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“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?
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
Cambridge Mathematics
1 Hills Road, Cambridge
United Kingdom
CB1 2EU
cambridgemaths.org [email protected] @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
