Sphinx

Task 166 ... Years 2 - 10

Summary

Use four sphinx shapes to make a sphinx. Easy to state and easy to start, but the solution, even for adults, does not usually come easily. Once found, applying the process of Working Mathematically opens an almost unbelievable wealth of investigation. This task has been chosen as the model which represents the possibilities in each and every task.

This cameo has a From The Classroom section in the form of a video made by the students of Pam McGifford's Year 8 class at Cressy District High School. It is a captivating report of their investigation into many aspects and properties of the Sphinx.

Sphinx also appears on the Picture Puzzles Shape & Measurement A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. It is the basis of two puzzles - one related to building shapes with sphinxes and tessellating, and the other related to measuring area and perimeter using an equilateral triangle as the unit.

Materials

• 8 sphinx shapes or cut your own from this Sphinx Board
(Suggestion: Print on thin card and laminate before cutting)

Content

• spatial perception
• tessellation
• logical strategies
• perimeter & area
• square numbers
• powers of 2
• proportion
• similar shapes
• value relations and fractions
• prime and composite numbers

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

It is best to never reveal the solution to Question 1. Students will find it eventually, but it often takes up to ten minutes. When it is found, mess up the pieces (including turning some over) and encourage one partner to put their hands behind their back (or sit on their hands), then give directions tell the other person how to remake the Size 2 Sphinx. Encourage students to record the language which was more helpful in this oral exercise.

What words are better/worse to use?
What words might a mathematician have used?
The solution is:

and, yes, there is only one solution. Why?

Question 2 does not require using a ruler (although the same relationship would result). It requires choosing a unit of measure and a straightforward one is the shortest side length of a Sphinx shape. Then:
• Perimeter Size 1 = 1 + 1 + 1 + 3 + 2 = 8 units
• Perimeter Size 2 = 2 + 2 + 2 + 6 + 4 = 16 units
So, it is not correct to assume that because the area has increased four times that the perimeter also does.

There are now several ways into the iceberg of this task:

• We have shown that four sphinxes make a sphinx, so ... four of these Size 2 sphinxes would make another new sphinx. This would be Size 4. And then four Size 4 would make... ?
• How many Size 1 sphinxes are needed for each new size?
• What is the perimeter of each new size?
• If we can make Sizes 1, 2, 4 what about Size 3?
• If it exists can we predict how many sphinx shapes would be needed to make it?
• Can we make it??
• Can we make any size sphinx??
• Return to the hunt for the Size 2 sphinx. Many other shapes can be made with four Size 1 sphinxes.
• If we restrict the search to those with whole or fractional sides matching, which shape has the shortest perimeter? Which has the longest?
• Are there any perimeters between the smallest and the largest that can't be made?
(Note: An example of a fractional join of sides means 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.)
• The Size 1 sphinx can be thought of as made from six equilateral triangles.
• If the equilateral triangle is worth 10 cents, what is the value of the Size 1 sphinx? Size 2? Size 4?
• If the Size 1 sphinx is worth 1, what is the value of the equilateral triangle?
• If the Size 1 sphinx is worth 1/2, what is the value of the equilateral triangle?
• A perceptive student once commented, No matter how big the Sphinx gets, the angles all stay the same. (See CubeTube link below.) Sphinxes make similar figures.
• Explore the ratios of their sides.
• A sphinx shape repeats itself using the rule that four sphinxes (arranged in a certain way) make a sphinx. So does a square - four squares make a square.
• Are there any other shapes that repeat themselves with a four rules? with a different rule?

Many of these possibilities are explored in more depth in the Iceberg of the Sphinx and Sphinx Album. But watch out! You are likely to get hooked. Probably better to get an overview of the possibilities of the Sphinx first from this Cube Tube Video made by students from Year 8 at Cressy District High School.

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

The ideas and links above offer many reasons for turning this task into a whole class lesson at a wide range of year levels. All you need is lots of sphinx shapes. You can make your own from a drawing supplied in the Iceberg link above, or you can purchase class sets of plastic sphinxes.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 25, Sphinx and Maths300 Lesson 99 What's It Worth?.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Sphinx task is an integral part of:

• MWA Space & Logic Years 3 & 4
• MWA Pattern & Algebra Years 7 & 8

The Sphinx lesson is an integral part of:

• MWA Space & Logic Years 5 & 6
• MWA Pattern & Algebra Years 7 & 8
In addition this task is included in the Task Centre Kit for Aboriginal Students and the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.