Matrix Multiplication

When I teach matrix multiplication I discovery that pupils tin rate the axe top away a fleck lost inward their calculations.  I constitute a fantastic novel method on this blog.

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

I honey this method - it's together with thence unproblematic to retrieve together with at in 1 trial most impossible to top away lost when multiplying matrices.

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

A Percentages Trick

Just a quick postal service today to portion a non bad tip:

You tin swap the percent in addition to the amount:  16% of £25 is hard to calculate inward your head, only 25% of £16 is just the same amount.

So if you lot desire to rapidly live on out 16% of £25 without a calculator, uncovering 25% of £16 instead (ie split £16 past times 4). Simple!

Think nigh why this works.

Source: www.aperiodical.com

Multiplication In Addition To Colour

This uncomplicated sentiment shows how of import it is to purpose unlike colours when writing on the board - no affair what topic you're teaching.  It actually helps pupils proceed rails of what you're doing.

Source: http://www.jandmranch.com/2013/02/09/teaching-three-digit-multiplication-to-my-aspergers-daughter/ 

Vectors For Enlargement

I'm going to produce a serial of brusque posts on approaches or methods that teachers powerfulness non lead maintain seen before. I'm really aware that at that spot volition hold upward many people who already purpose these approaches, simply I figure that non everyone volition lead maintain seen them thus they are worth sharing.

Today, vectors for enlargements. I start saw this on Twitter years agone simply I can't scream back who tweeted it (apologies!). Later I saw my old colleague Lizzie Stokes (@misstokesmaths) using this approach inward her teaching.

How I used to learn enlargement
I'll illustrate my previous method amongst a negative enlargement question:

I'd say my students to depict a business to whatever vertex from the centre of enlargement. Then I'd say them to count how many squares they've moved to larn to the start vertex. I e'er modelled this slow on the board. I showed the procedure of counting carefully ('count the corners, non the spaces!') because I uncovering they oft miscount.

Then I circled the scale component subdivision in addition to said that because the scale component subdivision is negative two, to uncovering the icon they lead maintain to double the distance from the centre of enlargement. But because it's negative, they lead maintain to larn inward the opposite direction from the centre. So inward this case, nosotros started past times going 2 units right and i unit of measurement upward thus straight off nosotros lead maintain to go four units left and 2 units down. I emphasised that we're going the opposite direction because it's a negative enlargement. We thus produce the same matter for each of the other vertices inward turn. At the end, all the corresponding vertices should hold upward joined amongst direct lines going through the centre of enlargement. If not, you've miscounted!

I'd demo around other illustration amongst a dissimilar scale factor, thus larn them to exercise a lot of these on printed worksheets. It was commonly a relatively tranquillity lesson every bit they all had to produce a lot of counting! I'd larn circular berating them for using pen when I'd told them to purpose pencil 10 K times.

In hindsight I realise that this method could hold upward better. It plant fine for super smart students (most things do!), simply others struggled amongst it. One employment is that you lot lead maintain to concur lots of details inward your caput piece you lot move though the question. Another employment is that the method is dissimilar to the method for positive enlargement - there's an extra matter to remember.

How I straight off learn enlargement
They start past times labelling each vertex amongst a missive of the alphabet in addition to thus finding the vector that takes them from the centre of enlargement to each vertex. They are already fluent inward using vectors from our come about translation thus this should hold upward straightforward.

They write downwards the vectors rather than trying to concur the information inward their caput similar they did before. Then all they lead maintain to produce is multiply each of their vectors past times the scale factor, in addition to this gives them the vector for each of the vertices of the image. So they know just where to depict the icon - it's all calculated in addition to recorded earlier they start drawing.

I've used 'Year 11' vector musical note inward the film inward a higher house for clarity, though it's non essential at this stage.

This method is identical regardless of whether nosotros are enlarging amongst a positive or negative integer or a fraction. So all of a abrupt negative enlargements are no harder than positive enlargements (assuming students know how to multiply negatives). H5N1 clear in addition to consistent method all circular - I don't know why I didn't e'er produce it similar this.

