Matrix Multiplication

When I teach matrix multiplication I respect that pupils tin expire a fighting lost inwards their calculations.  I constitute a fantastic novel method on this blog.

Let's tell nosotros convey matrix A together with matrix B.  We desire to calculate AB.  Write downwards the 2 matrices side-by-side.  Then move matrix Influenza A virus subtype H5N1 down. Now write your answer inwards the infinite at the bottom right.  I intend the adjacent measurement is self-explanatory from the film below, merely read the master copy blog post for a amount explanation.

I beloved this method - it's together with then unproblematic to retrieve together with instantly virtually impossible to expire lost when multiplying matrices.

And - on the dependent area of matrices - this made me chuckle:

Logs

I similar instruction logarithms but my students abide by them genuinely hard. There's a broad multifariousness of C2 assay out questions on logs then pupils genuinely bring to sympathise the topic, they can't but acquire a gear upwards of rules to acquire by. This is a proficient affair of course.

This weblog post from Sarah Hagan has absolutely loads of activities for instruction logs too her subsequent post has roughly follow-up activities.

There's a broad arrive at of instruction ideas inward this Logarithm Functions booklet from Mathematics Vision Project, which starts amongst an ordering activity:

Mathematics Vision Project

STEM Centre render four resources designed to exercise the basics too extend agreement of logarithms, summation roughly opened upwards ended questions for tidings or a mini-whiteboard activity. I've flora the truthful or imitation exercise especially effective for identifying misconceptions inward the past times (longer version amongst answers here).

Susan Wall - via STEM Centre

Here are roughly to a greater extent than of my favourite resources for instruction logs to Year 12:

• Teachitmaths.co.uk has a pick of logs dominoes - I especially similar these. 
• SRWhitehouse on TES ever provides first-class Influenza A virus subtype H5N1 degree resources. She has a great selection of resources for logs.
• Slides - 'An introduction to logarithms' past times Teachit Maths.
• C2 Target Board - Logs too Indices - hannahlees on TES.
• Student activities on introducing logarithms - Project Maths 
• Become Friends amongst Logarithms - notes too exercises past times Scott Surgent.
• RISP 31 - edifice log equations.
• Standards Unit A13 - Simplifying Logarithmic Expressions.

Factorising Harder Quadratics

My pupils panic at the sight of a quadratic alongside a leading coefficient greater than one.  I factorise these quadratics past times inspection (the 'guess as well as test' method) but my pupils aren't satisfied alongside this proposition - they desire a to a greater extent than structured approach.

Influenza A virus subtype H5N1 normally taught method inwards the Great Britain involves splitting the middle term inwards 2 (sometimes called the 'Grouping Method'). This is explained real clearly here (thanks to SRWhitehouse for this resource). Teachitmaths.co.uk has a PowerPoint explaining this method. It's worth watching James Tanton's video 'Splitting the Middle Term' too. He's non a fan!

'Grouping'. Source: Flat World Education

An alternative, which seems slow at commencement but paves the agency for a large publish of misconceptions, is the 'slide as well as divide' method. The method, as well as its associated problems, are nicely described inwards Nix the Tricks.

Nix the Tricks offers an interesting option - I've provided 2 examples hither but it's worth reading the majority for the total explanation.

It's also worth looking at this post past times Don Steward to run into his tap exceed method for finding factors as well as for lots of helpful practise questions.

Also worth a mention: when I commencement start didactics quadratic factorisation, I similar to role this well-designed sum-product worksheet from greatmathsteachingideas.com as a starter. It's proficient practise of an essential skill.

Coded Averages

Teaching coding inward Statistics i is a fleck of a pain.  My pupils simply don't similar it, in addition to it's difficult for me to larn enthusiastic most something that didn't actually be when I studied Statistics (perhaps because it's non solely necessary?). When I was revising this topic amongst my Year 12s this year, I used a actually uncomplicated real-life example:

Imagine I'm queuing upwards to purchase 4 items of clothing, which cost £29.95, £49.95, £59.95 in addition to £99.95. To hold upwards past times fourth dimension I determine to function out the hateful cost of these 4 items. To brand the numbers easier to bargain with, I add together 5p to each total in addition to and so separate past times 10, giving me £3, £5, £6 in addition to £10. It's really slow to function out the hateful of these 'coded' prices inward my caput - it's £6. And right away I tin uncode that hateful past times multiplying past times 10 in addition to so subtracting 5p (the inverse of the master copy coding).  That rapidly gives me the actual hateful cost of the clothes, which is £59.95.  

