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Some Thoughts on Locus - Its Place in the 2 Unit Syllabus
Reflections 
, November, 1993 -
Journal of the Mathematical Association of NSW
by JENNY TAYLER, Cheltenham Girls' High School
A considerable factor in the so-called 'real world' is the behaviour of certain chunks of space with respect to other chunks of space in the same general location. For example, if a chunk of space shaped like a BMW follows certain rules, in the same vicinity as another chunk of space shaped like a Mercedes following different rules, can each continue unscathed? If not, how serious is the result? For whom? Such considerations are essential in the design of roads, in particular intersections, clover-leaves and complicated freeways.
If, on the stage of the concert theatre in the Opera House, as set of points resembling Joan Sutherland performs according to her rules, accompanied by a (larger?) set of points called Luciano Pavarotti performing according to his - different - rules, can the show go on? Or will we be subjected to a display of artistic temperament as these two sets of points collide? This is the stuff of which choreography is made.
Clearly there are many instances where the relative behaviour of different subsets of space have very significant ramifications. Locus deals with predicting the behaviour of a given point, or set of points, under specified conditions.
Definitions of loci
- The locus of a point is the path traced out by the point when it moves according to a given rule (or rules).
- A locus is the path of a single moving point that obeys certain conditions.
- A locus is a set of points (x,y) which obeys certain conditions.
of these, 1 and 2 are the accepted versions. However, neither of them allows for the fact that a set of moving points can also describe a locus. A point moving such that it is always a fixed distance from a given point describes a circle. Meanwhile a line segment moving so that it is always a fixed distance from a given line describes a cylinder. Both the circle and the cylinder are loci. Any good definition needs to encompass both. I don't think that 3 is good enough; I find it vague in the extreme. Perhaps we need a double-barrelled definition, of the form:
(a) The locus of a single point is the path traced out by the point when it moves according to certain conditions;
(b) the locus of a curve is the solid traced out by the curve when it moves according to certain conditions.
Using this expanded definition, the concept of locus can be seen to include not only the fairly restricted topic entitled Locus in the syllabus, but also much of the significant Functions and Relations topic and all of the Coordinate Geometry topic.
The vast majority of the work on locus in the 2 Unit course deals with the type (a) locus, or the two-dimensional locus. Significant among these are the straight line, parabola, circle and hyperbola, all of which can be treated with an integrated locus approach. Others needing a little more imagination are the exponential, logarithmic and cubic. One which is not mentioned at all, but fits in very nicely with a locus approach and which I generally add (for a touch of verisimilitude?), is the ellipse.
Type (b) loci, the three-dimensional loci, are not covered by the course, nor are they specifically covered in the 3 Unit course. However, a few of them are very interesting, not to mention attractive, and are well worth spending a little time on, even if only to make the room look nice! The best of them are the cardioid, limacon, and nephroid, which are fun to draw and look very effective.
A couple of extra 2D loci, also superfluous to the course but nevertheless quite simple and, importantly, different, are the spirals - the equiangular spiral and the Archimedean spiral. A small problem with the latter is that it requires familiarity with radian measure, but this is by no means insurmountable. The rose spirals are a bit difficult, but could be demonstrated if the interest is there.
Some activities to try in working with loci
The average 2 Unit class contains a range of abilities and aptitudes that never ceases to amaze me. I am always hunting for different and practical ways to work with the course so that I do not 'lose' too many of the class. The locus topic, as identified in the syllabus, is one which has great potential for losing students. It is presented in a predominantly abstract fashion and lacks cohesion. In attempting to make it more approachable, and to involve the students physically as well as mentally, I have tried the following activities with varying levels of success.
1. The outdoor locus exercise
Use the students themselves as points.
Identify a particular student as the 'given point' -
there is room for imagination and humour here - and have other
students adopt physical locations which satisfy the conditions
of whatever locus you have chosen. Use 30m measuring tapes to
check their positions and then let them observe the pattern in
which they have placed themselves. If you have an area which
can be observed from above, this can be particularly effective.
Suitable loci for this exercise are the circle, parabola,
ellipse and hyperbola - the last two are not in the locus topic
but I think they should be covered.
2. The Cartesian version of #1 above
This is best done on a tennis or basketball
court where there are lines on the ground, or where you can
readily draw axes on the surface with chalk. Stand a student on
each integer on the horizontal axis. Then instruct them (for
example) to 'double yourself and add one' or
'square yourself' or 'take your reciprocal' and
then to move, preferably simultaneously for best effect, to the
appropriate ordinate. In the reciprocal example, the student on
zero has great scope for response!
This exercise is very successful with most of the functions whose Cartesian equations the students are supposed to know and whose graphs they are supposed to be able to draw! Acting them out seems to help the less abstract to grasp the functions better, and the student who doesn't really need the physical involvement enjoys it anyway. Pick a good day!
3. The cardboard circle demonstration for the
parabola
This relies on the fact that points on the
circumference of a circle are equidistant from the centre. If
you can place a circle such that one point on its circumference
touches the focus and the circle is tangential to the
directrix, then the centre of the circle satisfies the
definition of a point on a parabola. Using a number of
different sized circles produces a number of these points,
which, when joined, make a very convincing parabola.
4. Pencil and paper exercises
(a) Using the combination polar paper,
students can practise placing points in appropriate places to
satisfy the locus definitions of the parabola, ellipse and
hyperbola. These three are so nicely related that I think there
is no harm in exploring them all, including changing the
relative positions of the focus and directrix for each.
(b) For fun and interest, students can construct their own
'envelope' loci - cardioids, etc. They can also
investigate the spirals if they want to.
5. 'Find the function'
These group exercises in using clues to
identify an unknown function are really great. It is important
to stick to the rules, however, to ensure that it is a truly
cooperative exercise and that each student is involved.
The Cardioid: The locus of all circles, with centre on the circumference of a given circle, which pass through a given point on the circumference.
The Limacon:The locus of all circles, with centre on the the circumference of a given circle, which pass through a given point outside the circumference.
The Nephroid:The locus of all circles, with centre on the circumference of a given circle whose centre is the original, which are tangential to the vertical axis.
The Circle:The locus of a point which moves so that it is always equidistant from a given point.
The Parabola:The locus of a point which moves such that the ratio of its distance to a given point and a given line respectively equals 1.
The Ellipse:The locus of a point which moves such that the ratio of its distance to a given point and a given line respectively is less than 1.
The Hyperbola:The locus of a point which moves such that the ratio of its distance to a given point and a given line respectively is greater than 1.
