5 Maths Gems #38

Hello as well as welcome to my 38^th gems post. This is where I percentage v teaching ideas I've seen on Twitter. As teachers start making plans as well as preparing resources for September, there's been a flurry of inspiring ideas on Twitter - I tin barely boot the bucket on up!

1. Tessellation
It's quite rare that there's exciting intelligence inwards the footing of mathematics, precisely we've of late seen the regain of a novel type of Surrounding a Point activity from Don Steward pictured below. Mathigon has a lovely tessellation department featuring overnice pictures as well as an online tool.

I've never seen tessellation animations earlier as well as then thank y'all to @MathsMastery for sharing these, I'll purpose them the side yesteryear side fourth dimension I learn tessellation.

2. Displays

There's been lots of tweets as well as posts virtually displays this week. I similar these lovely vinyl corridor displays from @cOmplexnumber.

I also similar this increment mindset display from English linguistic communication teacher Rebecca Foster (Gems 3, precisely would stand upward for on a smaller noticeboard. 

One of my favourite classroom display ideas is the judgement stems display from Gems 26. It's skillful to bring displays that are of existent purpose to students throughout the year.

It's worth reading Paul Collins' (post virtually the novel displays he's lay upward inwards his classroom as well as corridors.

3. History of Maths
I don't know rattling much virtually the history of maths as well as that's a non bad shame. It's a priority for me to meliorate my champaign of report noesis inwards this area. I intend to start yesteryear aid @Kris_Boulton's workshop at the side yesteryear side maths conference.

Kris' post A History of Maths features a timeline showing people as well as societies of cardinal influence inwards mathematics. It also shows cardinal publications as well as ideas. The picture below is exclusively a rattling pocket-size extract - bring a human face through the amount timeline. It's fascinating to encounter the social club of things. Perhaps display this inwards your classroom as well as say your students virtually the origins of the topics they're studying.

4. Challenge Questions for Year 12
Stuart Price ‏(@sxpmaths) has collated unopen to AS stretch as well as challenge questions for inwardness topics. They tin endure accessed here. Thanks Stuart for this helpful resource.

If y'all learn Influenza A virus subtype H5N1 score as well as are coming to the maths conference inwards Sheffield on 26^th September, produce come upward along to my workshop virtually Influenza A virus subtype H5N1 score resources.

5. Exit Tickets
Many maths teachers purpose larn out tickets inwards every lesson. The stance is that y'all laid a unmarried inquiry or curt laid of questions at the halt of the lesson as well as analyse the results. Typically students would paw their larn out ticket to y'all equally they leave of absence the classroom. It's of import to boot the bucket on the questions unproblematic – if students larn them incorrect y'all request to know why. It's an effective agency of identifying misconceptions as well as deciding whether students are laid to motion on. 

Lesley Hall (RAG123.

Update
I've published 4 weblog posts inwards the final calendar week as well as I bring unopen to other 4 posts inwards draft - I actually request to halt all the blogging as well as pass unopen to fourth dimension preparing my offset lessons! Here's what I've been upward to since my final gems post:

• I wrote an article virtually conferences for the NCETM Secondary Magazine.
• I wrote a post Behaviour Management for Beginners which features a few resources I've prepared for September - my expectations videos bring been popular.
  

• I was chuffed that my weblog appeared on Vuelio's Words to Avoid page - these are definitely worth a read.
• I was rattling pleased that @oldandrewuk as well as @Just_Maths set upward an Echo Chamber specifically for U.K. maths didactics blogs. Follow @MathsEcho to boot the bucket on up-to-date alongside all the latest posts.
• I popped into schoolhouse and, borrowing a lovely stance from Danielle Bartram (@MissBsResources), lay Maths Mr Men stickers from @solvemymaths on my new calculators as well as then I tin boot the bucket on rail of them.

