Thursday, May 17, 2007

M4N4(d) - Property of Division

M4N4. Students will further develop their understanding of division of whole numbers and divide in problem solving situations without calculators.
d. Understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by the same number. (2050 ÷ 50 yields the same answer as 205 ÷ 5).

When we study properties of operations, we tend to focus on properties such as commutativity (a+b=b+a, or ab=ba), associativity [a+(b+c)=(a+b)+c, or a(bc)=(ab)c], identity [0+a=a+0=a, or 1xa=ax1=a], etc., which are typically associated with either addition or multiplication. Another important property is the distributive property of multiplication over addition/subtraction [(a+b)c=ac+bc and a(b+c)=ab+ac]. Not much about subtraction nor division is typically discussed. However, there is a useful relationship for both subtraction and division operations. That relationship concerning division is what is discussed in M4N4(d), which I have been calling the Equal Multiplication Principle (although you can multiply OR divide both the dividend and the divisor by the same number). The parallel relationship for subtraction is the Equal Addition Principle, which states that if the same number is added to (or subtracted from) both the minuend and the subtrahend, the difference remains the same.

So, when/where/how is the Equal Multiplication Principle useful? The example included in M4N4(d) is a familiar rule that says we can omit the same number of 0’s from both the dividend and the divisor. This is the specific instance of dividing both numbers by a particular power of 10. Clearly, this is a useful mental computation strategy. Moreover, for problems like 480 ÷ 15, doubling both (that is, multiplying both the dividend and the divisor by 2) will change the problem to 960 ÷ 30, which then can be changed to 96 ÷ 3 (which is the same thing as dividing the original numbers by 5).

In addition to being a useful mental computation strategy, the Equal Multiplication Principle plays a central role in division of decimal numbers. When you have a problem like 5.38 ÷ 1.6, we can multiply both the dividend (5.38) and the divisor (1.6) by 10 to change to problem into 53.8 ÷ 16. Once the divisor becomes a whole number, then we can simply use the long division to calculate the answer. Alternately, we could multiply both numbers by 100 to make both of them into whole numbers. Either way, the Equal Multiplication Principle says that the quotient we obtain is equal to the quotient for the original problem.

Furthermore, the Equal Multiplication Principle may also be useful while dividing a number by a fraction. For example, let’s look at 2/3 ÷ 3/4. We know that if multiplying both the dividend and the divisor by the same number, we will not change the quotient. So, how can we use the Principle here? Another familiar pattern of division is that the quotient of any number divided by 1 is the number itself. So, if we can change the divisor (3/4) to 1, then, we know that the quotient is the same as the dividend. To make the divisor 1, we have to multiply it by its reciprocal (that’s the definition of the reciprocal, isn’t it?). So,

2/3 ÷ 3/4 = (2/3 x 4/3) ÷ (3/4 ÷ 3/4) = (2/3 x 4/3) ÷ 1 = 2/3 x 4/3

So, we see that the division by a fraction is the same as the multiplication by the reciprocal of the divisor.

[By the way, you can also use a very similar argument to show that subtraction of a negative is the same as addition of its opposite.]

Anyway, this is a property of division that is not always emphasized in an elementary school mathematics. However, I hope you have a better sense of its potential usefulness.

So, why is the Equal Multiplication Principle true? One way you can see this is to use the measurement meaning of division (that is, how many groups of {divisor} are there in the {dividend}). So, if we have 2400 ÷ 400, we are asking how many groups of 400 can we make with 2400. If we look at both of these numbers using a hundred as a unit, we have “how many groups of 4 hundreds can we make with 24 hundreds?” Thus, the answer should be the same as 24 ÷ 4. It is important that we look at both the dividend and the divisor with respect to the same unit. You could think about 2400 ÷ 400 in terms of units of 5 if you want to. To do so, you need to know how many 5’s are in 2400 and how many 5’s are in 400. Of course, to find those answers, you have to divide 2400 and 400 by 5 – that’s the Principle.

One word of caution. The Equal Multiplication Principle says that the quotient does not change. However, it does not say anything about the remainder. So, for example, if you have 2500 ÷ 400, a common error is to say the answer of 6 remainder 1 because 25 ÷ 4 = 6 remainder 1. However, we need to keep in mind that we are using 100 as a unit. So, the remainder of 1 actually is telling us we have 1 unit of 100 remainder. So, the answer should be 6 remainder 100.

