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.
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
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,
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
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.
Thank 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.