# Frayer Models

Jonathan Hall’s resources are outstanding, Mathsbot is a go-to site for Mathematics teachers, I also really like his NonExamples.com which provides resources ideal for retrieval practice in Mathematics.

NonExamples.com – Jonathan Hall

I like all four formats used here. Seeing this site reminded me of the Frayer Model, I do like the clarity here, in one well-structured diagram we can have a definition and key characteristics which are illustrated very clearly by the use of examples and non-examples, so a great way to give notes and examples for students. As with all Jonathan’s resources, choices are available, many topics are available, or choose random for mixed questions.

On the subject of Frayer Models, I was most interested to see Peter Mattock’s announcement of his project on Frayer models. His website Frayer-Model.co.uk hosts a collection of Frayer Models in the categories Number, Algebra, Geometry and Measure and Statistics and Probability. These models have been created in Microsoft Word ™ so are fully editable.

Research has been undertaken on the effectiveness of the Frayer model for mathematics vocabulary instruction. See Effects of Mathematical Vocabulary Instruction on Fourth Grade Students by Eula Ewing Monroe and Michelle R Pendergrass. The paper states that a major implication of the study is that the CD-Frayer model is an effective method for teaching mathematical vocabulary. Note the examples on pages 21-24 illustrating the different models used in the study.

Students could be asked to generate examples and non-examples themselves, I suspect this would generate good discussions, certainly something I intend using this coming academic year. It seems to me that having students think about definitions rather than just been given them will help them remember them; see Daniel Willingham’s What will improve a student’s memory?

# Mathematical Miscellany #31

Underground Mathematics Review Questions
A project started recently to look at the various review questions available on Underground Maths to support teaching of the A Level specification now has further review questions added.

Read about the use of Review questions in the classroom on this Teacher Support page. The fact that these questions are designed to test understanding of one or more topics, exercise problem-solving and proof skills and help students make connections make them ideal for use in the A Level classroom. They can be used to introduce a topic or are also ideal for review, something which should be a regular part of linear courses.

For further questions to challenge your older students, see Challenging Mathematics Questions.

Posters and Displays

Looking at my blog statistics, I see the poster collection is very popular, it’s that time of year I think! I realised there were some missing gems and have added Ed Southall’s wonderful Mr Men collection (Note the two posts with poster packs and his delightful colouring book!

Also added is Clarissa Grandi’s display suggestions on her wonderful Artful Maths. Linking these two together, I do like the “Today I’m feeling…” collection!

Whilst there is much debate about classroom displays, see Mark Esner’s thoughtful TES article on the subject, we still may have corridors we would like to add some displays too or events where posters can be helpful. Many of the resources in the poster collection such as the Nrich collection  or Joey Kelly and Xi Yu’s Play With Your Math display very well on the whiteboard.

Puzzles

Earlier in the holiday I noted some puzzles in the post Puzzle Time; to add to those puzzles, some further puzzles for consideration from two favourite websites, Transum Mathematics and Mathisfun. Many of these would work as lesson resources.

For those still on holiday (the numbers are diminishing I know!) remember the #SummerMathsPuzzles from the Isaac Newton Institute for Mathematical Sciences. Each weekday throughout August the Institute publishes a maths-based puzzle.  Whilst the puzzles do not require sophisticated maths to solve, they are not be easy. A new puzzle becomes live each day in August and answers will be provided at the end of the month.

Mirror Writing
Visiting the Leonardo Da Vinci A Life in Drawing exhibition one sees many examples of his mirror writing. Curious to learn a little more, I came across information and additionally a lesson plan from the Museum of Science, Boston. Something a little different when teaching reflections perhaps?

UK Examination Results
I am keeping my Results 2019 page updated. The page includes links to all the results statistics and grade boundaries as well as links to ofqual’s apps and other news items.

