, in this situation, the 2 small rectangles (in blue) had to be split into 3 equal parts, so the horizontal segment of the blue segment of the quotient is 2/3 of 1/3-unit, or 2/9. Thus, the quotient is 2 2/9.

Now, in this model, each small rectangle you obtain has the horizontal dimension which is the unit fraction with the denominator for the dividend (4 in the first example and 3 in the second). The total number of the small rectangles in the dividend is the product of the numerator of the dividend and the denominator of the divisor. The number of horizontal column of the unit fraction can be calculated by dividing the total number of the small rectangles by the numerator of the divisor. Thus, the quotient can be expressed as:


In other words, to divide a fraction by another fraction, you simply multiply the dividend by the reciprocal of the divisor. Of course, this generalization may be straightforward for us, but it is extremely important that we analyze what mathematical ideas are involved in making that generalization. Then, we can decide whether or not this generalization is accessible to our students.

In any event, it does raise some questions about why the GPS asks students to model division (and multiplication) of fractions in Grade 5 without specifying the development of the algorithm in the same grade level. As I stated earlier, I believe the appropriate interpretation of the current GPS is that the algorithms are to be developed (and mastered) in Grade 6. However, it seems rather strange to separate modeling from the algorithm development, which is the generalization based on the models.

Finally, I would like to emphasize that the area models are useful when we know the operation involved. The area model cannot help students determine which operation to use. For that purpose, models like double number line are much more suited.

M5N3 Multiplication & Division of Decimal Numbers (3)

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

OK, this is the third (and hopefully the last) in the series of posts discussing multiplication and division of decimal numbers. In the last two posts, we discussed multiplying and dividing decimal numbers by whole numbers and multiplying by decimal numbers. We are developing these ideas using only our understanding of whole number multiplication and a powerful idea about our numeration system, relative size of numbers. What is left for us now is dividing by decimal numbers. Let's go back to our problem:

Problem 4
A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

This problem requires us to divide 3.6 by 2.4. We already looked at dividing decimal numbers by whole numbers, but we have yet to consider division by decimal numbers. In some curricula, fraction arithmetic is discussed first, so we can change this division to division of fractions. However, that line of reasoning is not available if we follow the GPS. So, what can students do?

Whenever students encounter a new problem, we would like them to ask, "What do I know that I can use?" or "How is this problem similar to what I have studied previously?" Such a habit is an example of what the authors of Adding It Up (National Research Council, 200?) call productive disposition. Again, a diagram might help us think about this problem.


One possibility is to think about 2.4 as 24 0.1's as we did before. But, what do we get if we divide 3.6 by 24? Let's see what the diagram will show us:

We can tell from this diagram that the result of dividing 3.6 by 24 is the weight of a 0.1-meter wire. So, how can we find the weight of a 1-meter wire if we know that a 0.1-meter wire weighs 0.15 grams? Since 1 meter is 10 times as long as 0.1 meter, the weight should also be 10 times as much. So, to find the weight of a 1-meter wire, we just need to multiply the weight of a 0.1-meter wire by 10. So, a 1-meter wire will weigh 1.5 grams.

With Problem 3, we also had another approach that considered 10 times of the multiplier. What would a parallel reasoning in Problem 4 be like? If we make the divisor (2.4) into a whole number, what does it mean? That means we are looking at a 24-meter wire, instead of a 2.4-meter wire. Again, it's 10 times as long, therefore, it should weigh 10 times as much, i.e., 36 grams. But if we know that a 24-meter wire weighs 36 grams, we can find the weight of a 1-meter wire by simply dividing 36 by 24. We don't have to do anything with the result since we haven't changed the weight of 1-meter of wire when we considered the weight of the 24-meter wire. A diagram might show this approach clearly:

This second approach may be more useful to generalize a paper-and-pencil algorithm. Basically what we did was to multiply the divisor by a power of 10 to make it into a whole number. Then, the dividend must be multiplied by the same power of 10 - since the length of the wire is now that many times as long, it should weigh also that many times as much. Then, we can simply divide the new weight by the new length, we can find the weight for 1 meter. Therefore,

Another way of describing this process is to move the decimal point of the divisor (the number outside of the long division symbol) as many places as necessary to the right to make it into a whole number. Then, move the decimal point of the dividend the same number of places to the right as well - annexing 0's if necessary. Then, we can perform long division as we have done previously - either a whole number divided by a whole number or a decimal number divided by a whole number. Again, this is the familiar algorithm, isn't it?

