So my blog frequency has become significantly lower recently - believe it or not I have been even busier than normal writing and sourcing resources for our new Year 7 mixed ability course, putting together topic tests for Year 7 and Year 11 (thanks AQA for all of your work putting your own topic tests together - I have stolen most of them!) and then writing the homework booklets for all three of my Year 11 schemes for term 1. All in all today is actually the first day since we broke up (bear in mind that Leicestershire broke up on 15/07/16) that I haven't been doing school work of some description - as a reward for finishing the homework booklets a day early I gave myself the weekend off!
One of the things that I have had to sort out as part of writing the tests and homework booklets is finding sources of questions on iteration and numerical methods for solving equations, so I thought I would share some of the better ones here, and also offer some tips on designing your own.
1) Check out A-Level worksheets - I dug through some of my old Core 3 resources (unfortunately I haven't taught A-Level for the last two years since moving to my new school) and found an ample supply of iterative formulae that were used. Some of them weren't suitable (too many natural logarithms and exponential functions) but many were with just some small adaptations. In particular a lot of A-Level questions ask pupils to show there is a root in a given interval using a change of sign approach and also ask pupils to justify why a given formula will converge to a solution. As far as I have seen the GCSE will not ask pupils to use a change of sign to show there is a root in a given interval,although to be fair it wouldn't be a bad thing to do with pupils as a way of tying roots of equations, graphs and iteration together. In addition it will definitely not require pupils to justify why a given iterative formula will converge, as this requires knowledge of calculus - although again it might be nice for the best mathematicians to look at this as a way of linking rates of change to iterative formulae. For some examples questions made from A-Level worksheets check out my Year 11 Higher or Higher+ term 1 and 2 homework booklets - there are a few pages on Iterative methods with a few exam style questions all taken from A-Level worksheets or similar.
2) Exam board website - we are using the AQA exam board and they have a multitude of resources available for use with iteration. If you don't know AQA's site http://allaboutmaths.aqa.org.uk/ it is well worth getting yourself signed up for it. Browse to the New GCSE (8300) and select the Numerical methods section under Higher GCSE Algebra resources and you will find worksheets with some decent enough questions, as well as their topic test with some more. The one I really like though is their 'bridging' material, which can again be found under the New GCSE (8300) page. They have a lovely document in there called Pocket 4, which is all about iterative formulae. Although billed as a KS3 bridging material I would definitely save some of the later activities and use them during the actual GCSE teaching.
3) Linked Pair Pilot - Although trial and improvement is not mentioned specifically in the new GCSE specifications, it is still being used under the guise of a numerical method. The Linked Pair Pilot papers, in particular the Applications 2 paper, has some nice examples of trial and improvement used to solve practical problems in geometry and other areas, which is nicely in keeping with the aims of the new GCSE. Often they have the tables printed on a separate page as well, which means you can feel free to not use them for the more confident mathematicians, just giving them the page with the question setup on instead.
4) Pixi Maths - If you haven't seen Pixi Maths TES shop yet, then I would definitely head over there (https://www.tes.com/teaching-resources/shop/pixi_17#). Pixi has created some lovely resources for a variety of topics, including iteration - https://www.tes.com/teaching-resource/iterations-11064012 although don't be fooled by the line that says trial and improvement has gone. Still there is a nice PowerPoint and activities which includes a jigsaw for the rearranging part of iteration and then a worksheet with some iterations to perform.
5) Design your own - It isn't actually that tricky to design iteration questions, although there are a couple of things to beware of to ensure the question will work. Start with a polynomial set equal to 0; cubics are good as they can't be solved using other GCSE techniques (except if it has an obvious factorisation) and are guaranteed to have at least one root. From here you can do one of two things:
(a) Use the Newton-Raphson formula:
One of the things that I have had to sort out as part of writing the tests and homework booklets is finding sources of questions on iteration and numerical methods for solving equations, so I thought I would share some of the better ones here, and also offer some tips on designing your own.
