GCSE New Content – Iterative Methods for Numerical Solution of Equations

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. 

Iterative techniques 1

AQA Bridging the Gap resources

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.
Excel - iterative solution of equations

From an AQA specimen paper, we see how this may be examined:

AQA Specimen Paper 2 Higher

AQA Specimen Paper 2 Higher

AQA specimen exam question a

AQA specimen exam question b

Mobile Puzzles – Algebra

Mobile Puzzles

Mobile Puzzles

The Transition to Algebra (TTA) project, an initiative of the Learning and Teaching Division at Education Development Center, Inc. (EDC) includes a wonderful collection of Mobile Puzzles. Visit solveme.edc.org to play SolveMe Mobiles (also available for the iPad.)

Looking at the menu, you will see categories with different levels of difficulty available from very simple puzzles to rather more complex puzzles which promote good mathematical thinking.

Menu

Students must determine the weight of each object shown which makes a good introduction to the skills required to solve equations, linear and simultaneous.

Looking at some of the Master level puzzles, you will find rather more complex puzzles:
Master Level

Note the menu in the corner of each puzzle page:
Play Menu

Selecting ‘Information’ provides extensive help; note that various tools are available so you can annotate puzzles and / or add symbols and equations.

create equation
Note that you can then drag a heart to subtract a heart from both sides:
puzzle demo
Note that under settings you can choose to show numbers in the mobile as in the illustration. If the solution is correct, the mobile will balance.

On the other hand….
puzzle demo wrong answer

 

GCSE New Content


Reading various documents on the new GCSE specifications I thought it would be useful to create a simple summary of new content. (This will be updated with additional information and resources in the coming weeks). Note that I have very recently updated (May 2015) the post on Venn Diagrams which includes several resources for teaching this new topic; currently I have added a link to Brilliant for combinatorics problems, Trigonometry demonstrations,  Nrich and TES resources for Frequency Trees and Desmos graph pages for inequalities including quadratic, circles and tangents to curves. See each section below.

Number
Ratio proportion and rates of change
Algebra
Geometry and Measures
Probability

Number
Structure and Calculation
5. apply systematic listing strategies including use of the product rule for counting

Resources: Combinatorics on Brilliant (create a free account to view)

Measures and Accuracy
15. round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding

Note that this is introduced in the new Key Stage 3 programme of study. (From La Salle education a very useful document which is annotated clearly with changes at Key Stage 3 can be found here. Note that you can also find information on changes to the Primary curriculum in this collection of resources).
KS3 Number

use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b

Ratio, proportion and rates of change
1. change freely between related standard units (e.g. time, length, area, volume/capacity, mass) and compound units (e.g. speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

9. define percentage as ‘number of parts per hundred’; interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics

11. use compound units such as speed, rates of pay, unit pricing, density and pressure

15. interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts

16. set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes.

Return to menu

Algebra
Notation, vocabulary and manipulation
7. where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function’; interpret the succession of two functions as a ‘composite function’.

Graphs
11. identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square

12. recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function, with x ≠ 0, exponential functions and the trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size

15. calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts

16. recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.

Resources: Trigonomery demonstrations
Use Desmos to explore tangents to a curve
Use Desmos to explore circles and tangents
BBC Bitesize – Finding the equation of a tangent to a circle 

Solving equations and inequalities
17. solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph

18. solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph

20. find approximate solutions to equations numerically using iteration

22. solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph

Resources: Desmos for inequalities – including quadratic

Sequences
24. recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a rational number > 0 or a surd) and other sequences
Return to menu

Geometry and measures
Properties and Constructions
2. use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line

8. describe the changes and invariance achieved by combinations of rotations, reflections and translations

Mensuration and calculation
21 know the exact values of sinθ and cosθ for θ = 00, 300, 450, 600 and 900; know the exact value of tanθ for θ = 00, 300, 450 and 600
Return to menu

Probability
1. record describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees

6. enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams

9. calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.

Frequency Tree

Frequency Tree TES resource – Alison Gilroy

Resources: Venn Diagrams
From Nrich Prize Giving and note the Frequency Tree representation
From TES: Frequency Trees

Aural Test – Statistics

StatisticsMy post on using mental tests for revision seems to have interested many readers so I thought I would follow this up. Having looked back in time to GCSE many years ago when an aural test was actually part of the exam (10%) I shall in future refer to these as Aural Tests. It was these tests that started me using the idea of an aural test on anything any time! They can be short and make ideal starters or plenaries or in the case of revision aural tests can last a lesson with lots of associated questions and discussion.

