We left off last time with an introduction to common denominators. First, we showed that a given number is unique and represented by a unique location on the number line, even if it has several different representations. For fractions, these different representations are often made manifest by representing the number in terms of different denominators. For example,

Next, we showed that it was always possible to represent two fractions, originally with two different denominators, as fractions with the same denominator. That is:

if

*m/n*and

*k/l*are fractions (where

*n*and

*l*are nonzero), then

*and*

**m/n = ml/nl****. That is, both now share the denominator**

*k/l = nk/nl**.*

**nl**For example, take 4/3 and 1/5. Both can share the common denominator 15: 4*5/3*5 = 20/15. 3*1/3*5 = 3/15.

Thus, we are now able to talk about these two fractions as having a common denominator. Once they are represented with common denominators, it becomes easy to compare fractions. Specifically,

*a/nl*is less than

*b/nl*if

*a < b.*

We now move on to how common denominators can help us add fractions.

Fundamentally, adding fractions is very similar to adding whole numbers on the number line. Recall that we could add 4 and 3 by making successive unit jumps on the number line: starting at zero, jumping 4 unit steps to the right, and then jumping 3 more units resulted in 7. Recall that a fraction

*m/n*is defined as the location on the number line after partitioning the unit segment into

*n*partitions, and then moving

*m*jumps to the right of zero. Similarly, the fraction

*m/n*could be represented as a segment whose left marker began at 0 and right endpoint was at the location

*m/n*. Further, it could be represented by the length

*m/n*--this length being the same no matter where on the number line it is located. The addition of fractions

*k/l + m/n*is the concatenation of two segments: the first segment whose length is

*k/l,*and a second segment whose length is

*m/n*.

This leads us to recognize that whole numbers are a special form of fractions: a whole number

*k*is just the fraction

*k/1*, where the unit segment is left as one segment, and we then make

*k*jumps to the right on unit segments. So a whole number is special case of a fraction, a fraction whose denominator is 1. And more importantly, the addition of whole numbers is really the addition of fractions with common denominators of 1.

Now, we start to think about the addition of fractions as we thought of the addition of whole numbers:

To add

*k/l + m/n*, we first partition each unit segment of the number line into

*l*pieces, and then concatenate

*k*segments to the right. Next, we create a new set of unit segments, whose origin is

*k/l*, and then partition each unit segment of this number line into

*n*pieces, and concatenate

*m*segments to the right. Here's an example of the sum of 2/3 and 3/4:

This method does, in fact, find the unique location on the number line that represents this fraction. That is, we've properly added the fractions and found the new location. But while this finds the unique location on the number line that represents this fraction, this location doesn't really help us to simplify our representation: we'd like to represent this new fraction in the standard form of

*a/b*. But that means that in the end, we'd like our new fraction to be easily represented by just one set of partitions: the

*b*ths. If we could find a way to draw both

*k/l*and

*m/n*on the same partitions, we would know how many total jumps we've made on one set of partitions, rather than on these two different partitions.

To do this, we need to rewrite

*k/l*and

*m/n*with common denominators.

To do this, we rewrite

*k/l*as

*kn/ln*,

*m/n*as

*lm/ln*, and now we can rewrite

*k/l + m/n*as

**k/l + m/n = kn/ln + lm/ln = (kn + lm)/ ln.**Applying this to our above number line, this says first, we partition each unit segment into

*ln*pieces. Then we make

*kn*jumps to the right, followed by

*lm*jumps to the right.

In the case of 2/3 and 3/4, we rewrite 2/3 as 2*4/3* = 8/12, and we rewrite 3/4 as

3*3/3*4 = 9/12.

The sum, then of 2/3 and 3/4 is 2*4/3*4 + 3*3/3*4 = 8/12 + 9/12 = 17/12.

Is it possible to show the 2/3 + 3/4 to an average 5th grader without confusing them more? Does showing how the unit segments don't line up help motivate why we are finding a new common denominator? Or does it just leave them wondering how in the world adding fractions is meaningful? From my perspective, it seems important to get them to see that the location on the number line is the same, but our new hash marks help us to write a simpler number. Fundamentally, we're saying that 2/3 + 34 IS 2/3 + 3/4. I would hope questions on homework would be phrased as "simplify 2/3 + 3/4" rather than "solve 2/3 + 3/4". We're

**not**solving an equation. We're finding an equivalent expression. I think when showing the students, the point to stress is that we want to add the fractions and turn the sum into our standard form where we have a single set of partitions.

Next, we'll discuss mixed numerals and summing before moving on to subtraction.