By Colleen Young

# Challenging Mathematics Questions

…for age 16+

The following questions come from various university Admissions Tests but will be useful for any students really wishing to challenge themselves in Mathematics, particularly those students going on to study Mathematics beyond school.

Advanced Problems in Mathematics, Stephen Siklos. Step Paper I 1993

You can download a free copy of  Stephen Siklos’ Advanced Problems in Mathematics:Preparing for University. Whilst Advanced Problems in Mathematics has been written to support students taking STEP examinations, it is excellent preparation for any undergraduate Mathematics course. One aim of the book is to bridge the gap between school anduniversity.

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 75 problems. Each problem has a title which is a direct link to the problem.

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. The specification is available on the Cambridge Assessment website.

You will find free STEP and AEA (Advanced Extension Award) solutions on MEI’s website. The STEP question papers are all available on the Cambridge Assessment website, note the STEP resources include a searchable database showing the topic of each question.

The AEA qualification from Pearson is based on the A Level specification and designed for the top 10% of students to help differentiate between the most able candidates. Note that AEA papers and mark schemes can be found here.

Underground Mathematics has STEP questions within their Review Questions. Each question comes with a fully worked solution.

See also, from Cambridge University, their STEP Support Programme. From the home page, access the resources, you will see STEP Support Programme Foundation modules, STEP 2 modules and STEP 3 modules. Any Mathematics students aiming for the highest grades, whether they are sitting university Admissions tests or not can benefit from studying for example the Foundation modules which provide a structured introduction to solving STEP problems.

From Nrich, Prepare for University.

To challenge your students, MAT,  STEP and AEA questions are all excellent. Dr Jamie 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.

For mark schemes see:

Further support for the Mathematics Admissions Test is available on Dr Frost’s site. The resources include topic based worksheets also a problem solving booklet.; the booklet has 151 problems which should certainly keep any students preparing for Mathematics university interviews busy!

Note the Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutions.

TMUA is a newer admissions test only one question is available on the Underground Maths site, however, papers from 2016 onwards and mark schemes here and there is much overlap between the specifications for the TMUA and other tests such as the Oxford MAT, so these questions should provide useful resources for students taking the TMUA examination. Interestingly, Durham University states that “Those students already registered for MAT may substitute those results in place of our own test, if they do not wish to take both.”

Warwick University advise taking one of MAT, TMUA or STEP. As Warwick state, these questions deepen students’ understanding of the A-Level material.

As already mentioned, TUMA papers and mark schemes are available from Cambridge Assessment and I would highly recommend the presentation introducing the test, from Julian Gilbey. As suggested – try the questions first (pdf file) before watching the presentation.

Talking to Julian Gilbey, he recommends for the TMUA, the importance of working through the Extended specification notes on the website, to learn about the logic side. (See Test Specifications for the specification and enhanced specification.) He also stresses that the more Maths you can do, the more you work on stretching problems and think hard about maths the better you will get at maths. Examples he mentions for resources are any questions on the Underground Maths website, (not just the review questions already mentioned here), UKMT and olympiad problems, STEP problems (probably just STEP I initially).

“And essentially your ability to ‘think mathematically’ and to solve mathematical problems is all that these tests are testing”

For further sources try UKMT Senior Maths Challenge Questions.

Dr Frost Maths.com

…and also see MadAsMaths with its many papers and solutions increasing in difficulty.

# Posters

There are many excellent sources of free Mathematics posters available on various sites.

Jenny Eather’s Maths Charts includes over 200 posters on a wide variety of topics. (Jenny Eather’s dictionary is also excellent).

Nrich have turned many of their excellent problems into attractive posters. Note the link to download a zip file of all the posters.

On Play With Your Math, Joey Kelly and Xi Yu have taken problems and adapted them so  in their own words…

everyone (and anyone) can play. We design posters and handouts that hook you visually and explain the problem in just enough words. The problems that we’ve picked require trying, struggling, failing, adjusting, and trying again until, finally, a discovery is made.

