Provide each pair with four Sphinx shapes and set the challenge from Task 166 of the Curriculum Corporation Mathematics Task Centre Project, which is to put 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.
As each group makes the new Sphinx, ask them to record the solution on triangle dot paper. Some students will attempt a 1:1 correspondence between the shape and the drawing. Others will choose the shortest Sphinx side to be one 'diagonal' unit on the paper and produce a scaled drawing.
It doesn't really matter which method the students choose as long as:
Begin by reminding students that while they were hunting for the larger Sphinx they created many other shapes.
At the end of the five minutes supply each pair with about 10cm of tape so they can secure their shape. Some will have corners touching corners at some points (rather than sides touching sides) which can be a little tricky.
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.
Ask students to leave their own shape displayed on the table and walk around to look at all the other shapes.
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 for 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.)
Ask students to write in their notebook in their own words under the following sub-headings:
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:
Refer to the display of shapes and ask:
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.
This is an opportunity to have a brief 'sideways' discussion about area only being measured in squares by agreement. The only necessary criteria for measuring area are:
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.
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.
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:
An alternative way to make use of the data is to search for:
Whichever path is chosen there is again the opportunity to summarise through the process of Working Mathematically. 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.
So 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 in note form.