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Ambivalent Equivalence

By Adam Williams. Ambivalent Equivalence. Equiva what? . The concept of equivalent fractions is difficult for many students to ascertain and fully understand.

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Ambivalent Equivalence

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  1. By Adam Williams Ambivalent Equivalence

  2. Equiva what? • The concept of equivalent fractions is difficult for many students to ascertain and fully understand. • Many of these difficulties can be attributed to a heavy emphasis on procedural knowledge and symbolic rules in early grades. (Tanner, 2008)

  3. Standards • Understand representations of simple equivalent fractions. (3rd Grade, 4th Grade) • Add and subtract fractions and mixed numbers with common denominators. (4th Grade) • Find equivalent fractions and simplify fractions. (5th Grade) • Explore finding common denominators using concrete, pictorial, and computational models. (5th Grade) • Add and subtract common fractions and mixed numbers with unlike denominators. (5th Grade)

  4. 3rd Grade • Fraction Kit (Burns, 2009) • The teacher gives the students five strips of construction paper. The students cut and label the strips into different sized fractions.

  5. Uncover • Students play in pairs, each starting with their whole strip covered by two halves. • They take turns rolling a six sided cube with the faces numbered one-half, one-fourth, one-eighth, one-eighth, one-sixteenth, and one-sixteenth. • The fraction that comes up tells the students what size fraction to remove to uncover the whole. • If they can’t remove a piece, they wait until their next turn or take the option of exchanging any of their strips for others in their kit, so long as it is an equivalent exchange. • The goal is to be the first person to completely uncover their strip. Playing the Game

  6. 4th Grade • Word problems such as these reinforce what the students have learned in previous grades. • They could easily use manipulatives to solve these problems, without forcing new material upon the students that may be frustrating to fourth graders. • Pattern blocks are a great way for students to visualize fractions and create better connections between addition and equivalency. I ate half a pizza and you ate one-eighth of the pizza. How much is left? My dog ate one-third of my sandwich, and my sister ate one-sixth. How much is left for me? A caterpillar ate three-fourths of an apple and a ladybug ate one-tenth more. How much is remaining? (Naylor, 2003)

  7. Fraction Clothesline • A task that evokes this higher level thinking must be extremely interactive and promote group speak. • To accommodate this type of learning, Tanner (2008) recommends setting up a few clotheslines throughout the classroom.

  8. Fraction Clothesline (cont.) • Each group should be given a set of cards that include several fractions (mixed and regular). In small groups, the students should take clothespins and put their fractions in order from least to greatest. • To incorporate equivalency, teachers can put equivalent fractions in with the others and observe how the students respond.

  9. 5th Grade • In the fifth grade, students move into the world of unlike denominators. • To help ease students into this process, manipulatives and visualization should be the first method of representing this operation.

  10. Unlike Denominators • Let’s say the problem provided is 1/3 + 1/2 = X. • First, the students would use a hexagonal block to represent a whole. • They would divide it into thirds by finding the correct shapes (the blue rhombi) and placing them on top of the whole. • To represent one third, they take away two of the pieces and have the first fraction of the problem. • The students repeat the process with one-half, using a hexagonal piece as the whole and a red trapezoid to close in the half. • In order to add, the students must combine the fraction blocks together.

  11. Unlike Denominators (cont.) • When the problem involves finding a solution greater than a whole, the students must take things a step further. • If the problem provided is 5/6 + 2/3 = X, the students will build their fractions much like in the previous problem. • When the students attempt to combine the pieces, they will notice that it is impossible to fit them all together. • The teacher should then guide the students into trying to create a whole using the provided pieces, and then use the remaining pieces to create the numerator and denominator of the mixed fraction. • Once they create a whole, they will use the excess parts to build a fraction to go along with it. In the case of this problem, it would be 1/2. This leads to the final solution of 1 1/2.

  12. Conclusion • In order to be comfortable with fractions in the later grades, students must build a firm foundation early on. • It is important for students to experience a differentiation of methods when learning about fractions, because there are multiple types of learners and many different levels of understanding.

  13. References Burns, M. (1999). Equivalent fractions and subtraction strategies. Instructor , 113 (5), 17. Duke, R., Graham, A., & Jonston-Wilder, S. (2008). The fractionkit applet. Mathematics Teaching Incorporating Micromath(208), 28-31. Naylor, M. (2003). Fill in the fractions. Teaching Pre K-8 , 33 (8), 28-29. Naylor, M. (2003). Putting the pieces together. Teaching Pre K-8 , 33 (5), 28-29. Rickard, C. (2007). Misunderstanding of fractions. Mathematics Teaching , 205, 32. Tanner, K. (2008). Working with students to help them understand fractions. APMC , 13 (3), 28-31.

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