# Learning through classification: What makes this number (or shape or graph) different from the others?

Which One Doesn’t Belong? (WODB) is a website with a very simple concept. It is “dedicated to providing thought-provoking puzzles for math teachers and students alike”. Basically, it presents four of something and you have to come up with a reason why each one of the four things doesn’t belong. But it is far more than a collection of brain teasers.

One way we can help students develop different ways of thinking in math is to have them work on activities where they have to classify mathematical objects. In working on these kinds of tasks, “Learners devise their own classifications for mathematical objects, and apply classifications devised by others. They learn to discriminate carefully and recognize the properties of objects. They also develop mathematical language and definitions”.[1]

WODB is a great resource for bringing this kind of rich, classification activity to students. The site is divided into three main categories – Numbers, Shapes and Graphs.

I started off with the Numbers section because it fit better with what we were doing in class. I decided to try out a few of the WODB puzzles with my class of adult education/HSE students as a warm-up to start class.

I gave out the following puzzle as an example to begin and we did it together as a “think aloud” so they could be clear on how the puzzles work.

As we talked though it, interesting things started to come out immediately. One student said, “I know 3 and 27 are both part of the three times table, and I know 31 isn’t, but I’m not sure about the 123. I don’t think it is.” Another student asked how she knew the 31 was not “part of the three times table” and she said, “Because I know that 30 is, because it goes 27, 30, 33, so it skips the 31”. Then another student had some memory of a divisibility rule for 3 (add up all the digits and see if it’s a multiple of 3) and we tried it with some numbers we knew were multiples of 3 and then we tried it on the 123. Then someone suggested we check and divided 123 by 3 to see if it came out evenly. All of that reasoning and calculation, just to make an observation about one of the numbers (the 31). I made sure to offer “9 is the only single digit number” and “123 is the only three digit number” as potential answers to help less confident students realize that any differences they saw were acceptable.

Once they all seemed clear on what to do, I put the following puzzle up on the board and asked them to come up with one quality that made each number different/unique compared to the other three.

Here’s the first one I gave students to work on their own:

The way I set up the activity was to draw the square pictured above on the board and have students copy it in their notebook and write the differences around it. Then as we went over it one number at a time, I had a few volunteers share what they observed. One math practice that came up immediately was the use of precision in explaining mathematical ideas. With me as the note-taker, I was able to encourage students to clarify their statements by asking other students to say it in their own words. When someone else understood something different from what they meant, we worked together to make the statement more precise. The next time I do this activity I think I will have the students come up and write their observations themselves.

Here’s what my students came up with:

The first thing that attracted me to the exercise was the way students at all different levels could approach the problem. There are so many different and correct ways to describe what makes each number unique that every student was engaged. As I walked around and looked at students filling the margins with calculations & observations and talking to their neighbors, I saw how effectively this activity could develop student fluency and comfort with numbers. I liked the approach of the student who said, “If you divide 16 by 2 it is the only one without a remainder”. They didn’t look at it and say “16 is the only even number”, but they were just playing around with the numbers and were curious as to what would happen if they divided each of them by 2. Then, with their observation up on the board, we talked about it a bit and connected it to the idea of even numbers.

Along similar lines, sharing observations was a great opportunity to introduce some vocabulary. In the board work above you can see that for 43, one student observed “No number can be multiplied by itself to get 43”. Which was a nice place to introduce “perfect squares“, which many of them recognized but did not know the words for.

Here’s the second one we did:

And here’s what students came up with:

We didn’t get to as many different observations for the second one, in part because it was second, but also because we had such a rich conversation about the first number we discussed. I started with the 17 and one student said, “It is the only one you can’t multiply anything to get”. We spent some time unpacking that (clarifying what he meant; coming up with the two numbers that can be multiplied to get 17 – 1 and 17; testing to be sure that was not true of the other numbers). Students were curious about this strange phenomenon of a number that “you can’t multiply anything to get” and it was a great way to start talking about prime numbers – a vocabulary word I introduced after all of our unpacking, along with composite.

