Multiplying Negatives

"A negative times a negative is a positive". It's a difficult ane to explain. We all learnt it at schoolhouse together with practised it to the indicate of fluency, but it's non until we're asked why it industrial plant that nosotros halt together with intend close it.

Numbers lines together with visualisations are real helpful when education the add-on together with subtraction of negative numbers. But alongside multiplication together with partitioning it's non then clear.

Let's await at a few approaches together with resources.

1 . Pattern Spotting
Draw a measure multiplication tabular array together with extend it backwards to include negative numbers. It's a straightforward designing that all students should hold upwards able to location together with continue. Get students to create this using Colin Foster's activity on page five of his Negative Numbers chapter.

2. Multiplication Grids
Take ii 2-digit numbers together with multiply them together using grid multiplication. For simplicity, let's accept 12 x 11:

Here nosotros direct maintain written 12 every 2nd 10 + 2 together with eleven every 2nd 10 + 1. But it would piece of employment simply every 2nd good if nosotros expressed those numbers differently. Instead, let's write 12 every 2nd fifteen - iii together with eleven every 2nd fifteen - 4. We should acquire the same answer:

This alone industrial plant if -3 x -4 = 12. 

Note that this explanation requires students to commencement sympathise that positive x negative = negative. This is relatively straightforward to explicate inwards damage of repeated addition. 

3. Proof
Here's a proof that is clear together with accessible to us experienced mathematicians. I'm non certain how accessible it is to Year vii students, but it's worth a go.

a together with b are positive

a + (-a) = 0 

[a +(-a)]•b = 0•b 

a•b + (-a)•b = 0 

a•b is positive. Therefore (-a)•b is negative 

b + (-b) = 0 

(-a)•[b + (-b)] = (-a)•0

(-a)•b + (-a)•(-b) = 0

Since (-a)•b is negative, nosotros conclude that (-a)•(-b) is positive.

Perhaps start alongside a numerical instance instead of a formal proof.

3 + (-3) = 0

Multiply everything past times -4

3(-4) + (-3)(-4) = 0(-4) 

 -12 + (-3)(-4) = 0 

 (-3)(-4) must equal 12 to brand this disceptation true. 

Further Reading
It's a practiced sentiment to read close a theme earlier y'all learn it, fifty-fifty relatively uncomplicated topics that you've taught many times before. Here are unopen to helpful links:

• The FAQ pages on Math Forum ever furnish interesting answers: Why is a negative times a negative a positive?
• I taste James Tanton's curriculum essays: Directed Number Gaps' activity - it industrial plant good alongside whatsoever twelvemonth group
• MathsPad has a slap-up attain of negative publish puzzles including arithmagons (some of these resources are alone available to subscribers)
• CIMT's negative numbers chapter has activities for practising multiplying negatives.

Colin Foster suggests that y'all enquire students to brand upwards 10 multiplications together with 10 divisions each giving an reply of –8 (eg –2 × –2 × –2 or –1 × 8 etc).

The squaring together with cubing (etc) of negatives is worth discussing - students should location that an fifty-fifty ability gives a positive value (eg what is the value of (-1)¹⁰⁰?).

It may hold upwards worth exploring reckoner deportment every 2nd good (ie unopen to calculators require brackets when squaring a negative). It's of import that students know how to run their reckoner properly. There's a slap-up resources from MathsPad for this - Using a Calculator: Odd One Out.

This theme is revisited inwards later on years when students are practising substitution. For example, if a = 3, b = -2 together with c = -5, detect the values of: abc; bc²; (bc)²; a²b³ together with then on. This Substitution Puzzle from mathsteaching.wordpress.com gets quite challenging.

Do allow me know if y'all run an interesting method or resources for education the multiplication of negative numbers.

