16+ Mathematics Resources

crashMaths Skills Check

crashMaths Skills Checks

crashMaths has been added to the 16+ series of pages. The growing library of resources now includes a set of very useful Skills Checks, with solutions, on Pure content from Year 1 of the A level specification. The site includes many practice papers and mark schemes.

With all A Level courses now linear, Retrieval Practice is essential. From crashMaths these AS Maths Key Skills Check worksheets will be very valuable for Year 13 in the second year of their A Level course. The Skills Checks are all on Pure Mathematics.

 

Further recent updates to Advanced Level resources include:

From Edexcel, A Guide to using GeoGebra when teaching AS and A Level Mathematics. This guide links to numerous GeoGebra files clearly mapped to the specification content.

Edexcel GeoGebra AS & A Level Mathematics

This guide has been added to The Use of Technology Page, also to the latest update of this file on technology resources mapped to content which can be found on The Use of Technology Page. The file includes links to instructions for the Casio ClassWhiz calculator.

Updates to the Further Mathematics content include this link to a post on Maclaurin Series for students which includes Desmos pages. The post also links to some useful notes and examples for students.
Maclaurin Series cos x

For Further Maths, this post on the new topic of Differential Equations may be useful. Again, notes and examples are linked to, also instructions on using WolframAlpha to check work on Differential Equations. WolframAlpha is also useful for checking work on Matrices.

The teaching of Polar Coordinates offers an ideal opportunity to use Technology.
Desmos polar slider

Proof Resources – Part 2

Edexcel proof question

Dr Frost Maths: Full Coverage – Algebraic Proofs involving Integers

On Dr Frost’s site, it is possible to browse all his excellent resources by topic so if, for example, we search on KS2/3/4 then Algebra, we see Algebraic Proofs. Under Proof the Year 9 file PowerPoint file is excellent for high ability students,  you will also see a very useful worksheet on counter-examples. I do like Dr Frost’ Full Coverage resources which are compilations of GCSE questions (GCSE – UK qualification taken at age 15-16). Answers are provided at the end of the document. Explore this outstanding site full of very high-quality resources, all Dr Frost’s clear indexing make the resources simple to find.


MathsBot

MathsBot – GCSE Questions

MathsBot is another superb site and very easy to find questions by topic, the GCSE Exam Style Questions are a good example. Select any filters and note the many question topics.


Maths4Everyone

Maths4Everyone – Algebraic Proof Workbook

This search of TES resources returns several highly rated free resources on proof.
Try Algebraic Proof – Expressions and Proofs from  James Clegg, the worksheet “teases out expressions to show certain situations (e.g. the sum of 2 consecutive odd numbers) and features options on an “answer grid” at the bottom of the page.” There are also some questions to try. From the excellent Maths4Everyone this Algebraic Proof (Workbook with Solutions) has numerous problems to try as well as very clear examples. Answers are provided – highly recommended (as are all the resources on Maths4Everyone by David Morse).

See also: Proof Resources


 

Proof Resources

CIMT

From CIMT who are one of my favourite sites for a reason – see this GCSE additional unit on Proof.  A favourite site because if you are ever short of examples it is highly likely you will find something on CIMT who have everything from resources for little people to Advanced Level and everything in between!

CIMT Proof

CIMT-Proof


Nrich CollectionNrich has this collection Reasoning, Justifying, Convincing and Proof for Lower Secondary. A search on Nrich on Proof returns a wonderful selection for all ages and stages. We have tasks to introduce ideas of proof to younger children through to preparation for STEP examinations.

Also from Nrich, try this Interactive Proof Sorter example which works on my phone as well as on my laptop. This would make a good starter – if you want to give out paper copies for students to work on as they come in, you can easily fit 4 copies to an A4 page!

Nrich - Proof


Teachit Maths, though a subscription site offers an extensive collection of activities as free pdf files. A search on Proof returns some great resources. I do like this Worksheet on Proof which has 20 varied tasks aimed at older students 16-18, though some would be accessible to younger students. Full teachers notes on solutions are provided. In the task illustrated here, a full proof is given and students asked to explain each step.

TeachItMaths

TeachItMaths – Proof Tasks


Don Steward

Don Steward – Multiple Proofs

From Don Steward on Median, we have many wonderful proof resources. Try always and never or multiple proofs. Why just multiply out brackets when we can dd a little proof?


 

Here’s the diagram…

What’s the question?

(This post is an update of my post from 2013 and now includes the more recent, excellent resource Goal Free Problems from Peter Mattock.)

Using diagrams as prompts like this is excellent for Retrieval Practice.

triangle diagram

Seeing this well-received resource, GCSE Question Prompts on TES reminded me that I have successfully used this idea myself before. For example for GCSE revision I have given students a selection of various triangle diagrams and asked them what the question might have been. This proved to be a useful way of revising several topics – some of which students sometimes mix up! For several of these triangles there are many possibilities and students can be asked which lengths and / or angles they could work out.


Further excellent examples come from Mark Greenaway – GCSE Visual Prompts for both Higher and Foundation. Mark’s resources (Prompts 1) show the diagram first and then also include the complete question. Prompts 2 provides 25 starter prompts for students to produce their own questions and answers using the given information.

