Back in May 2017 maths teachers around the country eagerly awaited the first exam for the new GCSE Mathematics syllabus. Proving geometrical relationships using algebra featured at grade 9. In Paper 1 of Edexcel’s test paper the last question of the higher tier looked like this.
Edexcel wrote about student’s performance on this question in their Examiners Report.
“This question was set as a differentiator for those aiming for grade 9, so it was not unexpected to find that many candidates did not answer the question at all.”
It’s because of this I decided to blog about how I teach proving geometrical relationships to students aiming between grades 7 to 9 in GCSE maths.
Whenever I set this the consensus in the classroom is I have not given enough information as students try to find the value of individual angles to add them together. However, after a couple of minutes of perseverance most students recognise the triangle at the centre and its vertically opposite angles are the key to this problem.
The main part of the lesson starts with some simple proofs which the majority of the students would have seen before. It is important when working through these questions to emphasise the need for clear annotations on the diagrams that flow with the algebraic notation. In my experience students often lose track when deriving proofs as they involve multiple stages of working.
I work through the questions on the second slide asking the class for prompts as I go. The questions on the third slide are like those on the second so most students can begin working independently. I encourage peer support throughout the remainder of the lesson and for everyone to sketch the diagrams in their books.
The plenary takes about 15 minutes for all students to complete. The most able generally complete it in under 10 minutes. I ask those who finish early to provide peer support for any who are struggling.
The most common approach is, we know that angles BAC and BDC are equal due to angles in the same segment. Angles ABC and DCB are both 90° due to angle at the centre being double the angle at the circumference (or angle at the circumference of a semi-circle). BD is the same as AC as they are both diameters of the same circle. Angle, Angle, Side proves congruency in this case.
I think it is important to encourage students to derive proofs as early as possible and when they first appear with a unit. For instance, in Key Stage 3 students prove each of the interior, alternate and corresponding properties of parallel lines and why the angles within a triangle add up to 180. In Key stage 4 students prove the various formulae associated to non-right angled triangles, each of circle theorems and the quadratic equation.
Proving geometrical relationships with algebra is the second lesson in the Mathematical Proof unit of work. It follows Algebraic Proof where students learn how to prove various numerical properties and precedes Proof with Vectors where students prove whether lines are parallel.
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