The Classic Mistakes site has a wonderful collection of those classic mistakes that teachers regularly see. These posters can be downloaded in colour or black and white, a podcast explaining the mistake is also available for each poster.

The Mathematical Moments site features many downloadable pdfs, posters that show the role that Mathematics plays in Science, Technology and Human Culture. A short or more detailed version of each poster is available and a search is provided.

On TES Resources Owen has created an excellent set of A4 posters inspired by Ian Stewart’s ‘17 Equations that Changed the World‘.
(You will need to register with TES (free) to download any resources).

You can read more about the 17 equations on The World Economic Forum where for each equation we have the following information:

• What does it mean?
• History
• Importance
• Modern use

The equations posters together with this information would make an interesting display.

Now available in 7 additional languages, try this wonderful collection of STEM Role Models posters from Nevertheless.

……

The excellent Maths Careers site includes many posters to download, scroll down the page to see a great collection. Note also the booklet on Where the Maths you learn is used.

Math Gems from Thinkzone has a collection of free posters described as an assortment of mathematical marvels – these won’t use up  your colour inks!

Ed Southall’s posters feel particularly appropriate as in 2021 as we celebrated 50 years of Mr Men and Little Miss.

Look at Clarissa Grandi’s display suggestions on her wonderful Artful Maths. I do like the today I’m feeling….idea! Or perhaps, Maths around us which includes a selection of images (all labelled for noncommercial reuse).

On David Morse’s excellent site, Maths4Everyone, we can see he is developing a section on Posters and Displays; in the meantime, there are some available on TES.

From Plus Magazine – a series of posters for display in classrooms.

Also, note lots of Maths for your walls!

For a collection of Numeracy Across the Curriculum posters, have a look at this collection which includes posters on Numeracy for a variety of subjects. I came across these on Twitter; looking at the address, thank you to Inveralmond Community High School.

Note Mark Horley’s comment below, Mark has created a Mathematical Timeline. Mark has made the timeline available as a Word, also pdf document.

I noticed Mark refers to Henry Dudeney on the 1917 and 1940 posters. For puzzle lovers, you can get a free puzzle book by Henry Dudeney, Amusements in Mathematics. I referred to this book on my free books page; having just checked, Amusements in Mathematics is still available free. The first set of puzzles will offer a trip down memory lane for those who remember money – pre-decimal! There are several categories of puzzles available.

From Tomorrow’s Engineers, you will find many careers resources including posters.

MEI (thank you) alerted me to this poster – 100 jobs in STEM!

You may want to make your own signs or posters, try this brilliant Mathematical alphabet from Sarah Carter on mathequalslove.net.

Or if you want any calculator symbols – try Casio’s collection of fonts.

For making your own vocabulary posters, using a word cloud generator such as Word It Out makes the job easy. Simply enter some text and create your word cloud. You can then share your cloud if you want to. It is also possible to embed it on your blog or website.

Note that you can use the tilde (~) character to keep words together.
A great feature for mathematicians – WordItOut can easily cope with the equals sign, division signs also – see for example this equation. It seems other symbols are possible also.

Various options are available, you can change the colours of your words and the background colour, you can also change the size of the font and the variation in size allowed.

Emma made our Year 7 class a poster of the vocabulary needed for our work on shapes and here is a very simple example on Linear Sequences.

Tagxedo is another possibility, this also allows a choice of shape.

….and on a lighter note, my son sent me this from GraphJam.

# Have you seen…?#8

#SummerMathsPuzzles from the Isaac Newton Institute for Mathematical Sciences?

Each weekday throughout August the Institute will be publishing a maths-based puzzle.  Whilst the puzzles will not require sophisticated maths to solve, they won’t be easy.

A teaser puzzle, £8.19, is available now, a puzzle where two players, both perfect logicians and strategists, play.

A new puzzle will become live each day in August and answers will be provided at the end of the month.