To give you a sense of the richness of the offerings, I’m including some samples from the Shapes and the Graph sections, which work the same way – you have to find a reason why each one doesn’t belong in the set – but with shapes and graphs.

Here are a few selections from the Shapes section:

Here are a few samples from the Graph section:

As I wrote, I used this as a warm-up at the beginning of class. After we finished up, I asked students to reflect on the activity and they all said or agreed with some version of the statements “It was interesting”, “It was challenging but not too hard” and “You really had to think about the numbers in different ways”.

A few of them asked where they could find more puzzles like these and I gave them the website to explore on their own. Several said they would try it out with their kids and report back on how it goes.

Please write a reply below and let me know what you think.

• How do you think students can benefit from categorizing numbers/shapes/graphs in this way?
• Would you try this with your students? Why or why not?
• Try one with your students and let us all know how it goes.

[1]  Swan, Malcolm., 2005. Improving Learning in Mathematics: Challenges and Strategies Department for Education and Skills

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Mark enjoys doing math problems that take weeks, family sing-a-longs and reading late into the night. At 16, he believed the next revolution would be waged through poetry. Now he believes it is adult basic education. But he still likes poetry. Mark has worked in adult literacy and HSE since 2001. He is a founding member of the NYC Community of Adult Math Instructors (CAMI). He was born and raised in Brooklyn where he lives happily ever after with his partner Sarah and their daughter Liv. Follow me on Twitter (@mtrushkowsky)

## 11 thoughts on “Learning through classification: What makes this number (or shape or graph) different from the others?”

1. Aydee Garcia says:

I like to use this type of exercises with my class. They are good warming activities that offer the students the opportunity to reflect on mathematical concepts and make them use their reasoning skills; they are non threatening and everybody ends with the feeling they can do mathematics.

1. Patricia Helmuth says:

This is a great resource for a multi-level classroom! Students quickly caught on that there were multiple answers to each number set and for those less proficient, the “fear factor” was marginalized when they understood that sometimes a number didn’t belong for a reason that they could identify, such as: it’s the only even number in the set. For my open enrollment class, this resource serves to reinforce math concepts and vocabulary for students who’ve been in class a bit, while at the same time, helping new students along. If you haven’t tried this yet in your classroom I highly recommend it. Even students who are not normally inclined to participate in class are drawn in to this activity.

2. This is a really cool resource that I would definitely use with my students! At first I just found it fun and really thought provoking. After reading the article, I can see so much value in using it to teach mathematical concepts and vocab. Words like prime, divisible, exponent, parallel, sum, etc. would be easy and interesting to learn!

3. Mindy Levokove says:

I am liking using these examples with my students. We’ve done a few, now and I think I’ll continue with them. They encourage participation from all the students, regardless of their math background and skill level. Yay!

4. Mindy Levokove says:

I liked using this in my classroom. Everyone had an opportunity to contribute and be involved. And everyone could feel like they understood what we were doing, and what the other students were adding. I also like how it works in general, in an abstract way, to build critical thinking skills, on any level and for any category. I think it involves a kind of thinking and analysis that is fundamental for building deeper, richer connections among ideas.
I have a new group, now, plus a few old timers. I’m going to do this exercise, using the numerical squares, again, to see how it goes, this time. I’ll write back again.

5. Jane Parkerton says:

In the first week of June, I was to proctor a Readiness exam [math, science and social studies]. i knew my students would be anxious and jittery. I usually give them a writing prompt, often just asking them to write about how they are feeling. Instead, after doing some of the WODB number exercises with Mark, I thought I would use one or two of these with my students. I thought the exercise would calm them, refocus their attention and wake up the brain cells! However, since I was not their math teacher, I used shapes.
I started with shape #3 because I thought something really simple would get them started. It was too easy or I didn’t push long enough. So I moved to shape #31, the puppy dogs. Again responses were laconic. We talked first about how they were similar. Then after I pushed, and left some silence, one person finally said, “Well one dog is looking a different way than the other 3.”
I should have gone with some of the number squares. Even if we had only 30 minutes, I think it would have generated more attention.
I am teaching this summer and intend to use some of these as warm ups.

