Transum Mathematics

Estimating Angles - Transum

Estimating Angles – Transum

Julie Morgan’s mention of Transum‘s starter on estimating anles reminded me of how often I use resources on Transum which is mentioned all over this blog! Staying with Angles, there is a further Angles Activity with 4 levels so you can practise as much as you want! And here are more Angles Activies including Starters.


I thought it would be useful to put together some of the resources I have used from this excellent site full of high quality resources. Transum has a superb collection of free resources. (Subscriptions offer additional features but the free model is superb.)

transum-starter-of-the-day
The appropriate place to start is of course with the fantastic collection of starters – Maths Starter of the Day. Note that there is a complete index of starters including the topic of the starter. Many of the Shine and Write activities would also make good lesson starters. For example are these statements true, sometimes true or false? As with many of the activities on this site you can refresh the page for further statements, you can also change the level. Or perhaps some Mathanagrams? I think it is through the starters that many teachers have come to know the site, but it has so much more!


Transum - Topic Index

Transum – Topic Index

The Transum site is easy to navigate, there is a clear Topic Index for Teachers also, for students a Maths Map with numerous activities to support their learning.

Transum - Maths Map for tudents

Transum – Maths Map for students


Venn Paint is included in my Venn Diagrams post.
transum-venn-paint

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


Systematic Listing Strategies includes Transum activities.

Transum - Combinations resources

Transum – Combinations resources

Transum Combination Starters

Transum Combination Starters


And many more mentions….

complete-square

Transum - Arithmagons

With Year 11,we practised Completing the Square with a Transum level 2 and level 3 activity. And in case you want some more Algebra – there’s a whole index of activities.

On Arithmagons the Arithmagon activity which has options for forwards and backwards problems on Addition, Multiplication and Subtraction displays very clearly on the Interactive Whiteboard.

Investigations Resources includes Investigations from Transum 

Happy Numbers Resources – an attractive statement of the problem on Transum

Mathematics Crosswords includes this Transum interactive crossword with the software from Eclipse.


I’ll finish (for now) with a Counter Moving Puzzle!
pu-wiang

 

Heading for High GCSE Grades

Following on from my previous posts on marking GCSE mock examinations, GCSE Mathematics – Marking Mock Examinations and GCSE Mathematics, one of the tasks on my To Do List is to write helpful reports for Year 11.
gcse-to-do

Something I do regulaly when writing reports is to ask the students if there is anything they think I should acknowledge (they know that I have to agree with the suggestions having seen the skill and/or learning behaviours in class!) I am pleased to say I know my students well but asking them can provide real insight into what is important to them and what they are proud of. I find Google forms excellent for getting feedback from students, my questions (clearly intended for my students in my school setting) to my students are here. The questions on this particular form are intended as writing prompts for my students. Their responses are thoughtful and highly articulate. I am so pleased at their understanding of their own learning; they are clearly aware of what they need to review further and which topics they are comfortable with. The question by question analysis certainly heps with this.

In my previous posts I mentioned that I gave them an analysis on the A01, A02 and A03 Assessment Objectives.

Providing able students with information works; many referred to the fact that they like the many resources we provide them with. Our school VLE has all the AQA Topic Tests for example and a large collection of resources they can use. One student commented that she did most of the AQA Topic tests, also past paper questions – a strategy which clearly worked as she achieved an exceptionally high mark. She also referenced the Assessment Objectives in her reflection.

Another student, also with an exceptionally high mark on AQA Practice Papers, Set 3 said that she needs to make make sure that she fewer silly mistakes. Her strategy: “Do even more practice questions when I revise than I do already.”
There’s a common theme of course –

do lots of questions, then do some more!            

dylan-wiliam

 

A student commented that she feels comfortable asking questions in class.
So very important, Dylan Wiliam puts it well.

