Exploring equivalent fractions: I’m 5 1/2 now!

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.

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A previous day’s work: Find 10 different ways to make 1 using halves, thirds, quarters, and sixths

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.

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Playing with the fraction circles

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.

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Completing the circle with tenths

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.

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Checking our theory

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!”

Right, kiddo.

“And you can’t divide 3 or 5 by half and get even groups!”

Precisely.

“And you can’t add any numerator to itself and get the denominator!!”

Amen, sister.

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!

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A hard day’s work

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.

Multiple representations of equivalence

A couple of math minutes today – one that laid out beautifully as I’d envisioned it, and one that gave me another reminder of how much I’m still chasing L’s abilities. Let’s start with the “fail,” shall we?

I’ve been keeping a clipboard in the car lately. Sometimes it has coloring pages (primarily from Dinosaur Train, but also color by number or other items I happen upon). Other times, it contains tasks that I estimate she can complete independently and quickly. This first math minute was one of those activities. It is a simple page I found online that is missing a sign (either subtraction or addition) in a series of number sentences. I passed the clipboard to L after she was buckled up in the Whole Foods parking lot. I pulled out of the parking lot, turned right, turned left, and entered the freeway. When I entered the freeway, I realized I hadn’t heard anything from L.

I asked, “So, what does that paper ask you to do?”

L replied, “I’m done.”

She passed me the clipboard. She was done.

#MommyFail

#MommyFail

Ok. Unexpected! I asked her how she knew what to do. She explained to me (as if speaking to a simpleton, by the way), that one could simply look at the answer and the first number. If the answer was larger, it would be an addition problem; a smaller answer would indicate a subtraction problem.

Fair enough.

Luckily, later this afternoon, I rebounded! I’ve been wanting her to think beyond simple mathematical equations (even multi-digit ones) because she’s pretty comfortable with the processes of addition and subtraction. I’d like her to begin thinking logically, rather than simply procedurally. I found this “balance the scale” page and decided to use it as the basis for this thinking. The goal of this paper would be to have kids identify which two addition sentences have the same answer and place them on either side of the balance.

I decided that I wanted her to explore this concept visually and kinesthetically, as well as have multiple ways to conceptualize the solutions. I pulled out three of our sets of math manipulatives:

All three models gave her the ability to combine terms as well as visually represent equivalence. She never saw the worksheet. I simply asked her to get the 2, 3, 4, and 5 inchimals. I prepared sets of 2, 3, 4, and 5 snap cubes. She got the 2, 3, 4, and 5 bananas. She then worked with her materials to determine that you could make two number sentences which equaled 7 (2+5; 3+4). Once she’d determined that, I asked her to show me on all three sets of materials so that she could physically see the balance points of the monkey and bucket balances, as well as see the same heights of the two stacks of inchimals.

Scene re-creation

Scene re-creation

We proceeded similarly for the next set. By the last two sets, I asked her to explain to me why the solutions made sense. She was able to verbalize that both sides equaled the same total (and give the accurate total). Job well done!

So, one very successful learning opportunity today, as well as another opportunity for mommy to remember that the tasks I anticipate as tricky are often not…