"Minus times minus results inwards a plus,
The argue for this, nosotros needn't discuss"
- Ogden Nash

Multiplying Negatives

"A negative times a negative is a positive". It's a difficult ane to explain. We all learnt it at schoolhouse together with practised it to the indicate of fluency, but it's non until we're asked why it industrial plant that nosotros halt together with intend close it.

Numbers lines together with visualisations are real helpful when education the add-on together with subtraction of negative numbers. But alongside multiplication together with partitioning it's non then clear.

Let's await at a few approaches together with resources.

1 . Pattern Spotting
Draw a measure multiplication tabular array together with extend it backwards to include negative numbers. It's a straightforward designing that all students should hold upwards able to location together with continue. Get students to create this using Colin Foster's activity on page five of his Negative Numbers chapter.

2. Multiplication Grids
Take ii 2-digit numbers together with multiply them together using grid multiplication. For simplicity, let's accept 12 x 11:

Here nosotros direct maintain written 12 every 2nd 10 + 2 together with eleven every 2nd 10 + 1. But it would piece of employment simply every 2nd good if nosotros expressed those numbers differently. Instead, let's write 12 every 2nd fifteen - iii together with eleven every 2nd fifteen - 4. We should acquire the same answer:

This alone industrial plant if -3 x -4 = 12. 

Note that this explanation requires students to commencement sympathise that positive x negative = negative. This is relatively straightforward to explicate inwards damage of repeated addition. 

3. Proof
Here's a proof that is clear together with accessible to us experienced mathematicians. I'm non certain how accessible it is to Year vii students, but it's worth a go.

a together with b are positive

a + (-a) = 0 

[a +(-a)]•b = 0•b 

a•b + (-a)•b = 0 

a•b is positive. Therefore (-a)•b is negative 

b + (-b) = 0 

(-a)•[b + (-b)] = (-a)•0

(-a)•b + (-a)•(-b) = 0

Since (-a)•b is negative, nosotros conclude that (-a)•(-b) is positive.

Perhaps start alongside a numerical instance instead of a formal proof.

3 + (-3) = 0

Multiply everything past times -4

3(-4) + (-3)(-4) = 0(-4) 

 -12 + (-3)(-4) = 0 

 (-3)(-4) must equal 12 to brand this disceptation true. 

Further Reading
It's a practiced sentiment to read close a theme earlier y'all learn it, fifty-fifty relatively uncomplicated topics that you've taught many times before. Here are unopen to helpful links:

• The FAQ pages on Math Forum ever furnish interesting answers: Why is a negative times a negative a positive?
• I taste James Tanton's curriculum essays: Directed Number Gaps' activity - it industrial plant good alongside whatsoever twelvemonth group
• MathsPad has a slap-up attain of negative publish puzzles including arithmagons (some of these resources are alone available to subscribers)
• CIMT's negative numbers chapter has activities for practising multiplying negatives.

Colin Foster suggests that y'all enquire students to brand upwards 10 multiplications together with 10 divisions each giving an reply of –8 (eg –2 × –2 × –2 or –1 × 8 etc).

The squaring together with cubing (etc) of negatives is worth discussing - students should location that an fifty-fifty ability gives a positive value (eg what is the value of (-1)¹⁰⁰?).

It may hold upwards worth exploring reckoner deportment every 2nd good (ie unopen to calculators require brackets when squaring a negative). It's of import that students know how to run their reckoner properly. There's a slap-up resources from MathsPad for this - Using a Calculator: Odd One Out.

This theme is revisited inwards later on years when students are practising substitution. For example, if a = 3, b = -2 together with c = -5, detect the values of: abc; bc²; (bc)²; a²b³ together with then on. This Substitution Puzzle from mathsteaching.wordpress.com gets quite challenging.

Do allow me know if y'all run an interesting method or resources for education the multiplication of negative numbers.

"Minus times minus results inwards a plus,
The argue for this, nosotros needn't discuss"
- Ogden Nash

Favourite Posts Of 2016

ten most viewed posts of 2016. Today I've written a dissimilar list. These are my personal favourites from 2016 - the posts that I most enjoyed writing, or am most proud of, fifty-fifty if they didn't acquire many views.

