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?

3

2

I’m the NUMERATOR. I tell you the number of

parts

I’m the DENOMINATOR. I tell you the total number

of parts

A fraction is formed by dividing a whole into a number of parts

One boxBroken into 3 piecesTake 2 of the 3

pieces

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

Fractions are always smaller than a whole

Fractions values are determined by counting parts

“It’s 1/3 because 1 part out of 3 parts is shaded.”

“4/3? That’s impossible!”

“A fourth is a little pie shape.”

A FRACTION is a number that represents a part of a whole.

Fractions are useful whenever you need to split things up.

1 Whole Cake

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.

“Each child also gets 1 third from the second candy bar. That’s 2 thirds for each person.”

Another Approach

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 areknocked 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?

2 pieces4 pieces6 pieces

(not equal)

Equivalent fractions

2

14

2

6

3

12

6

An equivalent fraction is one that has the same value and position

on the number line but has a different denominator

Equivalent fractions can be found by multiplying by a special form of 1

........5

5

4

4

3

3

2

21 etc

Multiplying By a Special Form of One

• Multiplying any number by 1 does not change the value 4 x 1 = 4, 9 x 1 = 9 ……….

• Any number divided by itself = 1.

Why does it work?

............12

12

11

11

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

21

Multiplying a fraction by a special form of one changes the numerator and the denominator but

DOES NOT CHANGE THE VALUE

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

17

17

25

25

20

20

50

50

125

125

Finding equivalent fractions

5

3

20

?Convert 5ths to 20ths

What do we multiply 5 by to get a product of 20?

That’s 4 so I must multiply by

4

45

3

20

12

4

4

5

3

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

18

12

6

6

3

2 1

3

2

3

2

18

12

618

612

3

2

Another way of doing this