Introduction to Fractions
If people do not believe that mathematics is
simple, it is only because they do not realize
how complicated life is
John Louis von Neumann
It is a mathematical fact
that fifty percent of all
doctors graduate in the
bottom half of their class
Unknown
Guiding Questions



What does it mean to understand fractions?
What kinds of problems help children develop
their understanding of fractions?
How do you use the details of children’s
thinking in your teaching?
What is a Fraction?
A fraction is formed by dividing a whole into a
number of parts
One box
Broken into 3 pieces
Take 2 of the 3 pieces
2
3
I’m the
NUMERATOR. I tell
you the number of
parts
I’m the
DENOMINATOR. I tell
you the total number
of parts
A common curriculum approach to fractions:
If children learn fractions by doing lots of
exercises like this one, what are they
likely to think about fractions?

How much is shaded?
Common Misconceptions

Fractions are pieces
“A fourth is a little pie shape.”

Fractions are always smaller than a whole
“4/3? That’s impossible!”

Fractions values are determined by counting
parts
“It’s 1/3 because 1 part out
of 3 parts is shaded.”
A FRACTION is a
number that
represents a part of a
whole.
1 Whole Cake
Fractions are useful
whenever you need
to split things up.
4 pieces are the
same as 1 whole cake
Fractions are useful
when you want to split
something fairly with
a friend.
Johnny and Sally were having a
snack, but there was a
problem. Sally’s mom only had
1 cookie left. What can they
do? Sally remembered
something that her teacher
said about splitting things into
equal pieces. Look to see how
Sally split the cookie...
Problem

3 children want to share 2 candy bars
equally. How much can each child have?
Sample children’s strategies
“I cut the candy bars in half, to see if it would
work and it did. Everybody gets a half. Then I cut
the last half in three parts. Everyone gets another
piece.”
Another Approach
“Each child gets 1 third from the first candy bar.
Another Approach
“Each child also gets 1 third from the second
candy bar. That’s 2 thirds for each person.”
Mental strategy
“I know that everyone can share each
candy bar and get 1/3 of a candy bar.
There’s 2 candy bars, so that 1/3, 2
times. It’s 2/3.”
2 ÷ 3 = 2/3
What fraction of the pie is green?
3
4
What fraction of the plates are
green?
1
6
What fraction of the pins are
knocked down?
3
10
What fraction of the musical
instruments have strings?
2
5
Fractions with Candy Bar

Believe it or not, teaching fractions can be both
educational and delicious


The Hershey's Milk Chocolate
Bar Fractions Book
Not everyone loves math, but surely everyone
loves Hershey's Chocolate Bars

Hershey’s are conveniently divided into 12 equal
squares, making them the perfect
manipulatives for demonstrating how fractions
work.
Pizza Story

A man walks into the pizza store
and orders a medium pizza.

The clerk asks “ Would you like
that cut in 8 pieces or 6?”

The man replied “ Make it 6. I
don’t think I can eat 8.”
Pizza Story

How can we split the pizza
into 6 equal pieces?
24pieces
6pieces
pieces
(not equal)
Equivalent fractions
An equivalent fraction is one that
has the same value and position
on the number line but has a
different denominator
6
2
3
1



6
12
4
2
Equivalent fractions can be found by
multiplying by a special form of 1
2 3 4 5
1      ......etc..
2 3 4 5
Multiplying By a Special Form of One
Why does it work?
• Multiplying any number by 1 does not change
the value 4 x 1 = 4, 9 x 1 = 9 ……….
• Any number divided by itself = 1.
2 3 4 5 6 7 8 9 10 11 12
1        


 ............
2 3 4 5 6 7 8 9 10 11 12
Multiplying a fraction by a special form of one
changes the numerator and the denominator but
DOES NOT CHANGE THE VALUE
8 25 2
8 25 2
3 17 10
3 17 10
9 6
9 6
20
20
5 125 4
5 125 4
7
11
11 7
50
50
Finding equivalent fractions
3
5
Convert 5ths to 20ths
?
20
3
4 of
so I must
multiply
by
WhatThat’s
do we 4multiply
5 by
to get a product
20?
5
4
3 4 12
 
5 4 20
Special form of 1
Simplifying Fractions: Cancelling
• Simplifying means finding an equivalent
fraction with the LOWEST denominator by
making a special form of 1 equal to 1
2
2
2 6
12
 
 1 
18
3 6
3
3
Another way
of doing this
12 12  6
2


18 18  6
3