**By Tina Cardone, Posted March 16, 2015 – **

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).

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.

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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Claire Wasserman- 7/9/2018 6:21:08 PMThank you for your article. Given the confusion of when and how to find equivalent fractions, do you recommend not introducing this operation until after initially teaching multiplication of fractions? This question is for community college remedial math class.

Claire

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Ebony Robinson- 3/22/2015 5:59:11 PM## Reply processing please wait...