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Last week, I was
playing with proportions on brilliant.org__,__
and I was excited to discover that the problems were crafted to be doable with
mental math, if you saw the relationship.

Proportion
problems are incredibly easy to write: Use a random number generator to fill in
three of the four slots and stick a variable in the remaining one. Done!
Solving random proportions is a mindless activity though, and proportions have
the potential to be so interesting. A well-crafted problem set can still be
solved using rote methods (such as cross multiplying), but students who are
paying attention approach them as puzzles to solve. Students think they are
being sneaky when they find ways around doing all the calculations, but
teachers know that these students are merely exercising their problem-solving
and pattern-sniffing (AKA look
for and make use of structure) skills.

Consider the
equations 4/10 = 32/*x* and 15/5 = *y*/4. Many students
would take a standard approach, such as cross multiplying.

This method works
whether the variable is in the numerator or the denominator. Students can
blindly apply it in all cases, but cross multiplying requires rewriting as
cross products (a mystery step: Where are the inverse operations or
relationships in a cross product?), multiplying, and then dividing. However,
students who recognize the relationships within or between the fractions can
complete the entire process without rewriting the equation.

These strategies
highlight the multiplicative relationships in sets of equivalent fractions, a
relationship that students need to see early and often. I recently told my
prealgebra students that they were working on a topic that my precalculus
students still struggle with: fractions. My juniors and seniors in honors
courses have less difficulty with new concepts than proportions; their
incomplete understanding of fractions is their downfall on a regular basis.

One of the goals
of the Common Core State Standards for Mathematics (CCSSM) is for students to
have fluency with math facts. The mental math that students use here is good
practice toward that fluency. If your students are new to this topic, using
different representations can help them see these relationships more clearly. Using
tape diagrams will allow students to visually align fraction representations to
see their equivalence.

Consider 3/4 = *x*/8; we can compare 3/4 with a unit
of the same size divided into eighths, then shade the equivalent amount.
Students may notice that there are twice as many parts when they switch from
fourths to eighths. Similarly, there are twice as many shaded sections. The
numerator and denominator have each been multiplied by two, which maintains the
ratio. With enough practice, students will recognize this relationship without
the need of a diagram.

The
representation of 2/3 = 6/*x* is
slightly less intuitive, but can be approached in a similar way. Take the
shaded region of 2/3 and re-partition it into six shaded sections. Then add on
same-size pieces to reach a unit of the same size. Some students might have an
easier time with this if, instead of drawing diagrams, they have a set of
fraction strips. Then students could compare 2/3 with other strips until they
found one with six sections. In either case, students can identify the
relationship of multiples of three: When there are three times as many shaded
sections, there will be three times as many parts.

One of the eight essential teaching practices in
Principles
to Actions emphasizes the connection of fluency and understanding. Using
a variety of representations builds understanding that students can return to
throughout their study of mathematics. These strategies continue to apply, even
as students encounter more advanced proportions, such as the example below.
Cross
multiplication may work, but it frequently adds steps to the overall process,
and it skips steps that are important for students to see. Math shouldn’t be
magic; math should make sense. Nix the tricks (cross multiplication); instead,
ask students to find their own shortcuts and justify them!

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.

## Comments:

Troy Whitbread- 2/27/2015 3:01 PMThank you. I enjoyed reading this article.