Patterns & Powers with Sphinx
The objective of this investigation is to explore powers of 2 and develop algebraic generalisation 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
2. Motivation
Gather the students at a table in the centre
of the room. Ask the students closest to the table to kneel. Place
Sphinx 1 on the table.
This is a Sphinx
shape.
(Pause)
Here are three more.
All you have to do is put the 4
Sphinxes together to make a new larger Sphinx shape.
This will occupy the students for some
minutes, but you will have to 'arrange' that active hands allow
others to take turns after an appropriate time. It is especially true
that boys may have to be prevented from dominating.
3. Exploration
When the students have been captured by the
task, which may not have been solved as yet, offer each pair a set of
cut out or plastic pieces. Very soon one pair will solve the problem and the
flood gates will open. As each group completes their Sphinx, give
them a strip of tape about 10cm long and ask them to join the pieces
 but not too securely because they will have to take the tape off
again later.
4. Recording
Ask the students to record their solution on
the triangle dot paper. Half a sheet per student is
sufficient.

Thorne Grammar, UK ... Boys tend to press to the front.
OK boys. Time's up. Girl's turn.

5. Extension
Gather the students at the centre of the
room again. This time use one of the students' Sphinxes made from
four cut out pieces. Compliment the students on solving the puzzle
and emphasise that we have used four Sphinxes to make a
Sphinx.
Pause and then repeat:
Mmm, four Sphinxes to make a
Sphinx ... So ...
It won't be long before someone suggests
that four of these ones could make the next size Sphinx. Send the
pairs off to find partners with whom they can join to make this new
size. Provide more tape as necessary.
Sequence of sphinxes 
Sizes 1, 2 & 4 
6. Sorting & Classifying
Bring the students back to the centre of the
room and display the sequence of Sphinxes made so far. There is
usually at least one group's shapes which couldn't be joined with
others to make the extension, so these can be used to display the
first part of the sequence.
Introduce a naming process for the
Sphinxes:
Let's look at what we have made
so far.
This is a Size 1
Sphinx.
This is a Size 2 and this new one
is Size 4.
Can anyone explain why I am
giving the different sizes these names?

The shapes are being numbered by the number
of Sphinxes that make the base of the Sphinx. It may take a while for
the students to realise this, so you may have to hint by pointing
along the base as you repeat the size names.
During this discussion someone may ask
What about Size 3? This is a great suggestion, so accept it as such and
indicate that ...we will return to that question later.
7. Predicting
Review the display of shapes again:
This is Size 2 Sphinx, but it is
four times bigger than the original because it takes four Size 1
Sphinxes to make it.
This is Size 4 Sphinx but it is
16 times bigger than Size 1 because it takes 16 Size 1s to make
it.
Now I am going to ask you to
predict two things.
What Size name will the next
Sphinx have and how many times bigger than the original will it
be?
The students are not likely to have too much
trouble with developing that hypothesis, nor should there be any
difficulty with checking it by combining the Size 4 Sphinxes which
are already in the room. (There is no need to tape the pieces of Size
8 together. Just making it is sufficient.)

Size 8 Sphinx

8. Recording
Ask students to put the heading
Sphinx and the date in their journal. Under the
subheading The Problem ask to record the original problem (creating
Size 2) in their own words and pictures. They can draw the original
Sphinx shape on triangle dot paper and cut and paste it.
Under the subheading What We Found Out,
students paste the drawing they have made of the Size 2 Sphinx,
followed by their explanation of how the investigation proceeded from
there.
Under the subheading Summary draw up this
table and ask students to copy it and fill in the blanks.
Sphinx Size

Number of Times Bigger
Than Size 1

1

1

2

4

4

16

8

64

...

...

Experience shows that many children
continue the table to at least the next level.
9. 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

10. Display
Some teachers use the growing sequence of
Sizes 1, 2, 4, 8 ... Sphinxes to make a display such as this one made
by Lee's class in Little Rock, Arkansas, USA.
10. Posing A New Problem
Refer to the earlier question about the
existence of a Size 3 Sphinx. Ask the students if they could predict
how many Size 1 shapes would be needed to make it if it did exist.
Set as a homework challenge the task of making a Size 3 Sphinx. You
will need to supply cut outs and triangle paper. Indicate to the
students that this is a more complex task than the Size 2, but if
they can do it you would also be interested in solutions for other
sizes. This is the type of homework challenge which extends over time.

For More Experienced
Mathematicians
Is it possible to prove that any size Sphinx
can be made?
The data so far suggests it is possible to make any
size, but that is not the same as proving that any
size can be made.
See Sphinx Album for many other sizes and an attempt at proof by one Year 9 student from Thorne Grammar.
