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Ambivalent Equivalence

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By Adam Williams

Ambivalent Equivalence

- 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)

- 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)

- 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.

- 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

- 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)

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

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.