Any others?
I promise that was helpful if you'd non seen it before!

I've blogged a break of times earlier almost choice methods for diverse topics (see my posts tagged 'methods'). If at that spot are whatever approaches or methods that you lot purpose that you lot mean value other teachers powerfulness non use, I'd dearest to listen them! Please comment below or electronic mail me or tweet me.

Factorising Yesteryear Inspection

In 2018 I decided to write to a greater extent than brusk posts well-nigh approaches or methods that teachers mightiness non guide maintain seen before. When I part these posts, I am good aware that in that place volition endure many people who already know the affair I'm blogging about, only I figured that it's notwithstanding worth sharing fifty-fifty if it's solely novel to a handful of people.

My outset shipping service inwards this serial was well-nigh using vectors for enlargements. Today's shipping service is well-nigh a agency to salve fourth dimension when factorising a non-monic quadratic aspect (ie a quadratic where the coefficient of x² is greater than 1) yesteryear inspection.

Back inwards September I had a shipping service published on La Salle's blog well-nigh methods for factorising non-monics. I showed that yesteryear looking at former textbooks nosotros tin give the axe encounter how methods guide maintain changed over time.

I explained that I prefer to factorise yesteryear inspection, only most teachers these days factorise yesteryear grouping (ie 'splitting the middle term'). In answer to that shipping service I had lots of teachers either (a) travail to convince me that grouping is amend than inspection (b) travail to convince me that some other method is amend than inspection. Often teachers debate that their method is non bad because their students genuinely liked it inwards the lesson, only I'm to a greater extent than interested inwards the extent to which methods 'stick' inwards the long term. When I used to learn factorising yesteryear grouping my students were e'er happy amongst it at the time, only a few months after they would scrap to scream upwards the amount procedure.

Factorising yesteryear inspection is intuitive in addition to logical, thence there's no physical care for to memorise. I appreciate that most teachers notwithstanding prefer the grouping method, in addition to I produce non intend to travail to convince anyone to modify their mind. But for those of you lot who similar factorising yesteryear inspection, here's a tip from Susan Russo (@Dsrussosusan):

Let's state you lot guide maintain to factorise 6x² + 17x + 12.

Factorising yesteryear inspection is super-quick i time you lot larn the hang of it, only hither both vi in addition to 12 guide maintain multiple factors thence this i mightiness accept a fleck longer than others.

If you lot listing all the possibilities in addition to banking concern gibe each one, in that place are twelve cases to check.
(6x + 1)(x + 12)
(6x + 12)(x + 1)
(6x + 2)(x + 6)
(6x + 6)(x + 2)
(6x + 3)(x + 4)
(6x + 4)(x + 3)
(2x + 1)(3x + 12)
(2x + 12)(3x + 1)
(2x + 2)(3x + 6)
(2x + 6)(3x + 2)
(2x + 3)(3x + 4)
(2x + 4)(3x + 3)

Yawn!

An proficient would likely operate out the right combination fairly speedily without writing downwards all the options. For a novice it's a hurting that in that place are twelve options to think well-nigh here. At outset it seems similar it mightiness accept a piece to pick out the right combination.

Susan pointed out something which should endure totally obvious only hadn't occurred to me before. We tin give the axe learn our students to refine their guesses inwards fellowship to brand this method to a greater extent than efficient. Here's the key: each bracketed aspect shouldn't incorporate whatever mutual factors. For illustration if you lot guide maintain a 2x in addition to then you lot can't position an fifty-fifty issue inwards amongst it.

