Download Exploring Geometry with a 9

Exploring Geometry
This activity is taken from the ATM publication Exploring Geometry with a 9-pin circular
geoboard written by Geoff Faux and available to order at www.atm.org.uk/shop/DNL098
Make this arrangement of bands
Are the bands parallel?
Are the bands parallel?
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April 2016
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Exploring Geometry
Angle Task 3
Learners need:
▪ a geoboard (at least one between two)
▪ elastic bands
▪ a supply of copies of the six circles resource page 29
▪ Some learners will find a use for their angle size data from Angle Task 2B.
Content
Process
Parallel line properties
Alternate angles
Vertically opposite angles
Corresponding angles
Noticing what is the same and what is different
Reasoning from known facts (e.g. alternate
angles equal means lines are parallel)
Some teachers may prefer to work at this task before going on to Angle Task 4. Others might
wish to go straight from Angle Task 2 to Angle Task 5, or even Angle Task 6.
Answering the question Are the bands parallel? is not easy. Different explanations will be
forthcoming, but adding bands to construct four isosceles triangles from a common vertex
often helps learners to see the band joining the common vertex to the centre as a line of
symmetry and therefore see why the lines are parallel.
A constructed set of isosceles triangles
Drawing in the line of symmetry is really
helpful
A trapezium with alternate angles equal (fact
from working on Angle Task 2B’)
The idea of converse lies behind the diagram on the right. In Euclidean geometry when two
lines are parallel the alternate angles are equal. A question that is not often asked is Does the
converse also apply?. When two alternate angles are equal does it follow that the two lines are
parallel? Is this idea of using converse something that all, some, a few, of year 7 learners will be
happy with? They will meet converse later in KS3 and more centrally in KS4, so how explicit
does a teacher make this idea here?
Here are some other examples of applying converse, only one of which works:
▪ The diagonals of a kite cross at right angles. If the diagonals of a quadrilateral cross at right
angles must it be a kite?
▪ In a right angled triangle the sum of the squares on the two sides beside the right angle
add up to give the square on the hypotenuse. If in a triangle the squares on the two sides
beside an angle add up to the square on the side opposite that angle, must the angle be a
right angle?
As a route to Angle Task 4 it might be worth spending some time writing in all the angles on
the isosceles triangle figure. The angles at the base of the various isosceles triangles will be
recorded from work in Angle Task 2B and so alternate angles on the various parallel lines can
be written in.
Extension How many different chord lengths are there on a 9 pin board? 8 pin? 7 pin? 12 pin?
April 2016
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