Central Tendency Puzzles — Reflection

On Twitter, Mark Chubb posted some amazing central tendency puzzles, and asked people to try them out.

I was intrigued, so I did. You can see the post here.

I just want to go through some of the student thought process, and reasons for making decisions I did while students worked through the puzzles.

This is a grade 6 class, we have covered central tendency, and so we needed to play with our understanding. I had noticed, despite building mean, for example, and distributing evenly, students still relied heavily on racing to the formula. Though they could say what the median and mode was, having them grapple with how it works with other measures of central tendency was going to be illuminating.

How it started

I decided to show the image that is in Mark’s blog without any instruction. As students entered from recess, they started talking about what they were seeing. They were asking each other what range was again, talking over each other to help, and then noticing a number at the end of each row and column.

Right away they connected to Skyscraper puzzles — which, honestly, that surprised me. I didn’t think they would, and so they started talking about how they had to relate the height and numbers from different views of the puzzle, and so I jumped in and asked how this might look with central tendency.

I started writing down what I was hearing as they talked, but they were going pretty fast!

Getting to work

So, after some conversation, they figured it out. Like Sudoku, across columns and rows would have to fit that particular number.

Giving out cards, they randomly headed to a white board and started playing with numbers.

The first puzzle had numbers to help, and students, before even writing, were drawn to the mean. They could tackle mean, and realized that once they had that, they could move from there.

“Ugh, ok — but that doesn’t work for the median”

Median was a point of conversation

I listened, and waited for the next person in the group to talk.

“Well, the median needs to be the exact middle, and this isn’t it”

“So, how far back do we have to go to fix it?”

There were pauses, and discussions, and many students went back to the beginning. I paused and brought us to the whiteboard, and we walked through what the group had done.

“They’re starting over again, they don’t need to”

“Tell us more,” I prompted.

“Well, they can just increase or decrease this number here, it will affect the mean, but then they can stick with the median. They can then adjust,”

Back to our whiteboards.

Second Puzzle

This one had no numbers to help, but students seemed to be more confident. Some thought this would be better — now they could choose any number they wanted, and not feel locked in. Others were daunted by this, and wanted the helper numbers. Those groups I let wander around to see what other’s were doing to get them started (and had them re-phrase why the other group’s did what they did).

Reflections

Median is a place these students generally needed to really work through; and, we started to place various numbers on number lines to reinforce the idea of middle. Students found the puzzles challenging and engaging (because some were going back to it at lunch), and were eager to try to make their own after this session.

I would love to do this earlier in the year before there is formal introduction to central tendency; as a place to play with number, number relationships, and have an exit ticket where students note the relationships between central tendencies.

Yes, we didn’t use linking cubes — and this was because I wanted to see how they would play with number on a whiteboard and with discussion.

Give these puzzles a try! They were great as a teacher to see where/which central tendencies were actually conceptually understood, and how they work/relate.