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STRATEGIES TO SUPPORT STUDENT LEARNING OF FRACTIONS Cynthia Santosuosso The Research… Only 24 % of thirteen and seventeen-year-old students identified 2 as the estimated sum for 12/13 + 7/8, while a greater percentage identified 19 or 21 as the estimated sum. Only 50% of eight grade students successfully arranged 2/7, 1/12, and 5/9 from least to greatest. Based on the NAEP (National Assessment of Educational Progress). Why are fractions so challenging? The way that fractions are written Classroom practices designed to help students make sense of fraction values mask the meaning of fractions. Students’ overreliance on whole number knowledge The many meanings of fractions, such as measure and ratio Fraction sense A deep and flexible understanding of fractions that is not dependent on any one context or type of problem Tied to common sense Allows students to reason about fractions Fraction Sense Workshop Format This workshop is based around teaching scenarios. From each of these scenarios will be drawn: key ideas about fractions how to communicate these ideas to students Scenario: Illustrating fractions Grades Taught: Third, fourth, and fifth Ms. Chu reviews some basic fraction concepts with her fifth graders. While examining their work, she notes that the drawings of more successful students varied, depending on the details of the problem. Students who were less successful relied solely on drawing circles (or pizzas), regardless of the problem type or context. Consider…. What can Ms. Chu do to expand her students’ drawings? The Math…Area Model Fractions are based on parts of an area or region Circles Most common Emphasize amount that is remaining to make up whole Not as flexible as other area or region because don’t allow for different-sized units Rectangles Pattern Blocks Paper Folding The Math…Linear Model Lengths are compared instead of areas Fraction Strips Cuisenaire Rods Number Line Important to note that each number on number line denotes the distance of the labeled point from zero, not the point itself Often difficult for students to view as comparing lengths The Math…Set Model Whole is understood to be a set of objects & subsets make up fractional parts Two color counters often used to model fractional parts Key Idea Understanding the relationship between representations of the same idea is core to fraction sense. Classroom Activities to Support this Idea Provide students with a fraction (1/10) and have them draw an example of each type of fraction model. For example: One cake shared among ten people (area model using a rectangle) A pizza divided into ten equal pieces (area model using a circle) One piece of gum from a package of gum that has ten pieces (set model) A segment marked 0 to 1 on a number line where the segment is divided into ten equal sections (linear number line) Classroom Activities to Support this Idea Have students write word problems based on Fraction Representation Cards. For example, if a student chose the “set model” card, he would write a fraction problem that could be solved using the representation on the card. Once corrected, another student could answer the problem Scenario: Modeling with fractions Grades Taught: Third, fourth, and fifth Students are using paper strips to make fraction kits. Ms. Alvarez notices Tyler labeling pieces as 1/7, which was not a directed step. He explained “When I counted them, there were seven, so I knew they were sevenths.” Consider… Is Tyler necessarily correct? Why / Why not? What action, if any, do you take? The Math… Tyler lost one strip (1/8). Determining the size of a strip in relation to the whole entails considering the number of copies of the strip needed to cover the whole. Key Idea The label on a fraction strip is not a fixed name, but a description of the relationship between the strip and the whole. Classroom Activities to Support this Idea Pattern Block Fractions Fraction Strips Scenario: Comparison of fractions Taught: Third and fourth grade Consider: Is he correct? After checking that Bruce understands what the “>” symbol means, what action, if any, do you take? The Math… No he is not correct. The correct equation is 1/6 < 1/4 because one sixth is less than one quarter. Key Idea: The more pieces a whole is divided into, the smaller each piece will be. Classroom Activities to Support this Idea Fraction Strips Classroom Activities to Support this Idea Stacked Number Lines Classroom Activities to Support this Idea Benchmark Fractions (0, ½, 1) 2/7______3/5 8/9______1/3 6/7______7/6 Scenario: Comparison of Fractions Taught: Grades 3 and 4 A group of students are investigating the books they have in their homes. Steve notices that 1/2 of the books in his house are fiction books, while Andrew finds that 1/5 of the books his family owns are fiction. Steve states that his family has more fiction books than Andrew’s. Consider… Is Steve necessarily correct? Why / Why not? What action, if any, do you take? The Math… Steve is not necessarily correct because the amount of books that each fraction represents is dependent on the number of books each family owns. For example… Key Idea The size of the fractional amount depends on the size of the whole. Classroom Activities to Support this Idea Provide opportunities for students to develop their understanding of the importance of context in fraction comparison tasks. Use materials and diagrams to demonstrate with clear examples, as in the previous tables. Question the student about the size of one whole: Is one half always more than one fifth? What is the number of books we are finding one fifth of? How many books is that? What is the number of books we are finding one half of? How many books is that? Scenario: Labeling fractions Grades Taught: Third and fourth Luke labels