Dividing by a Fraction: Trick or Treat?
By David Hedges, CORE Senior Education Services Math Consultant
It is almost Trick or Treat time, and I am reminded of my classroom days working with struggling Algebra 1 students. Fractions issues were the usual suspect! I used to tell my students that I was going to dress up as a fraction on Halloween and scare them all. Their choral response was usually something like, “Mr. Hedges, you are so crazy!”
Most of the math my students struggled with was not about Algebra 1 topics. The student struggles were mostly around a lack of fluency or understanding with earlier skills they had been exposed to on their math road to Algebra 1. Check out a prior blog from February 2018 written by Mary Buck, Senior Educational Services Consultant for CORE, about fluency within the Common Core Standards, Number Sense and Fluency.
I always found math to be easy as a K-12 student and went on to complete a BS in Mathematics at the ‘real USC’, University of South Carolina. As successful as I was with math, it was during a math methods course taught by Dr. Randy Philipp at San Diego State University that I had a fraction epiphany – meaning a whole epiphany about fractions.
During a class one evening Dr. Philipp asked us aspiring math teachers to draw a visual of something like:
Of course, I had been conditioned (think Pavlov) to ‘flip and multiply’ every time I was confronted with division by a fraction. I had to shake my head to clear the imprinting and allow my mind to explore the idea of a visual.
The breakthrough for me came by changing my approach from dividing by a fraction algorithmically to thinking about how many of the divisors are in the dividend. This builds on what students learn with whole number division. In this example, I needed to determine how many halves are in two-thirds.
Initially I drew a model of two-thirds and cut each third of the whole in half (see the diagram to the right). This created six equal parts of the whole, in which four equal parts of the whole represented two-thirds (shaded orange or orange-green) and three equal parts of the whole represented one-half (shaded green or orange-green). Three of the orange parts (one orange and two orange-green) equaled one-half of the whole. The left over fourth orange part filled up only a third of the one-half section. This provided a visual that illustrated there are one and one-third halves in two-thirds.
This can also be illustrated on a number line as shown below. The light shaded bar represents two-thirds while the two striped bars above this each represent halves. All of the first one-half-bar and just a third of the second one-half-bar on the top fit into the two-thirds bar below it. This reinforces that there are one and one-third halves in two-thirds.
After a few years I also realized that we can divide two fractions by first finding the common denominator and then dividing the numerators followed by dividing the denominators. The quotient of dividing the denominators of two fractions with a common denominator will always equal one! Therefore, having common denominators allows for a focus on dividing the numerators.
The drawing, the number line, and the common denominator methods are meaningful ways to “Apply and extend previous understandings of multiplication and division to divide fractions by fractions (6.NS.1)” and are a nice lead in to the algorithm described in the Common Core standards “(a/b) ÷ (c/d) = ad/bc” (National Governors Association Center for Best Practices & Council of Chief State School Officers, Common Core State Standards for Mathematics, 2010).
As for dressing like a fraction for Halloween, I never actually followed through with this, but each Halloween, I realize it is still on my bucket list. If you decide to dress as a fraction this Halloween, please send a photo 🙂