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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.
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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 I’m the NUMERATOR. I tell you the number of parts One box Broken into 3 pieces Take 2 of the 3 pieces I’m the DENOMINATOR. I tell you the total number of parts
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 “A fourth is a little pie shape.” “4/3? That’s impossible!” “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. Fractions are useful whenever you need to split things up. 1 Whole Cake 4 pieces are the same as 1 whole cake
Everyday Fractions Fractions are useful when you want to split something fairly with a friend.
Fair Share 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
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 2 pieces • How can we split the pizza into 6 equal pieces? 4 pieces 6 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 Equivalent fractions can be found by multiplying by a special form of 1
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. Multiplying a fraction by a special form of one changes the numerator and the denominator but DOES NOT CHANGE THE VALUE
Finding equivalent fractions Convert 5ths to 20ths That’s 4 so I must multiply by What do we multiply 5 by to get a product of 20? Special form of 1
Simplifying Fractions: Cancelling • Simplifying means finding an equivalent fraction with the LOWEST denominator by making aspecial form of 1equal to 1 1 Another way of doing this