Visual Prompts

Higher – Visual Prompts

Foundation_Tier_Prompts 1 ppt Foundation_Tier_Prompts2 ppt
Higher_Tier_Prompts 1 ppt Higher_Tier_Prompts 2 ppt
Foundation 1 pdf Foundation 2 pdf Higher Prompts 1 pdf Higher Prompts 2 pdf

Algebra Snippets 2

On a similar theme, not a diagram this time but an extract from a question: see Algebra Snippetts.


From Andy Lutwyche comes a very high-quality series of ‘The answer is …What was the question?’ resources. A variety of topics are covered and all answers are provided.
Here's the answer...


Peter Mattock has created Goal Free Problems, a site he set up, in his own words “to allow teachers to access and share goal free problems created by myself and others. Goal free problems have been proven to support pupils in improving their knowledge and understanding by removing the cognitive load of the goal and therefore not prompting means-end analysis of a problem.”

Here you will find hundreds of questions categorised by topic; there are also mixed questions available. A wonderful resource.
Algebra - goal free


Diagram - T SherringtonOn the subject of diagrams, I really like Tom Sherrington’s post “Empowering students to own their learning solves maths problems“; a great idea to start with a diagram with no labels at all as a way into a problem. I tried this with Year 10 (very able students), presenting them with only Tom’s diagram and was very pleased indeed with the outcome. I didn’t even give them the question – just the diagram (a small copy each) and we started by deciding what the question might be. We quickly got onto areas as a possibility so then answered Tom’s original question ‘what fraction of the shape is shaded?’. The class happily discussed how to solve the problem and a student asked ‘can we write on the diagram?’ which of course was perfect – absolutely they could write on it. We solved the problem, revising some basics and had the discussion about what to do when you don’t know what to do! I will certainly use diagrams with no labels again.

This idea could be used for a starter on just about any topic – provide students with an image or perhaps just an expression and ask them to write a question to go with the image.


MEI Starters

MEI have an excellent free collection of GCSE startersDesigned for the start of a GCSE lesson, the diagrams and questions are very clear and will display well on the IWB. There are several starters under the following headings: Mathematical Reasoning, Number, Algebra, Geometry and Measures and Statistics and Probability. Files with the answers and teachers notes are also provided. Many of the diagrams here could be used for students to write their own questions. It is not always possible to have the IWB up and running, particularly if you are coming from a different room and I do like to get students working straight away. Experimenting, I found that I could take a screenshot (I do like the Windows snipping tool) and fit eight to a page! I used a Word document with very small top and bottom margins and a two column layout.

Staying with MEI, Bernard Murphy has some great ideas here on using pictures in A Level trigonometry. Look at this diagram – all the trigonometric ratios!


How creative can you be? I wonder what they would make of something like this…

See also – Numbers – Visualizations

puzzle demo
For a wonderful introduction to equations using diagrams – see Mobile Puzzles – Algebra


Perhaps a photo from the Bad Maths collection on flickr


cc licensed ( BY ND ) flickr photo shared by Danny Nicholson

Click here for the complete collection.


or a holiday snap!

Neuwied

Neuwied – Germany, photo by David Young

 

Mathematical Miscellany #21

For A Level Teaching, Dr Jamie Frost has as always been very busy! These high-quality resources (slides/worksheets) provide complete coverage for the new specification.

Dr Frost KS5

Dr Frost Maths – A Level Resources

Whilst Dr Frost mentions Edexcel, the content for A Level Maths is the same for all examination boards; Further mathematics has some common Pure content.

Dr Frost has also provided very clear instructions for using the Casio FX99-1EX Classwiz calculator which you will find on his site. To really challenge your students Dr Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

There is a page on Dr Frost’s resources as part of the A Level (16+) Resources series of pages.


From the older students, let us turn our attention to the younger students.

Corbett 5-a-day Primary

Corbett Maths – 5-a-day Primary

The excellent Corbett Maths Primary includes 5-a-day Primary illustrated above. The page of resources for younger students has been updated with this brilliant addition.


Emma

5x8 Card

SERP – 5×8 Card

Emma McCrea alerted us to The 5×8 Card generated by a SERP Team. You can read more information on the background to this resource here.

Not having come across the SERP website before I investigated further and found more delights. I particularly like AlgebraByExample which is a set of Algebra 1 assignments that incorporate worked examples and prompt students to analyze and explain. These look like good resources for discussing common misconceptions.

SERP - Algebra by Example

SERP – Algebra by Example

Note the tabs at the top of the home page, from the Materials tab you can access all the resources. Note the free downloads at the SERP download centre where you can download individual resources, or very simply, use the menu on the left to download the complete workbook; answers are also available as are introductory guides for teachers and students.


With UK A Level results having recently been published, the Results 2018 page has been updated and checked. Links to grade boundaries and results statistics are provided and note the useful resources from Ofqual. GCSE Links will be checked on Thursday 23rd August.


It is still Summer holiday time for some of us, so I’ll end with this lovely puzzle I had not come across before. See plus magazine – Finding the nine. (There is a link to a very clear solution in the video). The