Sunday, 13 May 2018

ALG 14

This week's set of tasks was inspired by a task (below) from Doug French, that featured in his series The creative use of odd moments that appeared in Mathematics in School. (The jottings occurred during an interview with four high attaining Year 8 students, where we extended the task to find the equations of lines that formed other squares.)
 In our set of tasks we focus on the equations of the lines on which the sides of a square lie, and we look at what happens to the equations when the square changes in simple ways - in particular, its position, size or orientation.
It is of course possible to find each line's equation by using the formal procedure of finding the values of m and c in y = mx + c. However, the way the tasks have been designed should encourage students to think about the relationship between the coordinates x and y of points on a given line. In turn, the focus on the coordinates, especially of points on the axes, should enhance students' understanding of the formal procedure and of the common forms in which equations are expressed (ie y = mx + c, but also x + y = a, when it applies, and perhaps more generally ax + by = c).
MONDAY: We start with a nice little diamond that is fairly near, but not too near, the axes...
The lines have gradient 1 or -1, so we can find the equations by seeing where the various lines cut the y-axis (giving us the value of c in y = mx + c). However, we can also (or instead) work more 'locally', by looking for the relationship connecting the x and y coordinates - eg for points A (8, 8) and D (8–2, 8+2), we can see that x + y = 16. We might also notice that the line through D and C is 4 units 'above' the line through A and B (whose equation is y = x), ie the y coordinate has been increased by +4, for a given x coordinate, so y = x + 4 instead of y = x.
TUESDAY: We see what happens to the equations when we translate the square one unit up or across.
Here we provide both a check on Monday's task, and a chance to consolidate the ideas that arose, by varying the task in ways that lead to equations that are closely related to the previous ones.
For each line, we can again look for the (modified) relations between x and y, and/or visualise how the lines have moved and consider, in particular, where they cut the axes.
Notes: Regarding the first approach, we can do this in an empirical way, by examining actual pairs of coordinates, or we can adopt a more general argument. For example, consider what happens to the equation of the line through AD, ie x + y = 16, when the square is moved one unit up. We can examine new coordinate pairs like (8, 9) and (6, 11), which suggests their sum has increased from 16 to 17; or we can argue more generally that for any given value of x, the value of y has increased by 1, so the sum x + y has increased by 1, from 16 to 17.
Visualising the lines can be very helpful, but sometimes care needs to be taken in drawing conclusions from the way a line has changed. For example, when the line with equation x + y = 16 is moved up one unit, it intersects each axis one unit further up or across, which can easily lead to the conclusion that the equation changes from x + y = 16 to x + y = 18, rather than to x + y = 17.
WEDNESDAY: Here we change the size of our little diamond. This doesn't change the task substantially, except that we express the new square's coordinates and the resulting equations in a (more) general form. [Note that when it comes to the equations, our new unknown, e, is a parameter rather than a variable.] We can of course check our new equations by letting e take the value 2.
The last part of the task throws up an interesting, if rather arcane, issue. Is the square resulting from the value e = -4 on and above the line y = x, or on and below? If we derive the new square from the equations that arose in the first part of the task, then the answer turns out to be 'below'. Should we be bound by this, ie by the dictates of algebra?! Is this position for the square somehow more 'consistent' than placing it on and above the line y = x ?
THURSDAY: We consider a small square with a different orientation from the 'diamonds' we've considered so far - so no longer a little gem?
We're no longer dealing merely with slopes of 1 or -1. So a bit more of a challenge, whether we focus on pairs of coordinates and the relationship between them, or whether we consider where the lines cut the axes.
FRIDAY: We translate the square to a position far from the origin, so we are close to asking for general rules for our equations (which of course are already general rules...).

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