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 sub-headings:

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 non-metric 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.