Mathematical Miscellany #1

Just over three years ago I started writing compilation posts, a kind of newsletter style – a mix of mathematical goodies. I was inspired by Doug Belshaw’s Thoughts This Week – now Thought Shrapnel. It’s been on my blog to do list to revisit this idea again; how appropriate that Doug’s Thought Shrapnel today includes newsletters he enjoys. I was very happily distracted by some of these great compilation posts this morning and particularly struck by a post from Doug’s first suggestion Austin Kleon – important for any blogger, What to leave out and what to leave in. Some great quotes there including Elmore Leonard’s:

“Try to leave out all the parts readers skip.”

(Wondering which bits you are all skipping here!)

So I decided it was time for a new title for posts of this kind – hence ‘Mathematical Miscellany’. Previous compilation posts can be found under the category (note the drop down menu on the right for categories) Mathematical Miscellany.

In class
With my Year 10 GCSE class I used some ideas from a site I often use: Mudd Math Fun Facts. Squares Ending in 5 and Multiplication by 11 both made excellent starters, we looked at proofs as well as enjoying the mental Maths tricks! You will find more lightening arithmetic suggestions on the site.

Resources
OCR have published some very useful new resources; I have written before on their Check in tests. New tests for Foundation and Higher have been added. The Higher resources include Triangle mensuration, Language of functions and Algebraic expressions; these really are excellent resources and I will definitely be using all of them.

The GCSE New Content pages are updated regularly, note all the further resources at the end of that page, including from the wonderful Just Maths, GCSE questions by topic.

Assessment
Published this month we have the KS2 Mathematics 2016 teacher assessment exemplification, and also the KS1 version. Clearly teachers of secondary age children need to know what students are covering in Primary school. These documents have been added to the GCSE New Content page.

If you read just one blog post this week – from 2014, but Tom Sherrington’s 10 Silver Arrows: Ideas to penetrate the armour of ingrained practice is as valid now as it was then. I think these ideas are important for any teacher to reflect on – I’m pleased to see 6, a subject I have often written on (see Mini Tests).

“Tests are good.”

…and finally
It’s coming up to that time of year again, I see Valentines cards in the shops, save your money and send your loved ones (or anybody!) a math-o-gram!

Click on the image and move that slider!

In what happily seems to have become an annual tradition Desmos have provided you with the means to send a math-o-gram to the mathematicians in your life!

Geeky people you could even use the Desmos API …

Valentine’s Day seems an appropriate time to express love for Desmos!

Elsewhere – express your feelings for WolframAlpha!

Mathematics Stories

MEI M2 question

Working with my Further Mathematics students on a work / energy question based on the Jack and Jill nursery rhyme started a lively discussion on what other stories and rhymes we could turn into Mechanics questions! We all immediately thought of Humpty Dumpty! I think the examiner was having fun with this one – the question carries on with poor Jack falling over and hurting himself. Jill seems to get off lightly unlike the original! If anyone is interested this is question 4 on the June 2007 MEI M2 paper.

This theme also reminded me of a long forgotten resource I used to use with Year 7 – The Three Little Pigs as a mental test – this is definitely coming out again!
I found this many years ago on MathsStories.com as a free sample.

I cannot mention Mathematics stories without reminding everyone of  ‘John and Betty’s Journey into Complex Numbers‘ which has always made me smile.

John & Betty’s Journey into Complex Numbers

Staying with the subject of Complex Numbers, with the same students, another discussion we had recently when looking at Complex Numbers was the need to write clearly, a z can look like a 2 for example. See Mathematical Handwriting on Mathematics for Students. Also on Mathematics for Students I have updated the page giving links to A Level Exam papers to include some really useful solutions resources such as David Smith’s worked solutions.

O

Multiple Choice Questions

Daisy Christodoulou on Life After Levels – conclusion

Daisy Christodoulou on Multiple Choice Questions

From researchEd 2015 and highly recommended, Daisy Christodoulou discussed the removal of National Curriculum Levels. Read Daisy’s post and see her slides on her blog you can also watch the presentation.

In that presentation, Daisy discussed the use of Multiple Choice Questions, something I have always liked using in my teaching. Daisy’s discussion of making questions harder by changing the number of correct answers reminded me of the, in my opinion excellent, Multiple Choice A Level papers which the then London board included in their A Level Mathematics examinations (London Syllabus B) (see an example below).

In the meantime, happily we do have access to some Multiple Choice questions online and I thought it would be useful to bring them together in one place.

Diagnostic Questions

I have written several posts on the excellent Diagnostic Questions site which hosts thousands of multiple choice questions written by teachers designed to address student misconceptions. I have found it worthwhile to discuss the wrong answers with students so we are all aware of the kinds of mistakes it is easy to make. It is a site I use regularly, teaching Year 10 about the equation of a tangent to a circle at a given point for example, I created a Desmos page and a Diagnostic Questions Quiz using questions on circles (centre the origin) from Diagnostic Questions.
(pdf: quation of a Circle & Gradient of Tangent).