Resources

Here's a few resources I've establish for instruction this topic:

• Teachitmaths.co.uk has a lovely worksheet for helping pupils empathise the comport upon of coding. Just beware of the fault inward the extension questions - both information sets should accept 6 values, non five.
• Schoolworkout.co.uk has this sorting activity and this worksheet.
• Phildb on tes.co.uk has written this worksheet.

There's a express alternative hither so possibly I should write something myself - I'll add together it to my list!

Large Foursquare Roots

In C1, the non-calculator Influenza A virus subtype H5N1 marking module, my pupils sometimes perish stuck solving quadratic equations that are difficult to factorise. This powerfulness live because the leading coefficient is greater than i (see my earlier post) or it may but live because the numbers inward the quadratic are large.  I commonly propose that they persevere amongst factorising, but if they convey to resort to using the quadratic formula in addition to thence they may require to foursquare origin a large foursquare number. Here's roughly uncomplicated ideas for how to create thence - inward this representative I'm finding the foursquare origin of 324, though these methods run every bit good for larger foursquare numbers.

Note that if the quadratic equation can't live factorised inward the commencement house in addition to thence nosotros tin either purpose the formula or consummate the foursquare to solve it, in addition to nosotros volition halt upwards amongst a surd inward our lastly answer.  My adjacent post service is all close didactics surds!

Surds

Surds - what a fabulous theme to teach! I honey the abundance of resources available in addition to I’ve listed roughly of my favourites below.

Two things I ever create when didactics surds:

• Pupils listing the starting fourth dimension twelve foursquare numbers inward their books for reference throughout the lesson. They demand to readily recognise foursquare numbers inward guild to simplify surds in addition to then the to a greater extent than they practice listing them, the better.
• When manipulating expressions containing surds (eg expanding brackets) I brand comparisons to what they already know close algebra. For representative √2 in addition to √3 tin hold upwards regard of inward the same means as x in addition to y (ie non ‘like terms’) whereas 2√3 in addition to 5√3 tin hold upwards regard of inward the same means as 2x in addition to 5x (ie nosotros tin add together them to larn 7√3).

The NCETM suggests a distich of prissy ‘hooks’ for getting started didactics surds:

• Ask pupils to notice a means of drawing a occupation alongside a length of precisely √5 units (the hypotenuse of a correct angled triangle alongside sides 2cm in addition to 1cm) 
• Ask pupils to split the length of an A4 slice of newspaper past times its width. Repeat for A3 in addition to A5. What create they notice? (The response is ever √2)

This blog post has some to a greater extent than ideas for introducing the concept of surds. And I similar this on the history of surds.

Resources
Here’s roughly bully didactics resources (there's hundreds to a greater extent than online). Most of these travel as good at GCSE in addition to Influenza A virus subtype H5N1 level:

• Is it rational?
• Two interactive tasks from Mathspad
• Pairs activity 
• Introducing surds
• True or faux activity
• Surds -Applying in addition to occupation solving
• Multiplication squares
• Mr Barton’s activity
• Standards Unit N11
• Surds Connect 4
• Surds resources - leannegadsby on TES
• Manipulating Radicals
• Tripods
• And here’s roughly resources recommended past times the TES.

For a lovely laid of surds problems, run into my Problem Sets page.

Finally, if you're looking for something actually creative, cheque out this Wheel of Theodorus Art Project.

Edit 29/12/14: Thanks to @runningtstitch for suggesting an choice method for simplifying surds:

Radians

Radians scare my Year 12s. They oftentimes select to 'think' inwards degrees together with only convert their answers into radians. I wonder if I could produce a meliorate undertaking of introducing the concept.  I've only discovered this animation that is hence absolutely fantastic, I had to part it.

I works life it inwards this blog post but it originated here.  Both are worth a read.  

I honey this post service 'Degrees vs Radians' which presents an opened upward missive of the alphabet to students wondering, “Why Do We Use Radians Instead of Degrees?”.  

