Geometrical Reasoning in Key Stage 3

Geometrical Reasoning in Key Stage 3

This term I am introducing geometrical reasoning in key stage 3 to Year 7.  I was thinking about where to start the topic, what the key objectives should be and how I can challenge the various abilities.

Lower Ability students

We start the topic learning how to measure and draw acute and obtuse angles with a 180° protractor. Students have lots of practice to learn how to position the protractor correctly.

Understanding the types of angles is also key. If they can identify whether an angle is acute or obtuse they are less likely to use the wrong scale when measuring so 40° is not measured as 140°.

To challenge these students I would teach them how to draw and measure reflex angles using a 180° protractor. By separating the angle into a straight line and an acute or obtuse angle.  Some students are able to subtract the interior angle from 360°.

Geometrical Reasoning in Year 7

Calculating missing angles on a straight line and about a point is also a key skill

Middle Ability Students

Middle ability students begin with applying angle properties, such as angles on a straight line, vertically opposite and angles in a triangle. Little, if any time, is given to measuring or drawing angles.  Although I do make sure their knowledge with this is secure.

Being able to combine multiple angle properties within a single problem is key for this ability. A typical question is shown below.Geometrical Reasoning in Year 7

To challenge the middle ability students I like to introduce proof.  A nice way of doing this could be to prove vertically opposite angles are equal and move on to prove angles in a triangle using parallel lines.

More able students

More able students start the first lesson by recapping angles on a straight line.  Moving on to the proof of vertically opposite angles being equal and why angles about a point have a sum of 360°.  We then move on to proving angles in parallel lines and angles in a triangle.  Every angle question would involve at least two angle properties and emphasise the need to explain which angle properties apply.

Knowing where the angle properties originate from and being able to prove them is key for this ability.  Introducing proof at this point in their mathematics education ensures they are much more likely to prove formulae like the Cosine Rule or Quadratic formula at GCSE.

Geometrical Reasoning in Key Stage 3

Sum of the angles, x + y + 180 – x – y = 180.

For further examples of proof with geometry check out my YouTube videos.

Proof Cyclic Quadrilaterals Mathematics Revision

Alternate Segment Theorem 

Proof Angle at the Centre & Circumference

Angles in Polygons

Students are guided through how to construct a series of regular polygons to explore their interior and exterior angles.

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