Is 1/2 always greater than 1/3?

Lately I’ve been reading the book Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense by Julie McNamara and Meghan Shaughnessy.

I posted the following picture to Twitter while I read during my daughter’s swim class.

My colleague, Hedge, replied about being challenged by a middle school teacher on this very issue.

I let her know I was also challenged about this idea several years ago when I was a digital curriculum developer. The argument I heard back then was that using contexts to validate the correctness of fraction comparisons ran counter to the fact that fractions are numbers. As such, 1/2 is always greater than 1/3 regardless of the context. At the time, I wondered about it, but I still felt that bringing context to bear was important.

Flash forward to now and I have been mulling this idea over all day. I think I may finally understand why we have to be careful what we say about the role of context when comparing fractions. I may be completely off the mark, but I’m going to share my thoughts anyway and let you decide in the comments if you’d like to challenge my thinking or share an alternative point of view.

Let’s start with whole numbers. If I told you to compare 3 and 6, you would probably tell me, “3 is less than 6,” or, “6 is greater than 3.” That is how the numbers 3 and 6 are related.

Now, what if I were to show you these two pictures of 3 and 6: (As illustrated by my daughter’s toys.)

Three large dolls

Six small figurines

Technically, the 3 dolls are larger and therefore they amount to more stuff, but does that really mean 3 is now greater than 6? In the end, the number of dolls my daughter has (3) is less than the number of figurines she has (6). The context doesn’t fundamentally change the relationship between the numbers 3 and 6.

In this case, I don’t even know how I’d justify that she has more when referring to the dolls. Sure, they’re bigger, but she may prefer to have more things to play with and choose the 6 figurines even though they are less in total size.
Let’s continue by looking at this from a fraction perspective. Now I’m going to take 1/3 of the dolls and 1/2 of the figurines.

1/3 of the dolls is 1 doll

1/2 of the figurines is 3 figurines

In keeping with the idea that context should dictate when one number is greater than another, I should be convinced that 1/3 of the dolls is greater than 1/2 of the figurines because 1 doll is so much larger than the 3 figurines. Oh wait, or is it that I should be thinking that 1/2 of the figurines is greater than 1/3 of the dolls because I end up with 3 figurines which is a greater number of things than 1 doll? It’s not so clear cut, even though I’m trying to let the context dictate how to interpret the fractions.

What it boils down to is that fractions represent a relationship. If I think about the relationships each fraction represents, then 1/2 is always greater than 1/3 no matter how I try to spin it. Looking back at my examples, taking 1/2 of the group of figurines means I am taking a greater share of that group (that whole) than when I take 1/3 of the group of dolls (a different whole, but a whole nonetheless). The size of the things in my group (whole) doesn’t matter because the relationship represented by 1/2 is greater than the relationship represented by 1/3.

Now, does that mean we should ignore contexts altogether? No. There are still rich conversations to be had about who ate more pizza when one person eats half of a small pizza and another person eats a third of a large pizza. Context is still interesting to discuss and helps students use math to interpret the world around them. However, if our goal is to compare fractions, then 1/2 is greater than 1/3 every time.

That’s the argument I came up with today as I tried to understand the criticisms I’ve heard. Now that you’ve read it, what do you think?

Fraction Number Sense

This week I gave a presentation to grade 3-5 teachers from around my district. One of the points I wanted to drive home to them is that number sense does not apply to only whole numbers. Students can (and should!) have number sense with regards to fractions and decimals as well.

One area where I see teachers frequently take shortcuts and/or avoid number sense is with comparing fractions. Some teachers teach their students to use cross multiplication to verify whether two fractions are equal. Others teach their students to use one and only one strategy for comparing fractions: find common denominators. This strategy works, don’t get me wrong. However, it develops one skill, not number sense.

When comparing fractions, students should be mentally choosing from a variety of strategies. Why? Because students might notice they can make a comparison quickly and mentally. Why go to the trouble of creating common denominators, which likely involves making some notes on paper, if you can mentally make a comparison based on your understanding of fractions?

To that end, I put together three “books” that teachers could use to prompt some discussion and reasoning among their students. I’d love for their students to start realizing that fractions are something they can make sense of.

• John’s Pizza (10/5/2021 – Link updated)
• Lunch With Friends (10/5/2021 – Link updated)
• Eating Half (10/5/2021 – Link updated)

Each book focuses on comparisons around a particular strategy, but the strategy is never spelled out for the students. Instead, my hope is that by noticing, wondering, questioning, reasoning, and communicating, a class of students can make sense of the strategy in each book.

By no means are these books intended to fully develop students’ number sense with regards to comparing fractions. Additional experiences and practice are likely required. However, if you’re a teacher who wants a place to start this work with your students, give these a try and let me know how it goes.

Note: Each book is a PDF. I highly recommend downloading the PDF and opening it back up with a program such as Adobe Acrobat. The book really works best when each page appears one after another. If you open from Google, then you’re forced to scroll from page to page which isn’t as effective.