(See Assessment & Feedback in Mathematics for further information).

retrieval-practice-learning-scientists

Learning Scientists – Retrieval Practice

As we often do following a test, having reviewed the work we did a Mini Test, this was actually more of a maxi-mini test as I went through the mock papers making sure I asked questions on the most common errors and some facts students had forgotten as they had not studied those topics for some time. We marked this in class and students have annotated their work well, noting the need to review particular topics.
sr-book

Looking at their mini tests has been very informative, students have clearly reviewed their mock examination answers with the model answers. Interesingly though, some (far less) of the same errors were seen. A small number were unsure of exponential graphs and also of the more sophisticated manipulation needed with functions; for example reflecting a quadratic in the y axis or finding an expression for f(x-2) given x. We’ll be doing regular mini tests!

Mathsbot - Revision Grid

Mathsbot – Revision Grid

I will use revision resources they will be able to use themselves at home such as Diagnostic Questions and resources such as those referenced here. I think we’ll start next week with a little graph sketching!

exponential-function
I have created classes on Diagnostic Questions and gave my Year 11 class the class code to join yesterday – today I can see a student has already done of the AQA quizzes and got 16/16 – she got a 9 in the mock!

Mathematical Advent Calendars

It’s that time of year again…!

Nrich Advent Calendars

Nrich Advent Calendars

December means Advent Calendars and Nrich have published two Advent Calendars, one for Primary and one for Secondary each containing twenty-four problem-solving activities, one for each day in the run-up to Christmas. The primary Calendar tasks focus on encouraging mathematical habits of mind and the Secondary tasks have been chosen to encourage mathematical creativity.

You can in fact find a whole collection of advent calendars on Nrich and clearly the year doesn’t matter! Note the different themes available – a Sudoku for each day perhaps? Or a tangram? Maybe you want to play a game? 

Advent calendar Alex Pett

Advent Calendar by Alex Pett

Alex Pett created his advent calendar complete with history and problems for each day. Alex has provided a pdf version or use as a Google document. For an Activeinspire resource this version also has sound.

revision-calendar
Alternatively how about this Christmas Revision Calendar form Access Maths?

mathsbank

MathsBank Advent Calendar

And for an A Level Advent calendar, try this one from Mathsbank.

With Mathsbank you can display solutions step by step.

GCSE Mathematics

Marking GCSE Mock Examinations, I wrote myself a To Do List:

gcse-to-do

Completing the question by question mark entry for each of the three papers as mentioned in the previous post has shown some clear trends as to facts students have recalled incorrectly. I plan to check that students can accurately recall facts, terminology and definitions (A01) as a regular part of lessons, even for a few minutes. Clearly to achieve highly on questions requiring students to reason, interpret and communicate mathematically (A02) and Solve problems within mathematics and in other contexts (A03) students do need easy recall of all these facts.

mathscard

mathscard – Loughborough University

Resources to help with this aim include Flashcards on Tanner Maths, A4 or A5 size cards are available. For use in lessons I intend to project flashcards of my choice. Also useful here is Mathscard from Loughborough University; whilst written for A Level, several parts are useful for GCSE. The online version can easily be projected.

If we look at Trigonometry for example you will see that the graphs are included as are the standard triangles used to find exact values of angles.

trigonometry

mathscard – Loughborough University

I plan to ask the questions of students in the form of a mini test, so I will ask students the question and they just write down the answer, I can then display that answer. For trigonometry for example I will ask them to draw sketches of the standard triangles illustrated here. Appropriate flashcards can be displayed whilst asking the questions. An alternative would be the syllabus itself or a document such as AQA’s Teaching Guidance illustrated below. For example, from AQA we have:
trig-exact-values

sin-30-45-60

 

As Howard has pointed out in his comment below – note the structure of the values for sine and cosine!

 

 

 

 

Another possibility will be to use the Examination collections from Diagnostic Questions.
diagnostic-questions

These resources can of course all be made available to students so they can quiz themselves at home. This could in fact be a regular part of a homework.