1. Trigonometry Questions and Pythagoras Questions
In these 2 posts I featured roughly of my favourite novel GCSE questions for Pythagoras together with Trigonometry. There are roughly awesome questions hither which I actually enjoyed using inwards my lessons.

2. The Folder Experiment
I was surprised yesteryear how many people had a persuasion on using band binders instead of practise books with Year 11!

3. New GCSE: Capture-Recapture

My schoolhouse doesn't utilization Edexcel for GCSE so I don't remove to instruct this topic, but I had fun finding out most it. It's a lovely fleck of proportional reasoning. If y'all instruct Edexcel GCSE, y'all mightiness notice this post service helpful.

4. Multiplying Negatives

I wishing my weblog was 100% dependent area cognition posts. I'll attempt to produce to a greater extent than of them inwards 2017. Here I talked most diverse approaches to explaining why a negative times a negative is a positive.

5. Maths School Trips

When Legoland invited me to visit, I was conflicted. I direct hold strict rules most non advertising on my blog. But I actually wanted to become to Legoland, together with the chance to direct hold my household unit of measurement at that topographic point for complimentary (on my daughter's birthday no less!) was i I couldn't refuse. Thankfully their Mindstorms workshop was amazing, so I had no qualms most giving it a positive write-up (alongside other destinations for maths schoolhouse trips). I'm fond of this post service because it reminds me of my daughter's sec birthday.

6. Useful GCSE Questions from Linked Pair Papers
I was a fleck grumpy most having to instruct Linked Pair GCSE final yr but it turned out to move a useful experience. Linked Pair GCSE questions were frequently fantabulous together with I featured roughly of my favourites inwards this post.

7. Gem Awards 2016
Two together with a one-half years agone I joined Twitter whilst on motherhood leave. I was overwhelmed yesteryear the quantity of teaching ideas together with resources so I started writing a weekly gems post service to hold rails of it all. When I returned to move I couldn't maintain the frequency of the posts, but I yet write i or 2 gems posts each month. I regularly refer dorsum to them when planning lessons. In my annual Gems Awards post service I featured roughly of my favourite gems from the previous year.

8. A Level Reforms: First Thoughts

I wishing at that topographic point were to a greater extent than maths teachers blogging most H5N1 level. With the novel specifications launching inwards September, it's going to move of import to collaborate with other H5N1 score maths teachers during 2017. I promise to play a role inwards facilitating discussions together with sharing resources together with ideas. 

9. Handwriting

One of the best things most writing this post service was the tweet from the air traffic controller...

10. ♫ You order zero, I order nought ♫

Perhaps no i else is interested inwards whether people order null or nought, but I thought it was fascinating!

11. Worries
The shortage of maths teachers is causing problems inwards so many schools. It continues to direct hold a detrimental consequence on children all over the country. I wrote this post service when I was feeling especially anxious most the whole situation, together with flora that many of my readers shared my concerns. Later inwards the yr I wrote 'GCSE reform - volition it work?' in which I shared to a greater extent than concerns most the big issues inwards maths education. Nothing has changed since so - my concerns yet stand.

12. The Joy of Planning
Naveen wrote a TES article most how swell it is non to invention her ain lessons because it gives her fourth dimension to educate her dependent area knowledge. I sure as shooting empathise Naveen's hollo for - nosotros could all produce with to a greater extent than fourth dimension for dependent area cognition development. When the article was published, roughly prominent bloggers jumped on the thought of scripted lessons. On Twitter, the thought speedily developed into i where all lessons are standardised. I watched with sadness equally tweeters enthusiastically discussed taking away my favourite role of the job. I tweeted my occupation organisation - that teachers wouldn't enjoy delivering scripted lessons - together with I was defendant of selfishly putting my ain happiness earlier children's learning. This may good move true, but it upset me. Rather than argue, I decided to issue a positive post, 'The Joy of Planning', most how much pleasance I move out of the procedure of planning together with delivering lessons. I received lots of lovely comments on Twitter from teachers who besides actually relish planning lessons. It's overnice to know I'm non alone.

