© ATM 2011 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.
PLAYING AROUND
WITH 30° 60° 90°
TRIANGLES
Mike Ollerton invites Year 5 pupils to play with ideas.
In recent years I have become interested in the idea
of causing learners to ‘play around’ with ideas and
resources. For example playing around with
consecutive numbers using addition or multiplication or playing around with sequences: “If the first
three values of some sequences are 2, 3, 5... how
could different sequences develop?” This short
piece of writing is about playing around with some
30° 60° 90° triangles. This emerged from an invitation to work in a primary school to provide some
input into ideas for classrooms about ‘shape and
space.
Part of my brief was to spend the afternoon
with a Y5 class where we did a good deal of shapemaking through paper folding and considering
properties and names of the shapes as well as
creating tiling designs from A7 size paper.
The other part of my brief was to work with
staff at an after school meeting to continue looking
further at geometric ideas which teachers could
adapt for use with their pupils. One idea I offered
was based upon making shapes using 30° 60° 90°
The first fold,
EF, bisects AD
and BC
The second fold
goes through
corner B such
that corner C
folds onto line
EF
triangles. We began with ‘the’ standard way of
creating such a triangle through paper folding.
Now, I am fairly confident that if you are reading
this article you will already know the two folds
required to gain a 30° 60° 90° triangle... however,
just in case, the two folds described below can be
carried out on any rectangular size piece of paper.
Proving BFH and BCG are 30° 60° 90° triangles is probably beyond the scope of most primary
school age pupils, however the business of playing
around with several such triangles may not.
Once we have some 30° 60° 90° triangles we
can pose accessible problems such as:
• How many different shapes can be formed by
fitting any two of these triangle together by
same edge length?
• What about three triangles?
• What are the properties, names and symmetries
of these shapes?
Some more challenging questions could be:
• If we label the three lengths a, b and c, what
are the perimeters of the shapes?
• What shapes can be made if we abandon the
rule of joining together by same edge length...
and what conditions would we need to apply to
avoid infinity?
It was this final problem which occupied a few
minutes of my time last week; sitting at my desk I
happened to have some 30° 60° 90° triangles
cluttering up my workspace.
Possibly because I was in need of a few creative
moments, I started to play to around with them.
After a few minutes I came up with the following
designs (overpage) and I wondered if anyone might
be interested in seeing them and possibly coming
up with a lesson, or two, based upon using 30° 60°
90° triangles.
MATHEMATICS TEACHING 223 / JULY 2011
Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY
5
© ATM 2011 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.
Interestingly the triangles I used on this
occasion started off as rectangles with sides in the
ratio of 1:冑3, which I had only previously used for
another paper-folding type purpose. I had generated these rectangles by dividing pieces of A4 paper
into ten equal pieces as in the diagram below:
Mathematically speaking I should refer to these
rectangles as being in the approximate ratio of 1:冑3.
This is because dividing up an A4 sheet creates its
own problem which is to see what the percentage
error is of the smaller rectangles being in the ratio
of 1:冑3 given the A4 sheet is in the ratio of 1:冑2
If you are prepared to accept the percentage
difference is small enough not to worry about, in
terms of creating a piece of practical equipment
which is accurate enough to serve the purpose,
then we can get twenty 30° 60° 90° triangles out of
a sheet of A4 card without any waste.
The key issue, however, is less about playing
around the 30° 60° 90° triangles and more about
playing around with mathematics per se. Perhaps
the phrase ‘playing around with’ is synonymous with
exploring ideas; about seeing what happens when,
for example, learners play around with:
• square and triangular numbers;
• adding consecutive numbers;
a, b and c in the general
• changing the variables
quadratic y ax2 bx c;
• ATM regular polygon mats to create tessellations;
• Cuisenaire rods to make fractions;
• pegs and pegboards to make symmetrical
shapes by varying the line(s) of symmetry;
• Diene’s base 10 equipment to carry out
computations;
• patterns on a 100 square.
Something I am interested in, and which
frequently happens when I am fortunate enough to
work alongside teachers in their classrooms, are the
unexpected events which occur. These arise through
observations made by class teachers relating to
outcomes of unusual and surprising responses by
certain pupils. This, in part, is because the teachers
have opportunities to observe more and ‘teach’
less; it is also because the pupils are playing around,
exploring ideas and trying things out... and possibly
because there is never an objective written on the
board.
Mike Ollerton is a freelance consultant.
www. MT goes hexagonal
Always, sometimes, never true – a collection
of discussion questions taken from Thinking for
Ourselves for use in the primary classroom
Margaret Jones & Jill Mansergh
Money provides the means – a challenge to
solve a problem involving coins
Paul Stephenson
The Hexagon – a real, giant one, and some
curious transformations
Douglas Butler
6
Pivotal Tables – activities for pivotal tables
Sydney Tyrell
The craft of teaching – an activity unpicked to
show how teaching takes place
Doug Williams
Early Years – some suggestions for improving
mathematical skills and knowledge for primary
teachers
Anne Harvey
MATHEMATICS TEACHING 223 / JULY 2011
Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY
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