[IAEP] Looking for Concrete "Fraction Experiences"

[mokurai at earthtreasury.org]

On Tue, July 12, 2011 11:23 pm, Steve Thomas wrote:
> Looking for ideas on how we can give kids (and adults) concrete
> experiences with the concept of fraction.

You do not have to *give* them such experiences. You need to draw
attention to the experiences they have had all their lives.

You do eat slices of pie, cake, and pizza, and chocolate bars marked for
breaking, I trust. You use coins, and can get dollar coins, half dollars,
quarters, tenths, twentieths, and hundredths. You can talk about the
divisions of hours, minutes, seconds, yards, feet, inches, meters,
decimeters, centimeters, gallons, quarts, pints, cups, fluid ounces,
tablespoons, teaspoons, liters, milliliters, pounds, ounces, kilograms,
grams...

> Special bonus points for anyone who can come up with an example of
> division with fractions (ex: 1/3 divided by 1/2)

1/2 goes into 1 twice. In fact it goes into any whole number N by dividing
N objects into 2 pieces each, giving 2N pieces. Similarly, it goes into
1/3 twice 1/3, or 2/3. If you divide a circle into sixths, you can easily
see that a third of the circle (two pieces) is two-thirds of half the
circle (three pieces), in just the same way that, for example, two beads
is 1/4 of eight beads.

It has been done in detail, and is available on various OER sites, some of
which are given at

http://wiki.sugarlabs.org/go/Open_Education_Resources

I have written about this on other mailing lists. I will do a Turtle Art
version of this some time soon, after I do a bit more on the concept I
have been working on most recently, figurate numbers. I have several such
lessons at

http://wiki.sugarlabs.org/go/Activities/TurtleArt/Tutorials#Mokurai.27s_Tutorials

Tony Forster did a TA visualization for fractions that I plan to carry
further.

http://tonyforster.blogspot.com/2009/12/turtle-fractions.html

Others are welcome to join with Tony and me.

So here is the outline. You will have to take this more slowly with
children, of course.

* Cut a pie in pieces, and color some of the pieces, as Tony did. That
gives the basic idea of a fraction. Point out that when you cut a pie in,
say, 8 pieces, you are doing 1 divided by 1/8.

* Cut more than one pie in the same number of pieces each. This lets us
talk about "improper" fractions and mixed fractions (integer plus
fraction), and converting between them. We can also introduce rational
numbers at some stage of child development.

* Cut a pie in pieces, and cut the pieces into smaller pieces
(multiplication of the simplest fractions, such as 1/2 times 1/3). Some
fractions can be described using the bigger pieces, and some require the
smaller pieces. Talk about reducing fractions to lowest terms. (You will
need other materials in order to talk about Greatest Common Divisors. I'll
do something on that.) Take some time on multiplying fractions. Then
notice that, for example, if you divide a pie into sixths, three of the
pieces make a half. 3 times 1/6 is 1/2, so 1/2 divided by 3 is 1/6, and
1/2 (= 3/6) divided by 1/6 is 3. (Assuming prior understanding that if the
product of, say, 2 and 3 is 6, then 6/3 = 2 and 6/2 = 3.)

* Cut several pies. For example, cut two pies into three pieces each, and
then color pairs of pieces. How many groups of two pieces make two pies?
Congratulations, you have just divided 2 by 2/3.

* Work other examples, dividing whole numbers by fractions, then fractions
by other fractions, choosing cases that come out even to start with.

* Now look at examples where one fraction does not go evenly into the
other. What do you have to do to make sense of the remainder? Say you have
a pizza cut into 8 pieces, and you have hungry pizza eaters who want three
slices each. How many can you accommodate? Well, two, with two slices left
over. Two slices is 2/3 of three slices, so that comes to 2 2/3 portions.

None of this requires Turtle Art. You can cut pies or cakes, or pieces of
construction paper to do all of this. Oh, yes. How many pieces do the
local pizza parlors cut pizzas into? What fractions can you make from
those pieces? Can you find pictures of pizzas from directly above, so that
they appear as circles? (Yes.) What else? Craters on the moon? The whole
moon? Circular swimming pools, fountains, ponds?

It remains an open question whether the children will discover the
invert-and-multiply rule for dividing fractions by themselves, whether
they will need broad hints, or whether they will have to be told. It would
be interesting to me to hear how they would explain these ideas to each
other. I will be interested to hear your results.

> Thanks,
> Stephen
> _______________________________________________
> IAEP -- It's An Education Project (not a laptop project!)
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-- 
Edward Mokurai
(默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر
ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
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