Power of student questions & a central idea (angles)

As the children were returning in from lunch, I played yet another fabulous YouTube video created by Numberock: Numberock Angles to tune in to the idea we are are going to be thinking about angles. NumberRock's ( @Numberock ) videos are so engaging and not just a silly gimmicky song- they are really educational and helpful for children to understand key concepts. 



We began by examining our central idea and discussed what we thought it meant and what we might do in our unit to gain a deep understanding of it.

One student reminded us how we had already discovered that angles cannot exist alone and must exist with at least one other angle.  She told us how this already helps us with our understanding of our central idea.

She is spot on:

Then the power of a student question directed our learning.  

Doesn't only one angle exist with a revolution angle?

A great question that challenged our thinking.  

Eventually, a student suggested that even a revolution angle has two angles co-existing together. An angle in a clockwise direction and an angle in a anti-clockwise direction.   We wondered if that was true or not. 

This then led to another student question: Why is a circle 360°?

Why indeed?

I suggested how the mathematician who thought of this must have had a logical reason for it.  

Why wasn't 500 chosen or 41 or even 2 346 350?

Looking at those numbers, someone suggested an odd number like 41 wouldn't be a good choice because it is hard to divide odd numbers and we want to be able to divide a circle in many different ways.

That made a lot of sense to us. It was best to be an even number.

Suddenly an excited face explained her theory- 36 has so many factors!

By choosing 360°, it makes it easier for us to divide a circle in many different ways!

Does 36 have many factors?

We started thinking how many it had:

We agreed that 360 was a good number choice as it is divisible easily by many numbers.

That sparked another question: But wouldn't 100 be a better choice for the number of degrees in a circle? It would align with our base 10 number system like the metric system did. (So great to see these type of connections from our previous maths units)

So, we found out how many factors 100 has to compare:

They both have 9 factors we discovered.

So, why was 360 chosen for a circle?

Another student then sparked out:  Maybe there is an even better number with even more factors than 36 has!

So, we spent some time finding out.

Eventually, some shared that 48 has more so perhaps 480 would be a better choice than 360.

I love this type of thinking - 10 year olds challenging and trying to improve established mathematics!

One student drew a circle and showed us the different ways we could divide the circle if we said was 480°.  We liked this idea a lot.

Another challenged that and draw a circle having 100° and explained that she felt it was more logical and dubbed it the metric circle. Fabulous! 

Getting back to a circle being 360°, we relooked at our central idea and thought about how knowing a circle is 360° can help us find connections or relationships with angles.

We drew circles in our books to see what we could discover with a partner or individually.

Some of our connections we discovered:

So, what have we discovered about our central idea: 

      When angles co-exist, connections and relationships are formed?

Some of our oral reflections:

- Circles show us connections between right angles: 4 right angles equals a circle.

- All angles are connected to a circle being 360°

This sparked a question which we placed on our wonder wall: Can there be an angle larger than 360°?

- You can make 3 congruent obtuse angles of 120° to equal a circle.

- Semi-circles show us the connection of two straight angles equalling a circle.

- Even if you have one angle, it connects to a circle being 360°:

    

Thinking back to all this, it wasn't my intention for us to investigate the circle. It all came about from student curiosity.  

I think that's what makes enquiry-based maths so rewarding and exciting. Students have the best wonderings to explore.  When we as teachers value those and give them the time to explore, incredible maths understandings develop just like what we experienced today.  If I was steadfast on thinking of the original plan of what we would explore today, we wouldn't have made such great discoveries (and challenges to established mathematics) like we did.

This is what I think is the essence of enquiry-based maths. Children being given the power to direct learning and being given the opportunities to explore what they are curious about. 

Originally, I had an idea of us exploring the following activity (below) primarily to assess and assist those who needed support in how to measure angles with a protractor.  Magically, it tied in really well with our discoveries of the circle as children started commenting on how they find ways to measure the other angle simply by measuring one angle now that they knew a circle was 360°.

That's part of the beauty of open-ended maths activities - children are able to explore and discover a larger diversity of concepts than closed activities allow them.  Children also find open-ended activities a lot more engaging and rewarding to do.  

A sample:

When we had run out of time, based on how engaged everyone was with this activity, I asked whether we wanted to share what we had done or did we want some more time next week to continue exploring this. They definitely wanted more time.   Children love the challenge and the fact that they won't be deemed 'wrong' with open-ended maths activities like this one. Every child is made to feel successful whether they were able to find a lot or a few doesn't matter. They aren't pressured in typical 'right / wrong' maths questions and that leads to higher engagement and feeling of confidence as mathematicians. 

To wrap up, we looked at our central idea again and shared with our table partner what we understood about it.  It's important for children to be given these opportunities to voice their understandings of a central idea during a maths unit to remind them of the big picture understanding we are exploring. the central idea empowers them to create their own investigations which we are pretty much ready to do with today's key understanding.