1) Check out A-Level worksheets - I dug through some of my old Core 3 resources (unfortunately I haven't taught A-Level for the last two years since moving to my new school) and found an ample supply of iterative formulae that were used. Some of them weren't suitable (too many natural logarithms and exponential functions) but many were with just some small adaptations. In particular a lot of A-Level questions ask pupils to show there is a root in a given interval using a change of sign approach and also ask pupils to justify why a given formula will converge to a solution. As far as I have seen the GCSE will not ask pupils to use a change of sign to show there is a root in a given interval,although to be fair it wouldn't be a bad thing to do with pupils as a way of tying roots of equations, graphs and iteration together. In addition it will definitely not require pupils to justify why a given iterative formula will converge, as this requires knowledge of calculus - although again it might be nice for the best mathematicians to look at this as a way of linking rates of change to iterative formulae. For some examples questions made from A-Level worksheets check out my Year 11 Higher or Higher+ term 1 and 2 homework booklets - there are a few pages on Iterative methods with a few exam style questions all taken from A-Level worksheets or similar.
2) Exam board website - we are using the AQA exam board and they have a multitude of resources available for use with iteration. If you don't know AQA's site http://allaboutmaths.aqa.org.uk/ it is well worth getting yourself signed up for it. Browse to the New GCSE (8300) and select the Numerical methods section under Higher GCSE Algebra resources and you will find worksheets with some decent enough questions, as well as their topic test with some more. The one I really like though is their 'bridging' material, which can again be found under the New GCSE (8300) page. They have a lovely document in there called Pocket 4, which is all about iterative formulae. Although billed as a KS3 bridging material I would definitely save some of the later activities and use them during the actual GCSE teaching.
3) Linked Pair Pilot - Although trial and improvement is not mentioned specifically in the new GCSE specifications, it is still being used under the guise of a numerical method. The Linked Pair Pilot papers, in particular the Applications 2 paper, has some nice examples of trial and improvement used to solve practical problems in geometry and other areas, which is nicely in keeping with the aims of the new GCSE. Often they have the tables printed on a separate page as well, which means you can feel free to not use them for the more confident mathematicians, just giving them the page with the question setup on instead.
4) Pixi Maths - If you haven't seen Pixi Maths TES shop yet, then I would definitely head over there (https://www.tes.com/teaching-resources/shop/pixi_17#). Pixi has created some lovely resources for a variety of topics, including iteration - https://www.tes.com/teaching-resource/iterations-11064012 although don't be fooled by the line that says trial and improvement has gone. Still there is a nice PowerPoint and activities which includes a jigsaw for the rearranging part of iteration and then a worksheet with some iterations to perform.
5) Design your own - It isn't actually that tricky to design iteration questions, although there are a couple of things to beware of to ensure the question will work. Start with a polynomial set equal to 0; cubics are good as they can't be solved using other GCSE techniques (except if it has an obvious factorisation) and are guaranteed to have at least one root. From here you can do one of two things:
(a) Use the Newton-Raphson formula:
The examples of exam questions I have seen using this formula have had the subtraction simplified to give a single fraction as the iterative formula, however I cannot see any reason why pupils couldn't be given the formula with the basic substitution already done and told to do a 'show that', i.e.
(b) Rearrange - the classic method for generating iterative formula is to rearrange the equation
f(x) = 0 into the form x = g(x). This is being used a lot in the new GCSE practice and sample materials which include asking pupils to show how a given rearrangement can be arrived at:
If you use this approach to design your own question then a word of caution - not all possible rearrangements will find all of the roots. The best things do here is to check the graph of the rearranged function for the gradient in the locale of the root. The rule goes that if the gradient of the rearranged function around the root you are looking for is in the range (-1,1) then the formula will converge to the root there - if not then it wont. For example for the problem above the graphs of the original function and the rearranged function look like this:
where the red graph is the original cubic and the blue graph is the square root function. You can see that there are actually three roots to the cubic, corresponding to the three points that the root function intercepts the line y = x. However the given rearrangement wont find the root that is slightly bigger than 2, as the gradient of the root curve is greater than 1 around that point. The rearrangement will quite happily find the other roots in the intervals (0,1) and (-1,0) as the gradients are close to 0 around these points. It is definitely worth just checking this if you are going to design your own rearrangement questions as you wouldn't want to give your pupils rearrangement that doesn't work!
