Mathematics A level 1986

June 86
A trip down memory lane! (Edexcel’s Emporium has some papers from this era but not currently any of the multiple choice papers.)

Writing my earlier post on Multiple Choice Questions reminded me how much I used to like the first paper of the Mathematics A level from the University of London School Examinations Board – thirty multiple choice questions to complete in one hour, 15 minutes.

Multiple Choice Example

For questions 1 to 20, candidates had to select one answer from 5 and for questions 21-30 the instructions were as follows.
June 86 Mathematics 1

I will gradually upload these to create a collection of Multiple Choice questions, these could make good starters.

University of London June 1986 Mathematics 1

The pdf file here has the paper, followed by the exam board answers followed by notes from the 1986 version of me! These days I would illustrate with Desmos and/or WolframAlpha for example as well where appropriate.
Colleen Young answers

Note the comment from Graham Cummings below, there are further papers available in the Edexcel’s Emporium:

The Emporium has some 17 multiple-choice question papers from the period 1988-1992 – by no means a complete set, but they range across the Mathematics, Further Mathematics, Pure Mathematics and Applied Mathematics syllabuses. You can find them in the “Pre-C2000” cabinet within GCE AS/A Level.

 

 

Systematic Listing Strategies

Transum - Combinations resources

Transum – Combinations resources

Number

5. apply systematic listing strategies including use of the product rule for counting

The new UK GCSE specifications now include systematic listing strategies.

Some resources for this topic:

Transum Combination Starters

Transum Combination Starters

Transum Mathematics has many excellent resources including a section on Combinations, where you will find Combinations starters, activities and investigations. The Transum site includes links to some videos includes The Counting Principle which provides a clear introduction. Note Transum’s clear index for teachers and for students, Maths Map with numerous activities to support their learning.

Nrich - Fraction Combinations

Nrich – Fraction Combinations

Nrich has a sophisticated problem, Fraction Combinations which not only asks students to systematically list possibilities but work with fractions too.

AQA have a very clear Teaching Guidance document with many useful examples, I use these a lot in class. If you use or are considering using AQA you can register here (there seems to be a problem with the link on the Teaching Guidance document page).

AQA Teaching Guidance

AQA Teaching Guidance

And finally (for now!) Don Steward has this bracelets or necklaces problem.

Don Steward - combinations

Don Steward – combinations

By Colleen Young Posted in Number

Solving Linear Equations

A collection of resources to use to demonstrate and practise solving equations.

Student Exercises
I find Owen Elton’s worksheet, Balancing Equations on TES Resources very useful when introducing equations, the diagrams emphasise that we must do the same to both sides.
(See Diagrams in Mathematics, for more on the use of diagrams to help understanding).

Balancing Equations

Balancing Equations – Owen Elton

Linear Equations - Don Steward

Linear Equations – Don Steward

As an alternative to the balance approach, consider this doing / undoing approach, described here by Don Steward, this is an approach I use for finding inverse functions. The exercises Don refers to are here.

A2 - Mostly Algebra

A2 – Mostly Algebra

One of the Standards Unit resources, A2 Creating and solving equations (in Mostly Algebra) uses this approach, students create an equation and  then undo it; this is a great exercise for demonstrating notation. Another resource with this approach comes from the Mathematics Assessment Project (the design and development was led by the MARS Shell Center team at the University of Nottingham) Building and Solving Linear Equations lesson

For a superb collection of ideas and student exercises for solving linear equations see all Don Steward’s posts tagged linear equations. Many of these outstanding resources use a very visual approach with very clear diagrams to help students’ understanding.

Diagnostic Questions

Diagnostic Questions

There are numerous questions on linear equations on Diagnostic Questions.


