Fractions
are the mathematical equivalent of taxes:
If want to enjoy the benefits of living in a society, you have to deal
with them from time to time. The good
news is that working with fractions is a lot easier than trying to fill out a
Form 1040, even with problems like:
Penny used
2/5 lb of flour to bake a cake. She used another 3/4 lb of flour to bake
another cake. How much flour did she use
altogether? Draw a picture and/or write
the number sentence that proves the answer.
Simplify the answer.
Problems
like these are introduced in the fifth grade, though fractions themselves are
introduced towards the end of third grade.
The rectangular fraction model
is one of the more insightful ways to represent a fraction. We begin with a rectangle that represents the
whole amount, and divide it into equal parts.
Each part is a unit fraction.
For example, if our rectangle represented one pound of flour, we could
divide it into five equal vertical parts (on the left) or four equal horizontal
parts :

2/5 pound 3/4 pound
On the left, each of the small
rectangles is 1/5 pound and, since the shaded part is 2 of those rectangles, it
represents 2/5 pounds: the amount used
for the first cake. On the right, each
of the rectangles is 1/4 pound and, since the shaded part is 3 of those
rectangles, it represents 3/4 pound: the
amount used for the second cake.
The total amount of flour used is
just the sum of the two portions: 2/5 + 3/4. While we have
five shaded pieces altogether, they're
of different size and shape. But notice
that if we divide the first rectangle horizontally into four parts and
the second rectangle vertically into five parts (keeping the original
partitions and shadings), we obtain:

2/5 = 8/20 3/4 = 15/20
We haven't changed the size or shape
of either of the shaded regions, so the first still represent 2/5 pounds and
the second 3/4 pounds. But we have
broken both into smaller pieces, each 1/20 of the original rectangle. On the left, our shaded region now consists
of 8 pieces, each representing 1/20 of a pound:
thus it represents 8/20 of a pound, and we've drawn the fractional
equivalence 2/5 = 8/20. By a similar
analysis, the rectangle on the right shows the fractional equivalence 3/4 = 15/20.
Now, because all our pieces are the
same size, we can just count: there are
8 + 15 = 23 pieces altogether, each of which is 1/20 of the pound. Or, since 20 of these pieces make up one
pound, we can rearrange them to see that we have 1 3/20 pounds altogether:
and
2/5 pound + 3/4 pound

gives us
1 3/20 pounds
As with other aspects of the Common
Core, the goal is to promote conceptual understanding, which helps
learners avoid common errors and pitfalls.
As a single example, the algorithm for producing a fraction equivalent
to a/b is to multiply numerator and denominator by n: in this case, 2/5 = 2x4/5x4. A common error is to
only multiply the denominator and obtain the “equivalent” fraction 2/20. A common pitfall is
to obtain the correct equivalent fraction 8/20 but, since students have been trained to reduce fractions to
lowest terms, to immediately reduce this fraction back to 2/5.
The
rectangular fraction model avoids both error and pitfall.
Common Core material is available at EngageNY under a Creative Commons non-commercial license: in effect, school districts can use the materials free of charge, as long as they attribute the material correctly.
Jeff Suzuki teaches mathematics at Brooklyn College, and is one of the founders of the Mid-Hudson Valley Math Teachers Circle, a group of teachers, professors, and mathematics aficionados working to promote mathematics education in the Hudson Valley.
Read more answers to Common Core questions