This community is so badass that we have a group of teachers who have given up a bunch of there time just to get more people involved. I’m helping by doing what they ask me to do. So here’s my blog post for mission one:
Mission #1: The Power of The Blog
“What is one of your favorite open-ended/rich problems? How do you use it in your classroom? (If you have a problem you have been wanting to try, but haven’t had the courage or opportunity to try it out yet, write about how you would or will use the problem in your classroom.)”
Since I’m a maverick, and I can’t possibly choose a favourite, I’m going to write about a couple of tasks…
My New Patio – Nrich
Why I love it
Because it’s easy to explain to students. The entry point is so low. ‘Draw a square on your squared paper. Shade in one small square. Now fill the other space with more squares’. I love that the task lends itself to asking easy questions with hard answers.
Generic questions I like to ask:
- What’s the most tiles needed? Why?
- Can you write a rule for this?
- What’s the fewest tiles needed?
- Can you tell me a rule for this?
- OK, that was a really hard question; can you tell me a rule for [subset of squares]
- Are you sure that’s the best you can do?
Stuff to think about
This task benefits greatly from systematic thought and layout; recording strategies and group co-ordination. Is there any point in everyone working on the same size square? Is it a good idea to have everyone working on different sizes?
The ceiling on this task is pretty high. For one subset of squares you can prove that a certain type of tiling is always possible by induction. I’m not sure if there is a rule; every time I have done this task a student has come up with a way to foil any rule his/her peers could come up with. This might put some folks off; but how nice is it that a task that seems so easy can be so hard?? The one problem I have with this task is that it is difficult to link the task itself to specific curriculum points. But if that doesn’t matter to you; or you feel the Problem Solving Skills aspect is all it needs (as I do, sometimes…) Then crack on.
And if you find an answer; with a PROOF; let me know!
Pent/Hex/Polyominoes and Nets
I’ve used this task in many ways. I’m grouping it all together under one heading but I’ve done:
- Systematic finding and classifying of Pent/Hex/Polyominoes
- Arranging and tiling with Pent/Hex/Polyominoes
- Linking to nets of shapes, including other platonic solids
Why I love it
The low entry point for finding the different pentominoes means it can be explained quickly for learners of all abilities. The fact that they benefit from working systematically is great and the discussion over what defines ‘sameness’ and ‘different’ is hugely valuable; particularly if you are going to be studying congruence. The nets of cubes extension to hexominoes builds spacial reasoning and awareness. Getting learners to try to find tilings of rectangles with the pentominoes gets them working on a problem that puzzled recreational mathematicians (the horror on some students faces when they discovered that those exist!) for decades.
Questions I like to ask
- Are you sure you have got them ALL
- Compare with student X; they say they’ve found [number] more than you
- Are these the same?
- What makes them different?
Stuff to think about
This can take a LONG time. Make a decision beforehand as to whether you count reflections/rotations as ‘the same’. Think carefully about this; when rotations and reflections are not considered to be distinct shapes, there are 35 different free hexominoes. When reflections are considered distinct, there are 60 one-sided hexominoes. When rotations are also considered distinct, there are 216 fixed hexominoes. So be prepared for students to find hundreds if you allow rotations. Students trying to visualise whether their polyomino makes a net will probably ask for scissors to test it out physically. I always say yes; but be prepared for this.
Also be aware that conjectures come easily in this task, but proofs are few and far between; and VERY challenging. Maybe because of this, like the task above, curriculum links can be challenging. So go for Problem Solving Skills instead 😀
My (possible) New Favourite – Des-Man
I haven’t used it yet. But it looks awesome. This could easily become my favourite rich task. The step by step intro lowers the entry point and the fact students can use ANY function puts the ceiling WAAAAY up there. If you are starting transformations of functions, linear graphs, conics, anything involving a function in any way I suggest looking at this.