Monday, 18 January 2016

Can an angle exist alone?

To lead into our central idea (which we haven't looked at and unpacked yet):

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

We began by thinking about how we can define what an angle actually is.

I drew the following on the board. After each, the children were asked to think to themselves whether it was or was not angle and why they thought so.  

For the last image, they were asked to think whether those parallel lines could ever form an angle or not and why.

Students were then invited to share their thoughts.

A really rich discussion took place as children shared their ideas with supporting reasons. I stayed neutral not indicating who might be right or not throughout the discussion to allow more thoughts and ideas to flow more freely.  This strategy (I hope) also sends the message to children that theories are theories and it doesn't matter if they are proven wrong, but what is important is that we create them and are willing to share them. 

If I had immediately indicated who was correct or incorrect as students shared their ideas, the richness of what was shared would not have taken place. Instead, i simply kept saying, 'What other thoughts have we got?' That kept the theories and ideas flowing with students feeling unjudged by any 'incorrect' ideas they were harbouring. 

Eventually, this strategy helped us understand that angles are formed when two or more straight lines intercept.  With the images above, only b was an angle -actually four.  Image a had a wavy line so it is not an angle and c wasn't actually drawn to meet together. 

Regarding the parallel lines, students shared their understanding of what parallel lines are and so we concluded they couldn't ever make an angle no matter how long those lines extended.

This discussion helped me to pre-assess student understandings and also helped students to expand upon each other's thoughts which is something we are trying to focus on in our class of late. 

The following question was then posed:   Can an angle exist alone? (Or does it always have to exist with one or more other angles)

This really got the students wondering as they wrote their theories on a post it note.  Curiosity was rising very high with this one!

To share, we quickly came up, shared out theory and placed it on the continuum (see below).  Due to us recently doing mock U.N debates and learning about abstaining, a student suggested we should make that an option so we did. 

It was fascinating listening to how each student justified their reason. Eventually one student shared what we were searching for:

This student then used an angle from before to show us how when an angle is formed, actually another angle is formed. 

Interestingly, it made another student rethink her position before sharing with us her theory:

After we had all shared our theories, we discussed which seemed more plausible. Most of us felt swayed by the above student's theory that two angles are formed and so she explained that to us in more depth.  I gave a hand with asking what degrees we knew a circle to be and used her angle to measure what the exterior angle of the acute angle would be. (see below)

To expand upon these understandings and to help us with future investigations, we then started at 0° and drew, named and stated the possible measurements of each type of angle (below).  

When we got to a 180° straight angle, it was interesting that some children called it a straight line.  
- Is it a line? 
- Why is it an angle?
We discussed how it may look like a line, but what had we just discovered an angle is?  = two or more straight lines intercepting. Therefore, a straight angle is two lines joining NOT one line.  To illustrate, we drew one line, imagined the other line moving across to join that line.  That is why it is an angle.

Many were surprised to discover that an angle larger than a 180° straight angle existed.  Not knowing about reflex angles was a big reason why some of us thought an angle could exist alone one student remarked.

With the revolution angle, it was important for us to draw one line and then draw a second line on top of it to illustrate how it is two lines intersecting.  

A question arose:  When do we see revolution angles? Don't they just look like one line?

Such a great question.  I asked to look around our room for an example of a revolution angle.  = the hands on the clock!   At what times might we see a revolution angle with the hands.  

Children came up with numerous times (I was really surprised by how many they could think of so quickly!)

Another student then wondered:  What types of angles do we see the most and least?

Another wonderful question, so we decided that we would explore our homes tonight and do sketches of objects in them that show different types of angles. We predicted together that we thought we would find right angles the most and liked the challenge of trying to find reflex and straight angles.  Am excited to see what they share tomorrow. 

With these foundation understandings, most of the children are tooled up for some great investigations into angles over the next fortnight. 

We had 5 minutes to spare so we played a quick game:

I stood with my back to the class and counted one, two, three and then named a type of angle.   If I said 'obtuse angle' and a student was showing an obtuse angle with their arms, they were eliminated from the game.  We continued playing this till we had three remaining students who we declared our winners. A simple, fun game that hopefully helped those new to learning the different types of angles.....


  1. You're such an inspiration... I'm loving trying to use inquiry so much more now in my classroom. Teaching maths is so much more interesting when the whole class are switched on to their own path of learning.

    1. Hi Tracey,
      Thank you so much for your lovely feedback. It's really sweet of you to say!
      I'm really happy to hear you are experimenting with and enjoying inquiry approach to maths. I totally agree with your last sentence- the theories and connections children can come up with are amazing. I'm still always blown away by their thinking. Their own path of learning, I think, is key to allowing those great conceptual thoughts to formulate. Thanks again!

  2. You're such an inspiration... I'm loving trying to use inquiry so much more now in my classroom. Teaching maths is so much more interesting when the whole class are switched on to their own path of learning.

  3. Your site is a life saver - I love the approach and my students are enjoying maths this year. Thank you


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