the shaded portion of this fraction as ¼. Luke labels the shaded portion of this fraction as 1/3. Cont. He explains, “It’s just like the other one, only this time it’s one out of three, instead of one out of four.” Consider… Is Luke necessarily correct? Why / Why not? What action, if any, do you take? The Math… Luke is not correct. Representing a fractional part of an area using fraction notation involves determining the whole area and then considering the size of the shaded part of the area in relation to the size of the whole. Key Idea Students understand that the size of a fractional part is relative to the size of the whole. Classroom Activities to Support this Idea Provide opportunities for students to work with irregularly partitioned, and unpartitioned areas, lengths, and number lines. Solution: Measurement Strategy Consider the number of copies of the shaded part are needed to cover the whole. This is the denominator. One of the copies is shaded (the numerator). Solution: Part-Whole Strategy Divide the whole into parts of equal size. The whole is divided into equal parts by inserting the dotted line. Scenario: Equivalent fractions Taught: Third and fourth grades Which shape has 1/3 of its area shaded? Sarah insists that none of the shapes have 1/3 of their area shaded. Consider: Do any of the shapes have 1/3 of their area shaded? What action, if any, do you take? The Math… Key Idea: Equivalent fractions have the same value. Classroom Activities to Support this Idea Fraction Strips Classroom Activities to Support this Idea Stacked Number Lines Scenario: Addition of Fractions Taught: Grades 4 and 5 You observe the following equation in Emma’s work: 1/2 + 2/3 Is Emma correct? = 3/5 Consider… You question Emma about her understanding and she explains: “I ate 1 of the 2 sandwiches in my lunchbox, Kate ate 2 of the 3 sandwiches in her lunchbox, so together we ate 3 of the 5 sandwiches we had.” What, if any, is the key understanding Emma needs to develop in order to solve this problem? The Math… Emma needs to know that the 1/2 relates to a different whole than the 2/3 . If it is clarified that both lunchboxes together represent one whole, then the correct recording is: 1/5 + 2/5 = 3/5 Emma also needs to know that she has written an incorrect equation to show the addition of fractions. Key Idea When working with fractions, the whole needs to be clearly identified. Classroom Activities to Support this Idea Question the student about their understanding. The one out of two sandwiches refers to whose lunchbox? Whose lunchbox does the two out of three sandwiches represent? Whose lunchbox does the three out of five sandwiches represent? Use materials or diagrams to represent the situation. For example: Key Idea When adding fractions, the units need to be the same because the answer can only have one denominator. Classroom Activities to Support this Idea Use visuals to illustrate how to find a common denominator. Classroom Activities to Support this Idea Estimate the sum with benchmark fractions; check for reasonableness. 0, ½, and 1 can serve as reference points for estimation. ½ + 2/3= 1 1/6 ½+½=1 Scenario: Mixed Numbers Taught: Fourth and Fifth Grade Anna says 7/3 is not possible as a fraction. Consider….. Is 7/3 possible as a fraction? What action, if any, do you take? The Math… Key Idea A fraction can represent more than one whole. The denominator tells the number of equal parts into which a whole is divided. The numerator specifies the number of these parts being counted. Classroom Activities to Support this Idea Use fraction strips or a number line to model 1/3 1/3 1/3 =1 1/3 1/ 1/3 1/3 =1 1/3 =1/3 1+1+ 1/3 =2 1/3 Scenario: Multiplication of Fractions Taught: Fourth and fifth grades Katie solves the following problem: 2/3 x 5/7= 14/15 Consider… Is Katie correct? What is the possible reasoning behind his answer? What, if any, is the key understanding he needs to develop in order to solve this problem? The Math… Katie is not correct. She has written the cross product as the numerator of the product and the other cross product as the denominator. Key Idea Students must grasp the concept behind multiplying fractions, beyond the algorithm. Classroom Activities to Support this Idea Use grid pencils and colored pencils to demonstrate visually. Remember that “x” means “of.” 2/3 x 5/7 = 10/21 Scenario: Division of Fractions Taught: Fifth grade You observe the following equation in Bill’s work: 2½÷½= 1¼ Consider….. Is Bill correct? What is the possible reasoning behind his answer? What, if any, is the key understanding he needs to develop in order to solve this problem? The Math… No he is not correct. The correct equation is 2½÷½= 5 Possible reasoning behind his answer: 1/2 of 2 1/2 is 1 1/4. He is dividing by 2. He is multiplying by 1/2. He reasons that “division makes smaller” therefore the answer must be smaller than 2 1/2. Key Idea To divide the number A by the number B is to find out how many lots of B are in A. For example: 8÷2=4: There are 4 lots of 2 in 8 2 ½÷5=1/2: There are 5 lots of 1/2 in 2 1/2 Classroom Activities to Support this Idea Successful Strategies • • • • Varied visual representation (area, linear, set) Fraction strips Number lines Benchmark fractions Let’s Work on Our Own Fraction Sense! Fractured Numbers-Problem of the Month 3-2-1 Reflection 3 ideas or pieces of knowledge you have found useful. 2 things you want to try with students. 1 question you’re still asking. Sources Institute of Education Sciences. Developing Effective Fractions Instruction for Kindergarten Through Eighth Grade. What Works Clearinghouse. 2010. McNamara, Julie. Beyond Pizzas & Pies. Math Solutions. Sausalito, 2010. Spangler, David. Strategies for Teaching Fractions: Using Error Analysis for Intervention and Assessment. Corwin. Thousand Oaks, 2011.