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re: CalStateStudyOnManipulatives Over the past year I've used two kinds of manipulatives with Christopher, who is 10:
I didn't need play money and neither does anyone else. I got it only because I wanted to teach Christopher how to make change without a cash register, a lost art, and because . . . if I stacked up a pile of Real Money big enough to make change with, it was going to get raided for lunch money, bake sale money, field trip money (and that's just for starters). We are chronically short on ONES around here, let me put it that way. So I decided to make things easy on myself and buy some fake money.
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I'm a huge fan of fraction manipulatives. Christopher and I have spent quite a lot of time using a set of fraction tiles to illustrate:
the addition and subtraction of fractions
the addition and subtraction of equivalent fractions Nothing makes the idea that 2/12 is equivalent to 1/6 more obvious, IMO, than actually lining up two 2/12 tiles below one 1/6 tile and seeing that, yes indeed, 2/12 = 1/6. These are the fraction tiles I use. They cost $8.75 plus shipping: The same company, (Rainbow Resource, a homeschooling catalogue), also carries a set of extra fraction tiles without the tray that I wish I'd had when we first started trying to learn fractions. (I have them now, but we may be past the point of needing them. We'll see.) You need the extras because you really want the ability to demonstrate addition and subtraction of fractions with different denominators.
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There are lots of other fraction manipulatives out there, but I chose these after reading a comment from a mom on a homeschool forum somewhere. (I wish I'd kept the link.) She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these. At the same time, SAXON MATH uses circular manipulatives, so Christopher has been exposed to both, which I think is almost certainly ideal. A core principle in teaching math, from what I gather, is to teach the same material from different angles.
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Another terrific activity to do with fraction tiles:
Show how different combinations of fractions add up to 'one whole.' To do this you just have your child keep lining up fraction tiles on top of the bright red 'one whole' tile until he's covered the whole thing without anything hanging over the end. So, for example, he might put 2 1/12th tiles, 1 1/6 tile, & 2 1/3 tiles on top of the 1-whole tile, illustrating the fact that: 2/12 + 1/6 + 2/3 = 1 After awhile it starts to become obvious that you can put 6ths and 3rds & 12ths together evenly to make one whole, or 8ths & 4ths & halves, or 5ths & 10ths, . . . but you can't put 3rds and halves together, or 4ths and 5ths (not unless you have a bunch of 20ths, which you don't), and so on. You can see your child start to get a feel for multiples* and divisibility, whether he has explicitly studied multiples and divisibility yet or not.
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That's a whole other issue: is it useful to 'preview' concepts in this way? I have no idea, so offhand my answer is 'It depends.' That's one of the big gripes with constructivist math; the kids are constantly being exposed to advanced topics -- sometimes very advanced -- and then not taught the topics to mastery, because the book will be 'spiralling back' to the same topic the next year and the next year after that. Parents tend to hate this, but parents could be wrong. It happens. Let's just say that my perception, working with Christopher and the fraction tiles, was that he was developing an intuitive grasp of numbers that are multiples of each other versus numbers that aren't. This seemed like a good thing at the time, but who knows? I'm new at this. Come to think of it, I'm going to get the fraction tiles out again when I get back to teaching the Singapore Math lesson on Changing Ratios. (My neighbor and I team-taught this lesson to our kids two weekends ago, but it was over Christopher's head. Her son is a year older.) Singapore teaches changing ratios in the first half of 6th grade:
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Since I never remember definitions of even the simplest terms, I am including the definition of a multiple here: * multiple - The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 8)
New Study on Manipulatives Part 2
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Users must register to comment.I think multiples of 8 would be 8,16,24,32,etc. Your definition is good(8x1,8x2,8x3,8x4,etc.) Multiples of 2 would be 2,4,6,8. I teach my 5th graders that multiples of 8 would be just like counting by 8's. Hope that helps. Carolyn -- CarolynMorgan - 02 Jun 2005