Which one is bigger: 2/5 or 2/6?

In this post, we are going to talk about comparing fractions and to present a few models that might help us to do this work, especially for children who are developing their conceptual understanding of fractions. Before we start, I should make this disclaimer first: I really love fractions. Like… really. love. talking. about. them. I know I am probably a strange minority, but I am constantly fascinated by how children use fractions, establish their initial understanding of fractions, and eventually learn how to fluently operate on fractions.  When I tell my prospective teachers about my love of fractions, there is a distinct difference between our states of emotion:

My students are like:

It’s ok; I still love them (my students and fractions).

But back to my original question that was posed to me this week by my cousin who has a lot of mathematical experience. Her son, who is in the third grade, brought home a math problem that said “Compare 2/5 and 2/6. Which is bigger?”  When my cousin the problem, her first (and a natural) reaction was to say: “Wait! Are we now asking third graders to find common denominators? Is this a third grade math standard? What’s another way to answer this?” My cousin’s question is one that I’ve heard so many parents say before and is totally normal; Teaching math has changed over the years and it’s always good to take a pause and say “Wait what?”  As we unpack this specific problem problem together,  let’s establish a few ground rules:

1. You have the right to use your existing knowledge of division, fair-sharing, and any other knowledge related to fractions. If you look at 2/5 and 2/6 and think “well, I need to find a common denominator” that’s one way of solving this problem.
2. You likely have another strategy of comparing these fractions that I haven’t talked about explicitly. Sweet! (For the sake of time and space, I just wanted to unpack three models for this blog).
3. We should acknowledge that our kids already come to school with an INCREDIBLE amount of mathematical knowledge and can use this knowledge if we have teachers who are prepared to elicit and build upon those prior experiences. Our kids aren’t born to know the difference between a denominator and numerator are, but they sure are ready to talk about how to split things fairly so where everyone gets an equal-sized piece.

As we look at 2/5 and 2/6 to see which is bigger or smaller, we need to recognize that children might come with some assumptions about these fractions based on their knowledge of whole numbers:

1. For example, a child might think that 2/5 and 2/6 are the same (because 2=2).
2. Or that 2/5 is smaller because 5<6.

Each of these explanations make sense if you think about how kids are just learning how to see numbers like 2/5 as a complete and singular value, not just a 2 and 5 with a line that separates it.  Don’t dismiss this thinking– it really makes sense to kids! Let’s shift to discussing some ways to compare these fractions and to make a conclusion about the relative value of them, starting with the most complex way first that some of us have learned in school.

Strategy 1: Find Common Denominators, Get Equivalent Fractions, and then Compare Numerators:

As adults, we might find common denominators, find fractions with equivalent values, and then compare the numerators. If we do this, then we can compare 12/30 with 10/30.

This means the difference of 2/5 and 2/6 is only 2/30… which has a really small space for error especially if you are trying to draw an accurate area model. This is a valid strategy, but we know from the research that kids need visuals, concrete quantities, and/or actions to represent their thinking. #WhereMyCGIFolksAt?

Strategy 2: Part to Whole Models, Showing 2/5 and 2/6

In this strategy, a child can draw two rectangles of the same shape and size. To name the fractions as 1/5 and 1/6, the wholes should be the same size. Well consider how 1/6 of a large pizza is a different 1/6 of a small pizza. To compare fractions, we need to address whether or not the wholes from which you cut those pieces are the same size. So all the kids have to do is cut one whole into 5 equal pieces and another into 6 equal pieces and then shade two and compare? Sounds easy right? But before you take my word for it, try it. It’s no joke to try to draw by hand (NO RULER) a rectangle into 5ths and 6ths and be precise with that drawing! Super challenging because these fractions are really close in value to each other. (I cheated and used PPT table feature). This signals to me that the curriculum (whether intentional or not) might not have intended for kids to use an area model– too much room for error maybe? One that would have been more obvious might have been 2/10 and 2/100, but who the hell wants to cut a rectangle into 100ths. #BYE #RedDressLadyEmojiWalkoutNow

Strategy 3: Leveraging existing knowledge of 1/5 and 1/6 with fractions as division

Kids have experiences with dividing things up. It’s true. They’ve shared food or toys before and have experienced trying to make things “fair” for everyone. Although they are learning how to be precise with this (sometimes their “one half” portion can be quite “generous” compared to mine), this is knowledge that we should not dismiss or disregard. When sharing objects, you likely won’t hear a child say “Now I am going to do fractions as division, y’all” but that doesn’t mean they aren’t actually doing that math in front of you.  You might consider asking a child “Which kids would get more of a cookie:  one cookie shared among 5 kids or that same sized cookie shared among 6 kids? Which would you want to share so that you and your friends got the most amount of cookie?”

The book, The Doorbell Rang, is AN AMAZING example of this idea. When kids model fractions as division for themselves, they can begin to explore what this denominator really means if you have the same numerator. They can also make sense why the larger a denominator (like the 6 in our example) might not imply larger values if you have the same object to share. Shifting from seeing whole numbers to fractions is challenging and shouldn’t be taken for granted. And I can’t emphasize this enough:

We don’t need to wait until 4th and 5th grade to teach our students about fractions or the notion of “fairness.” They have a lot of experiences about dividing things up that they can leverage!

Once kids know that 1/5>1/6, then they can use that to see how 2/5>2/6 (Instead of sharing one cookie, now the kids are sharing two cookies). Problems like this lay the foundation for why although 5<6, we can show that 2/5 is greater than 2/6. It’s the size of the piece (or the number of sharers) that matters (at least in this context and the strategies presented).

What’s important about Strategy 3 is that is moves children away from focusing on what the actual difference is between 2/5 and 2/6 (which remember is 2/30) and moves them towards making sense as to what the numbers actually represent as it is connected to their experiences with division and fair-sharing. Are these the only ways to show a comparison of two fractions? Certainly not! I would highly recommend these books if you want to read up on some of the research (which is accessible reading for multiple audiences) here and here about how children learn fractions.

But you know that I am not just going to leave it as that, did you?

#BAHAHAHAHAHAAAAA

What if I presented you with this picture? I divided one rectangle into 5 equal pieces and another into 6 equal pieces. I shaded two of each rectangle to represent 2/5 and 2/6.

Did I just prove that 2/5 and 2/6 are the same amount? In what ways might there be something about this picture that could be misleading to kids? (The Shoe Task and this picture above are my more favorite pictures to show that generate such amazing conversations about fractions.)

I’ll leave you to think about that for a second.

References

Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.

Hutchins, P. (2014). The doorbell rang. New York, New York: HarperCollins.

McNamara, J., & Shaughnessy, M. M. (2015). Beyond pizzas & pies: 10 essential strategies for supporting fraction sense, grades 3-5. Sausalito, CA: Math Solutions. (This citation is for the newest edition).

Thanks to my cousin for letting me share our conversation with you all!