1. Introduction to Fractions

2. 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

3. 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?

4. What is a Fraction? 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 box Broken into 3 pieces Take 2 of the 3 pieces

5. A common curriculum approach to fractions:

6. If children learn fractions by doing lots of exercises like this one, what are they likely to think about fractions? How much is shaded?

7. 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.”

8. 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

9. Fractions are useful when you want to split something fairly with a friend.

10. 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...

11. Problem 3 children want to share 2 candy bars equally. How much can each child have?

12. 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.”

13. Another Approach “Each child gets 1 third from the first candy bar.

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

15. 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

16. What fraction of the pie is green? 3 4

17. What fraction of the plates are green? 1 6

18. What fraction of the pins are knocked down? 3 10

19. What fraction of the musical instruments have strings? 2 5

20. 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.

21. 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.”

22. Pizza Story How can we split the pizza into 6 equal pieces? 2 pieces 4 pieces 6 pieces (not equal)

23. 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

24. 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? Multiplying a fraction by a special form of one changes the numerator and the denominator but DOES NOT CHANGE THE VALUE

26. Finding equivalent fractions Convert 5ths to 20ths What do we multiply 5 by to get a product of 20? That’s 4 so I must multiply by Special form of 1

27. Simplifying Fractions: Cancelling Simplifying means finding an equivalent fraction with the LOWEST denominator by making a special form of 1 equal to 1 Another way of doing this