Note the various collections on the site.

Some Advanced Level Pure topics are also covered on Diagnostic Questions; see for example this quiz I created on  Logarithms and Exponentials.

Phil Bruce – Multiple Choice Starter Questions

On TES resources, author pbrucemaths has a brilliant collection of clearly indexed multiple choice starter questionsPhil Bruce has aimed to cover every objective on the GCSE syllabus.

Colin Billet – GCSE Multiple Choice Questions

Also on TES, Colin Billet has created multiple choice questions from old GCSE papers, one set is for the higher tier and another for foundation.

Mega Maths Quiz from Ben Cooper

For a superb collection of Multiple Choice questions, great for starters, plenaries or any time, head for Ben Cooper’s resources; see for example Mega Revision from Ben Cooper and 30 Number StartersResources by Ben can also be found on TES.

Don Steward – Rearrangement Steps

Don Steward has so many outstanding resources on his Median blog, including multiple choice quizzes, see this on rearrangement for example. Note that he also has a separate blog for GCSE practice resources. many of these quizzes are multiple choice.

From UKMT, the UK Maths Challenges provide a wonderful library of multiple choice problems; note the past papers where you will find questions and full solutions. Check also the Individual Challenges pages where you will find details of the challenge and see sample materials. Check the Junior Challenge for example. Note the addition for 2015 of extended solutions.

American Mathematics Competitions

Another brilliant set of multiple choice questions comes from another Mathematics Challenge, the American Mathematics Competitions (AMC). There are several competitions which increase in difficulty by student age. Problems and solutions are available. I do like this problem, a good one to use when teaching Systematic Listing Strategies perhaps.

Mathisfun has an extensive library of very clear diagrams and explanations and also multiple choice questions for most topics. Use the Index to find the topic you want and note that for most topics you will see some questions at the end.

mathisfun – Simplifying Square Roots – select image

As you can see from the Index all ages are catered for including older students; I have used the clear explanations and questions on finding the inverse of a 3×3 matrix with Further Mathematicians for example. Once you have selected an answer a complete solution is provided. Note the Question Database – some exploring to do I think!

The Who Wants To Be A Millionaire format can provide a fun way to present Multiple Choice quizzes. A Google search returns various resources and of course, provides you with PowerPoint Millionnaire templates you can adapt for your own use. Some highlights from that search, the Primary Resources files use a simple and clear format and William Enemy has described a resource on Great Maths Teaching Ideas which also uses that template; such a good idea to have all students answer all questions and add up their winnings! Another example – Algebra on TES.

For older students the old London A Level Multiple Choice papers could be very useful, I have scanned one of my old collection and will add more.

For an extensive collection of Challenging questions for Advanced Level students, we can turn to the Oxford Maths Admissions Test, note the menu on the right-hand side of the page, many papers and also solutions are available. I can never resist a quick graph on Desmos but note the very comprehensive solutions provided. For more Oxford MAT questions see this extensive collection on Underground Mathematics. These Underground Mathematics resources will provide so much more than the question in each case. Note the suggestions and solutions.

There are various options for creating your own multiple choice questions. That Quiz is simple but effective – all free and no adverts. There are many quizzes already available on a variety of topics, it is also possible to create your own quizzes. Teachers can register and add classes if they wish. You can search the many quizzes available, searching for Fractions, for example, led me to this quiz.

Testmoz

For an alternative way to set up a simple quiz try Testmoz. No registration is required. This has been written by Matt Johnson, an undergraduate student – the instructions are all very clear and you can check out the FAQ!  (I love those FAQ! For example: I lost my quiz URL can you retrieve it for me? Answer: No). Try this test on Directed Numbers – log in as a student, the passcode is cy090610

More to Explore!

In Daisy’s talk mentioned at the beginning of this post she mentioned British Columbia questions in her discussion of multiple choice questions, a quick search led me to this Pre-Calculus paper for example...

If you want to write your own questions, you may find the references in Learning Scientists‘ post on Multiple Choice Questions useful (Smith, 2016).
Bibliography:Smith, M. (2016) Weekly digest #4: How to write good multiple-choice questions. Available at: http://www.learningscientists.org/blog/2016/4/3/weekly-digest-4 (Accessed: 4 April 2016).

Algebra – expand three brackets

Following a recent test for Year 10 (UK age 14-15) I wanted to use colour to show clearly a way to make sure all terms have been included when three brackets have to be expanded (new to the UK GCSE qualification). (Colour can be so useful in Mathematics – something I have written on before).

I have created this example using Excel. This is just a simple example with positive coefficients only; I wanted to start by being clear on making sure that all combinations are included. There are various FOIL and quick methods available if you search but my own preference is a more systematic approach. An advantage of using Excel is the ability to trace precedents (Formulas menu).

The numbers could be changed and of course any examples could be made up and checked on WolframAlpha.

I have also collated some Algebra resources (many topics) – examples and exercises on Mathematics for Students. Note the very clear workbooks from Plymouth University

Functions – Plymouth University