:Teaching Notes:

Limitations of the applets in the coordinate geometry section

In all the interactive applets in the coordinate geometry section, the coordinates are displayed rounded to the nearest whole number. This may cause calculations done on rotated shapes to be slightly off.

The coordinates are rounded off for clarity and ease of teaching. If they had been displayed with say 6 decimal places, they would have made the display very cluttered, with coordinates running on top of each other. It makes it easier to teach because the integer coordinates are easy to do arithmetic with easily in class, and equal coordinates are easy to spot.

The problem

The problem comes when you try to calculate lengths and slopes. As an example, the image below is from the applet at Rectangles on the coordinate plane.

So for example if you try to calculate the lengths of the two diagonals, they will come out having slightly different lengths. This is because the coordinates shown are not actually the exactly correct values, due to being rounded off. Sometimes a value gets rounded up, sometimes down, producing this anomaly.

As another example, if you drag the rectangle to resize it, you may find that the calculated slopes of opposite sides are slightly different. This is for the same reason as above. (In this particular applet it turns out, by chance, that in the default initial size the slopes do come out matching.)

Orthogonal shapes are OK

If the shapes are not rotated, that is, all the lines are parallel to the axes, the coordinates will all produce consistent results. This is because they are forced to be that way by the internal applet algorithms.

Can it be fixed?

I don't think so. In a rotated polygon there is no set of displayable integer coordinates that will produce consistent lengths and slopes for all sizes of the figure. At least none I can see. The idea of displaying coordinates to many places would have fixed it, but that was soundly voted down by teachers.

A teachable moment?

Perhaps one good thing is to use this to introduce the effects of rounding errors. Have the class calculate both diagonals and discuss the discrepancy. Many students will end up programming computers so it is not an irrelevant discussion. Maybe someone will come up with a solution.

If so, I'm right here: John Page.