From the KS3 National Curriculum we see the above on algebraic notation, see also pages 56-66 of the Teaching mathematics at key stage 3 guidance. The guidance covers the entire KS3 curriculum and includescommon difficulties and misconceptions, examples for use in lessons, and suggested questioning and other strategies for teachers to use.
KS3 Guidance example
The following slideshow includes several resources you can use with students for practice in writing algebraic notation.
Included you can see Jonathan Hall’s Worded Expressions, as always with MathsBot resources we have lots of choices – for example, hide either the sentences or expressions. With the ability to generate new expressions we have an endless supply; also from Jonathan Hall, see his Forming Expressions, these resources are ideal for self-study as well as for use in class.
Mathsbot
Standards Unit – A1
Smile activity
Jo Morgan
Don Steward
Mathsbot
Andy Lutwyche
Maths4Everyone
Transum Mathematics
Corbett Maths
From Don Steward, we have translating English to algebra, expressions, see also translating English to algebra, relationships. Also included here is an activity, A1 from the Standards Unit on Interpreting algebraic expressions. This includes 4 card sets to match, ideal for looking at multiple representations, students match algebraic expressions, explanations in words, tables of numbers and areas of shapes. One of the goals of the activity is to help learners to translate between words, symbols, tables, and area representations of algebraic shapes. The Standards Unit resources can all be accessed without a login from the very clear to navigate University of Nottingham site linked to in the Standards Unit post.
One of Chris McGrane’s Starting Points MathsCurriculum Booklets – Algebra 1 from Phase 3 features some great activities for writing algebraic statements, featured on the slides you can see a Smile activity, and Jo Morgan’s lovely Introduction to Writing Algebraically – this is such a good idea, as Jo says in the resource description if they know how to do it with numbers, then they just do the same thing with the algebra.
Further excellent resources on this skill are available on Maths4Everyone.
On Transum, Writing Expressions is an exercise with a difference, listen to the audio then type in the expression.
What a great puzzle! Jonny Griffiths is aiming these puzzles (which always have a unique solution) at late GCSE/early A level Maths students. As Jonny says, this type of puzzle seems to engage students fast, making a great starter that revises important theory quickly; the aim of a Digitiser puzzle is to both practice and teach (or reteach) a piece of mathematics,
The image shows a simple sample task, you can find the solution on his website. The Digitisers pdf file is free. Clear instructions explaining the puzzles and notation used are given, we then have all the tasks by topic to help you pick your task – brilliant! Each task has a difficulty rating from 1 to 3 stars. Full solutions are provided.
This is a wonderful resource – puzzles like this go down well with students, but to have them all clearly by topic is perfect – thank you Jonny, for yet another amazing resource!
Digitisers – Jonny Griffiths – tasks by topic
Choose Completing the Square for example and we have:
Jonny Griffiths – Digitisers
Staying with starters for A level, for linear A Level courses Retrieval Practice is essential. From crashMaths, these AS Maths Key Skills Check worksheetsare very valuable for Year 13 in the second year of their A Level course. The Skills Checks are all on Pure Mathematics and make ideal lesson starters.
When working through solutions, take every opportunity to illustrate with technology.
Andy Lutwyche – Erica’s Errors series
Andy Lutwyche’s collection of Erica’s Errors where students must identify errors in solutions can be an ideal starter for either retrieval practice for an earlier topic or to consolidate learning for a current topic.
Stoke Maths
Stoke Maths MEP Starters are very attractively presented high-quality resources. Looking at the Spot the Mistake PowerPoints for example, as you can see in the image below there are a great collection of questions that include full answers. It’s great to see Mechanics and Statistics collections. The revision question starters provide very useful question sets.
On Transum try Advanced Starters, some of which I think could be useful for students aiming at the highest GCSE grades as well as for Advanced Level students. The problem, Find the Radius, illustrated in the tweet is very neat!
Hidden gems on the Transum website #25. Advanced Starters https://t.co/6uC2H8tp0c Although there are 366 Maths Lesson Starters available (One for each day of the year) you may not know that there are also Starters suitable for Year 12/13 or High School Senior level. #Mathschatpic.twitter.com/86xN4GJ3x7
Looking at the Main Transum Starters page I see at the foot of the page we have various categories of starters including the Advanced Starters. I see many starters here I like, looking at Coordinate Distance, I can never resist a Desmos page to illustrate a problem! This starter could be also be used to review some coordinate geometry – find the midpoint? Find the equation of the line?
Quad MidpointsJethwa Maths – 10 minute starters
On Jethwa Maths you will find starters for Mathematics and Further Mathematics A Level.
From OCR (MEI) their Foundations of Advanced Mathematicslevel 2 qualification covers arithmetic, algebra, graphs, trigonometry and statistics. Assessment is by a two-hour examination that consists of 40 multiple-choice questions. As OCR suggest these questions could be used for diagnostic tests.
The UKMT Maths Challenge questions are excellent for students of all ages. This post includes a selection of links and resources to access the Maths Challenges.
