## An Interesting Conics Question

My AS further maths class and I have finished the scheme for learning for the year, leaving oodles of time for review, recap and plenty of practice. Currently we are revisiting topics that they either didn’t score too well on the last mock in or that they have requested we look at again due to lacking confidence.

Today we were looking at conic sections, by request of the students, and this past paper question caught my eye:

We had recapped the topic quickly together then the students attempted so past paper questions. This one was one they rook to well, all competing the first 2 sections quickly and without trouble. Likewise, part c was mostly uneventful – bar one or two silly substitution errors and someone missing a difference of two squares factorisation.

Then they got to part d.

This caused them a little bit of worry so we worked together on it as a class, after we’d made sure everyone had a, b and c right.

Here’s what the board looked like when we’d done:

I asked the class what we should do to start, one suggested drawing a diagram *all this nagging about always drawing a diagram, especially if your stuck is paying off! *We drew it, but it didn’t help much:

Then one said, “if they’re parallel then one gradient is -1 over the other” – I refrained from scolding and calmly said “indeed, those gradients will be *negative reciprocals *of each other”. I think asked what the gradients were and quickly had “y1 – y2 over x1 minus x2” thrown at me.

So we worked out the gradient of the line join in n to the origin:

Then the gradient of PQ:

One then suggested we put the equal to each other, but he was corrected by another student before I could react. So we set one equal to the negative reciprocal of the other and solved:

A lovely question, with a lovely neat answer and a load of fun algebra on the way. I enjoyed watching the students tackle it and was glad I didn’t need to put too much input in myself.

We then discussed the answer and the class expressed surprise that 1 was the final answer and that a complicated algebraic journey could end so simply. It was a nice discussion and a nice way to start the day, and the week.

*This post was cross-posted to One Good Thing and Betterqs*

## A conic extention

This week has been a strange one, I’ve been trying to shake an illness but it keeps on getting worse, and my lessons have been disrupted a little bit by mock examinations. This, however, has given me a chance to work.woth small groups in some classes and to see how some classes are getting on with their courses.

In one of further maths lessons a student who takes maths but not further maths asked if he could.sit in the classroom and revise for his maths, asking if he needed help, I said it was fine and started the lesson,which was a review of the conic sections topic and an assessment sheet to identify any areas of weakness that may be apparent. I gave the none further maths student a spare copy of the assessment sheet, told him it was basically C1 skills but applied in a much more algebra heavy context, and asked him to have a crack.

Not only did he have a good crack at it, he answered it near perfectly. I was extremely pleased with his resilience in working through a question that has way more algebra than anything he’s looked at before and was glad he could make the links to the C1. His thoughts on the question were interesting, and I think they allowed the rest of the class to see more clearly how the conic section of FP1 fits with, and build on, the coordinate geometry sections in C1 and 2.

I think that in future I will use these parabola and hyperbola questions with all my high attaining AS maths students.

*This was cross-posted to the “One Good Thing” blog here.*

## What a fantastic puzzle!

When I logged on to twitter this evening I saw this tweet from Colin Beveridge (@icecolbeveridge):

Being the sort of person that seems a maths puzzle and finds it impossible not to have a crack at it I had a go.

*xy=3 x+y=2 what is 1/x + 1/y? *

My thoughts process was fairly straight forward:

*xy=3 so it follows that x=3/y and hence 1/x = y/3. Likewise xy=3 so y=3/x and hence 1/y = x/3. Thus, 1/x + 1/y = x/3 + y/3 = (x+y)/3 = 2/3 {as we know x+y=2}.*

It seemed a straight forward puzzle, I noticed some tweets including complex numbers and thought they were odd, “Professor Yaffle” (@adamcreen) then tweeted a much simpler solution:

*1/x + 1/y = (x+y)/xy =2/3*

Which I thought was lovely. Then Colin asked “how would your students tackle it?” I thought “Grrr, it’s the holidays so I can’t try it on them for a fortnight!” Then I about it a bit and decided that on the whole they would probable try to solve the simultaneous equations using substitution.

*x+y =2 so y=2-x *

*xy=3 so x(2-x)=3*

*2x – x^2 = 3*

*x^2 -2x + 3 = 0*

Hang on, there are no real solutions to that quadratic! My non-further maths students would stop there stumped, my Further Students would work through using complex numbers. I thought I check another substitution:

*xy = 3 so y=3/x*

*x+y=2*

*x+3/x=2*

*x^2+3=2x*

*x^2-2x+3=0*

Yep, that’s the same quadratic so I haven’t made any silly errors. I figured that if you followed this through with complex numbers you must end up with the same answer, but wanted to check:

*x=(2+(4-12)^1/2)/2*

*Or*

*x=(2-(4-12)^1/2)/2*

*So*

*x=(2+(-8)^1/2)/2*

*Or*

*x=(2-(-8)^1/2)/2*

*Root (-8) = i2root2 *

*x=1+i(2)^1/2 or 1-i(2)^1/2*

*Meaning y is the complex conjugate of x **in each case (by substitution back into original equations).*

*So 1/x + 1/y = 1/(1+i(2)^1/2) + 1/(1-i(2)^1/2) = (1+i(2)^1/2 + 1-i(2)^1/2**)/(( 1+i(2)^1*/2)(1-i(2)^1/2)

*Which, of course, simplifies to 2/3*.

What a delightful puzzle! There are no real values for x and y, but the answer is a lovely, real, rational number! I thoroughly enjoyed exploring it, and I hope my Further maths pupils will enjoy it too. I’m not sure whether to give it to my other A Level pupils or not, I will decide over the holidays. I am definitely going to give it to some of my year 11s though. We’ve just hammered algebraic fractions, and this is going to be an extension task!