I have a set of translucent plastic polygons that I use with my 5th graders. They come with the Connected Math unit “Shapes and Designs.” If you’re not familiar with it, the program is full of good problems – but this particular unit is underwhelming. It includes the kinds of activities no one would continue doing if the teacher and school suddenly vaporized (measure this, put these into groups… a collection of unconnected 30-minute tasks). I’ve been tinkering with ways to use these polygons – a set of mostly triangles and quadrilaterals – in a way that is harder. Harder in a good way. Whole ladder of abstraction way.
Last year, I elected to scrap several of the usual lessons in favor of what I decided to call Detective Work – students would spend over a week creating visual proofs for angles in polygons. The classes were lively and students worked out some of the most essential and important elements of the the unit without much input from me. I was impressed with their thinking and energy. (This, more than anything else, seems like a sign of a good lesson: that I am impressed with their work. Conversely, if I’m feeling disappointed in their work, it’s usually my own bad planning.)
Here is what we did and what I think made a difference:
- I kept the shapes out longer. We spent several classes categorizing and sorting. The students were able to get familiar enough with them that they could use them as tools. I always forget to account for this adjustment phase – I hand out manipulatives and expect that everyone will be as ready to use them as I am – an adult who has essentially graduated from fifth grade with flying colors every year for a dozen years in a row.
- The challenge was simple: Prove the measure of any of the angles in any of the shapes without measuring (estimates are helpful but don’t count). As a hint, I told them that they could use multiple copies of the same shape if they wanted to.
- I didn’t help much after that. Starting with the simplest polygons in the set (like the equilateral triangle which is called “shape A”), they tiled them around a point. Since three make a straight angle, each one must be 60 degrees, they reasoned. They moved on to more polygons – the various rhombuses and parallelograms, the three different trapezoids. They started to use the polygons they knew to fill in the ones they didn’t: “This one is a B and V, so it’s 90 + 30… 120 degrees!“
- I let this go on way longer than planned because they were enjoying themselves and discovering important things along the way. The parallelograms have two pairs of equal angles. You can use the shapes build up from an angle you know (like 90) or subtract away from one you know (like 180). They became very good at estimating the sizes of angles.
- It was intrinsically differentiated. Students chose polygons that were appropriately difficult for them, and they had the option to do as many or as few as they wanted. Students reshuffled themselves into the groups and partnerships of their choice, sometimes choosing to work alone.
- Solutions were public and communal. There was a real audience for the finished proofs (so much more persuasive than my vague urgings to be legible because I said so). We posted them on the bulletin board as we went along, and students compared and consulted with each other. There were at least five different methods for the regular octagon.
It was a week well spent. I’ll try it again this year.