Perimeter & Area with Sphinx
The objective of this investigation is to develop concepts and skills in perimeter and area in the bigger context of learning to work like a mathematician.


1. Materials
 At least 64 cut out Sphinx shapes or a class set of plastic shapes available from Mathematics Centre. If you need to make your own, use the drawing above. Save, import into a word processor, arrange multiple copies on the page and print. Printing onto thin card and laminating before cutting is an excellent option
 One roll of 'magic' Sellotape  the type used on photocopies
 Triangle dot paper
 A few strips of card about 10cm x 60cm and a thick marker
2. Introduction
Provide each pair with four Sphinx shapes
and set the challenge of putting the four Sphinx
shapes together to form a new Sphinx shape. It will take a few minutes for students to
achieve this and in the process they will make many other shapes with
the four pieces. These become the basis of the main part of the
lesson.
Of course, when one pair of students solves this initial puzzle they are likely to 'tell'. This can be turned to advantage. Give permission to anyone who wants to tell but the conditions are:
 They must be invited to demonstrate by someone else.
 They must keep their hands behind their back (or sit on their hands) and only give instructions.
3. Recording
As each group makes the new Sphinx, ask them
to record the solution on triangle dot paper. Check that student drawings are proportional to their new Sphinx and celebrate various ways of completing the recording.
One way to do this is to ask each pair to convince their neighbours that the drawing is an accurate representation of the model.
4. Exploration
Begin by reminding students that while they
were hunting for the larger Sphinx they created many other
shapes.
When you were searching for the
answer to the Sphinx puzzle you made lots of other shapes. I am going to give you five
minutes to make as many shapes as you can with the four small Sphinx
shapes. At the end of the time I will ask
you to choose your favourite one and make it for others to
see.
Accept all shapes, but indicate that you will particularly look for shapes where whole or fractional part edges of the unit Sphinxes match. For example, the short side of one can be put against the longest side of another, but only in three places, because the longest side is three times the shorter side.
At the end of the five minutes supply each
pair with about 10cm of tape so they can secure their favourite shape. Also ask them to record their chosen shape
on triangle paper. This time it would be good if everyone used the
scale of one short Sphinx length represented by one 'diagonal' unit
on the triangle paper.
5. Sorting & Classifying
Ask students to leave their own shape
displayed on the table and walk around to look at all the other
shapes.
When you have finished having a
look, I will be asking you which shapes belong together in families
and why.
The language the students use to answer the
question will suggest many directions to follow. This may include
opportunity to lead into symmetry, rotational symmetry, 'holes',
polygons, concave, convex, ...
As the discussion develops, bring the actual
objects to the front and sort them into categories. Make labels for
the categories using the card and marker. Use the students' shapes
as a display and add the agreed category names. (If the shapes were
originally produced on a range of coloured card, this display looks
very effective on a black background.)
6. Recording
Ask students to record in their notebook in
their own words under the following subheadings:
 The Problem
 What We Did
 What We Found Out
7. Have We Worked
Mathematically?
Remind students of (or introduce them for
the first time to) the components of the process of Working
Mathematically. Many teachers have these displayed on a chart in
their room. Ask students to identify which steps in the lesson
corresponded to which components of the process.
The components of the Working Mathematically
process are:
 working in context
 collecting and organising data
 seeking and seeing patterns
 discussing and recording
 using standard mathematical tools and
skills
 making and testing hypotheses
 publishing for others to learn
 posing a new problem
8. Posing New Problems
Refer to the display of shapes and
ask:
I've noticed that something is
the same in all these shapes and something is different. What do
think I have seen?
This will possibly lead to several
suggestions, but steer the conversation towards the areas of the
shapes being identical (four sphinx shapes) and the perimeters being
different.
9. Measuring Area?
This is an opportunity to have a brief
'sideways' discussion about area being measured in squares by
agreement. The only necessary criteria for measuring area are:
 one shape only is used
 the chosen shape must tessellate
In the imaginary land of Egyptonia, area
could certainly be measured in Sphinxes, and should this land ever
have to communicate with our world, the complications would be no
more than occur when the nonmetric United States has to communicate
with the metric remainder of the Earth.
10. Measuring Perimeter
Following the area measurement path is
secondary in this lesson to following the perimeter measurement path.
Suggest to the students that it would be possible to assign a
perimeter to each displayed shape even without a ruler, and ask them
how. Answers might include using pieces of string, but experience
with the drawing to scale suggested above will lead to making use of
the shortest side of a sphinx shape as a unit of measure. The other
sides of the sphinx are then either 1, 2 or 3 Sphinx units.
Ask students to work out the perimeter of
their chosen shape from their drawing. This should be checked by
another pair of students and then added to the display.
11. The Statistics Path
Teachers interested in introducing or
revising the concepts of representing and interpreting data now have
a window through which to enter. The collection of perimeter measures
is a data distribution, which can be extended by giving the students
more Sphinx shapes with which to explore. This data can be used to
discuss:
 appropriate ways of representing
the data
 making use of a spreadsheet to record
and display the data
 the statistics appropriate to describe
the data  range, mode and median seem appropriate, but the validity of
mean would need to be questioned in this context.
12. The Problem Solving Path
Another way to make use of the data
is to search for:
 shape(s) with the longest/shortest
perimeters
 symmetric or rotationally symmetric
shapes with the longest/shortest perimeters
 perimeters, if any, which can't be
made between the longest and shortest
13. More Working
Mathematically
In whatever direction you explore this investigation there is a further
opportunity to summarise using the process of Working
Mathematically. That is, to highlight that the most important part of the lesson is the reasoning, justification and communication skills a mathematician brings to an investigation.
After all, who cares about the perimeters of these
shapes ... or how to divide fractions for that matter? Could it be that
for students, tackling mathematical content makes more sense if it is
couched in the human endeavour of Working Mathematically.
Refer to the Working Mathematically chart
and ask students to tell you those parts of the lesson which
correspond to components of the process.
Finally the students can be asked to record
this discussion in their books. There may also be the interest among some students (all students?) to celebrate what has been achieved by publishing for others. This could be as a poster, PowerPoint, written report, photo slide show, comic strip... Activating this aspect of the Working Mathematically process at a time of keen interest in a problem has been shown to enhance student literacy.