Colin Beveridge has rigid feelings on the discipline - both teachers together with students would produce goodness from reading his post service 'Why Radians Rock (and Degrees Don't)', inwards which he convinces the reader of the unquestionable benefits of using radians.

Next Level Maths has a dainty explanation of radians, including an interactive activity.

Incidentally, if you're excited past times animations together with hence yous mightiness similar the other animations inwards this article '7 Animated GIFs That Will Make You Instantly Understand Trigonometry'.

A Degree Revision

I created these revision cards for a classroom activity, simply they're as helpful for pupils studying at home. These cards comprehend all the facts as well as definitions that pupils bespeak to larn for each module - I haven't included anything that is given inward the formula booklet.

These cards are based on the Edexcel specification.

Core ane as well as Core two revision cards
Core three revision cards
Statistics ane revision cards
Statistics two revision cards
Decision ane revision cards
Mechanics ane formulae revision  (thanks to my boss for creating the M1 cards!)

I promise to create C4 as well as FP1 adjacent year!

Hope they're helpful!  If yous desire to run out me a review or brand suggestions, delight create through TES or Twitter.

Skewness

Let’s verbalize S1, equally this year’s essay out is fast approaching.

I don’t similar the way my S1 textbook deals alongside skewness. It basically merely sets out methods for identifying skewness without actually explaining what it is. In this postal service I hash out an choice approach.

I usually piece of employment this starter interrogation inwards my get-go S1 lesson of the year:

The indicate beingness that when at that topographic point are outliers inwards a information set, the hateful becomes rather useless (unless of class you lot wishing to deceive people!).

Whilst education S1 I oftentimes uncovering myself referring dorsum to my Bibonacci starter (‘remember that CEO? He ruined the mean!’). So I decided to brand a worksheet out of it - come across below - which I used for the get-go fourth dimension this yr when education skewness (if you lot similar this thence delight acquire out me a review on TES!). My pupils enjoyed the activeness too it generated interesting discussions.

• Investigating skewness activity
• Investigating skewness instructor notes

There is a similar activeness here - from the website Illustrative Mathematics.

The substitution education indicate is that when information is skewed, the median is to a greater extent than meaningful than the mean. We convey diverse methods for identifying skewness inwards a information ready (summarised inwards these slides – sorry, master copy source unknown) - nosotros usually reckon the shape of a frequency diagram or box plot. Sometimes skewness is apparent merely from looking at the raw information (this sideways stem-and-leaf diagram is interesting). 

Pupils involve to recollect that positive skew way nosotros convey a few 'sparse' information points (possibly outliers) at the top end of the information ready – too similar inwards the Bibonacci example, this volition pull the hateful up. This volition assistance them conceptually empathise the diverse methods described inwards the textbook.

Finally, here's a keen play a trick on that I learnt from my colleague Lizzie... Imagine a cute puppy sitting on the left of the graph. H5N1 positive skew is going to come across the puppy (positive = yay, a puppy!) too a negative skew is going away from the puppy (negative = acquire away from this puppy!). It's a giddy but uncomplicated too memorable way of ensuring that students don't acquire the graphs mixed up.

---

Maths Painting Quiz

Spiked Math has approximately first-class Maths Movie Quizzes.  I chose these ones to entertain my Year 13s at Christmas.  Great fun for Maths teachers too!

Venn Activities

This post service is nearly using Venn Diagrams equally a educational activity tool, rather than educational activity the concept of Venn Diagrams themselves.  To quote the first-class Johnny Griffiths, 'The Venn Diagram amongst iii bubbles is the maths teacher's friend! ... To travel able to create amount inward an eight-region diagram shows a actually deep agreement of the distinctions that a practiced pupil has to travel able to make, spell the weaker pupil finds beingness faced amongst a painting exhibit such equally this much less threatening than a page of exercises'.

Before nosotros get-go using Venn activities inward Maths lessons, let's intend nearly whether pupils are familiar amongst how Venn Diagrams work. I acknowledge to beingness a fleck clueless nearly precisely what children report at main school. The primary here), merely I at 1 time prefer the simpler method described in linear equations activity. It helped consolidate their cognition of linear graphs too allowed me to banking concern check their understanding.

