Mastering Fractions: Key Concepts and Applications for Middle School Students

1. Fractions G. Donald Allen Department of Mathematics Texas A&M University

2. From the NCTM… • Middle school should acquire a deep understanding of fractions and be able to use them competently in problem solving. NCTM(2000)

3. From the NAEP… • Reports show that fractions are "exceedingly difficult for children to master. " • Students are frequently unable to remember prior experiences about fractions covered in lower grade levels NAEP, 2001 National Assessment of Educational Progress

4. Mathematics Proficiency • Conceptual understanding • Procedural fluency • Strategic competence • Adaptive reasoning • Productive disposition Adding it Up, - National Research Council

5. Bottlenecks in K-8 • It is widely recognized that there are at least two major bottlenecks in the mathematics education of grades K–8: • The teaching of fractions • The introduction of algebra

6. Student mistakes with fractions • Algorithmically based mistakes • Intuitively based mistakes • Mistakes based on formal knowledge. • e.g. Children may try to apply ideas they have about whole numbers to rational numbers and run into trouble Tirosh (2000)

7. Polyvalence, again • When it comes to fractions there are multiple interpretations. • What are they? • What do students think they are?

8. Multiple meanings • Parts of a whole: when an object is equally divided into d parts, then a/b denotes a of those b parts. • The size of a portion when an object of size a is divided into b equal portions. • The quotient of the integer a divided by b. • The ratio of a to b. • An operator: an instruction that carries out a process, such as “4/5 of”.

9. Definition of a fraction • A rational number expressed in the form • a/b --- in-line notation, or • --- traditional "display" notation where a and b are integers. This is simply the division of integers by integers.

10. Basic Fractions Equivalent Fractions Adding Fractions Subtracting Fractions Multiplying Fractions Dividing Fractions Comparing Fractions Converting Fractions Reducing Fractions Relationships Subtracting Fractions Fractions – Basic Syllabus

11. Comparing Fractions • Equivalent Fractions • Comparing - Like Denominators • Comparing - Unlike Denominators • Comparing – Unlike numerators and denominators • Comparing Fractions and Decimals

12. Converting Fractions • Converting to Mixed Numbers • Converting from Mixed Numbers • Converting to Percents • Converting from Percents • Converting to Decimals • Converting to Scientific Notation • Converting from Scientific Notation

13. Reducing Fractions • Prime and Composite Numbers • Factors • Greatest Common Factor • Least Common Denominator • Least Common Multiple • Simplifying

14. Relationships • Relating Fractions To Decimals • Relating Decimals to Fractions • Relating mixed fractions to improper fractions • Relating improper fractions to mixed fractions.

15. Equivalent fractions • Two fractions are equivalent if they represent the same number. • This means that if then • The common factor k has many names. This principle is the single most important fact about fractions.

16. Equivalent fractions • Why is • It’s just arithmetic! ? Productive disposition

17. Why are equivalent fractions important? • For comparing fractions • For adding fractions • For subtracting fractions • For resolving proportion problems • For scaling problems • For calculus and beyond

18. Addition • Addition • Addition - Like Denominators • Addition - Unlike Denominators • Addition Mixed Numbers

19. Addition - Like Denominators • Why is • It is by Pie charts? Fraction bars? Spinners? Blocks/Tiles? ?

20. Addition - Like Denominators • Answer. It’s just arithmetic! We know… • So,

21. Common mistakes Where??? College

22. How to add fractions, #1 • Definition of addition. In some sources we see… What’s wrong with this??

23. How to add fractions, #2 • Definition of addition. In other sources we see…

24. Example – no lcm

25. Example – with lcm lcm = 8

26. Go with the flow • Flow charting a process can reveal unnoticed complexities. • The difference between using the lcm and simple denominator multiplication is not insignificant.

27. Adding fractions process, #1

28. Adding fractions process, #2

29. Is this too difficult? • Remember this can be regarded as strictly a skill. • It will always be used as a skill – when it is used. • At what point – we may ask – is fundamental understanding suppose to kick in? Consider calculus – the accepted wisdom

30. Is this true? • Informal surveys among teachers consistently reveal that many of their students simply give up learning fractions at the point of the introduction of addition.

31. Tips for teaching fractions • Engage your students’ interest in fractions. • Stress the importance of fractions in the world around them and in successful careers. • Emphasize that fractions are used in a variety of ways.

32. Tips for teaching fractions • Practice understanding of fractions by using math manipulatives. • Practice basic words or phrases by giving students a problem and a list of relevant terms, e.g., "numerator," "denominator,“ • Practice fractions by having students observe their surroundings, e.g., what fraction of classmates have black hair, have brown eyes.

33. Tips for teaching fractions • Practice fraction problems by having students write their own fractions based on their own experiences. • Practice fraction problems by having students work in small groups to create their own surveys around fractions based on classmates' preferences http://www.meritsoftware.com/teaching_tips/tips_mathematics.html#3

34. Engaging students… • Pallotta, J. (1999). The hershey's milk chocolate bar fractions. Cartwheel Books. • Adler, D. A., & Tobin, N. Fraction fun. • Ginsburg, M. Gator Pie. • Leedy, L. Fraction Action. • Mathews, L. Gator Pie. Mostly elementary

35. Dividing Fractions • Division • Division by Integers

36. Multiplying Fractions • Multiplication • Multiplication by Integers

37. Division of fractions

38. Mixed fractions

39. Multiplication of fractions