Symmetry solves problems - Bob Vertes shows how to get maths lessons moving.
Pupils start on shape and space activities before number, and continue to enjoy the logic and challenge of geometrical two-dimensional puzzles - constructing shapes, both familiar and new, from given pieces.
Even simple jigsaws can make for good mathematical learning, especially if issues such as symmetry are considered, and (if appropriate for the age and ability of the child) examples are used to show that shapes with the same area can have different perimeters, and vice versa.
Take a 6 by 4 block - divided as shown in figure 1 into six smaller blocks each with an area of 6 squares. Notice that the big block is an enlargement scale factor 2 of one of the pieces - the 3 by 2 oblong shaded in the diagram. This is my four-in-one jigsaw.
Not only can you make an enlargement of the small block using the four pieces, but enlargements of the three other shapes are also possible.
Most people will separate the four pieces before trying to reform them into one of the new shapes, but this is not always necessary.
Holding down two of them where they are (the "C" and "Q") but rotating the other two together, you can obtain an enlargement of the "C" shape. Not undoing the whole problem before looking for alternatives is a useful strategy in other areas of maths - as in life. For older pupils a challenge may be to see if they can discover another set of four shapes with the same property.
Another problem-solving practical had its genesis in a primary school where I was working with a group of Year 56 pupils in the dining hall. The tables were half (regular) hexagons, alternatively viewable as three (equilateral) triangles (see figure 2). Initially five people sat round each table; so each person occupied one side length of an equilateral triangle.
The obvious way to place two such tables together - as a hexagon - is not the only one. We made a rule that the tables had to be put together in a way that still gave each individual one of those equilateral triangle side-lengths as they sat around the new arrangements. In other words, the unit lengths of these tables had to match up with unit lengths of other tables.
We investigated the maximum and minimum number of pupils that could sit around an arrangement of tables. Because we wanted to follow through some work the pupils had been doing on symmetry, the arrangements were restricted to shapes that had either line or point (reflection or rotational) symmetry.
Many arrangements are possible with even a few tables. Some of the shapes pupils found were delightful, including ones where some pupils were seated inside a shape as well as outside (figure 2). The children learned about reflections, rotations, enlargements, translations, symmetry and a number of issues from the often-neglected Attainment Target 1 - Using and Applying Mathematics.
Looking at the possible number of pupils who could be seated around perimeters of shapes comprising six equilateral triangles was a fruitful investigative activity - especially if the shapes were sorted according to which ones had only line, only point, or both types of symmetry. Some pupils went ahead and also compiled a list of arrangements with no symmetry, for completeness. This activity was accomplished by using triangular dotted paper to record answers, the discoveries having been supported by cutting up regular hexagons (we used ATM Mats), though if the noise level does not irritate the surrounding classes it can be great to start this using the real tables!
You can adapt the activity if you have "2 by 1" or "domino" tables in your dining hall, with squared or square dotted paper and hexagon shapes.
Bob Vertes is PGCE (secondary)mathematics course leader at St Mary's College, Twickenham.E-mail: firstname.lastname@example.org.ATM Mats are available from the Association of Teachers of Mathematics. Tel: 01332 346599. E-mail: email@example.com