All the questions were engaging but especially the last two. The discussion that led to proving that such shapes are impossible actually uncovered some new geometric ideas for us. We decided to ask students some similar questions:
1) Make a shape with three right angles.
2) Make a quadrilateral with three acute angles.
Most of the students used geoboards but a couple of pairs had access to an iPad. Some answered the first question with a square...
...as it has three right angles. More than three was the justification given. A simple tweak to the question "... exactly three right angles..." clarified things quickly enough. Our question was deliberately ambiguous to bring out such conversation.
The second question brought out a lot more thinking. Some groups looked to a create a kite and checked for acuteness by using a corner of a piece of paper:
Others came up with a chevron type figure:
Overall, the students engaged in some really good conversations and were using some very precise geometric language.
There were a couple of considerations though. One student I was watching found it very difficult to use the geoboard. When asked to make a pentagon, he didn't know where to begin and moved the elastic from peg to peg in the hope that some familiar shape would come up. Eventually he made a pentagon but didn't realise that he had done so. I then asked him how many sides his shape had. He proceeded to count all the pegs that the elastic was in contact with! I'd not seen this before so i handed him a pencil and paper and asked him to draw a pentagon. He did this quickly, and then drew another four all of which were irregular. For this student, the geoboard was not a good tool. Yet. Maybe it will be later on I don't know. But at least he knew that pentagons have five sides and could draw them.
Two other groups of students used the geoboard to make a 'pentagon' like this:
They argued it has 5 sides and traced these with their fingers. We shared this with the rest of the class who offered advice along the lines that 'It can't be a pentagon if the lines cross', or 'It's not a pentagon, it's three triangles.' After some debate the class agreed that polygons can only be made up of straight lines that don not cross and that all polygons must be closed.
Now I could have written this down on the board at the start of class and told the kids to learn this by heart. But this would have prevented any of the rich thinking and debate that actually occurred.