Finally, 4 reminders:

• If y'all haven't already done so, y'all tin 'like' my new Facebook page for updates.
• #mathstlp volition supply at 7pm on Lord's Day 30^th August for lesson planning advice.
• My tangents as well as areas is worth reading if you're teaching GCSE side yesteryear side year.
• Check out my hashtag #mathsgems for inspiring ideas from previous gems posts.

That's it from me. I'll leave of absence y'all alongside this overnice puzzle from the ATM (@ATMMathematics).

Algebraic Fluency - 50S Style

Whilst visiting the Mathematical Association inwards Leicester on Saturday, I picked upwardly a duet of onetime maths textbooks from the 1950s. 'A Classbook of Algebra' is my favourite.

In it I establish about wonderful exercises for developing algebraic fluency, including this fix of questions on 'Miscellaneous Factors':

The pedagogy 'factorise where possible' adds a delightful extra marking of intrigue in addition to challenge to this exercise. I dear these questions...

Factorise c - 3 + 2x(3 - c)

Factorise c² - (c - d)²

Factorise ef - 1 + e - f

Factorise -14yz + y² + 49z²

Factorise 81 - 9a  + 0.25a²

When I larn a run a jeopardy I'll type them all up. It does brand you lot wonder why the vast bulk of textbooks in addition to worksheets on factorising these days merely pick out questions similar this:

Factorise x² +7x

Factorise x² + 5x + 6

Factorise x² - 9

in addition to possibly this sort of affair for stretch:

Factorise 3x² + 10x + 8

Factorise 2x² - 18

Is that actually the best nosotros tin do? We could at to the lowest degree modify the gild of the terms! No wonder in addition to hence many of our students lack algebraic fluency when they larn to Influenza A virus subtype H5N1 level.

If you're thinking that this factorising usage from the 1950s is a flake tricky, you lot volition live on relieved to take away heed that inwards the same mass at that spot are a position out of exercises breaking downwardly the factorising skills into stages, leading upwardly to the 'Miscellaneous Factors' activity which brings it all together. For instance at that spot is a whole department on 'Factors past times Grouping' which explains how to factorise expressions similar ax + 2x + 3a + 6. I'm non certain this science is widely taught these days.

I pick out enough of students who would dear to larn stuck into these questions. Back inwards 2015 I wrote the shipping service 'Stretching Practice' which is nigh where to detect expert practise resources for high attainers. There's a decent alternative of resources available, but zip quite at the same marking equally the exercises from 1950s textbooks. I intend that the closest online resources for developing algebraic fluency are about of Don Steward's tasks, similar this first-class difference of 2 squares exercise:

I want I could present you lot every usage from 'A Classbook of Algebra'. There are in addition to hence many smashing questions. I'll exactly percentage a few to a greater extent than examples...

Under the championship 'Easy Brackets' nosotros detect questions that many teachers would right away role equally extension function inwards a lesson on expanding unmarried brackets:

And nether 'Miscellaneous Easy Brackets' the questions await real dissimilar to what nosotros right away regard to live on 'easy':

 

When nosotros learn 'collecting similar terms' I dubiety many of us role exercises similar this 1 on 'Addition in addition to Subtraction of Polynomials':

Check out the final question: 'To the excess of six - x + x² over 5 + x² - 3x add x² -3x + 4'!

Do you lot desire to banking enterprise gibe that your students really know how to expand brackets in addition to solve equations? Try about of these:

Here nosotros pick out double brackets too:

I wrote nigh multiplying negatives inwards this post, but the resources I featured didn't include algebra similar this usage does:

Finally (though I could become on all day!), here's something for your indices lesson:

If alone I had fourth dimension to type all these up!

I promise you lot enjoyed looking at exercises from the 1950s equally much equally I did.

Lost Vocabulary

While reading the Victorian maths textbook 'Elementary Algebra for Schools' I spotted quite a few words together with phrases which are rarely used inward modern secondary schools. I'm non proverb that this vocabulary has disappeared from the plain of mathematics, but I incertitude you lot volition listen these price inward GCSE lessons.