WolframAlpha solve equationA simple way to check a solution to an equation of any type is to simply enter your equation as a WolframAlpha query. Note that WolframAlpha includes a graphical illustration; it is so important for students to understand how equations may be solved graphically; I always illustrate graphical solutions when we are working with Algebra to help students make these links. Desmos of course, is ideal here.
Desmos & solving equations


Centre for Innovation in Mathematics Teaching
CIMT Interactive Resources
CIMT
have tutorials on equations: Linear Equations 1, Linear Equations 2, and Linear Equations with Brackets in their Interactive Resources


The following three resources work well for demonstrating the balance method of solving linear equations.

Duncan Keith’s Linear Equation Calculator is available on STEM Learning.


Choose the type of equation you require then the sequence of operations required to solve the equation.
Select Do it after each operation, for example -32 Do it were the keys selected to start the above problem.

The slideshow below shows how to use the calculator to solve equations where the unknown is on both sides.


Mathisfun

Mathisfun

Mathisfun

Mathisfun has this very clear and easy to use interactive illustrating the solution of linear equations.


Flashy MathsSolving Equations (select the swf file to play online).
This resource works in a similar way to the linear equation calculator.

There are 4 levels to choose from: the level 1 equations are of the type ax = b,
level 2, x±c = d, level 3, ex±f = g and level 4, hx±i = hx±j = k.
Choose the series of operations you require to solve the equation selecting apply at each stage to see the result of your chosen operation.
Choose numbers to add and subtract integers and letters to add and subtract variables.


 

Diagrams in Mathematics

Brilliant - Balances Warmup

Brilliant – Balances Warmup

Seeing this problem on Brilliant recently reminded me how useful diagrams can be in the study of Algebra. I solved the problem using Algebra (with a little colour for clarity!) as follows:

Brilliant Balances solution

select for larger image

Sybilla Beckmann’s paper, Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore is an interesting read on the subject of the use of diagrams.

Sybilla Beckmann paper

Sybilla Beckmann paper

I have recently referred to Colleen King’s wonderful Thinking Blocks on Math Playground. (See these two posts, Ratio and Fractions Resources)

One of the options on Thinking Blocks is the Modeling Tool. I used this to create a very simple diagram to illustrate this problem. This tool can be used easily to create the strip diagrams discussed in Sylvia Beckman’s paper.

Thinking Blocks Modeling Tool - Math Playground

Thinking Blocks Modeling Tool – Math Playground

Using this tool one can easily create models; note the many instructional videos to help you learn to use the tool.

Another option I found useful is use the extensive library of problems; these all come with models already prepared which you can choose to show or not. The slideshow has images to illustrate:

Master Level

Mobile Puzzles

For more Algebra with diagrams try Mobile Puzzles a collection of problems varying in difficulty for simple for young students to rather more complex.

Algebra Tiles Factorisation

NCTM Illuminations – Algebra Tiles

Algebra Tiles are such a good way to demonstrate algebraic manipulation, see the online demonstration on MathsBits.com.

Write x2+6x+11 in the form (x+a)2+b

Complete the square

Complete the square – use algebra tiles

And From Fawn Nguyen comes the brilliant Visual patterns, note the menu; the Gallery includes blog posts from teachers and students who’ve used visual patterns in their classrooms.

Visual Patterns

See also:
Here’s the diagram, what’s the question?

Here's the diagram, what's the question?

Here’s the diagram, what’s the question?

UKMT Mathematics Challenge – Skills Builders

UKMT Skills Builders Topics

UKMT Skills Builders Topics

From UKMT, try their free Skills Builders, on-line problem solving questions for both junior and intermediate level. Note the user guide, which applies to both Junior (age 11-13) and Intermediate (age 13-16); this offers clear step by step instructions with screenshots.

jmc

UKMT Maths Challenge (Junior)

Another possibility, generate a random quiz, using Mathster’s UKMT Mathematics Challenge Online Quiz. Choose Junior, Intermediate or Senior and one of three difficulty levels; you can also choose the number of questions, a time limit and the order the questions are presented in – random or in order of difficulty.

The UK Maths Challenges provide a wonderful library of multiple choice problems; note the these past papers where you will find questions and full solutions. Check also the Individual Challenges pages where you will find details of the challenge and see sample materials. Check the Junior Challenge for example. Note the addition for 2015 of extended solutions.

