Week 9

Last Week's Challenge 2 - Trig Colouring

This Week - Indices & Surds

I asked my team for suggestions of things they wished their students were better at when they started A-Level Maths and this was right up there... Please watch the video to refresh your memory on where the laws of indices come from. Then move on to tackle this week's problems. I have also added a couple of videos to remind you about how to manipulate surds.

Sorting and Indexing

A couple more activities to get you thinking about Indices and Surds from those nice people over at Underground Maths.

Index issues

Sort out the Surds

Indices and Surds in this very old A-Level Question

Expand the text for a hint for part (a) but you will either have to work it out or wait until next week for the solution.

Substitute the information into the equations, make 'a' the subject of both, and then when solving simultaneously, compare indices for the same base numbers.

An old A-Level Question requiring manipulation of surds

Expand for a hint.....

If 0<a<b

then

0<a^2<b^2

Note: This is not true for any numbers! a<b does not always imply a^2<b^2. For example, in the case where a=-5 and b=2 then first inequality is true but not the second... luckily everything in this question is positive!

Triangles Again

For each set of points determine what sort of triangle the three coordinates form (equilateral, isosceles, right-angled and scalene).

Hint/Solution

You need to use Pythagoras' theorem to determine some distances.

Solution

This Week's Test and Last Week's Solutions

Last Week's Test Solutions

This Week's Test

Fundamental Theorem of Arithmetic

This is a pretty good explanation of the fundamental theorem of arithmetic and how to prove it by contradiction. This method of proof is tricky to get your head round, but it works something along these lines: If I want to prove root 2 is irrational, I will first assume that it is rational (can be written as a simplified fraction) and then after doing some maths, I get a contradiction (it turns out the fraction wasn't simplified) so I have to reject the assumption that root 2 is rational. Like I said, it can be quite tricky to get your head around this style of proof but it's very useful once you do.

Solutions to this week's problems

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