Disappearing Act

  • Disappearing Act

    By Tina Cardone, Posted March 16, 2015 – 

    Cardone Art 1

    How many times have you seen this kind of mistake? How do we help students understand that they can’t just cross things out? Attending to precision in language while solving equations or simplifying fractions can be cumbersome. Even so, it is important.

    Describing what operation to apply to each expression rather than both sides of an equation requires more syllables but emphasizes the equality of two separate expressions. Showing students that despite their intuition, simplified fractions are equivalent (the same size) rather than reduced (smaller) is no simple task. Frequently, my language defaults to the word cancel when manipulating equations and fractions (habits developed over many years are hard to break!). Still, cancel is a vague term that hides the actual mathematical operations being used; therefore, students will not know when or why to use it. To many students, cancel is digested as cross-out stuff by magic, so they see no problem with crossing out parts of an expression or across addition.

    When simplifying fractions, students need to recognize the difference between additive structures and multiplicative structures. Students need time to explore these ideas, and precise language will help them identify what is new and different about fractions. Factors and multiples are key when manipulating fractions. Finding common factors or multiplying by the same factor allows students to identify equivalent fractions. Common multiples allow us to compare fractions and create like terms (via equivalent fractions) to add. Multiplication and division are key in maintaining the ratio in a fraction, but too often students expect to maintain that equivalence via addition or subtraction.

    Instead of saying cancel when manipulating fractions, require students to state a mathematical operation. Similarly, students should write a mathematical simplification rather than crossing out terms that cancel. In fractions, we are dividing to get 1. Students can say, “Divides to 1” and show that situation on their paper by making a big 1 instead of a slash to cross factors out. Emphasizing the division helps students see that they cannot cancel over addition (when students try to cross out part of the numerator with part of the denominator, for example).

    Cardone Art 2

    Another example of this is manipulating an equation with separate terms; in this case, we are adding the opposite to each expression. Students can say, “Adds to 0” and show that situation on their paper by circling the terms and thinking of the circle as a 0. Alternatively, students can circle the terms and write a 1 next to the terms for “dividing to 1” or write a 0 next to them for “adds to 0.”

    Whether working with equations or fractions, use the language of opposites and identities to precisely define the mysterious cancel. The more precisely we describe these processes, the more students will believe that math makes sense. The magical disappearing act can wait for the talent show.

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    Tina Cardone, @crstn85, is a high school teacher at Salem High School in Salem, Massachusetts. She is the author of Nix the Tricks and blogs about her teaching at DrawingOnMath.blogspot.com.

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    Ebony Robinson - 3/22/2015 5:59:11 PM
    I agree that students need to understand why we are changing the numbers, when it is acceptable to change the numbers, what is the correct process for changing the numbers and most importantly what is name for the process we are using. I am guilty of using the terms "cancel" and "reduce" across mathematical operations. I will now work on defining these actions in my classroom and encouraging my students to state the correct terminology when they are using these processes. Thank you for bringing light to this area.