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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