Q I am in my first year of teaching and have to teach about locus for the first time to my low-ability Year 9 group. I know that by its nature this is a practical activity and this is very much the way I approach the work with my group. I am lucky enough to have only 12 students in the group so it is quite easy to set up and run practical sessions. I wondered if you had any suggestions as to how I might approach this work with this group.
A End of term in sight, summer weather, and this sounds like an outdoor activity to me. Before venturing forth, the group needs to understand what is meant by locus, this being the path made by a point when following a set of rules or conditions. So, for locus, think about a path to be followed.
Taking the maths outside means that the lengths involved can be greater and it involves students in a measuring activity as well - they are not restricted to using just paper. Chalk and a hard surface or paint and brush on grass (with the permission of the caretaker or groundsman) could be used to map out the solutions.
Have a set of equipment available for each group: a ball of string, scissors, a tape measure, a large circular lid with a splodge of paint near to the outer rim (a fixed point on the object), and chalk or paint. Pupils have to think how they can use themselves to make the "fixed points" or straight lines.
Split the 12 into three groups - this means that you can move quickly from group to group to assist where necessary. Hints are always helpful. Make sure that there is a balance of ability within each group; in particular, ensure that one of the group is able to read the instructions to the other members: make this their responsibility.
To do the activity well the group needs co-operation - teamwork. Initially I had thought of making the groups rotate around each activity, but now think it would be better to give all five activities to each group. This means that they can then check if they have them correctly created without the need for pencil and paper for recording. This makes it more like solving puzzles and, I suspect, more fun. The lengths involved would be different for each group so that they don't just look over and replicate the answer.
The five activities would be to:
* find the locus of points that are a set distance from a fixed point;
* find the locus of points that are a set distance from a straight line;
* find the locus of points that are equidistant from two fixed points;
* find the locus of points that are equidistant from two straight lines that intersect,
* find the path made by a fixed point on a circle that is rotating along the ground.
Activity 1: Draw on the ground the locus of a point that is always 1.5m away from a fixed point. (Hint: Choose someone to be the fixed point and use your string.)
Activity 2: Show the locus made by a point that is always 1.75m away from a straight iron bar that is 4m long.
Activity 3: Create two fixed points 3m apart. What is the path made by a point that is always the same distant from each point?
Activity 4: Two straight lines, each 2m in length, meet at one end and have their other ends 1.6m apart. What is the locus of a point that is always the same distance from each of the lines?
Activity 5: Put a splodge of paint on the edge of the lid. Make a straight line with the string. Roll the lid along the line and draw the locus of the paint splodge. There is an animation of paint on a bicycle wheel at www.mathagonyaunt.co.uk
Following this lesson it is important to look at some practical applications. This could be things such as setting up a sound system for a gig - how many speakers might you need to cover a particular area? How much of a room can you reach from a single socket with a vacuum cleaner? Watering a garden using a sprinkler, where do you position it to get the best coverage? Lighting a street, how far apart should the lamps be to get the most cost-effective and ample light coverage? There are two great interactive resources for loci - one with a real-life application, and the other an open-ended tool enabling the creation and displaying of loci, atwww.ngfl-cymru.org.ukvtcngflmathsechalklociWeb A module about the teaching of loci can be found at www.standards.dfes.gov.ukkeystage3downloadsma_study015604_mod6.pdf
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
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