Revision will be a regular feature in lessons. As my Year 11 students are studying AQA’s Level 2 Further Maths, the overlap between this and the GCSE specification will help them.

Revision Resources can be found here and another very varied selection here.

For the formulae that need to be known and understood – see these slides.

GCSE Mathematics – Marking Mock Examinations

My next few posts will be on this theme.

For this first post – just some initial notes having marked the mock examination papers for my Year 11 class (UK age 15-16). This class is studying for AQA GCSE Mathematics and also the AQA Level 2 Further Mathematics qualification; they have completed the AQA specification and have taken a complete set of Practice Papers (Set 3) for their mock examination. (Teachers who offer, or are considering offering AQA maths qualifications can register for AQA’s All About Maths). AQA  recommend the use of these papers as mock exams as they reflect the balance and expected standard of their new assessment. The papers have been trialled with over 6000 students. Detailed mark schemes and  further marker training material is available to support teachers. Many of the points here apply for any specification.

ao123

Assessment Objectives

It is time consuming but provides a really valuable analysis and I feel time well spent on this new specification – For each paper, I am entering the marks for each question or part of a question into an Excel spreadsheet. I have used AQA’s Higher Papers AO grid so I and my students can see a breakdown of marks for AO1, AO2 and AO3. The grid is available with the Practice Papers.

AQA Higher Papers AO Grid

AQA Higher Papers AO Grid

 

 

This type of analysis clearly shows which topics and/or question types were more problematic for individual students as well as for the class as a whole. Using Excel with conditional formatting makes the information easy to interpret.

excel


Some AO1 marks have been lost simply because students have forgotten material or misread a question, perhaps reading too quickly and not noticing key words. Continuing to use mini-tests should help with this. It also strikes me that we can sometimes display some flashcards as a starter. We can also use starters for GCSE revision. Links to Flashcards from Tanner Maths and other high quality GCSE revision questions are in this post on Mathematics for Students. Last time I taught a year 11 class having done a similar analysis I made sure that questions on the most problematic areas from their mock examination featured regularly in our mini tests. We used Corbettmaths 5-a-day regularly which they found very helpful.

5-a-day


An understanding of graphs is crucial, it seems to me we should take every opportunity to provide a graphical representation where possible for all year groups. Multiple choice questions where students have to identify a graph could be used in KS3; students simply need a common sense approach: for example “if x is 0 what is y?” They need us to help them be confident that they can work it out – work it out, not everything is a memory test. Many students who were successful in identifying an exponential graph simply took a moment to work out a small number of coordinates. Encourage graph sketching in Key Stage 3.


On notation – something I have often seen before – students confuse notation for inverse functions with a reciprocal.


Naming angles and sides, it is so much clearer when students use what use what is given in the question – if an unknown side is BD then call it BD – certainly don’t call it c when there is a C elsewhere in the diagram! Encourage students to mark angles on diagrams – it provides very clear evidence.


Whilst the laws of indices may be understood, students can be thrown with algebraic examples, (32x)3 for example.


Constructions – one of my favourite quotes from a Principal Examiner, “No arcs, no marks!”


Some facts turn up so often, for example: 2 is the only even prime number, if you square a number it’s always a positive result – watch out for these – they turn up is so many different questions!


The value of a sketch: eg reflect a quadratic with a negative coefficient of xin the y axis and you’ll get another quadratic – also with a negative coefficient of x2.


I train my students in the ways of marking! They all understand isw – ignore subsequent working. So a harmless extra bit on the end of their answer may not matter. However there are times when extra information will lose you marks – for a question specifically asking about the spread of marks – don’t mention the median!


On my To Do list for my Year 11s and I’ll comment further on these in blog posts to come:

  • What are the most common things they have struggled to remember?
  • Plan the flashcards to display
  • Providing some model answers – annotating the questions, highlighting key words, clear explanations
  • Using the analysis of the papers, with other assessments to write helpful reports for Year 11 this term

Further posts will follow.