As we saw in the three recent posts, the familiar multiplication and division algorithms can be meaningfully derived using only our knowledge of whole numbers and the idea of relative size of numbers. In the Japanese standards, they discuss decimal multiplication and division first because the algorithms are essentially the same as those of whole number multiplication and division. Thus, when students study multiplication and division, they can focus more on extending the meaning of multiplication and division. Then, when students study multiplication and division of fractions, they do not have to worry about dealing with the new meaning of operations AND the new algorithms. It is not clear if the GPS writers had the same intent, but I hope you see how students can develop multiplication and division algorithm for decimal numbers without knowing multiplication and division of fractions.

M5N3 Multiplication & Division of Decimal Numbers (2)

M5N3. Students will further develop their understanding of the meaning of
multiplication and division with decimal fractions and use them.

In the last post, I discussed how the idea of relative size can be used to think about multiplying and dividing decimal numbers by whole numbers - M4N5(d). In this post, I want to continue to the next step, multiplying and dividing by decimal numbers. As I discussed in October, 2007, when the multiplier and the divisor is something other than a whole number, we must extend the meaning of division from an equal-group perspective to a more proportional one. Let's look at the two problems I left as "homework" last time.

Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

Problem 4
A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

Clearly, in Problem 3, we must multiply 2.4 by 1.8, while in Problem 4, we must divide 3.6 by 2.4. Since these situations involve a decimal multiplier and a decimal divisor, we can no longer use the equal group interpretation of multiplication and division - what does 1.5 or 2.4 groups mean? Rather, we must look at these situations more proportionally. In Problem 3, we are asking, if 2.4 is to 1, how much is to 1.8, and in Problem 4, if 3.6 is to 2.4, what is to 1? Alternately, if you use multiple comparison idea, Problem 3 asks how much is 1.5 times as much as 2.4, while Problem 4 asks 3.6 is 2.4 times as much as what?

Let's now think about how students can solve these problems using only what they have learned so far, which does not include how to multiply or divide by decimal numbers.

Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

One possible idea that students might use is to consider the multiplier, 1.8, in terms of the decimal unit using the idea of relative size. That is, 1.8 means there are 18 pieces of 0.1's. But what does that mean? A diagram might be helpful. Using a double number line (November, 2007), we can represent the problem like this:


When we say 1.8 is made up of 18 pieces of 0.1's, the diagram may look like this:


In other words, 1.8 meters can be thought of as a collection of 18 0.1 meter pieces. But, how does that help us find the missing number. We are not multiplying 2.4 by 18 - we don't have 18 groups of 2.4. What do we have 18 groups of on the top number line?



From this diagram, we can tell that what we have 18 of on the top number line is actually the weight of 0.1 meter wire. In other words, if we know how much a 0.1-meter wire weighs, then, we can find the answer. But, it's easy to see that the weight of a 0.1-meter wire can be determined by simply dividing 2.4 by 10, which is what students learned in Grade 4. Once we determine the weight of a 0.1-meter wire, i.e., 0.24 grams, then, we can multiply that by 18, which is also a Grade 4 idea. 0.24 x 18 = 4.32, so the weight of a 1.8-meter wire is 4.32 grams.

Here is another idea that students might come up with. Although we are looking for the weight of a 1.8-meter wire, let's first think about the weight of 18-meter wire, which is easy enough - simply multiply 2.4 by 18, a Grade 4 idea. However, since a 18-meter wire is 10 times as long a 1.8-meter wire, it should also weigh 10 times as much, too. So, in order to determine the weight of a 1.8-meter wire, we can simply divide that by 10 to find its weight. Since we already know how to divide decimal numbers by whole numbers, this last step should not be a problem. This line of reasoning may be represented on a number line like this:


Different students will feel more comfortable with different approaches. However, this second approach may be more useful to generalize into a written computation algorithm. In general, what we do in the first step is to make the multiplier into a whole number by multiplying it by an appropriate power of 10. Now, if the multiplicand is a decimal number, we end up multiplying it by a power of 10 to make it into a whole number as well (that's another way of thinking about the use of relative size). Now that we have two whole numbers, we can multiply them easily. However, this product is too big, and it must be divided by those powers of 10. For example,

Since multiplying by 10 means that the decimal point will move to the right one place while dividing by 10 means moving the decimal point to the left one place, we can describe what happened above this way: when we think of 3.7x4.26 as 37x426, we moved the decimal point 3 places to the right altogether, therefore, we have to move the decimal point to the left 3 places in the product of 37x246 to get the product for 3.7x4.26. And, this is (to us) the familiar multiplication algorithm for decimal numbers, isn't it?

Well, this has gotten a bit too long - of course, with actual 5th graders, you may need several lessons to get this much discussion done. Anyway, I think I must postpone the discussion of dividing by decimal number until next time. However, if you can think about how we solved Problem 3, you may find that Problem 4 can be solved in similar ways.

