Reasoning with Rational Numbers (Fractions) ‏ Originally from: Math Alliance Project July 20, 2010 DeAnn Huinker, Chris Guthrie, Melissa Hedges,& Beth Schefelker,

Your students and fractions… What specific difficulty have your students had or what overgeneralization have they made about fractions this year? Brainstorm individually for 1 minute and then discuss with your group.

Learning Intentions & Success Criteria Learning Intentions: We are learning to… Understand how conceptual thought patterns support the development of number sense with fractions. Understand how estimation should be an integral part of fraction computation development. Success Criteria: You will be able to… Justify your thinking when adding and subtracting fractions using concrete models and estimation strategies.

Solve the following CABS individually: 13a) Name a fraction that is between 1/2 and 2/3 in size: 1/2 < _________ < 2/3 13b) Justify (explain) how you know your fraction is between 1/2 and 2/3. NO COMMON DENOMI- NATORS!!!!

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

Fraction Strips: Each member at your table should make strips for the following fractions: Halves, Thirds, Fourths, Fifths, Sixths, Eighths, Ninths, Tenths, and Twelfths As you are making the strips, discuss the strategy you are using for each strip. How could these strips help you solve the CABS we previously worked on?

Making Connections to the Number Line Model Length or linear model – emphasizes that a fraction is a number as well as its relative size to other numbers 0 ½ 1 Where would ¾ fall on this number line? Why?

Benchmarks for “Rational Numbers” Is it a small or big part of the whole unit? How far away is it from a whole unit? More than, less than, or equivalent to: one whole? two wholes? one half? zero? 7 13

WAR!!! Deal out the fraction cards Each player plays one card The person who has the larger fraction played (and can justify why their fraction is larger) wins both cards. Keep track of the fraction pairs and strategies you used. What strategies did you use?

Conceptual Thought Patterns for Reasoning with Fractions 8/15 or 11/15 7/20 or 7/9 6/10 or 9/5 11/12 or 7/8 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 whole or one-half (residual strategy– What’s missing?)

= Estimate Don’t Know NAEP 13 yr 7% 24% 28% 27% 14% National Assessment of Education Progress (NAEP); MPS n=72) MPS % 9% 23% 41% 9%

Research Findings: Operations with Fractions Students do not apply their understanding of the magnitude (or meaning) of fractions when they operate with them. (Carpenter, Corbitt, Linquist, & Reys, 1981) Estimation is useful and important when operating with fractions and these students are more successful (Bezuk & Bieck, 1993) Students who can use and move between models for fraction operations are more likely to reason with fractions as quantities. (Towsley, 1989)

Representing Your Reasoning Split the two problems between the members at your table. Use estimation to reason through these problems. How did benchmarks help? – = =

Task: Estimation with Benchmarks Facilitator reveals one problem at a time. Each individual silently estimates. On the facilitator’s cue: Thumbs up = greater than benchmark Thumbs down = less than benchmark Wavering “waffling” = unsure Justify reasoning.

Review: Learning Intentions: We are learning to… Understand how conceptual thought patterns support the development of number sense with fractions. Understand how estimation should be an integral part of fraction computation development. Success Criteria: You will be able to… Justify your thinking when adding and subtracting fractions using concrete models and estimation strategies.

Math Misconceptions Open to page 34 (Understanding Fractions) and page 40 (Adding and Subtracting Fractions). Put a Post-it in these sections for future reference.