Looking through some old resources I came across a cassette (!) recording of myself reading the questions for a GCSE aural test I recorded for a correspondence college. I intend to transcribe that and will write a post on these old style tests in the near future.

Having successfully given my Year 13 students two aural tests on the Pure Mathematics C3 and C4 modules (after the first they requested the second) my wonderful colleague who teaches the group with me joined in the venture and gave them a third aural test on their  Statistics module. We and our students feel we have done some really useful revision in their last lessons for all three modules on their Advanced Level course.

So this week I have my last lessons with Year 11 (UK age 15-16) who are preparing for their GCSE. I want to look at their Statistics unit with them and have decided that an aural test should work well. Looking at various papers I have extracted some diagrams and asked questions around those. These are topics that I feel my particular class needs; I want to review various statistical diagrams. In case this is of interest I have made all the resources available here. Students need the answer sheet only. The teacher reads the questions and they have to listen very carefully and answer the questions. They will need to write answers in their exercise books or on paper as well as using the answer sheet. With these longer revision aural tests it is sometimes appropriate to give feedback after each question as opposed to waiting till the end to mark all questions. I use both techniques.

Creating the solutions reminded me once again of how useful colour can be to make solutions clear.

I would be interested to hear from teachers who try aural tests with their students; I find them useful for all ages.

Last Revision Lessons

Study leave is approaching fast for our examination classes so it’s time to think about those last lessons. I will be using a few mental tests with all my examination students as I find these work very well indeed. Last week with Year 13 we had a C3 (an A Level module – OCR MEI) mental test; at the end of the lesson they said that was really useful and requested a C4 mental test for this week. I think my favourite kind of lesson feedback is when students make requests like this!

These tests are simply short questions that test recall of the basic skills needed for the module; so for example some standard derivatives and integrals, graph sketches, changing the subject of the formula for expressions involving exponential functions and so on. Note that another possibility is to ask students to write down the expressions / calculations needed for a question (they can always come back and complete it later).

Basically, sit down with the syllabus in front of you and cover as much as possible. Although informal this is making the students individually recall material they will need – see Highlighting is a Waste of Time.

CIMT Teacher Resources

CIMT Teacher Resources

For some inspiration for mental tests have a look at the Centre for Innovation in Mathematics Teaching resources for GCSE (and for younger students note that mental tests are supplied for all the Key Stage 3 (UK age 11-14) units). In fact the GCSE resource shown above would also apply to AS level.

For the Teacher Resources scroll down to the end of the GCSE Resources page and you will find resources for each unit.

CIMT Mental Test

CIMT Mental Test – Using Graphs

For A Level students, questions such as Mohammed Ladak’s Essential Skills pack for Core AS or Corbett Maths A Level 5-a-day could be used / adapted. Questions do need to be short, recall type questions where just a short time is needed for any working out. I do find that because I use this idea regularly, I can just ask questions by looking at the specification and using my experience of what I know students forget!

Looking at the CIMT resources I noticed some more valuable revision resources; note the GCSE Revision pack; this has quick checks at Foundation, Intermediate and Higher Level and example papers with answers and mark schemes.

CIMT GCSE Quick Checks

CIMT GCSE Quick Checks

CIMT Higher Paper example

CIMT Higher Paper example

For revision tests by topic, each unit of the GCSE course has a revision test with answers. Whilst the vast majority of the material on the CIMT site is freely accessible, a few documents such as these revision tests are password protected. The password can be obtained if you send a request using your educational institution email address; CIMT also give the password to home educators.

I have added the CIMT GCSE Revision pack to the examination resources page which is part of the Revision Activities series of pages.

I Hate Top 10 Lists!

This morning I enjoyed reading Ross Morrison McGill’s thoughtful “Top 10 UK Education Blogs Or Not?”. It was the or not part of his title that caught my eye. Ross is quite rightly talking about the validity of the ranking but Top Ten lists have interested me for some time.

Top 10I have often been wary of so called Top (insert number) lists particularly when said list is simply a blog post and there may be little validity behind the choices. You will see that I always preface my own favourite lists with a reminder that the choices are personal to me – I am not claiming any authority.