Sunday, May 6, 2007

M3N5(b) - Meaning of Fractions

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.
b. Understand the fraction a/b represents a equal sized parts of a whole that is divided into b equal sized parts.

Teaching and learning of fractions continue to be a major challenge for both teachers and students. Many children (and teachers) think of fractions as parts of a whole [M3N5(a)]. However, M3N5(b) suggests that we look at fractions in a slightly different way as well. For example, according to M3N5(b), the fraction 2/3 means there are 2 pieces of 1/3. So, why is it important that children understand fractions in this manner?

In "Elementary School Teaching Guide for the Japanese Course of Study: Arithmetic (Grades 1 – 6)," the authors suggest that there are 5 meanings of fractions. For example, the fraction 2/3 may mean:

1. two parts of a whole that is partitioned into three equal parts,
2. representation of measured quantities such as 2/3 liter or 2/3 meter,
3. two times of the unit obtained by partitioning 1 into 3 equal parts,
4. quotient fraction (2 ÷ 3), and
5. A is 2/3 of B – if we consider B as 1 (a unit), then the relative size of A is 2/3.

Thus, M3N5(b) corresponds to the third meaning above. So, why is it not sufficient to think of 2/3 as 2 out of three equal parts (of a whole)? What advantages are there to think of 2/3 as two 1/3’s?

The most important reason for going beyond the part-whole view of fractions is that we want students to understand fractions as numbers. The part-whole interpretation of fractions is more about relationships, and it does not necessarily signify a quantity/number. When someone makes 6 out of 8 free throw attempts, the fraction 6/8 doesn’t signify a number. In fact, if he makes 8 of 10 attempts in the next game, we can say he was successful at 14/18 of attempts in those two games combined. This combination is NOT addition of numbers 6/8 and 8/10, in that case, we have to find a common denominator to find the sum. Rather, 6/8 and 8/10 are both ratios. The part-whole interpretation will signify a number if the whole we are considering is the number 1.

The part-whole interpretation is important, and may be a prerequisite, before students can consider 2/3 as 2 pieces of 1/3’s. For this interpretation to be truly useful, students must first understand 1/3 as a number – it is a number such that if you have 3 of them together, you will make the number 1. In other words, 1/3 is a number that is equal to the number in a group when 1 is divided into three equal sized numbers – 1 out of 3 of the number 1.

There are many places in the elementary school curriculum the interpretation of a/b as a copies of 1/b’s. For example, if students’ view of fractions is limited to the part-whole interpretation, they will have a hard time making sense of an improper fraction. After all, what does 4 out of 3 mean? On the other hand, if you consider 4/3 as 4 pieces of 1/3-units, then there is nothing different about 2/3 and 4/3. Or, consider the simple addition/subtraction of fractions with like denominators. For example, 3/5 + 4/5 means putting together 3 pieces of 1/5’s and 4 pieces of 1/5’s, giving us 7 pieces of 1/5’s all together, or 7/5. This reasoning is, in principle, the same as thinking of 30 + 40 as adding 3 tens and 4 tens, thus 7 tens.

The importance of looking at non-unit fractions as collections of unit fractions is not a Japanese idea. Thompson and Saldanha indicated in their chapter on fractions in the Research Companion to the Principles and Standards for School Mathematics, that this is a very important view of fractions. Unfortunately, they also note that this idea is rarely seen in US mathematics textbooks. As we begin implementing the GPS, therefore, it is important for us to remember this perhaps unfamiliar way of looking at fractions.

By the way, the fourth meaning is discussed in M5N4(a). The fifth meaning of fraction, i.e., fractions as ratios, are not treated until Grade 6 when students are introduced to the idea of ratios (makes sense, doesn’t it?). Moreover, in the Japanese elementary textbooks, the idea of a fraction of a set (or discrete model of fractions) does not appear until Grade 6 because they believe that the meaning of fractions in that context is much closer to the ratio meaning of fraction. [In fact, the part-whole meaning of fractions is very close to the ratio meaning of fractions.] This is an interesting contrast to GPS M2N4(a) and something Georgia educators must think about carefully.

I will be discussing models of fractions in another post.

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