d. Model the multiplication and division of common fractions.
M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
e. Multiply and divide fractions and mixed numbers.
In the previous post, I discussed similar statements about decimal number multiplication and division. These statements are very similar, but there is a significant difference. In M4N5, the GPS makes it clear that the focus is on multiplying and dividing decimal numbers BY WHOLE NUMBERS. In contrast, in M5N4, the GPS simply states “multiplication and division of common fractions.” Does this mean that students should understand the meaning of multiplication and division BY FRACTIONS?
As I discussed in the previous post, when we multiply or divide by decimal numbers, the meaning of multiplication and division must be expanded. However, the same meaning can be applied whether or not we are multiplying/dividing by decimal numbers or fractions. Thus, it is perfectly possible to discuss the situations involving multiplication/division by fractions in Grade 5 – as long as we are aware of this subtle yet significant difference. However, it would have been helpful had the GPS made the boundary of M5N4 a little more explicit.
There is another problem with M5N4 and M6N1. One of the goals of establishing the GPS was to reduce the repetition in the curriculum. That raises a question: what is the difference between these two statements regarding the multiplication and division of fractions? One obvious difference we can see is the use of the word, “model” in M5N4. This raises 2 questions for me. The first question is, how do we model the multiplication and division of fractions? As long as the multiplier is a whole number, we can (relatively) easily model the multiplication by making so many sets of a specific fraction (the multiplicand). So, 3x2/3 can be modeled by making 3 sets of 2/3 – with Pattern Blocks, Fraction Bars, Fraction Pieces, drawings, etc. How would you model if the multiplier becomes a fraction (or both the multiplier and the multiplicand are fractions), like 1/3 x 2/3? One possibility is to use the area model of multiplication, that is, 1/3 x 2/3 may be represented by the area of a rectangle with the dimensions, 1/3 unit by 2/3 units. We can do this by folding paper or by drawing picture like this:
What about the division? An easier way of representing division of fractions is to interpret the division as the measurement division – how many groups of (the divisor) can we make with (the dividend)? We can model using various fraction manipulatives, or drawing pictures. However, the difficulty arises when there is a left over piece, or the dividend is less than the divisor. In those situations, students must understand that the division is asking, “What part of the divisor is the dividend?” This shift is not always so simple for students. However, not all division situations involve the measurement division. How might we model those situations?
The second question is when should students develop the algorithms for multiplying and dividing fractions? Are the algorithms types of “models”? Many of you are aware that the GPS was heavily influenced by the 1989 Japanese national course of study. In the Japanese standards, the multiplication and division of fractions are Grade 6 topics. They do not touch on these topics in Grades 5 at all, though since they study adding and subtraction fractions in Grade 5, they can easily consider the situation with whole number multipliers. In any event, it is in Grade 6, the Japanese curriculum develops the algorithms. Did the GPS also mean that the algorithms were to be developed in Grade 6 – that is, “modeling” excludes the algorithms? This is another important point that the GPS could have made much more explicit.
By the way, the Japanese textbooks often use a model of multiplication and division called double number line. So, for a problem like, “If 5 1/3 feet of wire weights 2/3 ounce, how much will 1 foot of the same wire weigh?” they will model the problem like this:
This model is used not necessarily to find the answer but to help students understand the way the quantities relate to each other. By understanding the relationship, students can then decide what operation must be performed to find the missing quantity. What is important about this model is that this same model may be used to model both the multiplication and the division situation. So, if you have a problem like, “if you can paint 3 1/3 square feet with 1 pint of paint, how much area can you paint with 4 3/5 pints?” may be modeled like this:
Finally, it can also be used to represent the measurement division situation like, “if you car can travel 18 3/5 miles on 1 gallon of gasoline, how many gallons of gasoline do you need to go 180 miles?” will be modeled like this:
The strength of this particular model is that the same model may be used for both multiplication and division, thus potentially helping students to understand the relationship between those two operations. Furthermore, this model is also a powerful tool for representing proportional situations in general. One weakness of this model, though, is that it cannot be used to model area situations as those situations will not involve “per-one” quantities.
