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.
Sphinx - Size 1

Green Line

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.

Sphinx Year 7A
Thorne Grammar, UK ... Boys tend to press to the front.

Sphinx Year 7B
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.

Sphinx Sequence
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.)

Sphinx Size 8
Size 8 Sphinx

8. Recording

Ask students to put the heading Sphinx and the date in their journal. Under the sub-heading 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 sub-heading 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 sub-heading 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
Sphinx - Lee's Class

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.