The geometry specifications in the national curriculum give children the opportunity to work on some of the big ideas in maths and make connections while still working on number and underlying structure; that is the idea that in additive reasoning a whole is made up of two or more parts.
In this article, we’ll examine some of the subject content to do with angles and how children can become fluent with the basics and also reason and solve problems. The basic knowledge required to understand all that is necessary in the KS1 and 2 programmes of study is not particularly wide and so achieving fluency of recall and application of geometric knowledge is relatively straight forward. The issue in the past perhaps has been that insufficient curriculum time has been given to geometry as number fluency and calculations are prioritised over geometry. Of course if this happens, children don’t get the frequent practice necessary to fully internalise concepts, perhaps using terms without engaging in certain subject content. Geometry sets a context for basic number and calculation work and should be a valuable use of curriculum time.
The first realm of knowledge addressed here is the concept of turning – children often need to physically complete whole turns, half turns and quarter turns. Not just that, but they should represent this pictorially alongside the concrete task of actually turning their bodies:
Misconceptions can arise if we always start in the same direction though, so children must get plenty of practice in KS1 of representing turns from different starting points:
Children’s drawings don’t need to be as accomplished as those above, but what they must represent is the concept of turning and accuracy of the direction that they may be facing. To gain fluency, some children may need lots of practice but others may be able to move on to solve some problems. A good way of getting children to think deeper is to change the unknown. The standard question might be:
Face the front of the classroom. We’re going to complete a half turn…
In this example, the starting position is known, as is the degree of turn. The unknown is the end position. By telling children the start and end positions, the degree of turn becomes unknown. Similarly, by telling children the end position and the degree of turn, children could be encouraged to think backwards and find the starting position. By changing the unknown, children can be encouraged to work beyond basic fluency and instead reason.
Start position

Degree of turn

End position

NC aim

Known

Known

Unknown

Fluency

Known

Unknown

Known

Reasoning

Unknown

Known

Known

Reasoning

Explicitly teaching children these underlying patterns gives them the language to talk about and analyse problems and enables them to develop effective tools for problem solving.
As children move into KS2, they need to understand angle as the coming together of two lines and a description of the turn from one line to the other. As such this should be added to their knowledge of properties of shapes. The right angle comes first and this idea should be built on what children know and have represented in KS1:
By explaining what’s the same and what’s different in the two pictures, and getting children to do the same with similar pictures, they can be encouraged to make links. The same questioning can be used to show that angles are properties of shapes:
Children must become fluent in recognising right angles in the world around them and in shapes. Flash cards with shapes and other pictures including right angles in the environment can be made and used to give children regular opportunities to test themselves and practise recalling knowledge of right angles.
It is this sound knowledge of what a right angle looks like in different representations that enables children to begin to compare angles, saying whether a given angle is greater than on less than a right angle. Children need plenty of practice with sorting in order to achieve fluency.
When children come to ordering angles, an important misconception needs to be preempted:
When asking ‘Which angle is bigger?’ the misconception may be that because the lines in angle are bigger, children think that the angle is bigger.
As children get older, they begin to assign measurements and numerical value to the degree of turn and quickly need internalise facts (a right angle is 90°, a straight line is 180°, an acute angle is between 1° and 89° and an obtuse angle is between 91° and 180°). Again, using flash cards for regular testing can help to internalise this knowledge and keep it to mind when solving problems but that alone won’t lead to a deep understanding. Showing angle sizes on a number line can help to reinforce the increasing size of angles and the link between mathematical language and numerical value:
To get children beyond fluency and into reasoning, a point on the number line could be shown and children could be asked to select from a number of given angles which one is the most appropriate to be placed in that position on the number line. Similarly, they could be given a position on the number line and asked to draw an angle that is the appropriate size.
Children can then use their understanding of angles to solve some missing angle problems like so:
Here, children would need to have a clear understanding of additive reasoning and the idea that the whole is split into parts. In this example, the whole is 90° and it has been split into two parts: 30° and an unknown angle. They should be encouraged to interpret a picture like this as ‘thirty plus something is equal to ninety’ and that subtracting 30 from 90 will reveal the missing angle. These questions can become a little more complex:
Essentially the underlying pattern is the same though and will be for almost all missing angle questions: the whole is known (usually 90° or 180° or 360°) a part or parts are known and one part is unknown.
Work on angles provides many great opportunities to work on the basics of number and to develop the knowledge of underlying mathematical structures that is required to solve problems. It sets this number work in a context but not too much of a context that would hinder transfer of learning to other content domains.