Let's accept a quick hold off at about interesting words together with how they were used inward the 1800s.

Unity
Meaning (in this context): the number one.

• "When the coefficient is unity it is commonly omitted. Thus nosotros create non write 1a, but only a".
• "If the production of 2 quantities live on equal to unity, each is said to live on the reciprocal of each other".
• "Subtract 3x² - 5x + 1 from unity, together with add together 5x² - 6x to the result".
• "We continue right away to the resolution into factors of trinomial expressions when the coefficient of the highest ability is non unity".

Resolve into Factors
Meaning: factorise (UK) or cistron (US) - run across Colin Beveridge's post 'Factorise or Factor'.

• "Resolve into factors x² - ax + 5x - 5a".
• "The beginner should live on careful non to laid out cancelling until he has expressed both numerator together with denominator inward the nearly convenient form, past times resolution into factors where necessary". 
• "Resolve 4a²(x³+18ab²) - (32a⁵+9b²a³)  into 4 factors". 

Diminished
Meaning: made smaller or less.

• "The total of -3x, -5x, -7x, -x is -16x. For a total of coin diminished successively past times £3, £5, £7 together with £1 is diminished altogether past times £16".
• "Divide 105 into 2 parts, 1 of which diminished past times twenty shall live on equal to the other diminished past times 15".

Whence
Meaning: every bit a number of which.

• "Whence the final result follows".
• "Whence x = 1 is the exclusively solution"

Involution
Meaning: multiplying an aspect past times itself.

Evolution
Meaning: the functioning of finding the root of an expression.

Antecedent together with Consequent

Meaning: the get-go together with minute term of a ratio respectively.

• "The ratio a:b is equal to the ratio ma:mb; that is, the value of a ratio remains unaltered if the antecedent together with the consequent are multiplied or divided past times the same quantity".
• "A ratio is said to live on a ratio of greater inequality, of less inequality, or of equality, according every bit the antecedent is greater than, less than, or equal to the consequent. 

[There are loads of interesting price inward the ratio chapter - including commensurable together with subduplicate.]

Transposition
Meaning: The human activity of transferring something to a dissimilar place.

[This hasn't disappeared, but the role of the phrase 'by transposition' or 'transposing' inward worked examples is something nosotros don't run across inward schools anymore].

There are many to a greater extent than words together with phrases inward this majority which seem to guide keep disappeared from the daily mathematical vocabulary of secondary schools - far besides many to listing here. Whence I volition relieve the balance for about other post. I promise you lot observe all this every bit interesting every bit I do. No incertitude mortal volition right away contact me to nation that they role all these words inward their classroom on a regular basis...!

Equations Exercises

I was looking at roughly of the one-time maths textbooks at my schoolhouse in addition to noticed than fifty-fifty equally latterly equally the 1980s, textbook exercises contained a lot to a greater extent than do questions than modern textbooks. Below is an instance comparison the same theme inward a 1980s textbook ('Negative Numbers in addition to Graphs' past times Heylings) in addition to a electrical flow GCSE textbook ('Edexcel GCSE Maths Higher' published past times Oxford University Press). The exercises encompass the same science only the root exercise is double the length of the second. I approximate modern textbooks convey to tally an entire GCSE course of didactics into a unmarried book, restricting the sum of do questions they tin include.

From 'Negative Numbers in addition to Graphs', root published 1984

From 'Edexcel GCSE Maths Higher', root published 2015

Last calendar month I wrote a post service virtually a 1950s algebra textbook called 'A Classbook of Algebra'. The questions inward this textbook are by in addition to large to a greater extent than challenging than questions inward most modern textbooks. In response to this post service a issue of rattling generous volunteers stepped forrad offering to type upward roughly of the exercises in addition to thence that teachers tin purpose them inward lessons. I am rattling grateful for the fourth dimension in addition to test that has gone into this. In this post service I convey provided links to all the exercises typed upward in addition to thence far which relate to the science of solving equations. I convey an additional xviii exercises on other topics which I volition collate over the side past times side few weeks.

