Beyond Pizzas & Pies: Suppor.ng Frac.on Sense mathsolu.ons.com/presenta.ons Julie McNamara CMC‐ South/Palm Springs/November 2010 © 2010 Math Solu.ons Students’ Challenges with Frac.ons •  What are some of the biggest challenges students face with frac.ons? © 2010 Math Solu.ons th
8 Grade, NAEP 1996 Es.mate the answer to + Answer choices were: A.
1
7%
B.
2
24%
C.
19
28%
D.
21
27%
E.
I don’t know
14%
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“The difficulty with frac.ons (including decimals and percents) is pervasive and is a major obstacle to further progress in mathema.cs. . .” —Report of the Na,onal Math Panel, March 2008 3 © 2010 Math Solu.ons In one minute, write down everything that comes to mind when you think about or see: 4 © 2010 Math Solu.ons In one minute, write down everything that comes to mind when you think about or see: 9
5 © 2010 Math Solu.ons Compare Your Responses •  What similari.es do you see? •  What differences do you see? •  Any surprises or insights? 6 © 2010 Math Solu.ons Adjec.ves vs. Nouns (Adapted from Kathy Richardson, NCTM 2008)
•  Young children ini.ally consider numbers as adjec.ves or descriptors – –  9 bears –  6 cookies –  20 students •  Eventually, they come to understand numbers as nouns or concepts– –  9 is half of 18, –  It is 1 less than 10, –  It is 4.5 doubled, –  It is 3 squared, –  It is the square root of 81, –  ………? © 2010 Math Solu.ons 7 Adjec.ves vs. nouns (con.nued) •  Students need opportuni.es to transi.on from considering frac.ons as adjec.ves – –  1/2 of a pizza –  3/4 of an hour –  2/3 of a cup •  to considering them as nouns – –  5/8 is a liile more than 1/2, but less than 1 –  It is 3/8 less than 1 –  It is equivalent to 10/16 –  It is twice 5/16 –  It is half of 1¼ –  ………..? © 2010 Math Solu.ons 8 “It may be surprising that, for most students, to think of a ra.onal number as a number – as an individual en.ty or a single point on a number line – is a novel idea.” —Adding it Up: Helping Children Learn Mathema,cs, Na.onal Research Council, 2001 9 © 2010 Math Solu.ons th
4 Grade, NAEP 2009 Which frac.on has a value closest to ½? A.  5/8
B.  1/6
C.  2/2
D.  1/5
© 2010 Math Solu.ons th
4 Grade, NAEP 2009 Which frac.on has a value closest to ½? A.  5/8
B.  1/6
C.  2/2
D.  1/5
© 2010 Math Solu.ons 25% 6% 41% 26% Why are frac.ons so hard? •  Frac.on nota.on – numbers must be considered in new ways •  Prac.ces that simplify and/or mask the meaning of frac.ons •  Overreliance on whole number knowledge •  Many meanings of frac.ons © 2010 Math Solu.ons What is Frac.on Sense? “Frac.on sense implies a deep and flexible understanding of frac.ons that is not dependent on any one context or type of problem. Frac.on sense is .ed to common sense: Students with frac.on sense can reason about frac.ons and don’t apply rules and procedures blindly ‐ nor do they give nonsensical answers to problems involving frac.ons.” © 2010 Math Solu.ons Common Core State Standards “Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathema.cal understanding look like? One hallmark of mathema.cal understanding is the ability to jus.fy, in a way appropriate to the student’s mathema.cal maturity, why a par.cular mathema.cal statement is true or where a mathema.cal rule comes from.” © 2010 Math Solu.ons Common Core State Standards Number and Opera.ons – Frac.ons Grade 3: Develop understanding of frac.ons as numbers. Grade 4: Extend understanding of frac.on equivalence and ordering. Grade 5: Use equivalent frac.ons as a strategy to add and subtract frac.ons. © 2010 Math Solu.ons Beyond Pizzas & Pies: 10 Essen,al Strategies for Suppor,ng Frac,on Sense © 2010 Math Solu.ons #1 Provide opportuni.es for students to work with irregularly par..oned, and unpar..oned, areas, lengths, and number lines. © 2010 Math Solu.ons #2 Provide opportuni.es for students to inves.gate, assess, and refine mathema.cal rules and generaliza.ons. © 2010 Math Solu.ons #3 Provide opportuni.es for students to recognize equivalent frac.ons as different ways to name the same quan.ty. © 2010 Math Solu.ons #4 Provide opportuni.es for students to work with changing units. © 2010 Math Solu.ons #5 Provide opportuni.es for students to develop their understanding of the importance of context in frac.on comparison tasks. © 2010 Math