Wu's fractions: Common Denominators

This is the 3rd in a series of posts fleshing out the material written by Hung Hsi Wu in Critical Concepts for Understanding Fractions. See also Part I and Part II.

Last time, we left off after how to represent fractions on the number line. We showed that while a number is unique, and has a unique position on the number line, there were many different representations for that number based on how the number line was partitioned into pieces.

Recall we showed that 4/3 is the same as (5 x 4)/(5 x 3) with this example: locate 4/3 by its unique spot on the number line. We do this by breaking each unit into thirds, and then jumping 4 such 3rd-units to the right:


Now we partition each 3rd-unit into 5ths. Doing so immediately gives us 15-ths for each unit. Now we count how many 15ths our point 4/3 is: the answer is 20.



While our example was specific, our technique was not. Any fraction would have worked, and any new non zero partitioning. This demonstrated for us the mathematical fact: for all whole numbers k, m, and n, (such that n is not equal to zero and k is not equal to zero), m/n = km/kn.

In other words, m/n and km/kn are equivalent fractions. The equality symbol above tells us that these two quantities are the same. We can understand that to mean that there is one location on the number line that represents m/n, and it's the same location as that for km/kn, for any k, m, and n (such that n and k are nonzero.)

We note here that this is a good place to reinforce your comfort with commutativity of multiplication. We want students to feel comfortable recognizing that we could just as easily have said m/n = mk/nk and that would also be an equivalent fraction. This is where mastery of the underlying multiplication table is so important. The way to really believe the commutativity is to already know it's true for all of the natural numbers. We want to be able to multiply a fraction by k/k from either side without confusion.

This is also a good time to discuss a fairly beautiful fraction: k/k. Remember that our definition of a fraction is: the fraction m/n is the point on the number line, when we partition each unit into equal nths, and then make m jumps to the right of Zero on those new nth-hash marks. So the fraction k/k is the point on the number line when we partition each unit into equal kths and then make k jumps to the right of zero. This ALWAYS lands us at 1. So k/k is the same number as 1.

For students that are clicking this together, this is another way to see why m/n = mk/nk, but remember, we didn't resort to that explanation in the first place, because we'd like to become familiar with manipulating expressions like mk/nk WITHOUT reducing them. We are trying to build up more helpful denominators, to learn how to create new denominators, not just reduce them. Real mastery of k/k doesn't just lead us to say "that's 1, that cancels" but to say "we can replace 1 with k/k!"

Equivalent fractions are useful because they allow us to move quickly to compare fractions to each other, by use of this fact: Any two fractions may be represented as two fractions with the same denominator.

Mathematically, how would you do this? Well, take 2 fractions: m/n and k/l. m/n is the same as ml/nl. k/l is the same as nk/nl.
So now both m/n and k/l can be represented with the denominator nl.

Why do we care? Because now we can easily compare fractions, and we will now be better able to add and subtract them. Consider first the comparison. Two fractions with the same denominator can both be represented easily on the number line without confusion. Start with m/n and k/l. They become ml/nl and kn/nl respectively. ml/nl and kn/nl are represented by locations on the same sequence of (nl)ths. ml/nl is to the left of kn/nl if ml . By converting to a common denominator, we were able to solve the problem without having to graph out two different denominators. All we did was look at the numerators.

This leads to a specific method for comparing fractions. We convert each fraction to having the same denominator, and then look to see which numerator is bigger. In other words:

for all whole numbers k,l,m,n, k/l= m/n is equivalent to kn = ml .
This is just what we said above. It is also called the Cross Multiplication algorithm. (multiply the bottom of the left and the top of the right; multiply the bottom of the right and the top of the left. Which product is bigger?) In that case, we shorten the original procedure by ignoring that common denominator nl and just comparing the numerators. But indirectly, we are exploiting the fact that these two fractions are now represented by the same common denominator to determine which is larger or smaller.

Just as comparing two fractions is easier when you have common denominators, adding and subtracting them is easier too. We'll pick this up in the next post.

Update: above links fixed and should point correctly now!