## 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.