As well as a full solution which does not use the multiple choice answers further ideas are given for investigation.

UKMT Extended Solutions

UKMT Extended Solutions

The UK Maths Challenges are included on the Problems & Activities page.

Links for students, including advice for young mathematicians are provided on Mathematics for Students.

Team ChallengeFurther outstanding resources from UKMT include the Team Challenge resources. The UK Maths Challenges are included on the Problems & Activities page.
For more sources of multiple choice questions see this post.

Daisy C Multiple Choice

Daisy Christodoulou on Multiple Choice Questions

 

 

 

Mathematical Miscellany #3

Exam PowerPoint @westiesworkshop

Mr Westwater – TES Resources

With revision still in mind, for some excellent PowerPoint resources with A Level questions and mark schemes by topic try Mr Westwater’s resources on TES. (Twitter @westiesworkshop). Though Edexcel questions these are clearly useful for other examination boards also. There are six of these PowerPoint files available:
Core 1   Core 2
Statistics 1  Mechanics 1
Core 3
    Core 4

Diagnostic Questions

Diagnostic Questions

Craig Barton’s and Simon Woodhead’s Diagnostic questions site includes some excellent question collections for revision; as well as GCSE questions, there are quizzes for AQA’s excellent Level 2 Further Mathematics specification and for Core 1 at Advanced Level. Questions from the Oxford University Mathematics Admissions Test are also available.

 

From AQA this set of revision tests and mark schemes is very useful and from Oxford University Press a lovely set of free revision resources is available. Under Mathematics Revision Guides, choose the ZIP file Worksheets. Note that these are headed New GCSE – this is the old new specification, not the new new specification! These are clearly perfect for UK students taking exams this year but still excellent for future years.

AQA Revision Algebra

For many more revision resources note the Revision Activities series of pages.


GCSE Problem Solving: Steve Blades’ site www.m4ths.com has many excellent resources; on the GCSE page Steve has a section (near the end of the page) of eBooks, and one of those is on GCSE Higher wordy questions. See also Steve’s Think like a problem solver and mathematician book.
m4ths

For more Problem Solving Resources, see Problem Solving and Problem Solving 16-19.


Alec McEachran’s Furbles – a large data set!

Alec McEachran’s Furbles – a large data set!

Statistics – thinking about large data sets which we will need to for the new A Level specification – for an amusing large data set, how about the Furbles?! The images generated here are from the original 2003 version which is still available on Alec McEachran’s ptolemy.co.uk. Talk to students about summarising this data, perhaps ask for their impressions as to which colour is the most or least common. Data can be presented as a bar chart or a pie chart and you can choose to categorise in various ways. It is also possible to vary the number of Furbles, the maximum and minimum number of eyes and sides. The illustration here show the largest data set possible.

Alec McEachran’s Furbles

Alec McEachran’s Furbles


Visual Patterns

Visual Patterns – Fawn Nguyen

From Fawn Nguyen comes the brilliant Visual patterns, note the menu; the Gallery includes blog posts from teachers and students who’ve used visual patterns in their classrooms.


 

 

Top 100 Tools For Learning – 10th Anniversary

Jane Hart is the Founder of the Centre for Learning & Performance Technologies and as you can see from her blog, 2016 marks the 10th year of her annual Top 100 Tools for Learning list.

Jane has put together all the presentation slidesets as well as an alphabetical list of ALL the tools which have appeared on any of the lists.

The 2015 slideset is shown here.

Voting is now open for the Top 200 Tools for Learning 2016 and will close on Friday 30 September 2016, and the 10th annual list will be released on Monday 3 October 2016. You can find all the voting guidance and forms here. Note the 200, not 100; as you can see from Jane’s description she is making some changes to celebrate the anniversary.

From the voting guidance you can see that there are three methods of voting, my choice is:
3 Write a blog post about your choice and send the link to Jane Hart.

