# 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?

# Sink your teeth into data. Don’t just nibble.

Looking for math all around started as a challenge I made for myself and I’m realizing it’s becoming a full-fledged theme for my year. When I had to think of a topic to moderate this week’s #ElemMathChat, I started by asking myself, “What’s a topic we haven’t talked about since the chat started in August 2014?” After some brainstorming, I eventually came up with analyzing data. What a great topic for my theme! I don’t think I could throw a rock without hitting some data in the world around me.

In fact, as I was fleshing out the topic for the chat, I was regularly checking some real-world data online. After a long dry spell, we finally got some rain in Austin. And by “some rain” I mean a deluge. On a couple days last month it just kept pouring and pouring. Throughout each day it rained, I found myself checking our neighborhood weather station on Weather Underground to see how much rain had fallen. By the time October was over, we had received 10.3 inches of rain in my neighborhood! That simple piece of data became the catalyst for tonight’s #ElemMathChat.

I started digging into rainfall data for October, then rainfall data for other months, and finally I expanded my data dive into other cities in and out of Texas. When I was done, I had a spreadsheet full of various tables of data that I wanted to share in my chat. To make this chat work, I realized I needed to be intentional about how I shared the data in order to tell a coherent story. I also wanted to create a variety of data displays that would match the various data displays students encounter across grades K-5. As an aside, I think #ElemMathChat sometimes leans a bit heavy on content for grades 3-5, so I was trying to be mindful to show some graphs that could be analyzed in a Kinder or 1st grade classroom.

It took several nights to research, create graphs, and pull it all together to make a story, and in the end I’m proud enough of the final result that I wanted to capture it on my blog.

Before starting my data story, I shared the following guiding questions that tied into my primary goals for the chat.

My Data Story

Our story begins with the piece of data that started it all. I asked the participants to tell me what they noticed and wondered about this statement.

What do you notice and wonder?

Many people wondered how this amount of rain compared to other cities. Funny you should ask.

What do you notice and wonder as you look at this pictograph?

One thing I noticed is that I accidentally left the key off the graph. Oops! Each picture is meant to represent 1 inch of rain. Despite my mistake, several people liked that the missing key invited students to wonder about what the pictures represent. That sounds like such a wonderful conversation to me that I opted to leave the key off when sharing the picture in this blog post.

I had a little fun with this graph because I had to decide which cities to include. I decided to focus on other state capitals, but the question became, which ones? When I noticed how many start with A, I decided that was more interesting than picking random capitals. It just so happens that all the other capitals on this pictograph are all on the East coast, so I wonder if it would have been better to choose capitals with greater geographic diversity. In the end this is just a fun way to get our story started so I’m okay with what I chose.

Next we moved from cities outside of Texas to cities inside of Texas, specifically along the I-35 corridor from San Antonio to Waco.

What do you notice and wonder as you look at the October rainfall totals for these cities?

Now that I shared two different graphs, what questions could you ask students about these graphs? What math skills can students bring to bear to interpret and further understand the data in these two graphs?

One thing that we often do with graphs found in textbooks and tests is ask one question about them and then move on. How unfortunate! There’s so much rich information to dig into here. One of my key points for tonight’s chat was reiterating something I read by Steve Leinwand about mining data. Ask a variety of questions about data displays. Sink your teeth into them; don’t just take a small nibble.

The one thing that stood out to me and many others in the chat was how little rain San Antonio received. The difference between San Antonio and New Braunfels is quite striking considering how close they are to each other.

Other people felt that Austin’s rain wasn’t fitting with a general trend in the data. I didn’t want to get into it in the chat, but I’ve noticed the rainfall in my neighborhood tends to be less than other parts of the city. Our weather station recorded 10.3 inches for October but others in Austin clocked in at closer to 13 inches of rain. I thought about using the larger number, but because the catalyst for this whole story was my weather station’s data, I opted to stick with that. By the way, I don’t think it’s an issue with our weather station’s rain gauge. Over the years there have been many instances of rainfall in other parts of the city while my neighborhood in north Austin remains bone dry.

Now that we’ve looked at rainfall in and out of Texas, it’s time to drop a bit of a bombshell. With this new information, what story is the data telling so far?

Here’s what I see as the story so far: Austin received 10.3 inches of rain in October, which was a lot compared to areas outside of Texas, but fairly common for our area in Texas. Not only was this a lot of rain, but it also fell in a very short amount of time, 6 days.

Next, I asked for help. Now that you know it rained only 6 days in October, which data display would you choose to represent October rainfall?

Option 1

Option 2

Most people preferred option 2 because it shows the full picture of October. That was surprising to hear. In my mind, because we just saw the picture graph showing that it only rained 6 days in October, I didn’t feel option 2 was needed. I already know it didn’t rain on very many days, so why waste the space with all those days showing 0 inches of rain? Option 1 puts the focus squarely on analyzing the rainfall on the days where it actually rained. In the end there’s no “right” answer, it all comes down to how you justify showing what you choose to show.

We’re nearing the end of our story. There are two more graphs remaining. What does this next graph add to our story? What is one question your students could answer based on this data?

I love looking for relationships so here are the questions I came up with:

• Where do you see the relationship “three times as much” represented in this graph?
• Where do you see the relationship “half as much” represented in this graph?

I especially like wondering what students will come up with because both questions have more than one correct answer.

And now for the last graph. How does this close out our data story?

Here’s a follow up question for you. What could be the sequel to the story I just told? How could you and your students explore and tell the sequel? What other data stories could your students explore and tell?

I closed the chat, and I’ll close this post, with two key points I want everyone to take away from this conversation.