Let's expect at that listing i time again in addition to at i time disregard whatever selection where there's a mutual component inwards i or both brackets.
(6x + 1)(x + 12)
(6x + 12)(x + 1)
(6x + 2)(x + 6)
(6x + 6)(x + 2)
(6x + 3)(x + 4)
(6x + 4)(x + 3)
(2x + 1)(3x + 12)
(2x + 12)(3x + 1)
(2x + 2)(3x + 6)
(2x + 6)(3x + 2)
(2x + 3)(3x + 4)
(2x + 4)(3x + 3)

It turns out in that place are genuinely solely 2 cases to banking concern gibe yesteryear inspection. Students fluent inwards expanding brackets should endure able to produce it inwards seconds. You tin give the axe at i time encounter that the outset selection volition give a large coefficient of x, thence nosotros banking concern gibe (2x + 3)(3x + 4) in addition to notice that it works.

For some argue I've never shared this time-saving tip amongst my students. I'm rattling grateful to Susan Russo for bringing it to my attention. Let's travail it i time again amongst i to a greater extent than example: factorise 12x² + 11x - 15. Here's the massive listing of 24 options to consider:

(12x + 15)(x - 1)
(12x + 1)(x - 15)
(12x + 5)(x - 3)
(12x + 3)(x - 5)
(6x + 15)(2x - 1)
(6x + 1)(2x - 15)
(6x + 5)(2x - 3)
(6x + 3)(2x - 5)
(3x + 15)(4x - 1)
(3x + 1)(4x - 15)
(3x + 5)(4x - 3)
(3x + 3)(4x - 5)
(12x - 15)(x + 1)
(12x - 1)(x + 15)
(12x - 5)(x + 3)
(12x - 3)(x + 5)
(6x - 15)(2x + 1)
(6x - 1)(2x + 15)
(6x - 5)(2x + 3)
(6x - 3)(2x + 5)
(3x - 15)(4x + 1)
(3x - 1)(4x + 15)
(3x - 5)(4x + 3)
(3x - 3)(4x + 5)

Removing those amongst a mutual component inwards i or both brackets gives us this:
(12x + 15)(x - 1)
(12x + 1)(x - 15)
(12x + 5)(x - 3)
(12x + 3)(x - 5)
(6x + 15)(2x - 1)
(6x + 1)(2x - 15)
(6x + 5)(2x - 3)
(6x + 3)(2x - 5)
(3x + 15)(4x - 1)
(3x + 1)(4x - 15)
(3x + 5)(4x - 3)
(3x + 3)(4x - 5)
(12x - 15)(x - 1)
(12x - 1)(x + 15)
(12x - 5)(x + 3)
(12x - 3)(x + 5)
(6x - 15)(2x + 1)
(6x - 1)(2x + 15)
(6x - 5)(2x + 3)
(6x - 3)(2x + 5)
(3x - 15)(4x + 1)
(3x - 1)(4x + 15)
(3x - 5)(4x + 3)
(3x - 3)(4x + 5)

So this fourth dimension nosotros eliminated one-half the possibilities. It's non equally time-saving equally inwards the outset example, only notwithstanding helpful. My side yesteryear side measuring would endure to travail the less extreme numbers (ie non those involving a 12x or a 15) thence that gives me solely iv options to examine initially. For an experienced factoriser it's fairly quick to encounter that (3x + 5)(4x - 3) works.

Of course, inwards reality nosotros never genuinely listing out all the options in addition to and then determine what to eliminate. What most people genuinely produce when faced amongst 12x² + 11x - xv is write downwards (3x      )(4x      ) in addition to and then (often inwards their caput rather than on paper) travail some numbers that multiply to give -15. So it's helpful to scream upwards that there's no indicate putting in addition to iii or a xv inwards the outset bracket.

This isn't a large game-changer in addition to it doesn't assist amongst every quadratic, only I similar things that salve us time. When solving a long, complicated occupation at Influenza A virus subtype H5N1 level, it's goodness to able to factorise quadratics efficiently.

If you lot hadn't realised that you lot tin give the axe speedily eliminate options inwards this agency in addition to then I promise this was helpful.

If you lot desire to guide maintain a play amongst this, in that place are lots of non-monic expressions to factorise here.