Others have written well on this subject. So I present in no particular order and with no authority whatsoever some of my favourite articles on hating top 10 lists!

Returning to the list of Top 10 UK Education Blogs mentioned at the beginning of this post, regular readers will be aware I look regularly at Ross’s site – I’m a fan of the 5 Minute Series, particularly when it comes to Lesson Planning. On that list is a real personal favourite – David Didau’s The Learning Spy, a favourite because David constantly challenges my thinking. I am currently feeling guilty at the number of times I must have written about students being engaged when it comes to lesson observation! I try to be so aware of Robert Coe’s Poor Proxies for Learning. What I really mean by engaged is that students are indeed getting on with the task they have been set. I’ll only know more about their learning by
Poor Proxies for Learning

looking at other evidence. I recently noticed some of my own Year 11 students in a general revision session for the Year group successfully answering a GCSE question on Direct Proportion. I taught that topic to them some months ago and recall being pleased with the lesson at the time but it is so much more convincing seeing them answering problems some time later! (We have also picked it up again using resources such as Mr Corbett’s 5-a-day). I’ll be even more convinced of course when I look at the exam board question by question analysis later this year.

I read many Maths blogs of course, but think it is important to look at some of the more general education blogs too.

Happy Reading!

Final Revision

Revision
That quote from Robert Collier seems so appropriate when it comes to revision. This academic year I have used the day in, day out approach even more with my students, frequently reviewing earlier work even for short sessions. I am convinced this is important in our teaching and help makes things stick for our students.

Once again we are in the final run up to examinations, so I checked the various revision resources I have highlighted on this blog earlier this year and created a series of revision pages which I hope makes resources easier to find. I have recently updated these again. Before mentioning the resources though we should think about how best to use them.

The first page ‘Highlighting is a waste of time’ links to what I believe is a very important report on how students learn effectively; having used testing – even very short ‘self checks’ as they have come to be known in my classes I am convinced like the authors that this is very effective and we will be using testing in our revision classes, often short with immediate feedback so students can see if they can recall and apply information. Earlier this academic year when I asked my Year 9 students about good Maths teachers, one said:

A teacher who provides the student with the opportunity to see what they need to revise. Regular tests and quizzes do this.

So before we worry about amazing revision resources we must consider how we will use them so our students learn effectively. According to the report the two learning strategies with the highest utility are distributed study sessions (last minute cramming is not effective) and practice testing.

Interestingly, interleaved practice: though rated as just moderate utility gets a special mention for students’ learning and retention of mathematical skills. William Emeny has written on this see this post and a follow up on Great Maths Teaching Ideas.

So bearing these learning strategies in mind, many of the resources found on the series of revision pages could be used as mini tests with immediate feedback or several topics mixed up within a lesson and perhaps the trickiest topics revisited several times over the last weeks, even if briefly.

The revision activities can be found on the series of revision pages:

There have been recent updates, in particular to the examination questions page. I will certainly be using all the resources I have mentioned on that page. Resources in the collection allow for a mix it up approach but also provide examination questions by topic. A huge thank you to the teachers who so willingly share their resources – you are helping students everywhere. Correct attribution has been given wherever possible with the resources.

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Wishing teachers and students everywhere a successful final revision period.

Easter 2015

It’s Easter time again, so time for some Mathematical Easter treats, including an updated version of the best Easter eggs – those from WolframAlpha!

Also a post for students – a puzzle which is just an excuse to solve some simultaneous equations (and how to do it on Excel with the neat MINVERSE function!)

Google - easter

Google graph – click on the image.

A reminder that you can just type a function into Google and its graph will be returned!

Darth Vader curve

Darth Vader on WolframAlpha – click on the image

WolframAlpha of course can show you some graphs of Easter eggs!
I have noticed whilst using WolframAlpha random suggestions of queries popping up that somebody out there thought I might enjoy (very worrying how right they are!); this popped up – typing for example Darth Vader curve into WolframAlpha gives you just that! And my favourite Dilbert and associates are all there too!

Looking for Easter ideas and resources I came across these Easter games on the excellent mathsticks.com site which has an extensive collection of resources for younger students ( a site I have recommended for younger students).

On the subject of Easter eggs I must return to this definition. An updated edition of…
WolframAlpha – a little fun! 

Happy Easter to educators and their students everywhere!