So these are my own 2016 choices as top tools for learning because these are tools I use all the time in my job, both in my teaching and in my role as a senior leader. Note that we are not discussing subject specialist resource sites here (that’s another whole (updated) story), but tools for education generally; I think it is very useful to remind ourselves of Jane Hart’s own definition:

“Any software or online tool or service that can be used for your own personal learning or for teaching or training”.


CY 2016 votes Education Personal Learning & Productivity Place in Top 100 2015
Evernote x x  10
WordPress x x  8
Google (search) x x  3
Twitter x x  1
Slideshare x  20
PowerPoint x  5
Excel x x  56
Moodle x  15
WolframAlpha x  –
Desmos x  –

Looking at Jane’s alphabetical list of ALL the tools which have appeared on any of the lists, I thought it would be interesting to see which tools have been on all nine lists to date and also at popular tools from the last 5 lists, 2011-2015. I used a favourite tool, Excel to create these, the file is available if of interest: Jane Hart Top 100 Tools – Colleen Young analysis Note the tabs at the bottom, the first three worksheets show the tools which have been in every list, 2007-2015, the last two look at the longer list of tools which have been in the list for the last 5 years. You can read Jane’s fascinating analysis of trends for work place learning.

Top Tools 9 Years

Tools on the list every year since 2007

Top Tools - last 5 years

Average Rank 2011-2015

To elaborate further on my choices
Evernote
  – an outstanding note-taking tool and something I use every day. I have a notebook for each of my classes to which I upload any resources I want for that class; I also jot down any ideas I have for each class. I can use it on any PC, tablet or my phone. It is also a good way to share for example a list of websites with students – using a shared notebook. I have many notebooks both for my job and life generally; the search facility in Evernote is awesome and the ability to link from one note to another, very powerful. (Blog post on Evernote).

WordPress  – obviously – you are reading a WordPress blog right now! I have several other blogs, GamesStarters and for students I have created Mathematics for Students and something I am very pleased with is a blog I use to give the details of homework for each of my classes. I created ‘What was that homework?’ as a result of a survey of students across several schools where many students said that they would like homework details online. No student can ever say to me that they didn’t know what their homework was! I also have blogs on useful tools for students and teachers generally. (The very first post on this WordPress blog – which includes some useful WordPress links).

Google which is a vote for Google search. See Google help on search or this Google Guide

Twitter  – great for professional development – I have contacts in education all over the globe and have been led to many useful resources by my virtual colleagues! See Mathematics Conversations. and this post on Twitter.

Slideshare  – it is very easy to upload presentations to this (free) presentation sharing site; the two shows above are examples. Presentations for teachers or students could be uploaded for example.

PowerPoint I can’t really have Slideshare without including PowerPoint in my list because that’s where I start with my presentations, often with a little help from the interactive whiteboard software. It is interesting to see how popular PowerPoint was last year at number 5, other presentation software such as Prezi is also useful, but it’s so important to remember that it’s the content that matters! See this post, Presentations for some reading and resources on presentations.

Excel is something I use everyday in my job to analyse and present data; I also use many spreadsheets for teaching. Of all the applications in the Office suite this stands out for me, the changes from Excel 2003 to 2007 with the massive improvements to conditional formatting for example make this one outstanding application. There are also many Excel spreadsheets out there too to help in Maths lessons – see Maths Files for example.

Moodle We have a Moodle course for each year group in school; each course has links to any websites that we use in class so students can investigate further themselves if they wish. VLEs are sometimes criticised for being no more than ‘filing cabinets’; I would argue what useful filing cabinets they are – containing resources chosen by teachers for their students all in one organised place. For example we see that in the run up to examinations the relevant Moodle courses receive hundreds of hits as everything students need was available, not only syllabus information and papers but worked examples that we have uploaded. It is also of course possible to use forums and quizzes on a VLE and being a secure site for the school we can include information intended for our students only.

WolframAlpha.  WolframAlpha is not just all about Maths, it covers so many subjects and even though they would love us to pay for WolframAlpha pro, the free model still offers unlimited queries everyday!

