Monday, 25 January 2016

Student-owned Investigations into Angles' Connections within a Circle

We began by re-examining our central idea and what it means.

We then looked at the board from our previous thinking about angles and their relationships / connections with a circle and discussed what we had discovered and what it made us wonder about their relationship:

To make this student-owned as enquiry-based, we first began by writing down what we already knew about the connections and relationships that exist between angles within a circle. 

We spent about 5 minutes recording what we knew in the K column of our KCD charts (see below) and then orally shared our understandings with a our table partners.

The purpose of doing this was to help each child tune into what they might want to investigate further and to give them an opportunity to reflect on what they knew.

So, what were we curious about?

What did we want to investigate?

We shared some wonderings as a whole class to perhaps inspire others:

° Can angles exist like they do in a circle with other shapes?

° How many acute /obtuse / reflex etc angles can exist in a circle? 

° What would a quad-circle, penta-circle, hex-circle, octa-circle, nona-circle etc look like?

° What might a metric circle look like and the types of angles within it?

° Do super large and uber tiny circles have the same degrees?

° Is there a connection between the AREA of a circle, the PERIMETER of a circle and the DEGREES of a circle?

° Can we have angles larger than 360° (revolution)?

° Can angles exist inside other angles (excluding circles)?

With these ideas, we recorded some wonderings in the C column and then found partners who wanted to investigate the same thing. 

As we investigated, we recorded our discoveries in the D column.

Some samples:

These two students made a great connection. If we had a metric circle, then the degrees would align with how we divide a circle for %:

This was another interesting investigation.  They created a circle that had 100°, 1000 and 480° to see if one of those would be better than a 360° circle.

They shared with us that they felt a 480° circle should replace a 360° because 48 has more factors and so it is easier to divide the circle.

They took their thinking further by calculating how large an acute, right, obtuse etc angle would be if the degrees of a circle was different. 

At the end, we shared our discoveries or 'non-discoveries' with the whole class.  Some students shared how they had proved themselves wrong and others shared how by not making a 'discovery' they had actually made a discovery!  We connected this idea to how scientists conducting experiments- even if the experiment does not prove a hypothesis, it still proves something. It proves that the hypothesis is not correct and that in itself is a discovery.

From this discussion, we realised something very important as mathematicians: Even when we disprove a theory we might have, we have still made a discovery. Finding out why is was disproven or why our mathematical ideas don't work out as intended, we felt was really important for our learning. 

From a fairly simple tuning in, quick reflection on what we knew and the an opportunity to investigate what we were curious about, we were able to come up with some really impressive enquiry-based maths learning. 

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