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.
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.
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.
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!
Choose Completing the Square for example and we have:
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.
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!
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?
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.
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.
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 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.
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.
For a starter addressing common misconceptions try the excellent Classic Mistakes resources by Nigel Hopley.
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.
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.
It’s that time of year again and we can play the 2022 NCTM Year Game in our January lessons. Full rules are here; that’s a lot of 2s, multi-digit numbers such as 20, 220, or 0.22 are accepted for 2022 though note that students are encouraged to find solutions using only the single-digit numbers 2, 0, 2, and 2 rather than double digits like 0.2 or 22.. , students are also encouraged to challenge themselves to use the digits in the order 2, 0, 2, 2.
Playing this with younger students has been an opportunity to introduce the factorial function, and we tend to stray into double factorials as students are curious. A good exercise in algebra for your older students – can they find a relationship between the single and double factorial functions?
2022 is also an iban number – this has amused me for a long time – get your students thinking outside the box with the iban sequence: 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, 41, 42, 43, 44, 47, 70, 71, 72, 73, 74, 77, 100, 101…
We can also look at WolframAlphafor further information on the number properties of 2022 including what 2022 looks like in historical numeral forms. We could use the various historical numerals examples to learn how Babylonian, for example, numerals work. I have successfully used this as an interesting starter for January lessons.
The Babylonian system was a positional base 60 system, though interestingly uses ‘units’ and ‘tens’ symbols to create the 59 symbols needed.
For more on the Babylonian system including how fractions were represented see History of Fractions from Nrich.