So that's it - my favourite weblog posts of 2016. I thoroughly enjoyed writing these posts together with promise that I tin flaming hold to portion my experiences together with ideas inwards 2017.

Favourite Posts Of 2016

ten most viewed posts of 2016. Today I've written a dissimilar list. These are my personal favourites from 2016 - the posts that I most enjoyed writing, or am most proud of, fifty-fifty if they didn't acquire many views.

1. Trigonometry Questions and Pythagoras Questions
In these 2 posts I featured roughly of my favourite novel GCSE questions for Pythagoras together with Trigonometry. There are roughly awesome questions hither which I actually enjoyed using inwards my lessons.

2. The Folder Experiment
I was surprised yesteryear how many people had a persuasion on using band binders instead of practise books with Year 11!

3. New GCSE: Capture-Recapture

My schoolhouse doesn't utilization Edexcel for GCSE so I don't remove to instruct this topic, but I had fun finding out most it. It's a lovely fleck of proportional reasoning. If y'all instruct Edexcel GCSE, y'all mightiness notice this post service helpful.

4. Multiplying Negatives

I wishing my weblog was 100% dependent area cognition posts. I'll attempt to produce to a greater extent than of them inwards 2017. Here I talked most diverse approaches to explaining why a negative times a negative is a positive.

5. Maths School Trips

When Legoland invited me to visit, I was conflicted. I direct hold strict rules most non advertising on my blog. But I actually wanted to become to Legoland, together with the chance to direct hold my household unit of measurement at that topographic point for complimentary (on my daughter's birthday no less!) was i I couldn't refuse. Thankfully their Mindstorms workshop was amazing, so I had no qualms most giving it a positive write-up (alongside other destinations for maths schoolhouse trips). I'm fond of this post service because it reminds me of my daughter's sec birthday.

6. Useful GCSE Questions from Linked Pair Papers
I was a fleck grumpy most having to instruct Linked Pair GCSE final yr but it turned out to move a useful experience. Linked Pair GCSE questions were frequently fantabulous together with I featured roughly of my favourites inwards this post.

7. Gem Awards 2016
Two together with a one-half years agone I joined Twitter whilst on motherhood leave. I was overwhelmed yesteryear the quantity of teaching ideas together with resources so I started writing a weekly gems post service to hold rails of it all. When I returned to move I couldn't maintain the frequency of the posts, but I yet write i or 2 gems posts each month. I regularly refer dorsum to them when planning lessons. In my annual Gems Awards post service I featured roughly of my favourite gems from the previous year.

8. A Level Reforms: First Thoughts

I wishing at that topographic point were to a greater extent than maths teachers blogging most H5N1 level. With the novel specifications launching inwards September, it's going to move of import to collaborate with other H5N1 score maths teachers during 2017. I promise to play a role inwards facilitating discussions together with sharing resources together with ideas. 

9. Handwriting

One of the best things most writing this post service was the tweet from the air traffic controller...

10. ♫ You order zero, I order nought ♫

Perhaps no i else is interested inwards whether people order null or nought, but I thought it was fascinating!

11. Worries
The shortage of maths teachers is causing problems inwards so many schools. It continues to direct hold a detrimental consequence on children all over the country. I wrote this post service when I was feeling especially anxious most the whole situation, together with flora that many of my readers shared my concerns. Later inwards the yr I wrote 'GCSE reform - volition it work?' in which I shared to a greater extent than concerns most the big issues inwards maths education. Nothing has changed since so - my concerns yet stand.

