31
.
3 Circle theorems
© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original
SPECIFICATION REFERE
NCES
GCSE 2010
GM j Understand and construct geometrical proofs using circle theorems
FS Process skills
Select the mathematical information to use
Use appropriate mathematical procedures
Choose appropriate language…to communicate result
s and solutions
FS Performance
Level 1 Select mathematics in an organised way to find solutions
RESOURCES
Resources
Pairs of compasses, rulers
Links
http://lgfl.skoool.co.uk/keystage4.aspx?id=317
Concepts and skills
Find missing angles on diagrams.
Prove
and use the facts that
the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point
on the circumference
the angle in a semicircle is a right angle
the perpendicular from the centre of a circle to a chord bisects the c
hord.
Functional skills
L1 Construct geometric diagrams, models and shapes.
Prior key knowledge, skills and concepts
Students should already know how to state formally that
the angles in a triangle add up to 180°
the angles on a straight line add up to 18
0°
the angles opposite the equal sides of an isosceles triangle are equal
the exterior angle of a triangle is equal to the sum of the two interior opposite angles.
31
.
3 Circle theorems
© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original
Starter
Put on the board some diagrams, with unknown angle values that can be foun
d using what students
have already learned about isosceles triangles in a circle and tangents to a circle, e.g.
Work out the missing angles and give a formal reason for your answers.
(Answers:
a
= 35º,
b
= 27º)
Main teaching and learning
Tell
students they are going to learn some circle theorems, explaining that they need to be able to
formally prove each theorem.
Go through the three theorems in detail and then
go through some exercises.
Common misconceptions
Tell students that when they draw
circles using their pair of compasses, it is best not to hold on to
the pencil end as this tends to make the end move, resulting in an inaccurately drawn circle.
Plenary
Give students copies of some problems involving circle theorems and supply a partial
explanation.
Ask them to suggest better wording.
31
.
4 More circle theorems
© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original
SPECIFICATION
GCSE 2010
GM j Understand and construct geometrical proofs using circle theorems
FS Process skills
Select the mathematical information to use
Use appropriate mathematical procedures
Choose app
ropriate language … to communicate results and solutions
FS Performance
Level 1 Select mathematics in an organised way to find solutions
Key words
cyclic quadrilateral, proof, prove
Resources
Pairs of compasses, rulers
Concepts and skills
Find missing a
ngles on diagrams.
Prove and use the facts that
angles in the same segment are equal
opposite angles of a cyclic quadrilateral sum to 180º
alternate segment theorem.
Functional skills
L1 Construct geometric diagrams, models and shapes.
Prior key knowledge,
skills and concepts
Students should already know how to state formally that
the angles in a triangle add up to 180°
the angles on a straight line add up to 180°
the angles opposite the equal sides of an isosceles triangle are equal
the exterior angle of a
triangle is equal to the sum of the two interior opposite angles.
31
.
4 More circle theorems
© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original
Starter
Put on the board some diagrams, with unknown angle values that can be found using the circle
theorems already covered.
Work out the missing angles and give a formal reason
for your answers.
e.g.
(
a
= 53º, because the
(
b
= 34º, because the angle
(
c
= 57º, because the angle in
perpendicular from the
at the centre of the circle
a semicircle is a right angle.)
centre of a circle to a
is twice the angle a
t the
chord bisects the chord.)
circumference.)
Main teaching and learning
Tell students they are going to learn some more circle theorems, explaining that they need to be
able to formally prove each theorem.
Go throu
gh the three theorems in detail
and
then
go through some exercises.
Plenary
Give students copies of some problems involving circle theorems and supply a partial explanation.
Ask them to suggest better wording.
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