1. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved Chapter Ten any public performance or display, including transmission of any image over a network; preparation of any derivative work, including the extraction, in whole or in part, of any images; any rental, lease, or lending of the program. This multimedia product and its contents are protected under copyright law. The following are prohibited by law: Chapter Ten Developing Fraction Concepts

2. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-2 Children learn best through active involvement with a variety of concrete materials Most children need extended experiences with manipulative materials in order to develop mental images of fractions, and in order to reason and think conceptually about fractions. (Cramer, Behr, Post, & Lesh, 1997) Four Principles to Help Children Understand Fractions

3. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-3 Children benefit from opportunities to talk about their fraction understandings with each other and with their teacher Learning experiences should begin with helping children develop conceptual knowledge with fractions before moving to more formal work with symbols and computation. (Cramer, Behr, Post, & Lesh, 1997) Four Principles to Help Children Understand Fractions …Play with Fraction Wall and Advertising!

4. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-4 Understanding sharing situations Solve the following problem : 4 children want to share 3 candy bars equally. How much can each child have? How might a child solve this problem? Developing Fraction Concepts and Number Sense

5. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-5 Children must acquire the following abilities to work effectively with fractions The ability to represent fractions using words, models, diagrams, and symbols and make connections among various representations; The ability to give other names for fractions and justify the procedures used to generate the equivalent forms; The ability to describe the magnitude of fractions by comparing them to common benchmarks, giving simple estimates, ordering a set of fractions, and finding a number between two fractions. Number Sense with Fractions

6. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-6 Part-whole Quotient Ratio Multiplicative Operator Different Interpretations of Fractions

7. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-7 Emphasis in the elementary mathematics program Part-whole meaning of fractions comprises of different units. It can be: A region (an object to be shared or an area to be divided) A set of objects A unit of linear measurement Part-whole

8. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-8 Region model – the region is the whole and the parts congruent (same size and shape) Models of Fractions 2 6 What fraction of the rectangle is purple?

9. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-9 Parts-of-a-set model – the whole is a set of objects and the subset of the whole make up the fractional parts Models of Fractions

10. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-10 Measurement model – the length or measurement is compared in these models Models of Fractions 1 2 3 4 5 Length of model is 4 ⅓

11. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-11 Area model – the parts are equal in area but not necessarily congruent Models of Fractions

12. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-12 Ratio interpretation – a fraction can represent a ratio between two quantities Quotient interpretation – a fraction can express a division or sharing To support student understanding of this interpretation, use story problems in a familiar, real-world setting. Operator interpretation – a fraction can express a multiplicative operation (either division or multiplication). Other Interpretations of Fractions

13. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-13 Confusion between fraction names and ordinal numbers is common Early work should be completed without symbols Numerator = “Top Number” Denominator = “Bottom Number” Meaning of fractions is developed by considering different units. Continuous quantities Discrete quantities Fraction Names, Symbolism, and Units

14. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-14 Which is greater? How do you know? (Try to determine this by using materials or drawing a picture instead of using cross products, common denominators, decimals, or percents.) Comparing Fractions

15. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-15 Order the fractions from smallest to largest. Write how you know. Ordering Fractions

16. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-16 Which of these fractions are close to zero, one-half, or one? Relative Size of Fractions

17. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-17 Proper Fractions For every a/b, a < b Improper Fractions For every a/b, a > b or a = b Types of Fractions

18. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-18 Different fractions can represent the same amount Discuss the meaning of the word equivalent with students Discuss the similarities and differences of equivalent fractions Look at patterns and articulate rules for renaming a fraction Equivalent Fractions

19. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-19 All these fractions = 1 4444 33 5555 2222 7777 When the numerator & denominator of a fraction are the same, the fraction equals 1. mathjourneys.com/slideshows/equivalent_fractions.ppt

20. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-20 What do you get when you multiply a fraction by 1? You get AN EQUIVALENT FRACTION that makes adding & subtracting fractions possible. mathjourneys.com/slideshows/equivalent_fractions.ppt

21. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 10-21 mathjourneys.com/slideshows/equivalent_fractions.ppt