Solu.ons #6 Provide meaningful opportuni.es for students to translate between frac.on and decimal nota.on. © 2010 Math Solu.ons #7 Provide opportuni.es for students to translate between different frac.on representa.ons. © 2010 Math Solu.ons #8 •  Provide students with mul.ple strategies for comparing and reasoning about frac.ons. © 2010 Math Solu.ons #9 •  Provide opportuni.es for students to engage in mathema.cal discourse and share and discuss their mathema.cal ideas, even those that may not be fully formed or completely accurate. © 2010 Math Solu.ons #10 •  Provide opportuni.es for students to build on their reasoning and sense making skills about frac.ons by working with a variety of manipula.ves and tools, such as Cuisenaire rods, Paiern Blocks, Frac.on Kits, and ordinary items from their lives. © 2010 Math Solu.ons The Problem with Par..oning: It’s Not Just About Coun.ng the Pieces © 2010 Math Solu.ons What frac.on of the square is shaded? Tell me how you know. © 2010 Math Solu.ons Grade 4 student ‐ Hannah © 2010 Math Solu.ons Grade 4 student ‐ Jose © 2010 Math Solu.ons What frac.on is shown by B? © 2010 Math Solu.ons What frac.on is shown by B? © 2010 Math Solu.ons #1 Provide opportuni.es for students to work with irregularly par..oned, and unpar..oned, areas, lengths, and number lines. © 2010 Math Solu.ons Unpar..oned areas and number lines. © 2010 Math Solu.ons Unequally par..oned areas and number lines © 2010 Math Solu.ons Top or Boiom: Which One Maiers? © 2010 Math Solu.ons Circle the larger frac.on. Explain your answer. © 2010 Math Solu.ons “If the denominator is smaller, the piece is bigger.” © 2010 Math Solu.ons What frac.on of the square is shaded? Tell me how you know. © 2010 Math Solu.ons “This one is bigger because there is more pieces.” © 2010 Math Solu.ons #2 Provide opportuni.es for students to inves.gate, assess, and refine mathema.cal rules and generaliza.ons. © 2010 Math Solu.ons •  Is it always true? •  What do you think of ___________’s answer? © 2010 Math Solu.ons Understanding Equivalency: How Can Double Be the Same? © 2010 Math Solu.ons © 2010 Math Solu.ons © 2010 Math Solu.ons #3 •  Provide opportuni.es for students to recognize equivalent frac.ons as different ways to name the same quan.ty. © 2010 Math Solu.ons Measure the marker with brown rods © 2010 Math Solu.ons 1 and 2/4 brown rods © 2010 Math Solu.ons 1 4/8 brown rods © 2010 Math Solu.ons © 2010 Math Solu.ons Comparing Frac.ons: Do You Always Need a Common Denominator? © 2010 Math Solu.ons Use a common denominator to compare the frac.ons below. © 2010 Math Solu.ons #8 •  Provide students with mul.ple strategies for comparing and reasoning about frac.ons. © 2010 Math Solu.ons Using the Cuisenaire Rods to Par77on the Number Lines 0
1
Find a rod that fits on the number line exactly two ,mes. © 2010 Math Solu.ons Using the Cuisenaire Rods to Par77on the Number Lines 0
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Mark ½ on the number line 1
0
0/2
1/2
2/2
Con.nue using the rods to mark thirds, fourths, sixths, and twelwhs on the other lines. © 2010 Math Solu.ons Inves.ga.ng Benchmark Frac.ons © 2010 Math Solu.ons Frac7ons Equal to ½ •  What do you no.ce about all of the frac.ons that are equal to ½? •  How does the numerator compare to the denominator? © 2010 Math Solu.ons Which of these frac.ons can be compared by using ½ as a benchmark? © 2010 Math Solu.ons Using Benchmarks •  What other benchmarks could students use to compare 2 frac.ons? •  How could benchmarks help students with frac.on opera.ons? •  How does the use of benchmarks support the development of students’ frac.on sense? © 2010 Math Solu.ons Frac.on Sense Strategies for Comparing Frac.ons © 2010 Math Solu.ons Thank you! To access the slides for this presenta.on, please go to: mathsolu.ons.com/presenta.ons © 2010 Math Solu.ons What is Frac.on Sense? “Frac.on sense implies a deep and flexible understanding of frac.ons that is not dependent on any one context or type of problem. Frac.on sense is .ed to common sense: Students with frac.on sense can reason about frac.ons and don’t apply rules and procedures blindly ‐ nor do they give nonsensical answers to problems involving frac.ons.” © 2010 Math Solu.ons mathsolu.ons.com 800.868.9092 [email protected] © 2010 Math Solu.ons