Spot the Mistake

…and other updates.

Edexcel Model Answers example
Updating the page on Edexcel’s Teaching and Learning materials (part of the A Level (16+) Resources series) I have included their now complete set of GCSE to A Level Transition worksheets and also exemplar answers with examiner comments, a particularly valuable resource. These booklets look at questions from the AS and A level Sample Assessment Materials, which was used in the trial undertaken in summer 2017. Real student responses are shown together with commentary showing how the examining team apply the mark schemes. The commentary includes always useful notes on common errors. Noting that these could be used in class and students asked to find errors reminded me of some more excellent resources – time for an update of the Spot the Mistake collection.

Erica's Errors - A Lutwyche

Particularly excellent resources come from Andy Lutwyche, look at his excellent Erica’s Errors series for Spot the Mistake activities.

For more resources – see the Spot the Mistake collection.


MEI Making Sense of Information
Another updated page in the A Level series is on Statistics; this includes links to all the large data sets used by the examination boards as well as suggestions and resources for teaching. Note the September/October 2017 edition of MEI’s very helpful M4 magazine which has a focus on the teaching of Statistics and includes information and examples of updates on the large data sets for all the examination boards. The PowerPoint resource could also be used with younger students to get them thinking about the presentation and interpretation of data.


Other checked and updated posts include

Diagrams

 

Numbers – Visualizations

I have several references in various places on this blog to some great visualizations.
Time to put them all together!

Jeffrey Ventrella’s Composite Number Tree

Jeffrey Ventrella’s Composite Number Tree

From Jeffrey Ventrella this wonderful Composite Number Tree – I have used this successfully with many students. It makes a great starter. Students can work out themselves how the tree is being formed and comment on any patterns they notice.


Stephen Von Worley

Brent Yorgey

Brent Yorgey

 

 

 

 

 

 

 

Another excellent visualization, animated factorization diagrams comes from Data Pointed. And here is Stephen Von Worley’s blog post, Dance, Factors, Dance which tells the tale of the animation. Noting his reference to Brent Yorgey’s factorization diagrams led me to Brent’s own later post, More factorisation Diagrams. I love Brent’s use of colour here. If you want even more on these great diagrams he has more information and links on this page on his blog, The Math Less Traveled.


Visual Patterns

Fawn Nguyen – Visual patterns,

On the subject of Diagrams generally I have several posts on the subject. You can see Fawn Nguyen’s lovely Visual Patterns in Diagrams in Mathematics.

Venn Diagrams

gcse-content

The DfE document describing the GCSE Mathematics subject content is an excellent starting point for checking new content, all exam boards must include this content.

Note that only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content. See page 4 of the DfE document.

There are many excellent resources for teaching Venn Diagrams; investigate this collection.


Diagnostic Questions
venn-example

On Diagnostic Questions – Probability with Venn Diagrams.

Diagnostic questions now has over 21000 questions on Mathematics including wonderful collections of examination questions. The site is completely free (and promises to remain so). Plenty of help is available to help you learn how to use the site.


CIMT Venn Diagrams

CIMT is one of my Top >10 websites for a very good reason – when I want additional examples for any topic at any level I can always find them on CIMT! Venn Diagrams are no exception to this, you can find Sets and Venn DiagramsSet Notation and Logic and Venn Diagrams in the student interactive resources and the text chapter on Logic from the Year 7 text here; in sections 1.3, 1.4 and 1.5 of the text you will find examples and exercises on Set Notation and Venn diagrams. See also the additional Teacher resources for this unit (Unit 1, Logic) such as Additional exercises are also available as are Aural Tests. Other teacher resources include slides and Revision Tests (you will need the CIMT password for the Revision Tests).

Problem from section 1.4 CIMT Venn Diagrams

aqa-bridging-the-gap

AQA – Bridging the Gap

From AQA’s excellent Bridging the Gap resources, Sets and Venn Diagrams is superb, also see the OCR resources, from OCR’s Check In tests, see Combined events and probability diagrams. For further information on the changes to Probability at GCSE see this post.


Transum – Venn Totals

From Transum try Venn Totals 4 different levels of exercises which can be checked are available. Level 1 – Reading information from a Venn diagram containing two intersecting sets. Level 2 – Reading information from a Venn diagram containing three intersecting sets. Level 3 – Adding information to a Venn diagram containing three intersecting sets. Level 4 – Adding information to a Venn diagram containing three intersecting sets with some problem solving required. There are also exam style questions, to see the worked solutions a subscription is required.



