By Tina Cardone, Posted February 2, 2015 – 

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.  

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.