12. The Joy of Planning
Naveen wrote a TES article most how swell it is non to invention her ain lessons because it gives her fourth dimension to educate her dependent area knowledge. I sure as shooting empathise Naveen's hollo for - nosotros could all produce with to a greater extent than fourth dimension for dependent area cognition development. When the article was published, roughly prominent bloggers jumped on the thought of scripted lessons. On Twitter, the thought speedily developed into i where all lessons are standardised. I watched with sadness equally tweeters enthusiastically discussed taking away my favourite role of the job. I tweeted my occupation organisation - that teachers wouldn't enjoy delivering scripted lessons - together with I was defendant of selfishly putting my ain happiness earlier children's learning. This may good move true, but it upset me. Rather than argue, I decided to issue a positive post, 'The Joy of Planning', most how much pleasance I move out of the procedure of planning together with delivering lessons. I received lots of lovely comments on Twitter from teachers who besides actually relish planning lessons. It's overnice to know I'm non alone.

So that's it - my favourite weblog posts of 2016. I thoroughly enjoyed writing these posts together with promise that I tin flaming hold to portion my experiences together with ideas inwards 2017.

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.

5 Maths Gems #62

Welcome to my 62^nd update from the footing of Maths EduTwitter. Here I summarise unopen to of the latest ideas in addition to resources for teaching maths.

1. H5N1 Brilliant Problem
If yous cause got a Facebook concern human relationship hence it's definitely worth liking Brilliant.org's Facebook page for an ongoing period of time of lovely maths problems inward your intelligence feed. I especially similar the occupation below - I'd purpose it equally a starter inward a GCSE lesson.

2. Symbaloo
Emma Bell (@EJMaths) has created a real helpful Symbaloo of maths teaching websites. All yous cause got to produce is click on 'start using this webmix' in addition to register for a costless account. You tin learn inward your homepage hence whenever yous opened upward the cyberspace you'll cause got a lovely ready of maths websites, blogs, tools in addition to resources to lead from. This volition salve teachers lots of fourth dimension in addition to tending them disclose novel websites too.

3. Fraction Division
An approach to dividing fractions that doesn't involving 'flipping'... this method from @solvemymaths is worth a look.

4. Indices
@offpistemaths shared these indices questions from an Eton scholarship paper aimed at Year 8. I honey these problems - to a greater extent than of our students should last given the chance to endeavor challenging questions similar this.

5. Interactive GCSE Questions
Here's a novel costless resources from MathsPad. These interactive GCSE exam-style questions are available for both Higher in addition to Foundation. They would move good equally daily lesson starters on an interactive whiteboard. It's slowly to switch questions in addition to display answers.

Update
Summer is nearly over! H5N1 aeroplane in addition to GCSE results days were especially emotional in addition to exciting for me this yr (my get-go results at my novel school). It's weird to intend that this was our final GCSE results twenty-four hours alongside grades A* to U - side yesteryear side yr we'll last seeing numerical grades for the get-go time. It's also weird to intend that the coming schoolhouse yr volition last the final fourth dimension nosotros learn C1 in addition to C2, equally modular H5N1 aeroplane comes to an end.

You mightiness last interested inward seeing the 'hardest' enquiry on this year's Edexcel maths Higher GCSE, which was alone answered correctly yesteryear 2.6% of students (thanks to @MathsEmporium for sharing this).

Earlier this calendar week I went along for drinks at La Salle's #pieandmaths lawsuit - it was bully to lead manage of upward alongside unopen to Twitter friends in addition to run across unopen to novel people. I'm actually looking forwards to #mathsconf8 inward Kettering on 1st Oct (I've right away submitted a workshop proposal) - book now!

Posts
Did yous lead manage of my recent post almost Multiplying Negatives? If you're teaching improver in addition to subtraction alongside negative numbers, yous mightiness similar Classroom Displays

Kicking off Year 12

Your First Maths Lesson alongside Year 7

Words of Wisdom: Teaching Foundation GCSE

High Expectations

Complete the Rules

Icebreakers

Practical tips for a (newly) qualified maths teacher

Twitter...?
If you're non on Twitter hence the start of the novel schoolhouse yr mightiness last a proficient fourth dimension to join... The amount yous gain from it is good worth the pocket-size investment of time. Email me for advice if you're non certain how to learn started!