Matrix Multiplication

When I teach matrix multiplication I discovery that pupils tin rate the axe top away a fleck lost inward their calculations.  I constitute a fantastic novel method on this blog.

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

I honey this method - it's together with thence unproblematic to retrieve together with at in 1 trial most impossible to top away lost when multiplying matrices.

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

A Percentages Trick

Just a quick postal service today to portion a non bad tip:

You tin swap the percent in addition to the amount:  16% of £25 is hard to calculate inward your head, only 25% of £16 is just the same amount.

So if you lot desire to rapidly live on out 16% of £25 without a calculator, uncovering 25% of £16 instead (ie split £16 past times 4). Simple!

Think nigh why this works.

Source: www.aperiodical.com

Multiplication In Addition To Colour

This uncomplicated sentiment shows how of import it is to purpose unlike colours when writing on the board - no affair what topic you're teaching.  It actually helps pupils proceed rails of what you're doing.

Source: http://www.jandmranch.com/2013/02/09/teaching-three-digit-multiplication-to-my-aspergers-daughter/ 

Vectors For Enlargement

I'm going to produce a serial of brusque posts on approaches or methods that teachers powerfulness non lead maintain seen before. I'm really aware that at that spot volition hold upward many people who already purpose these approaches, simply I figure that non everyone volition lead maintain seen them thus they are worth sharing.

Today, vectors for enlargements. I start saw this on Twitter years agone simply I can't scream back who tweeted it (apologies!). Later I saw my old colleague Lizzie Stokes (@misstokesmaths) using this approach inward her teaching.

How I used to learn enlargement
I'll illustrate my previous method amongst a negative enlargement question:

I'd say my students to depict a business to whatever vertex from the centre of enlargement. Then I'd say them to count how many squares they've moved to larn to the start vertex. I e'er modelled this slow on the board. I showed the procedure of counting carefully ('count the corners, non the spaces!') because I uncovering they oft miscount.

Then I circled the scale component subdivision in addition to said that because the scale component subdivision is negative two, to uncovering the icon they lead maintain to double the distance from the centre of enlargement. But because it's negative, they lead maintain to larn inward the opposite direction from the centre. So inward this case, nosotros started past times going 2 units right and i unit of measurement upward thus straight off nosotros lead maintain to go four units left and 2 units down. I emphasised that we're going the opposite direction because it's a negative enlargement. We thus produce the same matter for each of the other vertices inward turn. At the end, all the corresponding vertices should hold upward joined amongst direct lines going through the centre of enlargement. If not, you've miscounted!

I'd demo around other illustration amongst a dissimilar scale factor, thus larn them to exercise a lot of these on printed worksheets. It was commonly a relatively tranquillity lesson every bit they all had to produce a lot of counting! I'd larn circular berating them for using pen when I'd told them to purpose pencil 10 K times.

In hindsight I realise that this method could hold upward better. It plant fine for super smart students (most things do!), simply others struggled amongst it. One employment is that you lot lead maintain to concur lots of details inward your caput piece you lot move though the question. Another employment is that the method is dissimilar to the method for positive enlargement - there's an extra matter to remember.

How I straight off learn enlargement
They start past times labelling each vertex amongst a missive of the alphabet in addition to thus finding the vector that takes them from the centre of enlargement to each vertex. They are already fluent inward using vectors from our come about translation thus this should hold upward straightforward.

They write downwards the vectors rather than trying to concur the information inward their caput similar they did before. Then all they lead maintain to produce is multiply each of their vectors past times the scale factor, in addition to this gives them the vector for each of the vertices of the image. So they know just where to depict the icon - it's all calculated in addition to recorded earlier they start drawing.

I've used 'Year 11' vector musical note inward the film inward a higher house for clarity, though it's non essential at this stage.

This method is identical regardless of whether nosotros are enlarging amongst a positive or negative integer or a fraction. So all of a abrupt negative enlargements are no harder than positive enlargements (assuming students know how to multiply negatives). H5N1 clear in addition to consistent method all circular - I don't know why I didn't e'er produce it similar this.

