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:


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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:




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Dividing Past Times Zero

I found this on Pinterest and now purpose it to assistance my pupils recollect the rules regarding segmentation as well as zero.  It seems to work!

I also similar to demo them a airheaded meme similar the i below to help the rule stick inwards their heads.  A search for 'divide past times zero' on Google Images brings upwards loads of these. I practise apologise to my pupils inwards advance for the bad language, which I intend makes it to a greater extent than memorable.


I was rattling pleased when one of my pupils late wrote this inwards a test. It's squeamish to run across that roughly of them pay attention!


Most calculators laissez passer on the message ‘Maths Error’ when a pose out is divided past times zero.  It’s crucial that pupils empathise why. For didactics ideas, I recommend this fantabulous weblog post 'Why can't you lot dissever past times zero?'.

We purpose this agreement when solving equations similar this:  x2 - 5x = 0. Influenza A virus subtype H5N1 mutual error is to divide past times x to give-up the ghost x - v = 0. This incorrectly gives solely i solution (x = 5). Knowing that nosotros shouldn't dissever past times x because x could travel null (unless we're told otherwise) allows us to empathise why the right method is to factorise ie x(x-5) = 0, giving the 2 solutions x = 0 as well as x = 5.

To stretch your students, effort this interesting activeness almost the value of 00, Evaluating a Special Exponential Expression from Illustrative Mathematics.

It's interesting that throughout history, many slap-up mathematicians (including Euler) possess got argued that segmentation past times null gives infinity. Most modern mathematicians accept it every bit undefined.  If you lot desire to practise roughly background reading on the ‘history’ of dividing past times zero, I recommend this article - it's an enjoyable read.



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Highest Mutual Element Together With Lowest Mutual Multiple

There are many methods for finding the Highest Common Factor as well as Lowest Common Multiple of ii or to a greater extent than numbers.  For years I've been didactics pupils to pose the prime number factors into a Venn Diagram, every bit described here. 

I recently discovered an option method that is impressively quick as well as simple.  It is described inwards this video every bit the 'Indian Method'. It's like to the 'upside downwards birthday cake method' but it's much quicker because at that spot is no requirement to operate primes.

Say nosotros desire to notice the Highest Common Factor as well as Lowest Common Multiple of 24 as well as 36.
Write down the ii numbers, then (to the left, every bit inwards my representative below) write downwards whatsoever mutual component (ie 2, 3, 4, half dozen or 12).  I've chosen 6.  Now separate 24 as well as 36 past times half dozen as well as write the answers underneath (4 as well as half dozen inwards this case).  Keep repeating this procedure until the ii numbers have no mutual factors (ie 2 as well as iii below).  Now, your Highest Common Factor is merely the production of numbers on the left. And for the Lowest Common Multiple, notice the production of the numbers on the left as well as the numbers inwards the bottom row. It's slowly to retrieve which is which - to notice the LCM, await for the L shape.


It's hence quick!  And simple!  Try it.

Don Steward features an option method inwards this weblog post.  He mentions that you lot tin notice a LCM past times dividing the production of the ii numbers past times their HCF ie inwards this example, (24 x 36)/12 = 72.

See my Resources Library for resources for didactics HCF as well as LCM.



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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.


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Angles Facts

I've e'er taught angles inward parallel lines every bit Fs, Cs in addition to Zs, simply I latterly discovered that pupils prefer to think of Fs, Us in addition to Ns, which operate inward just the same agency simply may live on easier to retrieve because they piece the discussion FUN.

(image from greatmathsteachingideas.com)

Of course of pedagogy there's nonetheless the challenge of remembering the to a greater extent than formal descriptions that they volition ask to role inward their GCSE examination (for event they are required to role the term 'corresponding angle' inward their answers, instead of F-angle).  I advise that they think of 'FC' (football lodge - every bit inward Tottenham Hotspur FC) for F angles = corresponding angles.  I besides advise they could retrieve that Z angles = alternate angles past times thinking of the commencement in addition to final letters of the alphabet. But this doesn't operate if I alter Z to N!  I'll accept to think of something else (or fifty-fifty better, larn them to come upwardly up alongside something).

Resources
There are loads of proficient resources for teaching this topic. Here are a few examples:
There are loads to a greater extent than resources for this theme inward my shape resources library. 

More Angle Facts
On a related note, here's closed to tips for teaching to a greater extent than basic angle facts:

Of course of pedagogy there's nonetheless the challenge of remembering the to a greater extent than formal descriptions that they volition ask to role inward their GCSE examination (for event they are required to role the term 'corresponding angle' inward their answers, instead of F-angle).  I advise that they think of 'FC' (football lodge - every bit inward Tottenham Hotspur FC) for F angles = corresponding angles.  I besides advise they could retrieve that Z angles = alternate angles past times thinking of the commencement in addition to final letters of the alphabet. But this doesn't operate if I alter Z to N!  I'll accept to think of something else (or fifty-fifty better, larn them to come upwardly up alongside something).

Resources
There are loads of proficient resources for teaching this topic. Here are a few examples:
There are loads to a greater extent than resources for this theme inward my mathsisfun.com has a distich to a greater extent than ideas - I similar their proffer that the C of the discussion complementary stands for 'corner' (ie a correct angle) in addition to the due south inward supplementary stands for 'straight' (ie 180 degrees on a withdraw line).
Interestingly, the discussion complementary comes from Latin completum (meaning completed) because the correct angle is persuasion of every bit a consummate angle.



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


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