On TES the GCSE topic specific resources include Probability resources. Included in this collection are several (all free) resources on Venn Diagrams.


Nrich too can always be relied on to provide resources – a search on ‘venn’ returns these resources.

Nrich Venn Diagrams

From Sums Mathematics come two very useful activities to illustrate Venn Diagrams. From the Index choose Sorting & dbases under Data Handling where you will find Venn Diagram activities for two sets and three sets.

Venn Diagrams

teachitmaths Venn diagrams

From teachitMaths, try Venn diagram dominoes (pdf versions of all the resources on this site are free).


mathisfun-questions

However, note that some of these questions refer to ‘difference’, examination specifications should be checked for notation, for example AQA’s helpful teaching guidance includes notation such as this illustration.

AQA Teaching Guidance

AQA Teaching Guidance

vivaxFor a useful way of displaying these regions on Venn Diagrams you could use the demonstration from the Venn Diagrams tutorial on Vivax Solutions. Geogebra or WolframAlpha can also be very easily used as shown near the end of this post.

Or returning to Transum, try Venn Paint.
transum-venn-paint

Level 2 has 3 set Venn Diagrams. Exam Questions are included also (solutions are provided for subscribers).
transum-3-sets


Included in Jonny Griffiths wonderful RISPS – see RISP 10.

www.risps.co.uk

fmsp-gcse-extension-tasksOn a similar theme – from the Further Maths network have a look at the excellent GCSE extension tasks, see NA1 for example.

From Craig Barton – see this whole collection of rich tasks with Venn diagrams.
venn-rich-tasks


Underground Mathematics

To really challenge your students combine Venn Diagrams and Algebra and try this review question from Underground Mathematics. (From a 1969 MEI O Level Additional Mathematics paper.

A search finds more problems – all resources on Underground Mathematics include complete documentation including suggestions, a full solution, printable materials and more.

Perhaps try Can we find how many boys study French, Latin and German? or Quadratic Solving Sorter – not a traditional Venn diagram problem but certainly a diagram as in the possible solution presented is a very good idea.


washing-upFrom Census at School combine Statistics and Venn diagrams, and check this task on Washing Up!


These three interactives from Shodor are a good introduction to Venn diagrams:
Venn Diagrams, Shape Sorter and Triple Venn Diagram Shape Sorter 

With the Triple Venn Diagram Shape Sorter you can either set the rules or guess the rules by selecting the appropriate botton:


Some excellent activities are available from the Illuminations site.

The Shape Sorter allows exploration of geometric properties of shapes.

Select Instructions and Exploration for clear information on how to use the resource.


WolframAlpha can be used to illustrate Venn Diagrams.
The slideshow here shows several examples.


geogebra
Alternatively – try GeoGebra or this UCLA  applet.


On a lighter note – a Twitter conversation on a fun idea!
Twitter Venn

Here’s some more on MailOnline

If you want to create your own there are plenty of tools to use – there is a good summary here on Cometdocs.

New Scientist Gallery - Venn Diagrams

…and to take Venn Diagrams to their extremes have a look at these wonderful images from New Scientist!

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

A clear diagram can be so helpful in understanding a problem, look what one of my Year 7 students did when asked for the nth term of a sequence having been given a diagram:
Seqences

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

For a very easy to use activity, try Jonathan Hall’s Algebra Tiles on his wonderful Mathsbot.com site.
Algebra Tiles mathsbot

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?

…and for some lovely visualisations have a look at this post.

data-pointed

Stephen Von Worley

 

 

Here’s the diagram….

What’s the question?

triangle diagram

Seeing this well received resource, GCSE Question Prompts on TES recently 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 GreenawayGCSE Visual Prompts for both Higher and Foundation. Mark’s resources show the diagram first and then also include the complete question.

Algebra Snippets 2

Algebra Snippets

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

Diagram - T SherringtonOn the subject of diagrams I really like Tom Sherrington’s post “Empowing 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 Starters – fit 8 diagrams to a page!

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 tale a screenshot (I do like the snipping tool in Windows 7) 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…

Data Pointed

Visualization from Data Pointed. Select image to see it on Data Pointed.

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