Any others?
I promise that was helpful if you'd non seen it before!

I've blogged a break of times earlier almost choice methods for diverse topics (see my posts tagged 'methods'). If at that spot are whatever approaches or methods that you lot purpose that you lot mean value other teachers powerfulness non use, I'd dearest to listen them! Please comment below or electronic mail me or tweet me.

Factorising Yesteryear Inspection

In 2018 I decided to write to a greater extent than brusk posts well-nigh approaches or methods that teachers mightiness non guide maintain seen before. When I part these posts, I am good aware that in that place volition endure many people who already know the affair I'm blogging about, only I figured that it's notwithstanding worth sharing fifty-fifty if it's solely novel to a handful of people.

My outset shipping service inwards this serial was well-nigh using vectors for enlargements. Today's shipping service is well-nigh a agency to salve fourth dimension when factorising a non-monic quadratic aspect (ie a quadratic where the coefficient of x² is greater than 1) yesteryear inspection.

Back inwards September I had a shipping service published on La Salle's blog well-nigh methods for factorising non-monics. I showed that yesteryear looking at former textbooks nosotros tin give the axe encounter how methods guide maintain changed over time.

I explained that I prefer to factorise yesteryear inspection, only most teachers these days factorise yesteryear grouping (ie 'splitting the middle term'). In answer to that shipping service I had lots of teachers either (a) travail to convince me that grouping is amend than inspection (b) travail to convince me that some other method is amend than inspection. Often teachers debate that their method is non bad because their students genuinely liked it inwards the lesson, only I'm to a greater extent than interested inwards the extent to which methods 'stick' inwards the long term. When I used to learn factorising yesteryear grouping my students were e'er happy amongst it at the time, only a few months after they would scrap to scream upwards the amount procedure.

Factorising yesteryear inspection is intuitive in addition to logical, thence there's no physical care for to memorise. I appreciate that most teachers notwithstanding prefer the grouping method, in addition to I produce non intend to travail to convince anyone to modify their mind. But for those of you lot who similar factorising yesteryear inspection, here's a tip from Susan Russo (@Dsrussosusan):

Let's state you lot guide maintain to factorise 6x² + 17x + 12.

Factorising yesteryear inspection is super-quick i time you lot larn the hang of it, only hither both vi in addition to 12 guide maintain multiple factors thence this i mightiness accept a fleck longer than others.

If you lot listing all the possibilities in addition to banking concern gibe each one, in that place are twelve cases to check.
(6x + 1)(x + 12)
(6x + 12)(x + 1)
(6x + 2)(x + 6)
(6x + 6)(x + 2)
(6x + 3)(x + 4)
(6x + 4)(x + 3)
(2x + 1)(3x + 12)
(2x + 12)(3x + 1)
(2x + 2)(3x + 6)
(2x + 6)(3x + 2)
(2x + 3)(3x + 4)
(2x + 4)(3x + 3)

Yawn!

An proficient would likely operate out the right combination fairly speedily without writing downwards all the options. For a novice it's a hurting that in that place are twelve options to think well-nigh here. At outset it seems similar it mightiness accept a piece to pick out the right combination.

Susan pointed out something which should endure totally obvious only hadn't occurred to me before. We tin give the axe learn our students to refine their guesses inwards fellowship to brand this method to a greater extent than efficient. Here's the key: each bracketed aspect shouldn't incorporate whatever mutual factors. For illustration if you lot guide maintain a 2x in addition to then you lot can't position an fifty-fifty issue inwards amongst it.

