I am currently teaching a math history class, as I mentioned in Three hour class!. In our last class, we spent time talking about early Greek mathematicians. In particular, we spent time learning about the Pythagoreans. Since the Pythagorean theorem is one that everyone has heard of, talking about Pythagoras and his followers really grabbed the attention of the students in class. While, it’s hard to determine what is myth and what is fact about someone that lived 2600 years ago, there are a few things that caught my attention that I wanted to talk about. Note that we are using A History of Mathematics in this class, so I refer to this as my source for reliable information.
One of the core beliefs of the Pythagoreans was that numbers could describe everything. At the heart of this was the belief that the integers could describe everything. While this worked with ratios, they didn’t treat distinct numbers like we do fractions and rational numbers. Instead, ratios were just a relationship between integers. In order for this to hold up, they needed that all numbers were commensurable, that is, that for any set of numbers there was a smaller number that divided equally into each of the given numbers. The process of finding such a number actually leads to an interesting way to compare fractions to see if they are equivalent.
Now, how can we find a number that divides into two numbers equally? Here, we will treat our numbers as lengths. So, suppose we have a length a and a length b. Our goal is to find a length c such that nc=a and mc=b for some integers n and m. If we suppose that a<b (if not just relabel) then we note that since c divides into a and b, then it also must divide in b-a. Therefore, if we cut the length a off of the length b, then two pieces of length a and a piece of length b-a. Now, we have two new numbers we have to find a divisor of. Therefore, the same argument holds, so we should cut the smaller of a and b-a from the other giving us a remainder that c will then have to divide. If you continue this process for any rational numbers a and b, then eventually, there will be some point where all cut pieces are the same length. When this first happens, we will be left with the greatest divisor, c, of both a and b.
Materials-While I’ve done this on my own, I haven’t done it in a classroom setting. It was fun for me, but I’m sure what materials would be best. To that end, you’ll need something that you can pull out, stretch, cut and it will hold its length. I used fishing line, because I had a lot of it, and it seemed to do well. Thread would be another option; however, it tends to fray at the end making it difficult to tell exactly how long you cut it to. Play-doh is another option because you can remold it and use it for multiple examples, but it also doesn’t hold its length very well. Another option may be Balsa wood which holds its length well, but won’t bend if you want to find a curved length. You will also need something to cut your medium with. I used an exacto knife, but scissors would also work depending on what you choose to use. For the following, I will assume you are working with fishing line and an exacto knife.
Measuring- At some point you will have to measure out lengths of line and cut them. If you don’t want to let your students work with a knife or scissors, you can pre-cut the line to appropriate sizes. If you would like to focus on the comparison of ratios, I would suggest using a ruler to measure out lengths of line in terms of inches or centimeters depending on your preference. On the other hand, you can use a compass and straight-edge constructions starting with an arbitrary length to construct your distances (these constructions appeared in Euclid’s Elements). I will assume you are using a ruler with inches as I continue.
The first example I would start with is to cut two lines of length 1. If we compare these, they are the same, so the ratio would be 1/1 or just 1.
Then cut a line of length 2 and length 1. In order to compare these to find the greatest divisor, we would lay them out next to each other.
Then cut the longer one at the end of the line of length 1. This should then give you three lines of length 1.
Since these are all the same length lay them out so that they reconstruct your original distances of 1 and 2.
We then notice that we have a ratio of 1/2 since there is one piece in the first part and 2 pieces in the second. You can also get a ratio of 2/1 if you switch the order.
Now, cut a line of length 2 and a line of length 4. Follow the same process as before, and you will be left with 3 lines of length 2. Therefore, if you reconstruct your original lengths, you will have 1 piece of length two compared to 2 pieces of length two, so the ratio is 1/2 or 2/1 depending on which is first. For the students, note here that despite the difference in length of the pieces the corresponding number of pieces is the same between these examples, so we say that 1/2=2/4 or 2/1=4/2.
You can further do this for any other fractions you want to compare, perhaps using a 3/6 comparison to the ones above, or compare lines for 2/3, 4/6 and 6/9. The nice part would be that each of your pieces would be integer multiples, so comparing fractions or ratios would reduce to dealing with integers which are more comfortable.
If you would also like to work with negative numbers, you can work put the lines going from one starting point in different directions. Then if the strings go in different directions, you can say the ratio is negative.
Sums and differences of fractions- When using lengths of line to represent numbers, addition is as simple as putting the two lengths of lines end to end. To subtract, we put the line we want to subtract from the end of the line in the other direction and cut off the left over, just as we did in our first portion, so finding a sum or difference doesn’t introduce any new technique. However, we can then look at the sums of rational numbers in the following way.
For the first example cut a line of length 1, length 1/2 and length 3/4. If we want to add 1/2 and 3/4, we place these lines end to end.
In order to determine this sum we then have to compare this new length to the length of 1. Since 1/2+3/4 >1 we would cut a line of length 1 from the new formed line.
This distance will be the smallest portion, so we will then cut this portion off the line of length 1 and the new line we had formed.
For this example, we will continue cutting the same length off ending up with pieces all the size of the first piece we cut off.
If we reconstruct the length we made and the length 1, we will have 5 pieces of line and 4 pieces of line. Hence, our new length is 5/4.
From here I would work with a few more examples using both sums and differences of given lengths. Then follow the same process to find the dividing length of 1 and the created length to determine the new ratio. Note that by doing this, you will end up with your sum without having to go through the arithmetic process of finding a greatest common divisor and adding. Thus, again it reduces the problem of fraction to an equivalent problem working with integers.
Other examples- Depending on the level of class, you can continue to work with more complicated examples. One such example is looking at the Pythagorean theorem. Here you could cut a line of length 3 and length 4, then place them so that the lines are perpendicular. Cut a new line that completes the triangle and have the students find the length. They can do this by measuring directly, or by using the previous method showing it would be 5/4 of one side and 5/3 of the other.
Another example you can work with is you can cut 2 lines of length 1. Then, in the same way place these perpendicular to each other and cut a line to finish the triangle. At this point, if you use the previous technique, you in fact will continue to cut the line forever (or at least enough that you get really small lengths that you cant tell the difference between). Note that while the student may blame their own cutting, this is a case where you will never find a common divisor because the length √2 is not rational, and therefore it is not commensurable with 1. If followed through rigorously, this will also provide a proof that not all numbers are commensurable, hence the main belief of the Pythagoreans that the integers can define everything is incorrect.
We have therefore been able to talk about fractions using integers. This always for an understanding of more complicated numerical topics by only using things we already know about an easier number system. While the process does fall apart for irrational numbers, I think it will help someone making the initial journey into fractions. If you try this by yourself or in a class, let me know in the comments how it worked for you, what level of student you were working with and what materials you ended up using.
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