1. © TUNING UP FRACTIONS LINDA WEST LWEST@SINGAPOREMATHTRAINING.COM SMARTTRAINING, LLC

2. © BIG IDEAS IN MATHEMATICS - COMPETENCIES –Visualization –Making Connections (looking for patterns in order to generalize) –Communication  Problem Solving  Number Sense Based on the Theories of: –Jerome Bruner –Zoltan Dienes –Richard Skemp

3. © NUMERATOR An adjective The counting number Tell how many The “numberator”

4. © DENOMINATOR A noun A label Tells what kind of unit The “deNAMEnator” An Ordinal number

5. © Let’s Compare

6. © What makes these types of numbers so difficult for students?

7. © Notation confusing “bigger” numbers indicate “smaller amount” Same number indicates different amount Multiplication sometimes yields smaller result while division sometimes yields a larger one The “whole” must always be held in the mind Fractions, decimals and percents traditionally taught as disparate topics

8. © FRACTIONS Fractions Quantity Proportion Percentage is exclusively used for Proportion Decimal is exclusively used for Quantity

9. © Whole vs. Fractional Number Sense Whole NumbersFractional Numbers  Counting units  RELATIONSHIPS TO OTHER NUMBERS More/Less 5 & 10 benchmarks Part/Whole relationships  COUNTING UNITS  RELATIONSHIPS TO OTHER NUMBERS More /Less 0, ½, 1 Benchmarks Part /Whole Relationships Equivalency

10. © Whole Numbers Fractional Numbers EQUAL UNITS BASED ON 10’S WITH RELATIONSHIPS TO EACH OTHER CONNECT CONSTRUCTS WITH ABSTRACT NOTATION AND OPERATIONS WHOLE UNIVERSE OF “UNITS” THAT ARE LESS THAN 1, BUT ARE RELATED TO EACH OTHER! CONNECT CONSTRUCTS WITH ABSTRACT NOTATION AND OPERATIONS

11. © Keys to Understanding Concepts and Problem Solving before Rules and Drill Connections before Calculations

12. © The Dangerous Rush to Rules None of the rules help students think about the meaning Rules give students no means of assessing whether an answer is reasonable Surface mastery of rules is quickly lost Algorithm rules do not immediately apply to every situation Incredibly defeating for students

13. © 2 ÷ = ? Please: 1.Solve. 2.Draw a picture. 3.Write a word problem.

14. © 12 ÷ 4 = ____

15. © Our real question is: How many halves are contained in 2 ¾?

16. © WOULD ANYONE LIKE TO SHARE THEIR WORD PROBLEM? Jacqui ran 2 ¾ miles. This was ½ the distance that she runs each day. What is the total distance that Jacqui runs each day?

17. © NOW TRY THIS ONE 2 ½ ÷ ⅓ Use some of the pattern blocks on your table to solve this problem. Verify with algorithm. Write a word problem that can be solved using this equation.

18. © 2 ½ ÷ 1/3 2 ½ ÷ 2/3

19. © FUN WITH FRACTIONS += Using digit tiles 1 – 9, once only in each equation, how many equations can you create?

20. © HOW DO WE DEVELOP VISUALIZATION SKILLS WITH FRACTIONS? Take out the yellow hexagon. Cover it with as many different pattern blocks as you can until the entire hexagon is filled. How can we express what we have done concretely in a number sentence?

21. © HOW DO WE USE Logic To explain The fraction division algorithm?

22. © Let’s go to an easier problem: 6 ÷ 3 = ____ What does that mean? How can I read that differently? How many threes are in six? 6 is 3 of what number?

23. © Who can read the equation differently? How many ½’s are in 6? 6 is ½ of what number?

24. © DEVELOPING THE CONCEPT

25. © DO YOU SEE A PATTERN? How many halves are in 1? (2) How many halves are in 2? (4) How many halves are in 3? (6) Let’s focus on thirds: –How many thirds are in 1? (3) –How many thirds are in 2? (6) –How many thirds are in 3? (9) –How many thirds are in 4? (12)

26. © DO YOU SEE A PATTERN? LET’S FOCUS ON FOURTHS: –How many fourths are in 1? (4) –How many fourths are in 2? (8) –How many fourths are in 3? (12) –How many fourths are in 4? (16)

27. © DO YOU SEE A PATTERN? LET’S SWITCH IT UP A LITTLE BIT: –How many halves are in 1? –How many thirds are in 1? –How many fourths are in 1? –How many fifths are in 1? –How many halves are in 2? –How many thirds are in 2? etc., etc.

28. © AREA MODEL TO REAL LIFE We need multiple embodiments. (Zhang 2012)

29. © DIVISION BY A FRACTION 0 1

30. © USING BAR MODELS WITH FRACTIONS Divide 3 by 2/3 2/3 There are four 2/3’s. and another half of a 2/3 in 3. So there are four-and-a-half 2/3’s in 3. 3 ÷ 2/3 = 3 x 3/2 = ?

31. © What is your word problem?

32. © He travelled of the total journey on the last day.

33. ©

34. © A PICTURE IS WORTH … Find ¼ of 48. Find ¾ of 48. 1/3 of a number is 48. Find the number. 4/5 of a number is 48. Find the number.

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