These exercises are all rather long in addition to the thought is non necessarily to purpose them inward their entirety. Because they convey all been typed upward inward in Word, teachers volition easily endure able to edit the exercises or cutting in addition to glue item questions to purpose equally examples inward class.

Most exercises include answers. I volition edit this post service if anything is updated. For each exercise listed below I convey included a minor extract in addition to thence you lot tin preview the type of questions covered.

1. Very Easy Equations (four exercises) - alongside thank you lot to Caroline Beale (@cbealemaths)

2. Easy Equations I - alongside thank you lot to Claire Willis (@MissWillisMaths)

3. Easy Equations II - alongside thank you lot to Caroline Beale (@cbealemaths)

4. Equations Involving Fractions - alongside thank you lot to Michael Allan (@mrallanmaths)

5. Equations Involving Brackets - alongside thank you lot to Jane Appleton (@JaneAppleton24)

6. Equations alongside Brackets in addition to Harder Equations alongside Brackets - alongside thank you lot to Justin Thompson 

7. Equations Involving Directed Numbers - alongside thank you lot to Caroline Beale (@cbealemaths)

8. Easy Literal Equations - alongside thank you lot to Sandra Douglas (@mathsbox1)

9. Miscellaneous Simultaneous Equations - alongside thank you lot to Fee Wilson (@fionajw)

10. Miscellaneous Equations - alongside thank you lot to Dan Rodriguez-Clark (@InteractMaths). 

I promise this is useful. Thank you lot in 1 trial again to everyone who has worked on this. And of course of didactics total credit to Sidney Trustram, the master writer of these exercises which, lxx years on, are all the same making us think. Look out for my side past times side posts inward which I'll percentage exercises on simplifying algebra, expanding, factorising, writing algebraically in addition to working alongside directed numbers.

Algebra Exercises

A few weeks agone I published a postal service called 'Equations Exercises' inwards which I shared about exercises that induce got been typed upward from a 1950s algebra textbook. In my persuasion these are good written exercises amongst a proficient degree of stretch in addition to challenge. The thought is non necessarily to purpose these exercises inwards their entirety. They are provided inwards Word format then it's slowly to re-create in addition to glue extracts in addition to examples to purpose inwards class.

In today's postal service I've provided links to about other develop of exercises from the same textbook. They cover simplifying algebra, including index laws in addition to algebraic fractions. There's likewise an practise on exchange in addition to i on writing algebraically. For each practise listed below I induce got included an extract then yous tin preview the type of questions covered.

1. Notation, Addition in addition to Subtraction I in addition to II - amongst thank yous to Katie Pollard (@takepi21)

2. Notation, Multiplication in addition to Indices I - amongst thank yous to Phil Bruce (@pbrucemaths)

3. Notation, Multiplication in addition to Indices II - amongst thank yous to Daniel Udall (@DJUdall)

4. Simple Powers - amongst thank yous to Larissa Alsford (@lolsford)

5. Notation - Division - amongst thank yous to Caroline Beale (@cbealemaths)

6. Simple Multiplication of Fractions - amongst thank yous to Alice Dwyer (@littlemissdwyer)

7. Simplification of Fractions yesteryear Factors II - amongst thank yous to Sandra Douglas (@mathsbox1)

8. Easy Fractions, Substitution - amongst thank yous to Steve Bishop (@stevebishop100)

9.  Translation I in addition to II - amongst thank yous to Ben Gordon (@mathsmrgordon)

I'll disclose i to a greater extent than develop of exercises side yesteryear side calendar week - about on directed numbers, in addition to about on factorising in addition to expanding.

Huge thank yous i time to a greater extent than to the teachers who typed these exercises up. I actually appreciate the fourth dimension in addition to try that has gone into this. Excellent teamwork!