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Developing Fraction Concepts

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  1. Developing Fraction Concepts Math Alliance July 13, 2010 Beth Schefelker, DeAnn Huinker, Chris Guthrie & Melissa Hedges

  2. Learning Intentions and Success Criteria • We are learning to… • deepen our knowledge of fractions and rational numbers • We will be successful when… • we can apply knowledge of fraction benchmarks and conceptual thought patterns to reason and compare fractions.

  3. 34 • How do students see this fraction? • Students often see fractions as two whole numbers (Behr et al., 1983). • What are ways we want students to “see” and “think about” fractions?

  4. Different models offer different opportunities to learn. Area model – visualize part of the whole Use the grey triangles to cover ¾ of the octagon. Length or linear model – emphasizes that a fraction is a number as well as its relative size to other numbers 1 ½ 2 Where would ¾ fall on this number line? Why? Set Model – the whole is set of objects and subsets of the whole make up fractional parts. 3/4 of the smiley faces are blue

  5. What is a fraction? What is a rational number? Are they the same?

  6. Rational Number vs Fraction • Rational Number = How much?Refers to a quantity or relative amount,expressed with varied written symbols. • Fraction = NotationRefers to a symbol or numeral used to represent a rational number. • (Lamon, 1999)‏

  7. Reasoning About Fractions Name the fraction shown in the shaded region of the figure below: Share your responses. What do you notice? What do I need to consider as I decide on an answer?

  8. Who’s right? Go to a poster that has a different answer than yours. Defend why that response could be correct. Now come up with a second argument to defend your answer. “The study of fractions offers many delightful and challenging opportunities to practice mathematical reasoning.” p. 65 Beckmann

  9. 2 ¼ ¾ 9/12 9/4 9 • Consider each response above as you respond to each question: • What’s the whole? • What are the parts? • Big Ideas • A fraction tells us the relationship between the part and the whole. • A fraction is always a fraction of some whole. The whole needs to be understood while working with the fraction even if it is not made explicit. • Models help clarify ideas and visualize the relationships between numerator and denominator.

  10. What does the research say about how students use fractions? • A majority of U.S. students have learned rules but understand very little about what quantities the symbols represent and consequently make frequent and nonsensical errors. • (NRC, 2001)‏ • Share one error you’ve seen your students make.

  11. Reason with “Rational Numbers” and Use Benchmarks Is it a small part of the whole unit? Is it a big part? More than, less than, or equivalent: to one whole? to one half? Close to zero?

  12. 11 24 16 85 Finish these fractions so they are close to but greater than one-half. Finish these fractions so they are close to but less than 1 whole. 9 15 12 21

  13. Comparison of Fractions • Consider ways to reason with benchmarks when comparing these fractions. • 5/7 or 3/7 • 3/8 or 3/4 • 5/4 or 8/9 • 15/16 or 9/10 • 1 1/3 or 6/3

  14. Conceptual Thought Patterns for Comparing Fractions More of the same-size parts. Same number of parts but different sizes. More or less than one-half or one whole. Distance from one-half or one whole (residual piece).

  15. Ordering Fractions on the Number Line Deal out fraction cards (1-2 per person). Allow quiet time to think about placements. Taking turns, each person: Places one fraction on the number line, and Explains his/her reasoning using benchmarks and conceptual thought patterns. Warning: No conversions to decimals! No common denominators! No cross multiplying!

  16. Fraction Cards 3/8 3/10 6/5 7/47 7/100 25/26 7/15 13/24 14/30 16/17 11/9 5/3 8/3 17/12

  17. Using Representations to Conceptualize Fractions How did you think about the fraction 8/3? What does 8/3 mean? • Develop a real-life context for 8/3? • Make a representation for your story that helps develop an understanding of 8/3.

  18. Reflect • As you placed the fractions on the number line, summarize some new reasoning or strengthened understandings.

  19. Walk Away Fractions as quantities. Benchmarks: 0, 1/2, 1, 2 Conceptual thought patterns.

  20. Homework • Beckmann • Read pp. 65-70 • Class Activities: p. 33 1 and 4 • Also recommended though not required “Practice Problems for Section 3.1” p. 70 #7 • Using reasoning other than finding common denominators, cross-multiplying, or converting to decimal numbers to compare the sizes (greater than, equal to, or less than) of the following fractions: • 1/49 1/39 • 7/37 7/35 • 13/25 5/8 • 17/18 19/20

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