Geometrical Gemsmei.org.uk/files/conference17/Session-A8.pdf · geometry is actually such a...

Geometrical Gems

Rachael Horsman rachael.horsman@cambridgemaths.org

“Geometry is grasping space . . . that space in

which the child lives, breathes and moves. The

space that the child must learn to know, explore,

conquer, in order to live, breathe and move better

in it”

Freudenthal 1973: 403

Spatial intuition or spatial perception is an

enormously powerful tool and that is why

geometry is actually such a powerful part of

mathematics – not only for things that are

obviously geometrical, but even for things that

are not. We try to put them into geometrical form

because that enables us to use our intuition. Our

intuition is our most powerful tool . . ."

Atiyah, 2001:658

Parallelograms

The framework

Cavalieri’s Principle of Shearing

Cavalieri’s Principle of Shearing

Cavalieri’s Principle of Shearing

Cavalieri’s Principle of Shearing

Cavalieri’s Principle of Shearing

Cavalieri’s Principle of Shearing

Cavalieri’s Principle of Shearing

Design a

parallelepiped

with a volume of 240cm3

Triangles

Area game

Mid-points

The mid-point and quadrilaterals

The Hoop Game • Identifying circles

• Drawing circles

• Finding the centre of a circle

• Reconstructing circle pictures

Circle Patterns

Visualisation

Say what you see!

Half the group need to turn around

Say what you see!

Describe what you see to your

partner. They have to reproduce this

diagram exactly – but you can’t see

what they are drawing!

Use words like…

square isosceles bisect

right-angle vertical

horizontal mid-point

Turns and Angles

• Unlimited rotations

• Limited rotations

• I-Hinges

• V-Hinges

• X-Hinges

• Bends

What turning/angle situations can you identify in the room?

Multi-purpose model

Multi-purpose model

32cm

Multi-purpose model

2a

3

4

b

Perceptions

Perceptions

Am I a nerd or a geek?

Am I a nerd or a geek?

Espressos

We have three brand new

research documents written and

live on the website, with plans to

add a new one every month

“Thank you so much - a very

helpful summary of different

positions, terminology and

research” @emmasarkar

“Love this

new series of

'Espressos'

from

@Cambridge

Maths -

bitesize

maths

education

research”

@mathsjem

Blogs

We continue to publish one

high-quality blog a week on

subjects ranging from PISA

tests to imaginary geometry

“This is amazing, and the title

image is totally messing with my

brain!”

@alqualin

“Read this all you teachers

and heads”

@alqualin

@CornwallMaths

Rachael Horsman

rachael.horsman@cambridgemaths.org

Cambridge Mathematics

1 Hills Road, Cambridge

United Kingdom

CB1 2EU

cambridgemaths.org info@cambridgemaths.org @CambridgeMaths LinkedIn

Okazaki, 1999

1 right angle 2 right angles

3 right angles Un-equal diagonals

4 right angles Equal diagonals

Opposite angles are equal Diagonals bisect each other

At least one pair of parallel

sides Diagonals meet at 90º

2 pairs of parallel sides Two pairs of adjacent equal

sides

4 equal sides Two pairs of opposite equal

sides

No lines of symmetry Rotational symmetry order 1

2 lines of symmetry Rotational symmetry order 2

4 lines of symmetry Rotational symmetry order 4

Scalene Isosceles

Equilateral One line of symmetry

One obtuse angle Two lines of symmetry

Two obtuse angles Three lines of symmetry

One acute angle One right angle

Two acute angles Three acute angles