For superb resources for the Oxford Admissions test multiple choice questions see these Underground Mathematics Review Questions where you will find not only the questions but suggestions and complete solutions.
At the start of my teaching career I really liked the first paper of the Mathematics A level from the University of London School Examinations Board – thirty multiple choice questions to complete in one hour, 15 minutes.
For questions 1 to 20, candidates had to select one answer from 5 and for questions 21-30 the instructions were as follows.
The pdf file here has the paper, followed by the exam board answers followed by notes from the 1986 version of me! These days I would illustrate with Desmos and/or WolframAlpha for example as well where appropriate.
Note the comment from Graham Cummings below, there are further papers available in the Edexcel’s Emporium:
The Emporium has some 17 multiple-choice question papers from the period 1988-1992 – by no means a complete set, but they range across the Mathematics, Further Mathematics, Pure Mathematics and Applied Mathematics syllabuses. You can find them in the “Pre-C2000” cabinet within GCE AS/A Level.
Signing up to Mathematics Emporium is highly recommended, note that it is a free website intended for the use of teachers of mathematics in secondary schools, regardless of what board you use. Register for an accountand ensure you supply a correct centre e-mail address in your name for verification, your centre name and centre number.
On Transum Mathematics, we now have Equatero which John Tranter created having seen the TV game show, Lingo.
Transum Mathematics
The rules are simple – find the calculation – any numbers or symbols in the correct place are shown as green, any numbers or symbols that are correct but in the wrong place will be yellow and any numbers or symbols that are not included in the calculation will be red. Rather an addictive game – but plenty of thinking! This has been added to the Number collection in the Puzzles & Games series.
It’s great to see Jonathan Hall adding manuals for his great resources on MathsBot. Here is Bar Modellingfor example.
See post on Bar Modelling for problem examples and resources.
Thinking Blocks – Math Playground
It’s really helpful when sites have a clear ‘What’s new?’ type section or you can easily search by recent additions and/or subscribe to a newsletter. Just a few examples from the many excellent sites for Mathematics:
On Transum see Breaking News where you will find information on new and updated resources. Current news includes the resources mentioned above, Equatero and Vocabero.
What’s new at Desmos? This is really comprehensive – a great way to keep up to date.
On Dr Austin Mathsselect New from the Menu, I see some more of her always high quality resources – I’m a fan of Fill in the Blanks (as well as all her other resources!) For a whole collection of Fill in the Blanks type resources from various sources, see this post.
I must start this post with something Dan Meyer said at the MEI Conference 2021 that really struck a chord with me, he said “There are no mistakes or misconceptions, just takes and conceptions.” Dan Meyer mentioned WW Sayer who said:
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong’. Rather he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole art of teaching.
WW Sayer
Nevil Hopley’s excellent Classic Mistakes site.
For a starter addressing common misconceptions try the excellent Classic Mistakes resources by Nigel Hopley.
diagnosticquestions.com
A superb resource to use in class (or for students to use at home) to address misconceptions is of course Craig Barton’s and Simon Woodhead’s Diagnostic Questions site. The site has many thousands of questions with carefully designed multiple-choice responses to address common misconceptions.
The Insights feature is so helpful for learning about misconceptions, suppose we look at a White Rose Quiz on Algebraic Notation, for example, looking at the Insights we can see for any question the number of responses for each option from the many students who answered this question.
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NCETM – resources for supporting KS3 Maths Topics
From NCETM, these videos and resources for teaching Key Stage 3 maths topics include common misconceptions and pitfalls; looking at Directed Numbers for example we find slides and a pdf support document including as illustrated here, “What things typically go wrong?”
Some years ago a website, counton.org which is now no longer available published a very useful document on misconceptions. In 22 sections, in each section misconceptions are given along with the correct version. Further explanations are also provided and also follow up exercises with answers.
The above pdf document includes all 22 sections. The first 8 of these documents, by Ilan Samson & David Burghes, are on the CIMT website.
Malcolm Swan’s excellent ‘Improving Learning in Mathematics‘, includes a section (5.3) on exposing errors and misconceptions. An activity suggested there is to let your students become examiners and mark the work of others, this works very well, I have highlighted some excellent resources for this on the ‘Spot the mistake!‘ page.
On the SERP website before see MathByExample and AlgebraByExample which is a set of Algebra 1 assignments that incorporate worked examples and prompt students to analyze and explain. These resources can provide prompts for discussing common misconceptions.
From Michael Pershan, see his Math Mistakes site, to quote the Author:
The purpose of this site is to collect, organize and make sense of the mistakes that students make while doing math. I’m also increasingly interested in using mistakes to help us create worked examples that students can learn from.
All the examination boards publish helpful material which addresses common misconceptions, such resources can promote very useful class discussion as can examiners’ reports.
Edexcel’s A Level scheme of work which is freely available on their website includes for each section, Common misconceptions/examiner report quotes. The point made in the notes which follow is such an important comment, I have seen many errors that would have been avoided if only students had looked a picture of their work.
