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