The situations we've chosen are fairly straightforward, but the game we're playing is mathematically quite sophisticated. It's similar to starting with a familiar statement like 8 + x = 10, which works in natural numbers, and asking what happens in a case like 8 + x = 5: this can be made to work if we stretch our ideas about number, ie if invent new ones - the integers.
Note: From an RME perspective, we could say that we are engaged in horizontal mathematisation (expressing a 'real' situation mathematically) and then in vertical mathematisation (developing the maths).
MONDAY: Here we play with an area formula, for a shape that can vary in size. We start with a straightforward application of the formula but then consider a case which only works if we allow (or invent) edges with negative lengths.
This is what the shapes in parts i and ii look like (if you allow negative lengths in part ii):
TUESDAY: We look at another familiar area scenario, namely area of a trapezium (and, to keep it simple, a trapezium that is right-angled).
One way to find the area geometrically is to divide the trapezium into two triangles. For part i we can, for example, divide the trapezium into two triangles of area 30 and 15 square units (top-left diagram, below).
The top-right diagram shows what happens to the trapezium as point P moves until it is 5 units to the left of the formerly top-left vertex. The trapezium 'twists' over itself to form two triangles whose areas, we can argue, are 20 and -5 square units.
The two diagrams at the bottom of the slide, below, show an alternative interpretation for part ii. The yellow triangle corresponds to the 30 square units triangle in the top-left diagram. The green triangle corresponds to the 15 square units triangle in the same diagram, except its base has changed from 5 units to -5 units. If we 'cancel' the region where the yellow and green triangles overlap, we are left with the regions with area 20 and -5 square untis shown in the top-right diagram.
WEDNESDAY: Things get really interesting.... What's a 2-and-a-half sided regular polygon?!
You may recognise the form of the instructions if you are familiar with LOGO and Turtle Geometry. If you don't have a turtle to hand I hope you will have enacted the instructions yourself and traced the resulting paths on paper or in your head! This is what they turn out to look like:
THURSDAY: We again make the shift from whole numbers to fractions, this time on a familiar number grid.
Here are positions for the T-shape for parts i and ii.
Note: if we accept the principle behind the part ii answer, of allowing fractions of a square, we can find infinitely many positions for the T-shape, for parts i and ii, by moving the shape vertically (maybe just a tiny bit) as well as horizontally. What we're doing here, in effect, is to change the discrete 2-D grid into a continuous Cartesian plane.
FRIDAY: Here we consider square grids made of matchsticks - or parts of matchsticks.
Note: The slide below shows some interesting attempts to structure the grid by three Year 7 students (from a 'low attaining' set: set 3 of 4). I've written this up in chapter 3 of the Proof Materials Project report, Looking for Structure.
NEXT WEEK: We revisit the phenomenon of Letter as Object, ie we look at contexts where a letter represents a pure number (of objects) or a quantity (the price or length or mass or some other numerical quality of an object), but where there might be a temptation to use the letter as a shorthand for the object itself (as in 'a stands for apple, b stands for banana' - the classic fruit salad algebra). We will see that sometimes one can slide harmlessly between letter as object and letter as quantity, but that sometimes it leads to a completely fake algebra.