Desmos, the outstanding graphical calculator deserves a vote in my opinion, it is wonderful for learning mathematics, accessible for young students yet has the sophistication required for university students. Brilliant. It is entirely free, very simple to use even for young students and brilliant for projecting in lessons. There are also free apps for both Android and iOS.

So that concludes the voting from Colleen as 10 tools are needed for a valid voting entry!

UK A Level Reform

I thought it would be useful to collect the information on A Level reform in one place so have reorganized the existing pages under UK Assessment which will provide, I hope, easier access to information. The various pages, which you can see on tabs near the top of the page have been developed from areas which have proved popular.

The series of pages includes the following

From MEI comes this excellent presentation relevant for any new specification whichever examination board is being used. A strength of this presentation is its clarity. Simple statements of fact are given in very clear language and as well as stating what we know we also have a very important reminder of what we don’t know!

Further references and reading
Note that the documents mentioned in the presentation above are included here.

From the Further Mathematics Support Programme:
2017 A Level Mathematics and Further Mathematics
This is really helpful – clear and simple statements. Note the link to the very helpful document on changes to AS/A Level Mathematics & Further Mathematics.

A Level Subject Content for teaching from 2017

GCE Subject Level Guidance from DfE, this guidance is intended to help awarding organisations understand how to comply with the GCE subject-level conditions and requirements for mathematics. The guidance includes further information on the use of Technology and large Data sets. Making Statistics Vital from Jonny Griffiths has some tasks which could be excellent, look at the this task on World Wide Statistics for example which includes the task with answers and a spreadsheet with data for 191 countries. Census at School is an excellent source of data and you can generate data for your own school too.

Making Statistics Vital

Making Statistics Vital – Jonny Griffiths

How demanding are questions in the new A level maths? Ofqual is planning to conduct a research study to support the accreditation process for new A level maths qualifications. Taking place in July this year, the research aims to compare the level of difficulty of exam boards’ sample assessment materials by having judges compare pairs of exam questions. An interesting method of comparison and something I have come across before – see No More Marking on Comparative Judgement.

The A level mathematics working group report is a very valuable document and one I referred to in a presentation on GCSE Problem Solving – I do believe we should always look both backwards and forwards to inform our teaching; where have our students been and where are they going? I have also created a new version of the problem solving presentation: Problem Solving 16:19.

Underground Mathematics

The Cambridge Mathematics Education project (CMEP) mentioned in the presentation has now been renranded as Underground Maths, a superb site which will I feel be very important in our work with our Mathematicians students.


Timelines

For each current school year group (2015-2016) – changes to GCSEs, AS and A Levels

New GCSE and AS & A2 courses for teaching September 2016

New A Level Maths courses – revised start date               BBC News Story

 

Is That A BIG NUMBER?

is that a BIG NUMB8R

Twitter (quite rightly!) suggested I might like to follow IsThatABigNumber. The aims of the site are as follows:

  • Put numbers in context
  • Promote numeracy
  • Share number facts
  • Develop Number Sense

Is that a big number menu

Exploring the menu at the top, note as well as the home page, we have other options, we could ask for example how big is 5km? The question returns a very comprehensive answer with comparisons to many other numbers! I do like the various quotes that appear randomly, as I was exploring I saw (very appropriate for someone responsible for Assessment) from Charles Babbage: “Errors using inadequate data are much less than those using no data at all”.

Other options as you can see include conversion, we could convert 1cm to 1 m (though I can already do that with WolframAlpha which returns rather more comparisons, and can also do it myself, I hasten to add!). Alternatively look at some random Number Facts or an interesting option – try a quiz

On the home page we can the site is in its early stages, (the authors welcome feedback). Numbers in the News is a good idea; always looking for other sources too I discovered the BBC’s #bbcgofigure. Some links on the home page currently seem to link to simply a Google search page; I suspect this is a work in progress and I wish the authors well – promoting numeracy just has to be a good thing!

On the subject of big numbers, I have recently completely checked and made major updates to my post on Standard Form resources.