 

Circles & Teddy Bears!

Select Posters & Flyers for poster

Select Posters & Flyers for poster

Cambridge University’s Underground Mathematics is an outstanding resource for teachers of students age 16-19 and I believe will be an important source of ideas for teaching the new Advanced Level specifications.

There are various posters and flyers available including these teddy bears discussing circles!

 

Iteddy-bear regularly feature favourite resources; here’s a great way to look at circles! The teddy bear! As with all the resources on Underground mathematics much more than just the problem is available; note the printable/ supporting materials for the teddy bear problem.

I can never resist creating a Desmos page!

Further posts on Underground Mathematics.

teddy-bear-desmos

Diagnostic Questions

To use the links in this post you will need to be logged in to the brilliant Diagnostic Questions site. Create an account if you have not already done so as this site with thousands of high quality diagnostic questions and additional analytical features is free. If you scroll down the page you’ll see that Diagnostic Questions are giving “you, the teacher in the classroom, a promise that Diagnostic Questions will always remain free.”

Diagnostic Questions provide a way of assessing your students’ knowledge and understanding, they are excellent for identifying misconceptions. Try for example the collections of GCSE 2017 examination questions from AQA, OCR and Pearson Edexcel.(scroll down each of the pages linked to for numerous quizzes on different topics on the GCSE syllabus).

Diagnostic Questions GCSE 2017 Collections

Diagnostic Questions GCSE 2017 Collections

Diagnostic Questions - GCSE examples

Diagnostic Questions – GCSE examples

You will find excellent coverage of topics new to the GCSE specification. You can also search all questions for a topic of your choice, for example a search on iteration will lead you to the whole collection of Trial and Improvement and Iterative Methods questions.

When you are logged in to Diagnostic Questions, you can easily return to the menu using the menu icon on the left.
menu2
Returning to the collections, there are many – scroll down the page and you will see collections such as GCSE Maths Takeaway – 111 mini topic-specific quizzes covering all the content on Higher and Foundation GCSE (keep scrolling down the page for all the quizzes). These quizzes are ideal to use as baseline assessment before revising a topic, or as a measure of progress following the teaching of a topic.

For schools teaching AQA’s Level 2 Further Maths specification, the AQA Level 2 in Further Maths collection has 12 sections of very useful questions for this specification.

You will see choices for each quiz including the very useful option to download the questions as a pdf.
quiz-options

quiz-pdf
For example I created a quiz on Circles and Tangents, downloading this as a pdf creates this file. See the guide mentioned below for instructions on creating quizzes.

Extensive help is available to help you learn how to use the site. The Getting Started with Diagnostic Questions Frequently asked questions guide is very helpful for new users.

This post has looked at some of the questions available but note all the other features – start exploring!


 

 

 

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

diagnostic-questions

Diagnostic Questions

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. Also


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.


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 try Transum’s Venn Paint.
transum-venn-paint

Level 2 has 3 set Venn Diagrams. Exam Questions are included also (solutions are provided forsubscribers).
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!

Wisweb Applets – HTML5

A happy discovery on Twitter, a conversation with Christian Bokhove of Southampton University led me to another home for the Wisweb widgets. Try the Digital Mathematics Environment from the Freudenthal Institute.

To access the resources mentioned here:

  • Choose Open DME for student
  • Login as guest.    
  • Explore Secondary Education (also the DME Widget list)

A guest login has restrictions but you will be able to explore and use resources, many of which make excellent demonstrations.

A site very well worth exploring to become familiar with what is available and one I’ll return to.
Some examples follow:

wisweb-algebra

Choose Secondary Education, Algebra and also the Algebra Widgets to explore resources such as the following :

Algebra Arrows

Algebra Arrows

Algebra Arrows illustrated above is excellent for exploring functions. Form inputs, operations and output by dragging them onto the main workspace, connect them up and optionally connect to a graph. Click inside any of the elements to change the content.

