# More free books and Iterative Techniques again

World Book Day last Thursday inspired me to update the page on free books page; working further on that page has inspired me to create a further page.

Nelson Thornes GCSE Texts

STEM learning has an extensive library of free resources for Mathematics (and also for Computing, Design and Technology and Science) including textbooks; you can search the collection in various ways, a search for textbooks for ages 11-18 returns these results. You will find a real treasure trove in this collection hence this new page. which can be found on the Reading Menu. Do have a look at all the goodies available, although some of the books are old many have chapters on topics such as Venn Diagrams; GCSE books developed for Intermediate students could be rather useful as the new Foundation course looks more like the old Intermediate level!

Some resources will take older readers on a trip down memory lane – anyone for Porter’s Further Elementary Analysis!

Several books are available – answers included. Perhaps these will be handy when we get the new A Level specifications!

Checking some of the Nuffield National Curriculum books proved a distraction as the following activity in one of the texts (Number and Algebra) reminded me that I wanted to revisit iteration for a revision session with my Year 10 GCSE students.

. I thought these questions would provide a way to revise several topics – simultaneous equations, solving equations graphically and iteration. Graphical solution to equations is something that seems to puzzle students and it does not come naturally to them that you can solve an equation by rearrangement.

I decided we will look at question IV, first we can use algebra to form a cubic equation in x, then solve the cubic by trial and improvement – a familiar method that seems more intuitive to students than rearrangement.

Then we can rearrange the equation – plot the graphs and show that we still get the same result. The next step is to impress them with using the rearrangement in the form of an iterative techniques question – it’s a lot quicker than trial and improvement. This is a particularly able set of students so I have gone well beyond what we need here; I checked the rearrangement on Autograph hence some of the additional slides in the file I have included below in case it is useful to anyone.

I thought I would include this AQA practice question for the new GCSE as when we looked at iteration earlier this year the subscripts really bothered students; my colleague said exactly the same of his set.

In case it’s useful this pdf file shows the examples fully worked with the graphs to illustrate: Solution of equations graphical & iteration

pdf file

Returning to a Nuffield text I mentioned earlier, I found this question in one of the exercises (an isolated question, I couldn’t see any more). Now what’s the probability of that?!

## 5 comments on “More free books and Iterative Techniques again”

1. Unless a method is specified, valid solutions will get full credit.

• The reason I asked is that just today I was reading a post from a math enthusiast about rotations about the bimedians of a tetrahedron (bimedian is a line joining mid points of opposite sides) and went through a long rigmarole using analytic geometry. I thought about the situation and saw that for a regular tetrahedron the bimedians had to be concurrent, and pairwise orthogonal, ie a set of axes. Proof – Symmetry – not even one whole line !!

2. Two points:
1: in the x+y=12, x^2=y+10 problem it is much easier to substitute for y in the first equation
y=x^2-10 so x+x^2-10=12, a nicer quadratic !
2: The iterative method you describe is all very well, but a poor rearrangement can lead to a non converging sequence. viewed as two equations one can see that success depends on the slopes of the two graphs at the point of intersection. i wonder what the kids will do if they hit this problem.

• Re iteration – yes indeed and we teach that at A Level – one can look at the gradient at the point of intersection of y=x and relevant function but clearly that is well beyond GCSE. At GCSE students are given the rearrangement to work with and no clues about where they come from – I thought I would tell my students a little more – we may even try a rearrangement that doesn’t work.

Re the quadratic – agreed – in fact I’ll probably do what we often do and look at various possibilities! I have updated the file to include both! We can discuss in class.

• I was idly wondering if someone who used a bit of trig for the first question of all would get zero.