## United Kingdom Mathematical Challenge (UKMT) – Multiple Solutions Part 1

The year 9 and year 10 students at school recently took part in the IMC (Intermediate Mathematical Challenge), a one hour test provided by Leeds University. The best thing about this challenge (along with the JMC and the SMC for younger and older students respectively) is that the questions require the students to apply their knowledge and skills in new and interesting situations. Not to mention the fact that the students get a pretty decent looking bronze, silver or gold certificate if they do well!

There are always 25 questions which increase in difficulty. Students generally struggle with the last 10 questions and often gain most marks on the first 15. I quite like to have a go at the last 10 questions in every paper before looking at the solutions because you sometimes find that you’ve solved a problem in a different way than is presented in the solution booklet. Take for example the last question on the IMC in 2011:

The solution given by the IMC concerns the use of similar triangles and area scale factors.

I did it in a slighhtly different way using a co-ordinate grid:

It is easy to see that the diagonal line DE has equation: y = -x + 1                     (1)

Since the point E is half way up BC,  the line AE has equation: y = 0.5x          (2)

To find the point of intersection of DE and AE just put them equal to eachother and solve:

-x + 1 = 0.5x

x =  2/3

Sub this value into equation (2) to get y = 1/3. Hence the point of intersection of DE and AE  is (2/3, 1/3).

Now that we have the point of intersection, it is quite simple to find the area of each section. Omitting units we have:

Area of trinagle A =0.5(2/3 × 1) = 1/3 = 4/12

Area of triangle B = 0.5(1/3 × 1/2) = 1/12

Area of triangle P = 0.5(1/3 × 1) = 1/6 = 2/12

Hence area of section Q =  1 – (A + B + P) = 5/12

Hence the ratio of areas P to Q is 2/12 : 5/12 or 2:5.

The great thing about multiple solutions is that you can refer back to the question when you’re covering similar triangles/area scale factors and also when you’re covering intersection points of curves. Simply making a point that questions can be solved in different ways is not acceptable. We have to show students multiple solutions and more importantly give them the opportunity to come up with different solutions by themselves. It’s only in this way that they will learn that mathematics is a creative subject in which different ‘tools’ can be used to approach different problems. ( Go to Multiple Solutions Part 2: Celebrating Different Methods (Lesson 1))