A Problem
I'll move out yous alongside this lovely ready of problems shared yesteryear @Mathematical_A, taken from 'A Square Peg inward a Round Hole'. Find the fraction of the surface area of the quadrant occupied yesteryear each semicircle.

5 Maths Gems #62

Welcome to my 62^nd update from the footing of Maths EduTwitter. Here I summarise unopen to of the latest ideas in addition to resources for teaching maths.

1. H5N1 Brilliant Problem
If yous cause got a Facebook concern human relationship hence it's definitely worth liking Brilliant.org's Facebook page for an ongoing period of time of lovely maths problems inward your intelligence feed. I especially similar the occupation below - I'd purpose it equally a starter inward a GCSE lesson.

2. Symbaloo
Emma Bell (@EJMaths) has created a real helpful Symbaloo of maths teaching websites. All yous cause got to produce is click on 'start using this webmix' in addition to register for a costless account. You tin learn inward your homepage hence whenever yous opened upward the cyberspace you'll cause got a lovely ready of maths websites, blogs, tools in addition to resources to lead from. This volition salve teachers lots of fourth dimension in addition to tending them disclose novel websites too.

3. Fraction Division
An approach to dividing fractions that doesn't involving 'flipping'... this method from @solvemymaths is worth a look.

4. Indices
@offpistemaths shared these indices questions from an Eton scholarship paper aimed at Year 8. I honey these problems - to a greater extent than of our students should last given the chance to endeavor challenging questions similar this.

5. Interactive GCSE Questions
Here's a novel costless resources from MathsPad. These interactive GCSE exam-style questions are available for both Higher in addition to Foundation. They would move good equally daily lesson starters on an interactive whiteboard. It's slowly to switch questions in addition to display answers.

Update
Summer is nearly over! H5N1 aeroplane in addition to GCSE results days were especially emotional in addition to exciting for me this yr (my get-go results at my novel school). It's weird to intend that this was our final GCSE results twenty-four hours alongside grades A* to U - side yesteryear side yr we'll last seeing numerical grades for the get-go time. It's also weird to intend that the coming schoolhouse yr volition last the final fourth dimension nosotros learn C1 in addition to C2, equally modular H5N1 aeroplane comes to an end.

You mightiness last interested inward seeing the 'hardest' enquiry on this year's Edexcel maths Higher GCSE, which was alone answered correctly yesteryear 2.6% of students (thanks to @MathsEmporium for sharing this).

Earlier this calendar week I went along for drinks at La Salle's #pieandmaths lawsuit - it was bully to lead manage of upward alongside unopen to Twitter friends in addition to run across unopen to novel people. I'm actually looking forwards to #mathsconf8 inward Kettering on 1st Oct (I've right away submitted a workshop proposal) - book now!

Posts
Did yous lead manage of my recent post almost Multiplying Negatives? If you're teaching improver in addition to subtraction alongside negative numbers, yous mightiness similar Classroom Displays

Kicking off Year 12

Your First Maths Lesson alongside Year 7

Words of Wisdom: Teaching Foundation GCSE

High Expectations

Complete the Rules

Icebreakers

Practical tips for a (newly) qualified maths teacher

Twitter...?
If you're non on Twitter hence the start of the novel schoolhouse yr mightiness last a proficient fourth dimension to join... The amount yous gain from it is good worth the pocket-size investment of time. Email me for advice if you're non certain how to learn started!

A Problem
I'll move out yous alongside this lovely ready of problems shared yesteryear @Mathematical_A, taken from 'A Square Peg inward a Round Hole'. Find the fraction of the surface area of the quadrant occupied yesteryear each semicircle.

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.