The Digital Mathematics Environment has much more than the original Wisweb applets – a quick glance at Secondary Education shows we have resources to explore.

Choosing Secondary Education/Algebra/Exercises – Equations/Linear equations led me to another favourite. A whole series of these exercises is available. I like the way the steps and working are clearly shown

Equations

There are useful demonstrations that could work well in class:

Uising Machines

Operations with Letters – Using Machines

Test Yourself - Algebra

Operations with letters – test yourself

Linear Graphs

Formulas for Graphs – Linear Functions


Choosing Secondary Education – Geometry will give you the following choices:

wisweb-geometry

I am very happy to see widgets such as Building Blocks again. This is useful for demonstrating plans and elevations. I discovered I could clear a block by selecting both left and right mouse buttons simultaneously.
Building Blocks

Building Blocks

dme-widgets
Statistics and Probability widgets includes a widget illustrated here on the Normal Distribution; try experimenting with the various variables.

Normal Distribution

Probability Trees could be useful for creating diagrams as the basic diagram is very easy to set up – simply enter the number of branches you require.
Probability Trees

(If you are using one of the older browsers and have up to date Java, you can still use the Java versions of the Wisweb Applets.)

 

Wisweb Applets

wisweb
These versions of the excellent Wisweb applets from the Freudenthal Institute do require Java which is a problem as most modern browsers are moving away from plugins and toward standard HTML5. There are certainly issues with Chrome and Java, Oracle say “If you have problems accessing Java applications using Chrome, Oracle recommends using Internet Explorer (Windows) or Safari (Mac OS X) instead.” This too is of limited value as Microsoft Edge does not support Java, though see further information. This includes the statement that “Internet Explorer 11 and Firefox will continue to run Java on Windows 10”.

If you still get Java error messages even though you have an up to date version of Java installed, check that Java is enabled in your browser. Additionally, it may be that you need to Configure Java (see your programs list) and under the Security tab add an exception site, eg: http://www.fi.uu.nl/wisweb/ I left my security setting as high and this did solve the problem.

If you want to look ahead beyond Java then try the Digital Mathematics Environment from the Freudenthal Institute.


Try  Algebra Trees for example. Form inputs, operations and output by dragging them onto the main workspace, connect them up and optionally connect to a graph. Click inside any of the elements to change the content.

Once you have tried a few of these applets you will find them intuitive to use.

Algebra Arrows for example could be used to compare different orders of operations. Build a tree, make the input x and note the output generated:

How to use Algebra Arrows

(I used Screencastomatic to create the thrilling video! This is very easy to use.)

Try this applet which shows how a solid is formed from a net, just move the red slider from 0 to 100.

There are several applets which are excellent for showing plans and elevations.  The applets work well on the interactive whiteboard for demonstrating to students, they are also ideal for students to explore themselves

Cube houses shows several models with their elevations, select drawing then 3d-model to give a model you can rotate  to generate different views.

Readers familiar with the excellent Improving Learning in Mathematics materials may recognise the applets for building houses; these are described in SS6 – Representing 3D shapes which has suggested lesson activities and describes the applets (see pages 4 and 5).

Building Houses allows you to create buildings and see the plan, front and side elevations as you build. (If that link does not work – try this).
You can add (build) or remove (break down) bricks and control the size of the square base.

Building houses with side views challenges students to construct 3D models given the plans and elevations; the task is made more challenging by specifying that as few cubes as possible should be used.

Note that in order to achieve the minimum number of cubes, ‘floating’ cubes are needed.

Note that these resources have been added to the ‘explore‘ series of pages on the companion blog for students.
Update – these resources worked well with my students – they particularly enjoyed the challenge of trying to build models using the minimum number of cubes!

Readers interested in the Improving Learning in Mathematics materials note the other resources including Interactive Whiteboard resources Malcolm Swan’s excellent Improving Learning in Mathematics – Challenges and Strategies.