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”.
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:
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.
Thanks for reading!
 Swan, Malcolm., 2005. Improving Learning in Mathematics: Challenges and Strategies Department for Education and Skills