February 29th

Mathematics for Students

Firstly, A little more on why we have a leap year from The Telegraph, and/or watch this video from Numberphile.

Leap Year Mathisfun Mathisfun – Leap Years

We can also find details on Mathisfun

Leap YearThe Google Doodle for today celebrates Leap Year

For any Excel users this formula will determine whether or not a year is a leap year:
=IF(OR(MOD(A1,400)=0,AND(MOD(A1,4)=0,MOD(A1,100)<>0)),”Leap Year”, “NOT a Leap Year”)
Leap Year Excel

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World Book Day

Image – Klara Kim on flickr

Image – Klara Kim on flickr

On Thursday 3rd March it was World Book Day.

On the subject of books I have completely updated the free books information; this includes new additions. Note in particular Colin Foster’s Instant Maths Ideas – lots of ideas you can try in the classroom.

We could bring books and Mathematics together with some Statistics (at any time); have a look at this resource from TES, World Book Day Maths Data Investigation where students analyse word length and sentence length in some book extracts. UK readers who remember Statistics coursework, this brings back memories of AQA’s coursework task ‘Read All About It’ where students considered various newspapers and magazines for readability.

AQA - Read All About It

AQA – Read All About It

The extension task for the TES resource above considers the reading age of a text, you may wish to consider further readability formulae; if you paste some text to this site, Readability Formulas you can easily check statistics for your chosen text and generate a reading age according to the various tests.

WolframAlpha can be used for Words and Linguistics, note the various examples given, including number names.and document length.

 

 

Writing - word lengths

Analyze My Writing

Alternatively, try Analyze My Writing. Simply paste in some text for a comprehensive analysis including basic statistics, word and sentence length and readability. It is also possible to create Cloze tests.   You can read more about this resource on Richard Byrne’s always impressive “Free Technology for Teachers”.

It seems appropriate to check some world records on books!

I tried the text of this post for readability – college level!

Mathematics Proof Resources

Twitter chat

Maths CPD on Twitter – NCETM

Following a Twitter #mathscpdchat I thought a quick post noting some resources for proof would be useful. The chat is summarised on NCETM here.

Proof Resources - Mark Greenaway - Soffolk Maths

Proof Resources – Mark Greenaway – Soffolk Maths

From Mark Greenaway‘s Suffolk Maths site comes this excellent and varied set of resources on Mathematical Proof

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!

Nrich have 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.

Institute of Education - Proof Survey

Institute of Education – Proof Survey

Maths & Medicine

Maths & Medicine

Jo Morgan (Resourceaholic) alerted us to a very interesting resource from the Institute of Education, University of London, a proof survey with some great questions for Year 10. Looking for more information about this, I found another for Year 9!

This was a lively chat with many contributions giving ideas and resources – the chat is clearly summarised as mentioned above on the NCETM website.

Statistics

Following last week’s post on Probability, time to check the Statistics content for the new GCSE specification. To start at the beginning again we can look at the Mathematics GCSE subject content and assessment objectives for Statistics; we should also check the Key Stage 3 content to see what we should have done already!

KS3 Statistics Content

KS3 Statistics Content

KS4 Statistics Content

KS4 Statistics Content

Note that at GCSE, all students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content. Only the more highly attaining students will be assessed on the content identified by bold type.

Each examination board has clear GCSE specifications, all included in the Further resources section of the GCSE New Content – Resources page. Specimen and Practice Papers provide us with important clues as to how content will be assessed and thanks to the wonderful Mel Muldowney of Just Maths we can see examination questions by topic, which of course includes Statistics. These questions have been collated by Mel as the basis for a GCSE working party set up by the GLOW maths hub. (You can see more on the work of the Maths Hubs here). Whatever we are teaching, referencing the specification, any exemplification documents and also examination questions is essential in understanding how students will be assessed.

There are some changes in content for Statistics if you have been teaching the old specifications; note we have lost specific reference to the Data Handling Cycle as well as Stem and Leaf diagrams. Also missing, I see no reference to stratified sampling for example. We have gained tables and line graphs for Time Series data.

The DfE Key Changes to GCSE Maths document explains ‘No mention of statistical problem solving / data handling cycle’ by saying that it is ’embedded into other subjects such as science and geography where it is explored experimentally’. (I find the phrase no mention of statistical problem solving puzzling).

Looking for further clues on Time Series data, AQA’s Teaching Guidance states that students should be able to:

  • understand that a time series is a series of data points typically spaced over uniform time
    intervals
  • plot and interpret time-series graphs
  • use a time-series graph to predict a subsequent value

The sample available (the full document is available to registered members of All About Maths) of AQA’s excellent Teaching Guidance document is in fact the section on Statistics.

