Tuesday, April 7, 2015

Using Pattern Blocks to Reason and Prove

As part of one of our Ministry's Numeracy initiatives, we tried out the following task in a Grade 6 classroom. 

Find the size of each of the angles in the pattern blocks shown. Protractors are not allowed!

Our Minds-On discussion showed that the students had a good grasp of what acute, obtuse and right angles were. At the same time there was some disagreement as to what a polygon was: it was only after a fair bit of using counter-examples to their definitions that they agreed that polygons are closed 2-D figures made up of entirely straight lines (which begs the question, are pattern blocks polygons?). 
When the students began the task proper (and after they asked me a dozen times if I really meant they couldn't use their protractors!) it was interesting to see their strategies.
Some just estimated angles (below). It was a quick fix to show these students that some of their answers were contradictory and that they needed exact answers instead. 

Some recalled that there are 180° in a triangle so this meant that each angle in the equilateral triangle was 60° and progressed from there. Others put compared three equilateral triangles to two right angles (or squares) as shown:
Others benefited from marking the angles they were trying to find actually on the pattern block as this helps them see the amount of 'turn' between the two sides (check out this post for some ideas on how angles measure turn):
As much as possible, I am learning to hang back and not jump in and show students what to do: when I do this, I realise that it is me doing the maths, not them; it is me reasoning and proving, not them. There certainly are some awkward pauses when I do this but this is most likely because the students are mulling over what they could do.
And sure enough, they could solve the problem without my help:

And for fun, they could verify their solutions in more than one way:
So when it comes to problem solving, I really like the philosophy from Singapore's Ministry of Education:
Teach less, learn more.

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