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” 713 • 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?
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?)
12 7 13 8 += Estimate NAEP 13 yr7%24%28%27%14% MPS 6-7-813%9%23%41%9% • 1 • 2 • 19 • 21 • Don’t Know National Assessment of Education Progress (NAEP); MPS n=72)
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)
14 56 34 1112 – = + = 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.