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September 28, 2014

An interesting detour...

I usually go over addition of fractions with equivalent fractions. The idea being that you can only add fractions when the pieces of the pizza (so to speak) are the same size...

The children had worked with this for three days and learnt to notice that when the denominators are the same they could use that fraction to replace the original fraction. In order to reiterate that doing equivalent fractions was to get pieces of the same size and look at the number of pieces we would have for that size I started comparing fractions using a similar method i.e. do equivalent fractions till the denominators are the same (same size of pieces) and then compare the numerators (number of pieces).

We did a few examples in class and I asked the class to write fractions of their own and do a similar comparison at home. Homework is always interesting, it gives you a look into how children are processing the material and often the kind of short cuts they have thought of. The children came back with fractions that they knew how to compare. Other than the pieces they had learnt well about by playing fraction games (1/2, 1/3, 1/4, 1/6, 1/8). None of them had used the equivalent fractions. We started looking at the kind of fractions that are easy to compare and don't need an equivalent fraction.

1) Fractions with the same denominator (obvious)
2) Fractions with the same numerator
This started with fractions with a numerator of 1 and it was nice that the children were able to connect with division stories and notice that if the denominator was larger than the size of the piece was smaller and the larger fraction was the one with a smaller denominator!
This was then generalized to other numerators as long as they were the same. 
3) I decided to push the envelope and introduce another rule to get the kids to think about it. How about comparing 4/5 and 5/6 and coming up with a logic for that.
I spelled it out each of them is a pizza with one piece missing, the first one has a bigger piece missing (1/5) the second a smaller one (1/6). What does that mean?
Two kids in class got it immediately, I use peer learning significantly in class and I asked the kids who got it to reiterate what they understood, after a couple of them said what they understood (even if the words they use are the same as what I used, more kids get it each time!). A good part of the class had got it and I asked children to practice 10 comparisons with each of the rules.

An alternative I use for 'did you do your homework' is asking 'if anyone wants to show me their homework'. This time when the kids got back, I asked the kids what fraction pair they still found difficult to compare. The pair they put on the board gave an indication of how much they were able to assimilate. Each student said they still had a pair that they found difficult to compare. One by one they put 17 fractions on the board. Thankfully, there were no fractions of the 1st and 2nd kind. 

I asked the class if they could find a fraction pair on the board that they did know how to compare. If they could they would come and confirm with me that their logic was ok and then explain it to the class. There were some of the third rule, then things got interesting.
- They were able to extend the third category with fractions like 6/8 and 13/15. Both are missing two pieces, but the second is a smaller piece. Neat now they were thinking are there others that can be compared like that? There were 5 of this kind.
- The next set that the children noticed noticed where the fraction numbers where the denominator was large, but the numerator was smaller i.e. The piece is smaller and there are less of them 3/25 vs 2/27.
- The last was special, one child pointed out that one of the fractions was less than half and the other more than half. 17/42 he pointed out was less than 21/42 which was half. The other fraction 9/16 was greater than 8/16 which was half. So 9/16 was greater than 17/42 even though there were more pieces of a smaller kind.

With these rules 15 of the 17 fractions on the board were knocked off. There were only two left these were pretty close to call with and the children who could compare them had done so using equivalent fractions.

Then we moved on to was nice to have started from something that I was required to do - teach addition of fractions  and move to something that I wanted to do, a feel for fractions, creating abstract rules from real situations and being able to think creatively.