Fly on the Math Teacher's Wall Squashing Fraction Misconceptions

I love fractions!  Today I am linking up with some of the best math bloggers out there to bring you the Fly on the Math Teacher's Wall Blog hop.  Last time, we talked about place value and this time we are talking about squashing fraction misconceptions.  One of the biggest misconceptions I had when I first started teaching is that finding a common denominator is the only way to compare fractions.  Boy was I wrong.  After reading a great teaching book and listening to my students share their invented strategies, my misconception has been cleared up.  Today I am going to share with you 5 different strategies for comparing fractions.  

Common Denominators

Yes, you can compare fractions with common denominators.  However, this isn't always the most efficient way of doing things and it involves a lot of steps and a lot of calculation which means there is a lot of places where you can make mistakes.  The good news is, it works every single time and sometimes you just can't figure out which fraction is larger without it.  

Common Numerators

The long lost twin of common denominators, finding a common numerator is just like finding a common denominator.  However, sometimes the numerators already are the same and sometimes it can be more efficient to calculate a common numerator than a common denominator depending on the numbers in the problem. 

The numerators already match!  Use this to help you compare the fractions instead of finding a common denominator.  

Comparing these two fractions is tricky because they are very close together! Finding a common denominator would work but look how much easier it is to find a common numerator for this problem because the numerators are much friendlier numbers to work with then the denominators.  Most kids will instantly know the LCM of 3 and 5 but I bet they won't know the LCM of 17 and 27! 

Comparing to a Benchmark

This is a great strategy that can be very efficient on the right numbers.  If your fractions are close to a benchmark number like 0. 1/2 or 1, this can be so quick and easy!  

These two fractions are great to compare using a benchmark because one of them is a bit less than 1 and the other is a bit more.  

One of these fractions is a little more than one half and the other is a little less than one half.  This makes them easy to compare using a benchmark!

Draw a Model

Model drawing is so important in the development of fraction understanding.  I certainly don't want to leave my fifth graders in a place where they need to draw a model every single time they need to compare fractions but it is an excellent stepping stone and one that should not be skipped.  When students draw models, they develop some big ideas about fractions and help make a visual model in their head that they can refer to later if needed.  I spend a lot of time teaching good model drawing in second and third grade.  There are many ways to draw models, but I like to focus on using rectangles because they are easy to partition and if you partition them all in one direction, it is a quick jump from a rectangle model to using a number line.  

The farther apart two fractions are, the more reliable model drawing can be.  When the fractions get very close together, small model drawing inefficiencies can lead to students getting the wrong answer or concluding that the fractions are equal when they are now.  The student who can use a rectangular model like this one is just one step away from really understanding number lines.  

This student used a number line to compare these fractions.  Notice that if the fractions were really close together, this model drawing might not work.  It also takes some time to set up and draw accurately.  Partitioning into equal pieces is definitely a conversation to have with students as you work on model drawing.  I introduce the number line model in grade 3.  

Unit Fraction Reasoning 

Unit fraction reasoning is often one of the first strategies to develop.  It starts in first grade when you are partitioning rectangles into halves and quarters and a student notices that one half is bigger than one quarter.  It develops from there and as kids get more comfortable with using unit fractions it can lead to some great ideas when comparing fractions.  

This student used the fact that each of these fractions is missing a pieces that is a unit fraction to help him figure out which fraction was bigger.  Don't let the writing fool you about the amount of time the student took to figure this out.  He just looked at them and knew each was missing one piece and the one missing the smaller pieces would be the bigger fraction.  The writing was done during the sharing of strategies and is an attempt to capture his thinking for the other kids to see.  

If you want to see what strategies your students have for comparing fractions, here is a quick little worksheet that will give you an idea of some of the strategies your students have.  The numbers were chosen strategically to illicit a range of strategies.  

Ready to learn more about squashing misconceptions?  Head on over to Beyond Traditional Math to read more about the importance of the whole!