This is the first in a series of posts about the mad love we have in our home for math manipulatives.
I’ve spent some quality time
thinking obsessing about how to deal with L’s math brain. She doesn’t deal well with repetition and likes to think “big picture” things. I have had to deal with the idea that her automaticity will come through repeated use of facts rather than memorization simply because she shuts down if we attempt to do anything resembling math facts. There’s another blog post brewing in my head right now about the approaches we’ve tried…
In any case, our idea right now is that we are exploring in-depth topics that she seems ready to absorb.We’ve been playing with fractions already, especially as we cook and bake pretty frequently so she’s used to seeing them in recipes and on cooking tools.
Recently, L turned 5 1/2. When I was a kid, we celebrated my summer birthday at the 6-month mark at school. L is also a summer birthday, and at the awesome play-based nature school she attends two days a week, they celebrate half birthdays.
In the car, we began discussing how L was 5 1/2 and how there were other ways to express that same idea. I didn’t delve into the conversation at that moment, but knew what the next day’s lesson would be.
I started by layout out for her the conversation we’d had the day before, telling her that there were lots of fractions that would express the same amount. I wrote out for her a variety of fractions with 5 as a whole number and 3, 4, 5, 6, 8, 10, and 12 as denominators (with no numerators). I then asked her to use her fraction circles to figure out which of those denominators could be made into a fraction that showed the same amount as 1/2. I very specifically wanted to include denominators that wouldn’t work to help her think about what fractions mean, not just how to work with them.
She used a few really interesting approaches, including laying pieces on top of the 1/2 piece (so putting the 2 1/4 pieces on top of the 1/2 piece to show they were the same) as well as using the additional pieces to complete a circle along with the 1/2 piece.
She was pretty quickly able to record 2/4, 3/6, 4/8, 5/10, and 6/12 as representing the same amount as 1/2. I told her she’d discovered a really cool idea called “equivalency” and asked her what word she could find in the beginning of “equivalent”. She found “equal” and we discussed how equivalent fractions represent the same amount as one another.
L showed me how the 1/3 pieces couldn’t be made to be the same amount as 1/2. She showed me how one 1/3 piece was too small to fill up the same amount of space as a 1/2 piece while two 1/3 pieces filled up too much space. We set those aside. She repeated the same reasoning with two 1/5 pieces and three 1/5 pieces. We put those in a pile we called the “crying, sad fractions” and crossed them out on the paper =)
We then looked for a pattern that emerged between which fractions could be made equivalent to 1/2 and which couldn’t. I introduced the words “numerator” and “denominator” so we could talk. We looked at the denominators and noticed that each of the denominators in fractions that could be made equivalent to 1/2 were even. We decided to test this theory by taking all of the pieces of each type of fraction and putting even amounts of them on each side of a dotted line.
After getting the even fractions divided into twos, we then considered the third and fifth fraction pieces again. L put one 1/3 on one side of the line and another 1/3 on the other side of the line. She sat there holding the third 1/3.
I wrote out for her 1/2 = 6/12 = 5/10 = 4/8 = 3/6 = 2/4 and we discussed again the pattern. This time she noticed that if you added the numerator to itself, you would get the denominator. I told her again what a cool discovery this was, and then directed her to this question:
If you notice that all of the denominators in the fractions equivalent to 1/2 are even, why can’t thirds or fifths be equivalent to 1/2?
It was a big question. She sat there. She put the final 1/3 underneath the dotted line. I worried that I’d pushed too far.
“Their denominators are odd!! They aren’t equivalent to 1/2 because they are odd!”
“And you can’t divide 3 or 5 by half and get even groups!”
“And you can’t add any numerator to itself and get the denominator!!”
As we were high-fiving, I decided to double check her understanding of the underlying concept, so I wrote up a quick challenge sheet: Can you make an equivalent fraction to 1/2 out of sevenths? fourteenths? ninths? twentieths? What’s another denominator that you can make an equivalent fraction to 1/2 of?
She spent a minute puzzling over the first four, figuring them out correctly. Then, I heard it in relation to that open-ended question: “Well, I’ll just start in the thirties. 30, 32, 34, 36, 38, 40… any even number would do it.” Eureka!
In true form, though, we got to discuss one more great math term. L said, “So I could say that I am 5 6/12, but I never would.” I asked her why, and she told me, “It’s too hard to understand that one. 5 1/2 is easy.” I reminded her about the word simple, which she told me meant the same thing as easy. It was a quick hop over to writing down “simplification” and defining that as a fraction in the form of the smallest numerator and denominator that represents the same amount.
All in all, a solid day’s work.