I often ask my students to find all possible shapes made of five squares - the pentominoes. There are twelve of them. Another good task is sorting them into two groups according to some criterion.
I saw John Golden was getting students to do this.
What do you think? Which one is the odd one out and why?
Normally I don't think it's useful for me to propose answers, because what I value most of all in this task is the maths basic, creativity - looking for yourself and deciding for yourself how to compare them. It's this that makes me return to WODBs weekly.
But Christopher Danielson does something useful in his teacher guide to Which One Doesn't Belong? He gives the background to the shapes on each page of his shape book, and talks through likely responses.
You could run WODBs without knowing background. You could just record students' responses and it would still be useful. This is especially true if you're happy with uncertainty, thinking on your feet and coming back to things later; but it helps to have thought through the possibilities. You're more likely to understand what the students are seeing, and the significance of it.
So, let's touch on some of the responses you might have to this. I got lots on Twitter that helped me to see a lot more than I had before, thanks to Vincent Pantoloni, John Golden and his students, Becky Warren and Rod Bogart.
One of the nice things about using photos rather than drawings is that you get extra aspects you might not have been thinking about. In this case, the shapes are made out of straight strips of wood; you can see the joins. So you could sort them by how many strips are needed.
Symmetry: the ones on the right have no symmetry, the top left has one line of reflective symmetry, and the one on the bottom left has four lines of reflective symmetry, and order four rotational symmetry.
How many 'ends' are there? These are squares with only one neighbour. How many 'branches' are there? These start at squares with three or four neighbours. And is there a part where four squares are touching in a square? How long are any branches (or 'appendages', or 'limbs')? How long are the longest straight lines? How many pieces could you leave by removing one square?
Number of vertices. Number of side lengths. What size rectangle would it fit into?
Negative spaces: what shapes are left by the concave part of the shapes? In all but the top right, these concave shapes are isosceles triangles.
Convex/concave: The bottom right is the only convex shape; the others are concave.
[edit: Why did I write that - it's not true. Thank you Justin Lanier for noticing, and for helping me build something out of this. The bottom right is 'more convex' in some way; as Justin puts it, it has the 'least amount of notch'!
Perimeter: The bottom right pentomino doesn't have a perimeter of 12.
Orientation on the page/screen: the X is the only one lined up with the edges of the image.
Pentomino addition: You can think of pentominoes being made by adding a square to a tetromino.
The bottom left pentomino can't be made from the L-tetromino by adding a square. You could go back to trominoes and think which can be built from those.
I made another WODB with my classroom pentominoes:
'Seeing' the whole shape from inside it: If the shape were the shape of a room, is there a point you could stand in to see the whole room? Not for the W.
And there are no doubt plenty more. Maybe you could comment with other ways that you see?
It all goes to show that a simple image can be the starting point for students' own ideas - there are a lot to choose from, and they don't have to be second-guessing the teacher or the maker of the image.
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And it made me think a Which One Doesn't Belong might be the thing to uncover other criteria for sorting. I made one with my wooden set at home.Ways to divide pentominoes. What are their rules? pic.twitter.com/3E8pQMNWiv— John Golden 🔗 (@mathhombre) September 11, 2017
Normally I don't think it's useful for me to propose answers, because what I value most of all in this task is the maths basic, creativity - looking for yourself and deciding for yourself how to compare them. It's this that makes me return to WODBs weekly.
But Christopher Danielson does something useful in his teacher guide to Which One Doesn't Belong? He gives the background to the shapes on each page of his shape book, and talks through likely responses.
You could run WODBs without knowing background. You could just record students' responses and it would still be useful. This is especially true if you're happy with uncertainty, thinking on your feet and coming back to things later; but it helps to have thought through the possibilities. You're more likely to understand what the students are seeing, and the significance of it.
So, let's touch on some of the responses you might have to this. I got lots on Twitter that helped me to see a lot more than I had before, thanks to Vincent Pantoloni, John Golden and his students, Becky Warren and Rod Bogart.
Symmetry: the ones on the right have no symmetry, the top left has one line of reflective symmetry, and the one on the bottom left has four lines of reflective symmetry, and order four rotational symmetry.
How many 'ends' are there? These are squares with only one neighbour. How many 'branches' are there? These start at squares with three or four neighbours. And is there a part where four squares are touching in a square? How long are any branches (or 'appendages', or 'limbs')? How long are the longest straight lines? How many pieces could you leave by removing one square?
Number of vertices. Number of side lengths. What size rectangle would it fit into?
Negative spaces: what shapes are left by the concave part of the shapes? In all but the top right, these concave shapes are isosceles triangles.
Convex/concave: The bottom right is the only convex shape; the others are concave.
[edit: Why did I write that - it's not true. Thank you Justin Lanier for noticing, and for helping me build something out of this. The bottom right is 'more convex' in some way; as Justin puts it, it has the 'least amount of notch'!
]I see. Are you thinking something like "least amount of notch", aka "fewest number of squares to add (or subtract!) to make convex"?— Justin Lanier (@j_lanier) September 17, 2017
Perimeter: The bottom right pentomino doesn't have a perimeter of 12.
Orientation on the page/screen: the X is the only one lined up with the edges of the image.
Pentomino addition: You can think of pentominoes being made by adding a square to a tetromino.
'Seeing' the whole shape from inside it: If the shape were the shape of a room, is there a point you could stand in to see the whole room? Not for the W.
And there are no doubt plenty more. Maybe you could comment with other ways that you see?
It all goes to show that a simple image can be the starting point for students' own ideas - there are a lot to choose from, and they don't have to be second-guessing the teacher or the maker of the image.
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