Let's expect at that listing i time again in addition to at i time disregard whatever selection where there's a mutual component inwards i or both brackets.
(6x + 1)(x + 12)
(6x + 12)(x + 1)
(6x + 2)(x + 6)
(6x + 6)(x + 2)
(6x + 3)(x + 4)
(6x + 4)(x + 3)
(2x + 1)(3x + 12)
(2x + 12)(3x + 1)
(2x + 2)(3x + 6)
(2x + 6)(3x + 2)
(2x + 3)(3x + 4)
(2x + 4)(3x + 3)

It turns out in that place are genuinely solely 2 cases to banking concern gibe yesteryear inspection. Students fluent inwards expanding brackets should endure able to produce it inwards seconds. You tin give the axe at i time encounter that the outset selection volition give a large coefficient of x, thence nosotros banking concern gibe (2x + 3)(3x + 4) in addition to notice that it works.

For some argue I've never shared this time-saving tip amongst my students. I'm rattling grateful to Susan Russo for bringing it to my attention. Let's travail it i time again amongst i to a greater extent than example: factorise 12x² + 11x - 15. Here's the massive listing of 24 options to consider:

(12x + 15)(x - 1)
(12x + 1)(x - 15)
(12x + 5)(x - 3)
(12x + 3)(x - 5)
(6x + 15)(2x - 1)
(6x + 1)(2x - 15)
(6x + 5)(2x - 3)
(6x + 3)(2x - 5)
(3x + 15)(4x - 1)
(3x + 1)(4x - 15)
(3x + 5)(4x - 3)
(3x + 3)(4x - 5)
(12x - 15)(x + 1)
(12x - 1)(x + 15)
(12x - 5)(x + 3)
(12x - 3)(x + 5)
(6x - 15)(2x + 1)
(6x - 1)(2x + 15)
(6x - 5)(2x + 3)
(6x - 3)(2x + 5)
(3x - 15)(4x + 1)
(3x - 1)(4x + 15)
(3x - 5)(4x + 3)
(3x - 3)(4x + 5)

Removing those amongst a mutual component inwards i or both brackets gives us this:
(12x + 15)(x - 1)
(12x + 1)(x - 15)
(12x + 5)(x - 3)
(12x + 3)(x - 5)
(6x + 15)(2x - 1)
(6x + 1)(2x - 15)
(6x + 5)(2x - 3)
(6x + 3)(2x - 5)
(3x + 15)(4x - 1)
(3x + 1)(4x - 15)
(3x + 5)(4x - 3)
(3x + 3)(4x - 5)
(12x - 15)(x - 1)
(12x - 1)(x + 15)
(12x - 5)(x + 3)
(12x - 3)(x + 5)
(6x - 15)(2x + 1)
(6x - 1)(2x + 15)
(6x - 5)(2x + 3)
(6x - 3)(2x + 5)
(3x - 15)(4x + 1)
(3x - 1)(4x + 15)
(3x - 5)(4x + 3)
(3x - 3)(4x + 5)

So this fourth dimension nosotros eliminated one-half the possibilities. It's non equally time-saving equally inwards the outset example, only notwithstanding helpful. My side yesteryear side measuring would endure to travail the less extreme numbers (ie non those involving a 12x or a 15) thence that gives me solely iv options to examine initially. For an experienced factoriser it's fairly quick to encounter that (3x + 5)(4x - 3) works.

Of course, inwards reality nosotros never genuinely listing out all the options in addition to and then determine what to eliminate. What most people genuinely produce when faced amongst 12x² + 11x - xv is write downwards (3x      )(4x      ) in addition to and then (often inwards their caput rather than on paper) travail some numbers that multiply to give -15. So it's helpful to scream upwards that there's no indicate putting in addition to iii or a xv inwards the outset bracket.

This isn't a large game-changer in addition to it doesn't assist amongst every quadratic, only I similar things that salve us time. When solving a long, complicated occupation at Influenza A virus subtype H5N1 level, it's goodness to able to factorise quadratics efficiently.

If you lot hadn't realised that you lot tin give the axe speedily eliminate options inwards this agency in addition to then I promise this was helpful.

If you lot desire to guide maintain a play amongst this, in that place are lots of non-monic expressions to factorise here.