Labelling angles is difficult.
Even asking kids what angles actually measure is difficult.
Seeing a turn instead of a line or distance takes practice, and pupils need to be nudged to reckon with these differences. In the task below, pupils need to order the angles presented.
There are a lot of different things going on here, from line length, to arc radius and even whether the arc is major or minor, to the actual size of angles themselves.
With our keyworker kids that are in school, I adding arcs to angle B and extended the lengths of the line segments in G to hammer home that the angle has not changed.
Explicitly picking apart each aspect of what we’re talking about takes longer, but it’s more successful than just hoping they catch on, or trying to catch them out.
The gif below begins to demonstrate angles explicitly as turns. I went round and round on how to talk about direction, as it’s not that direction ‘doesn’t matter’, but it isn’t necessarily what we’re interested in right now.
Pupils really struggled with the idea that line and angle labels are more like instructions that names. In the notation lesson slides, the following definitions use a simple structure of ‘what we say’ and ‘what we do’, as shown below.
The key thing is absolutely loads of practice. Using whiteboards as below and getting pupils to construct as per my instructions helps them focus just on translating the words into geometric figures. For example ‘sketch line segment AC’, ‘sketch triangle BDE’, etc.
It’s also worth throwing in something like ‘sketch triangle ABEC’ and discussing how it’s an invalid instruction. Or ‘sketch rectangle ABCE’. This gives pupils a real focus in clear and effective mathematical communication.
Asking pupils to find the equivalent angles to angle 2 in the diagram brought up 12 different possibilities.
Again, this brought attention to the fact that angle labels are there as descriptors rather than arbitrary names.
Pupils asked if this meant any angle with an A in the middle, which is why I added angle GAB to demonstrate that this isn’t the case. Likewise, the quick sketch I made on the board below shows why angle A isn’t enough.
It took a lot of convincing for pupils to accept there are in fact six angles in this image! This became clearer once writing them out (helpfully impressing on them the importance of specific labelling!)
We then did some mini whiteboard questions (which are shown in the gif below as continuous even though they are animated in the powerpoint resources at the top of the post to move only when clicked).
Pupils really benefited from the explicit modelling of BC and BD being equivalent in terms of their effect on angles from B, and the use of dashed grey lines to show implied lines helped foster awareness of geometric figures being pliable rather than fixed. This is something older pupils really struggle with in higher geometry, whereas as maths teachers will lob a load of lines down on anything.
The gifs below demonstrate overlapping angles and using what they know about addition and subtraction in a bar model form to find missing angles.
These tasks take that idea and provide examples for practice, especially around identifying appropriate sections of diagrams in the second page. Pupils found this quite challenging until they turned their rulers over on their narrow edge, placed it on AB and then turned it clockwise to BC themselves. Likewise for the second sheet, they’re aware of what dashes mean from our perimeter work previously, so it was easy for them for extend this idea to angles too.
The sheet below also deals with formal notation and asks students to consider statements in sentence form and notation form before deciding in they’re true or false from the diagram notation.
And finally, this matching activity really hones in on details, with only minimal differences between each trapezium.
[All the activities in this post are linked at the top, and are a mishmash of different things made for different classes at different times that I drop into lessons as appropriate. They aren’t designed or presented as a sequence of lesson content.]