(Updated May 2017 with Transum resources in the Further Resources section, March 2017 with Edexcel’s New Content Resources and further TES resources and November 2016 – Diagnostic Questions added to Further Resources /Questions section.)
Looking at the new content for UK GCSE Mathematics a completely new entry on the specification is “find approximate solutions to equations numerically using iteration”.
For some more information on this AQA have some very useful resources, including their Bridging the Gap resources which look very useful for students who have studied the 2007 Key Stage 3 Programme of Study and will be studying a new Mathematics GCSE specification. The resources include examples on iterative methods for solving equations numerically.
Students can be reminded to use the ANS key on their calculators; it seems to me that this will be a good opportunity to show students how useful Excel can be for such techniques and will enable teachers to quickly generate results with different starting values.
From an AQA specimen paper, we see how this may be examined:
In case you are wondering about that flowchart, Newton-Raphson is the method being used.
a little algebra and we see what AQA are up to in their flowchart.
I do love my graphics tablet!
(See Writing Maths Online)
Looking at Edexcel’s Content Support you will find very helpful resources for teaching new content. For new content, information, examples and exercises (with answers) are given. This includes Iteration. Scroll down the list for a zipped file.
Further Resources / Questions
In this post I have included fully worked examples and related graphs; this includes an example (note the pdf file) where an equation has been solved using trial and error and then rather more efficiently using an iterative technique.
On the brillant Diagnostic Questions site you will find excellent coverage of topics new to the GCSE specification. You can also search all questions for a topic of your choice, for example a search on iteration will lead you to the whole collection of Trial and Improvement and Iterative Methods questions. (You will need to be logged in to the site to follow the link. Create an account if you have not already done so as this site with thousands of high quality diagnostic questions and additional analytical features is free. If you scroll down the page you’ll see that Diagnostic Questions are giving “you, the teacher in the classroom, a promise that Diagnostic Questions will always remain free.”
From piximaths – see Iterations.
Also on TES Resources dannytheref has a very clear PowerPoint and accompanying worksheet on how to answer iteration questions.
This search on TES returns some further free GCSE Iteration resources including a very clear introduction, Iteration and Square Roots from Owen134866 and a Tarsia puzzle from Jill Hillitt which provides several examples.
On Just Maths we have so many wonderful resources including 9-1 questions by topic, looking at the Higher Tier questions, note that under Algebra (scroll down) Iteration – questions and solutions are available. Questions from Edexcel, AQA and OCR are included.
The Maths Genie website includes numerous questions and solutions; on this page, scroll down to the last section for Higher and note that under Algebra we have for Solving Equations Using Iteration, revision examples, examination questions and solutions. Looking at the examination questions, the first three questions use Trial and improvement, questions 4 to 7 are on iterative techniques.
For more questions Transum Mathematics offers the exercises shown below on Iteration:
Level 1 – Generating sequences using next term rule.
Level 2 – Rearranging equations.
Level 3 – Using flowcharts to define iterations.
Level 4 – Solving Equations to 1dp.
Level 1 will be useful for KS3, students can find the notation very unfamiliar for sequences.
Why don’t they put a question which does not converge and why not?
How would you work out xn+1 = square root of 29-(4x-squared) to find x3 to 2 decimal places when x0 is 1.
Do you have the original source of this question, so we can see the complete question? At the moment it looks like x2 would be undefined.
Any thoughts of what grade this kind of question would be?
I think we have to be very wary of the grading of individual topics / questions Fiona; if I had to give a rough guide I’d say 7+.
See this informative blog post on Better Maths ‘How will the new grading work?’
I don’t understand why you used (2x^3 + 8)/(3x^2 + 5) for x^3 + 5x – 8 = 0. Could you please enlighten me?
Hi Iva – a good question! That’s AQA not me but it’s the Newton-Raphsson method – have updated the post. Thank you for the question.
Thank you. I clicked on the link. I get that the denominator is the first derivative of x^3 + 5x – 8, but I still don’t understand where the numerator, 2x^3 + 8, comes from.
Sorry Iva – I think I uploaded the wrong WolframAlpha image originally – try again – see post now!