Johnny Griffith's first-class RISPs (Rich Starting Points for H5N1 Level Core Mathematics) includes the Venn activities Risp 10 (coordinate geometry) too Risp 4 (periodic functions).  In the instructor notes for Risp 10 Johnny shares his views on the usefulness of Venns.

A Venn action for Solving Quadratics yesteryear Factorising yesteryear aka nka on TES. 

A GCSE extension action on quadratic graphs from MEI.

I haven't tried this however merely Don Steward also gives us this lovely here), merely I at 1 time prefer the simpler method described in number activity.

Interactive Venns

Venns piece of occupation good on the Interactive Whiteboard - here's about examples.

• Nrich has about interactive Venn diagrams on types of number.
• MathsPad has about lovely Venn activities (most involve a subscription). I've successfully used this Factors, Primes too Multiples Activity with my Year 8s.   
• Teachitmaths.co.uk has interactive Venn activities - for instance for shapes and prime factors, or y'all tin create your ain (subscription required).

Other Uses

• When I innovate complex numbers inward FP1 nosotros create a Venn Diagram during cast give-and-take nearly types of numbers (real, rational, integers etc) too their Greek symbol notation.  The NCETM has a similar Venn action inward this resource sheet.
• Up until lately I used Venn Diagrams to learn Highest Common Factor too Lowest Common Multiple (method explained here), merely I at 1 time prefer the simpler method described in this post.
• Quadrilaterals tin travel classified using a Venn Diagram similar this one.  We tin hence answer questions such as 'Is every foursquare a rectangle? Is every rhomb a square?' A 'family tree' also plant well.

In determination - Venns are everywhere! But inward secondary Maths they are alone formally taught to those studying iGCSE (set theory) too Statistics 1 at H5N1 score (probability). I'm pleased to encounter that Venn Diagram methods inward probability are on the new Maths GCSE Syllabus so volition travel to a greater extent than widely studied inward future. But inward improver to beingness taught equally a topic inward their ain right, Venn Diagrams are a versatile educational activity tool amongst numerous applications inward the classroom. And they convey the added bonus of beingness created yesteryear a British Mathematican! Well done Mr Venn.

Teaching Amongst Desmos

I discovered Desmos final yr through @ColleenYoung's fantastic weblog Mathematics, Learning as well as Web 2.0. It speedily became my favourite slice of 'teachnology'. Before Desmos I occasionally used Omnigraph or Autograph, but I straightaway occupation Desmos regularly because it's merely so tardily to use. It actually couldn't endure simpler. As it's web-based, I encourage my pupils (particularly at Influenza A virus subtype H5N1 level) to occupation Desmos at dwelling family to back upwards their studies, for event to depository fiscal establishment agree graph sketches.

Desmos on the Interactive Whiteboard
When instruction exponential functions to Year 12, I used Desmos on my IWB to demo the graphs of functions such equally y = 2^x as well as y = 3^x. We discussed our expectations for graphs such equally y = 2^-x as well as used Desmos to check.  When nosotros plotted y = (-2)^x nosotros got an interesting resultant - which had me stumped for a infinitesimal - tin terminate you lot explicate it?

Desmos is especially helpful for demonstrating concepts inward Further Maths (such equally the directrix as well as focus of a parabola) - there's roughly skilful examples in this blog.

Desmos inward the information technology Room

My schoolhouse doesn’t ain tablets as well as information technology rooms are rarely available, then my pupils don't oft acquire the adventure to occupation technology during their Maths lessons. But I was then bully to part Desmos amongst my Year nine class, I made a big endeavor to acquire organised. I produced this worksheet for them to occupation inward an information technology room, amongst the aim of introducing dissimilar types of graph (quadratic, cubic, reciprocal) as well as elementary transformations. After a really curt demonstration of Desmos, the degree spent 2 whole lessons working independently. I was pleased amongst their progress as well as I promise that they'll recollect Desmos as well as maintain to occupation it equally a learning tool both at dwelling family as well as at school.

I loaded my worksheet onto TES today - along amongst a graph transformation activity that I occupation at Key Stage iv - as well as it got me thinking close whether whatsoever other resources are available for instruction amongst Desmos.  I flora a few examples on TES, such as Investigating request lines amongst Desmos yesteryear Tristan Jones and Using Desmos to learn Coordinate Geometry (KS3) yesteryear phildate. I human face that the make of instruction resources available volition grow equally increasing numbers of teachers notice as well as comprehend this fantastic tool.  