6. For more WODB resources, search for the hashtag #wodb on Twitter and follow @WODBmath!

7. Cathy DeCapua says:

Why you should teach Which One Doesn‘t Belong.

• It starts a conversation about math and math vocabulary. Students learn about odd and even numbers, squares, primes, and inverse operations.

• Students have different approaches that they share with each other and different strategies for figuring out how each number is different from all of the others. Their combined efforts yield eyeopening solutions. For example, one student literally said, “Oh my G-d” when one of his classmates revealed that all of the digits in the 2-digit numbers in the first square (the 9, 16, 25, 43 square) add up to 7, making the 9 the number that does not belong.

• These squares can be used across disciplines and even across different types of classes. ABE students can use these squares in all subjects* and ESL students can use them as well to isolate parts of speech, for example.

*One example would be science. When you are teaching chemistry– chemical and physical changes, to be exact–you can write three chemical changes and one physical change (or vice versa) and ask students to try to find the one that does not belong.

8. Cathy DeCapua says:

I sent students home with a few questions to answer regarding their experiences with WODB. After attending an ABE conference, I learned about “Math Writing”, where students must write about their math experiences. Many students believe that math and writing are two distinct subjects that have nothing to do with each other. This is far from true. What I have discovered is that one subject can help to improve the other. After you have done a math problem, if you are forced to sit down and write about it, this not only helps to build your math skills but also helps to develop your writing skills. Writing about the process of solving a math problem requires clear recall of the steps you’ve taken to solve it as well as the ability to organize those steps. Further, this type of writing requires you to be able to express yourself clearly so that others who read what you have written will have a clear understanding of your experience.

In addition to questions about how they proceeded, I asked students other questions, such as what their initial reaction to WODB was and what they learned from it. I also asked if it helped to improve their math skills.

Here are some of the noteworthy things that students contributed:
• “WODB is very interesting and challenging. It teases and refreshes the brain. It helped widen my thinking skills.”
• “When I first saw the paper with the squares and numbers I didn’t know what to do, but I knew that it was about math so I tried. I started with the third square because it was about money. I know about money. I was able to do this square. Then I felt more confident and started to look at the other squares. I learned that you can take any number and find what is different about it. Working with these squares can help me improve my math skills and maybe help me along the way in life.” (I asked the student to explain how working with these squares might help her along the way in life, but she did not respond.)
• “I learned from this experience that math problems can sometimes be challenging, but if I give it a little more time and effort I will understand better. Working with these squares was helpful because they made me try my best to figure them out.”
• “When I saw this math I was interested to learn. I was also scared. I said, OMG! By the time I worked through all of the squares I learned that each number is different for a different
reason. The experience was great!”
• “My first reaction when seeing these problems on the sheet was that this is lazy, boring work. It looked like a difficult puzzle to solve. When things involve a lot of thinking, I sort of lose interest in working hard and feel like giving up. I started with the picture square [the money square] because it was the most attractive, and in my head math in money is less difficult. I then proceeded to do the other squares. I learned that there can be a few, more than one correct solution. After reviewing it in class, I saw other solutions that hadn’t occurred to me. I felt bad that I did not try harder to figure out other solutions. I could see the other ways of solving the squares now and learned a lot about math from this experience.”
• “When I first looked at the problems I was very confused. I didn’t know what to do. I felt fear. Once it was explained I started with the first square and worked from there. What I learned from the experience is that if you put your mind to it you will learn so much, and I learned how good my math skills are.”
• “I feel like this math is magic. Each number doesn’t belong, so I needed to ponder carefully. You need to further study it because there is not only one answer to each square….the math is magic!”
• “It was an exciting experience and I had fun doing it. I learned that numbers can have so many different values to them. This was helpful in many ways because I was using images to do math and working with different numbers using multiplication, division, addition, and subtraction.