Tuesday, February 11, 2020

Fraction Models to Review with Your Eighth-Grader

Cover your ears, Friends, because this is me saying it again louder for the people in the back:


I mean, everybody knows this if they think about it. Everybody likes a good graph or chart or 3D model or virtual model or LEGO model or hand-drawn diagram or Google map. Everybody likes to see visual representations of information, and to fiddle with stuff to figure it out.

It drives me bonkers, then, that math manipulatives very rarely make their way out of the early elementary classrooms, especially when they're so useful for exploring and explaining a wide variety of more sophisticated concepts.

Ask me sometime about my handmade, take-apart model of the binomial and trinomial theorems!

In her intro to algebra curriculum, Syd is currently learning more about rates and proportions (handy timing, as that's what I use to allocate sales to my Scouts who work Girl Scout cookie booths--guess who's going to do all my cookie booth math for me this weekend?). In order to focus on problem-solving with rates and proportions, one should already have a competent working knowledge of calculating fractions.

Here, by the way, is one of those places where a kid can get lost in math for life. Math builds upon itself, and a curriculum that's currently teaching a kid how to graph proportional relationships is not going to hyperlink a review of every type of calculation that she should have mastered in order to solve the problem.

It should, but that's a different gripe.

So if graphing a set of proportional relationships requires a kid to divide fractions and she doesn't remember how, she's not going to be able to graph that set of proportional relationships. And then the next set that she's asked to graph, she still won't be able to do it, because she didn't learn how during the previous set, because she couldn't do the necessary calculations.

And then when the class moves on to scaling figures, if any of that work assumes that the kid can graph proportional relationships, then, well, she's not going to learn how to scale figures, either. And so on and so on until she gratefully taps out of math completely after Algebra 2 and spends the rest of her life telling people she's bad at math.

She's not bad at math, Friends. YOU'RE not bad at math, even if you think you are. You just got lost somewhere and nobody helped you pick yourself back up.

So when Syd was doing unit rates the other day, and for a moment she couldn't remember how to turn an improper fraction into a mixed fraction, even though she looked at the fraction for a second and then was all, "Oh, right, I've got to divide," I said, "Do you know why you have to do that?"

She said, "Because that's how you do it."

When a kid says that, it's because they don't KNOW why a certain algorithm works, and if they don't know WHY the algorithm works, then they haven't mastered the concept behind the algorithm. A kid who hasn't mastered the concept behind the algorithm that's really just a calculation shortcut is on shaky ground, mathematically, because memorizing algorithms isn't math, or at least it's not what math ought to be.

When I'm in charge of making sure people understand math, I make sure they understand the actual math, so that they can apply it to new situations, incorporate it into patterns, problem-solve using it, think creatively about it.

The best way that I've found to that level of understanding is seeing it and touching it. And that's how I found myself bringing out the fraction circle manipulatives and Cuisenaire rods and reviewing some fraction models with my eighth-grader.

First up: that tricky little improper fraction to mixed fraction conversion! You need to do this a LOT in algebra, and forgetting the algorithm or being unsure that what you're doing is correct will distract you from the algebraic concept you're trying to master, or just cause you to get the wrong answer so that you can't master the concept at all.

This model is an easy reminder of what you're doing when you convert an improper fraction to a mixed fraction, and vice versa.

To model how to convert an improper fraction to a mixed fraction, give your kid a pile of the same fraction from a set of fraction circle manipulatives, and ask her how many she has. Let's say she has 7 1/4s. She can write that as the improper fraction, or she can go a step farther and write 1/4+1/4+1/4+1/4+1/4+1/4+1/4=7/4.

Then, ask her to assemble the fraction manipulatives, seeing how many wholes she can make. She'll be able to assemble one whole, with another partial circle next to it. Write the new fraction, which is 1 3/4.

Walk through that in reverse, and you're instead modeling how to convert a mixed fraction to an improper fraction. Remind them of the algorithm, and have THEM show YOU how the algorithm matches the model at each step.

If they need some drill to cement the algorithm, here's a worksheet builder.

Here's another example, still with unit rates (I tell you, algebra has a LOT of fractions in it!): Syd missed a problem because she multiplied a fraction when she was supposed to divide it. When I pointed out to her that she'd multiplied instead of dividing, she of course knew that what she'd done had to be wrong, but she couldn't at first figure out what exactly was wrong about it, because she also correctly remembered that there WAS multiplication involved when you divided fractions.

After thinking for a bit, Syd remembered that you of course have to invert the divisor before you multiply it, but because that step didn't make any sense to her, it was easy for her to forget.

Time to get out the Cuisenaire rods for a review of dividing fractions!

I have a very thorough step-by-step showing how to model fraction division using Cuisenaire rods here, so I mostly took these photos for fun.

Here's Syd doing the first step every time you work with Cuisenaire rods. Got to build the stairs to remind you which color represents which number!

Syd always likes doing this, too. Ahh, those number bonds for ten!

By the time we got out the Cuisenaire rods, Syd had remembered on her own how to work the algorithm for dividing fractions, so I had her work the algorithm first, then prove it using the Cuisenaire rods:

I think that the visuals here make the mathematical process so interesting. I'm fascinated at the way that the numerator of the divisor becomes the denominator of the quotient, and how elegant is the way that it becomes so.

And you'd never see it without these colorful math manipulatives in your hands!

Also, I DARE you to work with Cuisenaire rods for more than five seconds and not get distracted making fun patterns with them:

Again, if you do need some drill work afterwards, this is my favorite worksheet generator.

Speaking of multiplying fractions... that's also a good one to review models for, because doesn't it break your brain that multiplying fractions makes the product SMALLER?!?

I really like this fraction multiplication model that uses colored cellophane:

This is more enrichment than review, but middle school is a great time to play with the Fibonacci Sequence:

Need even more ideas for making fractions fun and real? Check out my Fractions, Decimals, Percents and Ratios Pinboard.

Because of COURSE I have a Fractions, Decimals, Percents and Ratios Pinboard!!!

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