Revisiting M4N5(d) & M5N3 - Multiplication and Division of Decimal Numbers (1)

M4N5. Students will further develop their understanding of the meaning of decimal fractions and use them in computations.

d. Model multiplication and division of decimal fractions by whole numbers.

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

I have discussed these standards previously (October, 2007). However, I had an interesting conversation with a colleague from another university in Georgia recently. She also teaches mathematics courses for prospective elementary school teachers at her school. We were discussing how I incorporate Japanese curriculum materials to discuss decimal multiplication and division. She then shared with me that, when she teaches this topic, she relates it to fraction multiplication and division. Her approach is perfectly valid and viable; however, in the GPS, decimal multiplication and division are discussed before fraction multiplication and division. So, I feel it is important that our future elementary school teachers experience how decimal multiplication and division may be developed WITHOUT the knowledge of fraction multiplication and division. So, I want to discuss how multiplication and division of decimal numbers can be approached using the knowledge of whole numbers only.

Let's start with multiplying and dividing decimal numbers by whole numbers.

Problem 1

One meter of wire weighs 5.7 grams. How much will 3 meters of the same wire weigh?

In the last post, I discussed the idea of relative size. If you use that idea, 5.7 grams can be considered as a collection of 57 0.1 g pieces. Thus, in 3 meters, we have 3 sets of 57 0.1 g's. That means we can use the calculation 57x3 to find out the total number of 0.1 g's in 3 meters. If there are 171 pieces of 0.1 g's, again using the idea of relative size, we know that is the same as 17.1 grams.

In general, when you are multiplying a decimal number by a whole number, you can just consider the given decimal number in terms of its smallest decimal place value as the unit. For example, 0.37 is 37 0.01's, 0.824 is 824 0.001's, etc.. You will then have a whole number as the multiplicand (in terms of a decimal unit). Since the multiplier is a whole number, we can use multiplication to find the total number of pieces of the decimal unit. We can then convert the final result into a decimal number by using the idea of relative size. Using the familiar paper-and-pencil algorithm, it basically means that the decimal point for the multiplicand (the number on top) and the product are in the same place.

Now, let's look at division of decimal numbers by whole numbers.

Problem 2

A wire that is 3 meters long weighs 5.7 grams. How much will the same wire weigh if it is 1 meter long?

Again, using the idea of relative size, we can think of 5.7 grams as a collection of 57 pieces of 0.1 gram. Since 3 meters of this wire include these 57 pieces, and you want to know how much 1 meter will weigh, you simply need to divide 57 by 3, which is just a whole number division problem. The quotient, 19, tells us the number of 0.1 gram pieces in each group. Again, using the idea of relative size, we can conclude that 1 meter of this wire will weigh 1.9 grams.

In general, just as in the case of multiplication of decimal numbers by whole numbers, when you are dividing a decimal number by a whole number, you can consider the decimal dividend in terms of its smallest decimal place value as the unit. Then, we can simply use whole number division to find out how many pieces of the decimal unit will be in each group. The final quotient can be found by using the idea of relative size. Using the familiar long division notation, this suggests that the decimal point for the quotient and the dividend will be in the same place, i.e., the place values for the dividend and the quotient should line up, just as they did with whole numbers.

One point of complication we need to pay attention is what if the dividend (whole numbers pieces of a decimal unit) is not evenly divisible by the divisor. So, for example, what can we do if the weight of the 3-meter wire was 5.8 grams. When we divide 58 by 3, we have the remainder of 1. But, this is also the number of 0.1 gram piece. So, we can say that the answer to the division of 5.8 by 3 is 1.9 with the remainder of 0.1. In other words, the remainder is the number of the decimal unit. Thus, in the long division notation, the decimal point of the dividend and the decimal point for the remainder must also line up.

Of course, another option is to divide on - the remainder of 1 can be thought of as a collection of 10 0.1's (actually, the remainder is 0.1, so we are really talking about 10 0.01's), then we can keep dividing. Sometimes, this will result in a terminating decimal, while in other cases, you will have a repeating decimal. With this understanding of dividing on, when students learn about the quotient meaning of fractions they can then understand that every fraction can be re-written as a decimal number by simply dividing its numerator by the denominator.

OK, this post is already rather long. So, I will have to wait till the next time to discuss multiplication and division by decimal numbers. In the meantime, I encourage you to think about how students can reason about to solve the following problems, which involve multiplication and division by decimal numbers. Keep in mind that they have not learned how to multiply or divide by decimal numbers or fractions.

Problem 3

One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

Problem 4

A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?