introduction to fractions. if people do not believe that mathematics is simple, it is only because...
TRANSCRIPT
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