Algebraic Notation

Algebraic notation – National Curriculum

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 includes common 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.

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.

From Corbett Maths
Algebra: expressions – forming     Video 16    Practice Questions    Textbook Exercise

16.     Algebra: expressions – forming     Practice Questions answers       Textbook answers

From Andy Lutwyche – Algebraic Expressions Spiders

Here’s an interesting query type on WolframAlpha – simple word problems. See more examples of Word Problems (and All Examples by Topic).

WolframAlpha – simple word problems

From my post on Bar Modelling see The Mathenæum from Ken Wessen which includes Modelling Word Problems.

The Mathenæum – Ken Wessen

A Level Starters

From Jonny Griffiths a new publication – Digitisers.

Digitisers – Jonny Griffiths

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 worksheets are 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.

crashMaths Skills Check

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.


From Ben Bently, comes a collection of low stakes quizzes using questions from Diagnostic questions.


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!

Transum Mathematics

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 Midpoints

Jethwa 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 Mathematics level 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.

Choose Past Papers, mark schemes and reports. Papers and mark schemes from June 2007 are freely available.

OCR Foundations of Advanced Mathematics
OCR Foundations of Advanced Mathematics

We could of course illustrate the solution well with the use of a little use of technology!



A source of multiple choice A Level questions, particularly for Pure Mathematics is Diagnostic Questions; see for example this quiz I created on  Logarithms and Exponentials.


DrFrost UKMT Algebra
Dr Frost Maths

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.



A trip down memory lane! (Edexcel’s Emporium has some papers from this era.)
University of London June 1986 Mathematics 1

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.

Multiple Choice Example

For questions 1 to 20, candidates had to select one answer from 5 and for questions 21-30 the instructions were as follows.
June 86 Mathematics 1

University of London June 1986 Mathematics 1

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.
Colleen Young answers

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 useRegister for an account and ensure you supply a correct centre e-mail address in your name for verification, your centre name and centre number.

Edexcel Emporium

Mathematical Miscellany #63

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.

We also have a Maths words version – Vocabero.


MathsBot – Bar Modelling Tutorial

It’s great to see Jonathan Hall adding manuals for his great resources on MathsBot. Here is Bar Modelling for 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.

From NCETM, see News and Features.

Nrich – sign up for their Newsletter.

What’s new at Desmos? This is really comprehensive – a great way to keep up to date.

On Dr Austin Maths select 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.

MathsBot was mentioned earlier, keep up with Jonathan Hall’s latest MathsBot resources.

Jonathan Hall – MathsBot

An absolute favourite resource Author – Andy Lutwyche, search his resources by newest on TES. I see some brilliant new Erica’s Errors on Mechanics.

Andy Lutwyche – Erica’s Errors series

On Purposeful Maths from Phil Bruce and Martin Green, see What’s New; we see for example added in December, Adding Negative Numbers.

Purposeful Maths

Starting Points Maths is arranged by date, most recent first, you can also look at Latest Tasks by topic area.

Starting Points Maths

The task continues with further questions and students draw a graph; I really like the way students see the multiple representations here.

From MEI – on Integrating technology into your scheme of work, we see published in December, GeoGebra resources for Further Maths A level.

MEI GeoGebra Tasks

Misconceptions in Mathematics

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

classic mistakes 2
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.

….


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.

Andy Lutwyche

See for example from Andy Lutwyche, his excellent Erica’s Errors series for Spot the Mistake activities or, also, on TES, Andy’s Clumsy Clive series. Andy’s many Spot the Mistakes resources.

Many of the outstanding resources from the Standards Unit address common misconceptions; see PD2, learning from mistakes and misconceptions.


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.

SERP – Algebra by Example

From this page you can access all the resources.


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.

Michael Pershan

m4maths
m4ths.com

Steve Blades’ site www.m4ths.com has many excellent resources; on the GCSE page we see under ‘Miscellaneous Worksheets‘, several documents including 18 Common Misconceptions.


All the examination boards publish helpful material which addresses common misconceptions, such resources can promote very useful class discussion as can examiners’ reports.

From Cambridge University, see Common Errors in Mathematics.


Happy New Year 2022

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?

Excel has a function for computing double factorials, illustrated here. I like to show my students a few examples and see if they can work out what is going on.

Have a look at this article from Wolfram Math World and check this journal article – Henry Gould, and Jocelyn Quaintance. “Double Fun with Double Factorials.” Mathematics Magazine 85, no. 3 (2012): 177–92. https://doi.org/10.4169/math.mag.85.3.177.


And so to number properties of 2022…

Number Gossip

We can always check Tanya Khovanova’s Number Gossip site for properties of 2022, the common properties of 2022 are shown here. All Number Gossip properties are detailed here. I have come across these before, but we have not had an untouchable number since 2010; untouchable numbers are those that are not the sum of the proper divisors of any number. A chart of the first 600 untouchable numbers is available from EasyCalculation.com. Here’s a great post from Iva Sallay on Find the Factors – 516 is an Untouchable Number.

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…

How many ways can you write 2022 as a sum of squares? And can 2022 be expressed as a difference of two squares? I came across this idea from a new year (2009) maths item of the month from MEI, we could be more general – are there any years which cannot be written as the difference of two squares? Here’s a superb resource for this problem on Nrich, What’s Possible? Possible questions and approached in class are included in the article with a printable worksheet for students.

WolframAlpha – 2022hsitorical forms

We can also look at WolframAlpha for 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.

babylonian-2017

For more on the Babylonian system including how fractions were represented see History of Fractions from Nrich.


We could look back and use the excellent MacTutor History of Mathematics Archive from the University of St Andrews, Scotland. We could check today or any day for Mathematicians who were born or died on that day.

The site is searchable in several ways, including the comprehensive index of History Topics.


Teacher Resources on Line
trol calendars

On the subject of dates and the new year, from trol, Teacher Resources on Line, we can make a calendar for 2022. I do like the fold and tuck models – no glue required.

Wishing educators and students everywhere a very Happy New Year.

geoGreeting