We can also see some questions from Edexcel via Just Maths.

From OCR, the very useful series of Check in tests (select link and scroll down the page) includes Interpreting and Representing Data and Analysing Data. OCR’s collection of GCSE Teaching and Learning resources also includes a (provisional) Delivery Guide for Statistics which includes suggestions for activities and resources in teaching Statistics.

Nrich - StatisticsFrom Nrich see the short problems on Handling Data which are based on the UKMT Junior and Intermediate Maths Challenges; problems are available on Processing and Representing Data and there a small number of short problems on Interpreting Data. Longer problems are available in both categories, see Processing and Representing Data and Interpreting Data. As always from Nrich these are excellent resources to get your students thinking, see M, M & M for example, accessible to all and a great example for demonstrating the need for being systematic. Note the follow on problem, Unequal Averages to extend thinking on averages.

Standards Unit S4The outstanding Standards Unit resources which really help understanding includes some excellent Statistics activities, see S4, S5 and S6. I have often used S4 on Understanding Mean, Media, Mode and Range.

Note the above link includes a PowerPoint to introduce the activity and also gives the solutions. I used the associated software to generate the images for the solutions.

There are numerous examples of Statistics resources in various posts and pages on this blog. David Millward’s PowerPoint collection includes presentations on Statistics.

Guess the correlation

Guess the Correlation – Omar Wagih

….and to finish, some games!

Probability

The new GCSE specification has some changes in the content for Probability. Time for a look at that content and some resources we can use to teach it.

To start at the very beginning we can look at the Mathematics GCSE subject content and assessment objectives for Probability; we should also check the Key Stage 3 content to see what we should have done already! It is worth noting that Probability has been removed from the Primary curriculum so this will be new at Key Stage 3.

KS3 Probability Content

KS3 Probability Content

GCSE Probability Content

GCSE Probability Content

Note that at GCSE, all students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content. Only the more highly attaining students will be assessed on the content identified by bold type.

Each examination board has clear GCSE specifications, all included in the Further resources section of the GCSE New Content – Resources page. Specimen and Practice Papers provide us with important clues as to how content will be assessed and thanks to the wonderful Mel Muldowney of Just Maths we can see examination questions by topic, which of course includes Probability. These questions have been collated by Mel as the basis for a GCSE working party set up by the GLOW maths hub. (You can see more on the work of the Maths Hubs here). Whatever we are teaching, referencing the specification, any exemplification documents and also examination questions is essential in understanding how students will be assessed.

AQA Teaching Guidance

AQA Teaching Guidance

Looking at AQA’s excellent Teaching Guidance document for example and searching for Probability, as well as exemplification questions we can see suggested links to other areas of the syllabus. Probability could be assessed with N5, Apply systematic listing strategies for example, or perhaps with Ratio or Algebra problems. This is a document I personally reference constantly in my teaching for the new GCSE specification, the questions are excellent and show clearly the changes in emphasis on the new specification. Note that there is a sample available and the full document is available to registered members of All About Maths.

Probability - New Content

Probability – New Content

Formulae

GCSE: Formulae

If you have been teaching GCSE for a number of years then perhaps the biggest changes will be the inclusion of Venn Diagrams and Frequency Trees and of course the emphasis on Problem Solving including the kind of question like the now well known Hannah’s Sweets question from Edexcel combining Algebra and Probability. Also note that although the formulae shown here are not specified in the content they should be derived or informally understood by candidates. These formulae will not be given in the examination.

As well as the resources already mentioned, Venn Diagrams and from TES, Frequency Trees we have Prize Giving from Nrich, note the Frequency Tree representation.

Further resources follow:

AQA Venn Diagrams

AQA – Bridging the Gap

From AQA’s excellent Bridging the Gap resources: Set notation, number lines and Venn Diagrams

From OCR’s Check In tests, we have for the basics, Basic probability and experiments and then to follow, Combined events and probability diagrams.

 

Nrich Short Problems

Nrich Short Problems

From Nrich see the resources below; the short problems are based on the UKMT Junior and Intermediate Maths Challenges:

 

Standards Unit

Standards Unit

The outstanding set of Standards Units resources includes, S1, S2, S3 and S7 on Probability.

From Kangaroo Maths, for younger students (KS2, KS3) see the ‘Build-a-Mathematician’ tasks; the stage 8 and stage 9 resources include Probability and Tree Diagrams respectively.