I have an ongoing fraction-positive crusade with my older pupils. Embrace
the fraction! Learn to love it, and it will be your bosom friend. I'm not sure
what it is about fractions that frighten and intimidate, but they do, and while
calculator created decimals might seem nicer, they aren't. Decimals can be
messy, tricky, inexact creatures that can get a person in trouble.
At it's base, a fraction is simple: The bottom (denominator) number tells you
how many pieces you've cut your item (cookie, pie, cake) into, and the
top number (numerator) tells you how many of those pieces you have.
Sure, there are rules about adding/subtracting, multiplication, and division,
but there is just so much you can do with a fraction!
In the hands-on, visually driven world of Common Core, fractions are
introduced using egg cartons and clocks. Why? Well, they are comfortable
items most children have seen, they are cheap, and they contain units of twelve.
Twelve is a very friendly number when it comes to fractions and
equivalent fractions (different looking fractions which have the same value).
(This blog post is part of an on-going effort to given parents an understanding
of how mathematical elements are being taught currently.)
To start, we'll practice dividing a dozen eggs into equal parts.
Here they've been divided into two equal parts (or halves):
Three equal parts, or thirds:
Four equal parts, or fourths:
Six equal parts, or sixths:
And twelve equal parts, or twelfths:
Now, you'll notice that I've skipped, 5ths, 7ths, 8ths, etc.
That's because 12 can't be evenly divided into those parts.
Once students can recognize the shape of the different fractions,
we can start playing with equivalent fractions.
The fraction 1/2...
is the same as 2/4.
When I work with my students, I often use legos, instead of
eggs. I have more freedom to use fractions not included
in the family of twelve (Hello, 8ths!). Also, they take up less
space, and all of my students like playing with legos. And, I have
bricks that are set units, so it is easier to pick up 1/2 and hold it in
my hand and manipulate it. It's also easier to show that 1/3 and
1/4 are different sizes, so I can't directly add them.
Here I'm using the same basic set up of twelve as the
egg carton, but I've expanded how I've represented the fractions.
Now, what else can I do with this model?
I can add fractions. I can take a 1/4 piece and show that
if I'm going to add it to a 1/3 piece, I need to break both
sections into twelfths to do it. Once I do that, I would
have 7/12 for an answer. Similarly, subtraction can be
introduced this way.
We can also explore the idea that 1/4 of 12 is 3. Or that
5/6 of 12 is 10. To do this, I count the eggs or the pips (the
circles on top of the legos) contained in each fraction.
Improper fractions (where the numerator is larger than the
denominator) and their relationship with mixed numbers
(whole numbers with fractions) are also easy to manipulate
with this modeling.
This method can also be used with clock faces, as they are
conveniently divided into twelfths. That pathway opens up
pie shaped graphs, geometric elements, and time work.
I find that my students really like this method of fraction
manipulation and exploration. It works for most of them.
They have a deep understanding of why you need common
denominators to add, and how a single fraction can have
multiple representations. There is often a stumbling point about
why 1/4 is 3 eggs, and 1/3 is 4 eggs (out of twelve), but we sort
My main complaint with this model (and with many other
visual based methods) is that I wish the mathematical
part of fractions was more integrated. Most of my 4th and 5th
grade students memorize the equivalent fractions in an egg carton.
They don't actually know how to mathematically reduce fractions.
Their knowledge of fractions is tied directly to the fraction families
you can make in a base of 12. If you give them 8ths, they don't
know what to do. Do they have a "deep understanding"? Yes.
But without the numerically driven (non-model) base, it isn't
as flexible as it needs to be. In my experience, this void isn't
addressed effectively in 5th grade, and then in 6th grade I have
students who want to draw fraction out in a model. None of
my high school students (those who fear fractions, and turn
to decimals as their (faulty) saviors) had the complete Common
Core base. It will be interesting to see if the students who
grow up completely immersed in Common Core, and who
have that model based knowledge at their heart, are more
comfortable with fractions. I hope so.