Angles on 'straight lines' - tackling a key misconception.

As mentioned in my blog post a couple of days ago (I know, two in a week!) I have been teaching angle properties to a couple of different year groups. One misconception I kept bumping into is surrounding pictures like this:

The key misconception I am talking about is this, "123 + c + a = 180 because those angles lie on a straight line".

For me, it is easy to sympathise with this, as of course these angles do "lie on a straight line". I think there are a couple of issues here and have been trying to deal with both through the topic. The first is a language issue, and the second involves the diagram.

The first issue is the idea of angles that 'lie on' a straight line. To me talking about angles on straight lines actually helps reinforce this misconception, rather than preventing it. Instead I think it is better to talk about angles that "form a straight line", this allows you to demonstrate that the angle 123 and c form a straight line, but that a is not needed to form the line, it is further down the line.

The other approach I have used alongside this is to get pupils to actually mark the point where the angles come together to form a straight line. Of course with the pupils being encouraged to look for angle properties rather than chase after particular angles (see my previous post), the conversation goes something along the lines of, "where do you see angles forming a straight line", followed by "can you mark where they form the straight line", which leads to pictures like these:

These sorts of pictures really help show why the two (or three in the case of the upper line) angles are the ones that form the straight line, and that a is not involved.

My advice would be that when teaching angle properties, consider how the language you use supports pupils in identifying angle pictures correctly, and ways in supporting pupils on securing the correct angles as part of the correct pictures.

Angle properties - don't go chasing angles...

Recently I have been teaching angle properties and calculations to Year 7 and Year 9. Particularly in Year 9 we have been exploring problems that require multiple properties and steps to arrive at a solution such as the problem below:

The approach I am taking here is not to focus on finding a particular angle, but rather than trying to focus pupils on the sorts of pictures they see. This means that instead of asking questions like "can you tell me the size of this angle?", I am asking questions like "Can you see any straight lines in the picture?". I am also modelling this process in examples, for example when we went through this example:

rather than trying to find h and then trying to find i, we instead just went through the different angle properties we knew and found angles that fit, including completely useless facts like 46 +90 + 44 = 180. Altogether we wrote down:

h + 46 + 90 + 44 + 61 + i = 360 (full turn)

h + 46 + 90 = 180

46 + 90 + 44 = 180

44 + 61 + i = 180

61 + i + h = 180 (all straight lines)

h = 44

i + 61 = 46 + 90 (both vertically opposite).

Only when we had written all of this down did we talk about and look at which bits of information may be useful in helping find h and i (quickly identifying multiple ways of finding both h and i) and eventually writing down the values of both angles.

This approach is definitely having an impact in terms of pupils working through these sorts of problems as they are less hung up on the fact that they can't immediately find values of an angle and are correspondingly (nice use of terminology!) more ready to make an attempt at these problems. This, coupled with a visualisation of walking down the paths that the diagram shows (more on this in a blog to come) seems to be a real support to pupils in working with these sorts of diagrams.