## Saturday, February 21, 2015

### 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.

 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.

1. Love the idea of common numerators! Excellent post. So important that students have multiple strategies based on number sense! :)

Donna
Math Coach’s Corner

2. Thank you for your ideas. These are ways we are supposed to teach comparing fractions at my school. Some of them are confusing for my students (we're 3rd grade) but benchmark numbers are very important to us. Thank you for sharing.
Daisy at Not Your Mother's Math Class

1. My students "invent" all of these strategies on their own. We do a lot of strategy sharing and it is amazing what kids will come up with when given a good problem in context.

3. Your post illuminates the idea of number sense around fractions rather than learning "the way to get the answer". We spend so much time in the primary classroom developing number sense around whole numbers but, as students get older and the curriculum comes fast and furious, time to develop number sense with fractions is kept to a minimum. These strategies favor thought around fractions and what you know in order to compare rather than just "doing 'x' to compare fractions. Great post!

1. I love the idea of thinking about this as developing number sense with fractions. Before we ask kids to operate on whole numbers we make sure they have number sense. We need to do the same thing with fractions.

4. Thanks so much for your wonderful post! I will be passing your post on to one of my former fifth grade teaching friends. I too love fractions, and I have been fortunate to be able to teach them at various levels. I love to see that same excitement in my students when they truly understand! Thank you again, Tara!

Smiles,
Sarah

1. Fifth grade is such an important year for fractions!

5. I love this post. I am going to be teaching upper grades next year and looking at all of these ways to compare fractions has been so helpful to me.

1. What an exciting opportunity for you to expand your knowledge. It is great when an upper grades teacher has experience with younger kids because it makes you naturally good at intervention!

6. Great post Tara! These strategies really get students to think and explain how they got their answers. I can imagine the math discussion.

Greg
Mr Elementary Math

7. This is exactly what we're working on right now. Perfect timing! And reading your post tonight gave me an aha moment... thank you!!!

8. Love these types of posts with many strategies for attacking a problem.

One strategy I sometimes use is converting to percentages, even rough conversions. Of course, this depends on having fractions where I have a rough idea what the equivalent percents are, otherwise other strategies are likely to be faster.

When talking about multiple strategies, I like to include comments to encourage the kids to find their own and talk about them. That way (a) they are encouraged to compare and contrast strategies and (b) you can hear misconceptions when they come up with strategies that don't work.

Lastly, there seems to be a slightly tricky distinction between fractions as an action ("fraction of") and fraction as a value. Elsewhere, I saw a tip to encourage showing the number line representation as a key to establishing the connection between fractions and their values.

9. Love these types of posts with many strategies for attacking a problem.

One strategy I sometimes use is converting to percentages, even rough conversions. Of course, this depends on having fractions where I have a rough idea what the equivalent percents are, otherwise other strategies are likely to be faster.

When talking about multiple strategies, I like to include comments to encourage the kids to find their own and talk about them. That way (a) they are encouraged to compare and contrast strategies and (b) you can hear misconceptions when they come up with strategies that don't work.

Lastly, there seems to be a slightly tricky distinction between fractions as an action ("fraction of") and fraction as a value. Elsewhere, I saw a tip to encourage showing the number line representation as a key to establishing the connection between fractions and their values.

10. I enjoyed reading your blog from a college student's perspective. I am currently in a math methods course at the University of Wisconsin-Oshkosh and we've just recently covered fractions. We, as a class, have been currently reading from the Van de Walle text, Teaching Student-Centered Mathematics, where we have been emphasizing the importance of understanding the relationship between the part and the whole, of a fraction, rather than the size of the whole itself. I strongly agree with your statement that using models to build the understanding of fractions is a great stepping stone to becoming more fluent when comparing fractions in the future. I think that your last strategy of "Unit Fraction Reasoning" shows evidence of a mastery level and exemplifies where a student can go when they have achieved a greater understanding of fractions. With such a wide variety of strategies that students are able to use, it doesn't make sense to only use common denominators when comparing fractions.