Resources Anonymous

'Hello. My elevate is Jo together with I'm a resourceaholic'. Yes, I confess, I'm utterly addicted to searching the network for maths instruction resources.

It all started when I saw a PGCE educatee photocopying this blog. Ooh, novel websites! Heaven for a resourceaholic.

I haven't had a run a endangerment to facial expression at all these websites yet, only here's around of my top resources recommendations from the websites I've looked at together with therefore far. I've focused on H5N1 marker because that's where I intend it's hardest to detect interesting instruction ideas.

Statistics 1 
Illustrative Mathematics has around fantastic ideas for instruction S1. Here's a pick of examples:

• Correlation together with Regression: a lovely activity on java shops together with crime. Includes calculating a correlation coefficient, interpreting a regression equation together with considering causality together with outliers.
• Discrete Random Variables:  a brusk activity Sounds Really Good! (sort of) which features a real-life purpose of expectation. Bob's Bagel Shop is similar.
• Probability: the carte du jour action Describing Events is an splendid introduction to probability together with the activity Venn Diagrams together with the Addition Rule is skilful too. I also similar this brusk Titanic action on independence.
• Data:  These activities on Haircut Costs together with Speed Trap focus on comparison box plots. Describing Data Sets alongside Outliers and Identifying Outliers are almost outliers together with skewness inwards information (see my related postal service on instruction skewness).
• Normal Distribution: brusk contextual exercises blog. Ooh, novel websites! Heaven for a resourceaholic.
  
  I haven't had a run a endangerment to facial expression at all these websites yet, only here's around of my top resources recommendations from the websites I've looked at together with therefore far. I've focused on H5N1 marker because that's where I intend it's hardest to detect interesting instruction ideas.
  
  Statistics 1 
  Illustrative Mathematics has around fantastic ideas for instruction S1. Here's a pick of examples:
  
  • Correlation together with Regression: a lovely activity on java shops together with crime. Includes calculating a correlation coefficient, interpreting a regression equation together with considering causality together with outliers.
  • Discrete Random Variables:  a brusk activity Sounds Really Good! (sort of) which features a real-life purpose of expectation. Bob's Bagel Shop is similar.
  • Probability: the carte du jour action Describing Events is an splendid introduction to probability together with the activity Venn Diagrams together with the Addition Rule is skilful too. I also similar this brusk Titanic action on independence.
  • Data:  These activities on Haircut Costs together with Speed Trap focus on comparison box plots. Describing Data Sets alongside Outliers and Identifying Outliers are almost outliers together with skewness inwards information (see my related postal service on instruction Logs. 
  • Quadratics together with Cubics: I similar this short Cubic Graph task (answers here) from Mathematics Assessment Project. There's around great Forming Quadratics activities hither too.
  • Circles: Mathematics Assessment Project also has amount lesson plans - I've successfully used the splendid materials from Equations of Circles 1 and Equations of Circles 2 alongside my Year 12s. 
  • Surds: See resources listed inwards my before spider web log post, blog. Ooh, novel websites! Heaven for a resourceaholic.
    
    I haven't had a run a endangerment to facial expression at all these websites yet, only here's around of my top resources recommendations from the websites I've looked at together with therefore far. I've focused on H5N1 marker because that's where I intend it's hardest to detect interesting instruction ideas.
    
    Statistics 1 
    Illustrative Mathematics has around fantastic ideas for instruction S1. Here's a pick of examples:
    
    • Correlation together with Regression: a lovely activity on java shops together with crime. Includes calculating a correlation coefficient, interpreting a regression equation together with considering causality together with outliers.
    • Discrete Random Variables:  a brusk activity Sounds Really Good! (sort of) which features a real-life purpose of expectation. Bob's Bagel Shop is similar.
    • Probability: the carte du jour action Describing Events is an splendid introduction to probability together with the activity Venn Diagrams together with the Addition Rule is skilful too. I also similar this brusk Titanic action on independence.
    • Data:  These activities on Haircut Costs together with Speed Trap focus on comparison box plots. Describing Data Sets alongside Outliers and Identifying Outliers are almost outliers together with skewness inwards information (see my related postal service on instruction Surds.