Looking for the most elegant solution!

Original post by hecandothatfromran

Very short solution....
Consider the line parallel to BC through. You have two identical triangles....

huh? You mean similar triangles? The same one I posted?

Reply 21

AspiringGenius

15

Original post by TenOfThem

Perhaps the choices could offer different marks

Maybe a choice of 5 questions in each section 2 of which are worth 10 marks, 1 worth 15, one worth 20, and 1 worth 25

Candidates can select so an A* person may choose 3x25 and 1x20, whereas a C grade candidate might go for 4x10, perhaps on at 15 if they are good in one type of question

The 10 mark questions may have lots of little questions but the 25 mark question would require problem solving skills

How would you feel then

That's better as it differentiates between lower graded candidates without just slapping them all with Us.

Reply 22

english students are generally worst at maths than their international peers because of this idea that you should be able to pass just by memorizing stupid techniques.

Reply 23

Phichi

Original post by moritzplatz

english students are generally worst at maths than their international peers because of this idea that you should be able to pass just by memorizing stupid techniques.

What a stupid and invalid opinion? You're basing this off nothing, and if anything, people from different cultures and ways of teaching, are more likely to just follow the ways taught to them, in an English classroom, seeing as everything is new to them, a different language, way of teaching, examination system. I've seen plenty of international students, doing alot worse than the English students, infact, i've only ever seen a few number of international students out preform a whole class of English students.

(edited 10 years ago)

Reply 24

Original post by Phichi

What a stupid an invalid opinion? You're basing this off nothing, and if anything, people from different cultures and ways of teaching, are more likely to just follow the ways taught to them, in an English classroom.

thanks.

I am talking specifically about maths.

I base my opinion on what some academics told me and on my own experience in a well respected maths course.

Reply 25

Phichi

Original post by moritzplatz

thanks.

I am talking specifically about maths.

I base my opinion on what some academics told me and on my own experience in a well respected maths course.

Re-read my post. On the majority, the international students that come to English schools are from the Eastern parts of the world (School and courses dependent obviously). Usually you find it's harder for them to learn certain materials, due to the fact its different from what they originally know, is also taught in a foreign language, and focus more on perfecting what they are taught, not working outside the box (from what i've seen, and a generalization). Prehaps you could also consider saying, that international's that come for maths courses are more serious about it, then say an English student who is readily given the opportunity, but there is no way you could suggest English students have the idea of passing just by memorizing the course, there is a large population, with all different mindsets.

(edited 10 years ago)

Reply 26

Original post by Phichi

Re-read my post. On the majority, the international students that come to English schools are from the Eastern parts of the world (School and courses dependent obviously). Usually you find it's harder for them to learn certain materials, due to the fact its different from what they originally know, is also taught in a foreign language, and focus more on perfecting what they are taught, not working outside the box (from what i've seen, and a generalization). Prehaps you could also consider saying, that international's that come for maths courses are more serious about it, then say an English student who is readily given the opportunity, but there is no way you could suggest English students have the idea of passing just by memorizing the course, there is a large population, with all different mindsets.

I was talking about university level maths, maybe I wasn't clear.

I am not saying it's about the mindset of english students, it is about the way exams and A-level courses are structured.
you can get an A* in maths without having good problem solving skills.

this is not the case in most other school systems, and this is reflected by the fact that in oxford (and I guess in other top unis as well) english students regularly underperform internationals

Reply 27

Jkn

Original post by m4ths/maths247

Hi
This is a question I gave some pupils/teachers in school today and there were 4 different approaches all with the same answer.
I'm interested to see if there are any other ways the great students of TSR can think of getting the answer. :smile:

(see attached)

I can think of a few ways: finding all the angles before splitting it into two triangles and evaluating, considering different ways to calculate area to gain information, calculating overlapping areas and setting up equations by assigning a letter to each region, using the cross product, using vector methods to find co-ordinates before calculating in the typical way, using the centroid of a triangle, using similar triangles above and below, using similar triangles to the left and right, using co-ordinate geometry to find the co-ordinates to calculate area, making use of integration by setting up a cartesian plane integrating using parametrisation and/or polar co-ordinates, numerical methods combined with the squeeze theorem, de-generalising to any of a number of problems in applied maths to then apply other methods like infinite series and/or sophisticated theorems (e.g. probability, statics (mechanics)), etc.. etc..

Of course, you could combine a number of these to form even more, though there's a point where it becomes 'reverse engineering' (where you will learn nothing new).

Here's one of the simpler ones:

The intersection between AC and DM can be seen by placing a temporary origin at A (AB is positive x direction and AD positive y direction) and hence getting

\displaystyle x=1-2x \Rightarrow x=\frac{1}{3}

. Considering the triangle AOB (

\frac{1}{4}

of the total area, by symmetry), we know that we can obtain the area by subtracting the areas of the two white triangles (of height

\frac{1}{3}

, as found). Hence Area

=\frac{1}{4}-\left( \frac{1}{2} \times \frac{1}{3} \right) = \frac{1}{12} \ \square

.

Here is the best one I can think of:

Height of kite is

\frac{1}{2}

. Splitting the kite down the middle, we know that it is similar to the large triangles on the left and right. As the height is halved, we see that the width of the two surrounding kites are each double the width of the central one and so the width of the kite is

\frac{1}{3}

. Hence Area

=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{3} = \frac{1}{12}

by the simple formula for the area of a triangle

\square

.

I think this is the best method as it really shows where the value of

\frac{1}{12}

comes from in a very raw and obvious way. It can also be seen by symmetry and so is accessible to someone who has never learnt maths. Which were the 4 the pupils/teachers came up with? :smile:

Oh and inb4 "your proofs aren't rigorous" (they are outlines :smile:

)

(edited 10 years ago)

I don't know if this solution has already been purposed but this is what I did.

picture003.jpg184.9KB

Reply 30