good at math: Negative Numbers

Showing posts with label Negative Numbers. Show all posts

Thursday, October 24, 2019

5 Maths Gems #7

Common practise of successful departments' too 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 marker maths.

Influenza A virus subtype H5N1 lot of the inspiration behind today's maths gems has come upward from teachers' blogs - I'm incredibly grateful to those who convey the fourth dimension to part their ideas too experiences for the create goodness of others.

1. Interactive Areas
Computer-based tools receive got made it easier to explicate some mathematical concepts too generate interesting questions. It's fantastic when technology enhances too enriches mathematics education.

I of late tweeted this colourful example which shows that nosotros tin cutting a circle into sectors too rearrange them to create upward one's hear the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this vivid applet which illustrates the same matter inwards really clear interactive steps. Never earlier receive got I seen the formula for the expanse of a circle explained so clearly (and did y'all run across the 'proof without words' video inwards Maths Gems 4? That was skillful too).

Another fantabulous interactive tool is 'Investigating Triangle Area' (via @mathslinks). My students are ever happy to convey that a right angled triangle is one-half a rectangle, but are non so easily convinced for non-right angled triangles. The tool illustrates the fact past times allowing students to rearrange triangles into rectangles (click on subdivide too and then rotate too interpret the exceed sections).

2. Estimation

Examples from estimation180.com

I've seen a lot of American teachers tweet close Common practise of successful departments' too 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 marker maths.

Influenza A virus subtype H5N1 lot of the inspiration behind today's maths gems has come upward from teachers' blogs - I'm incredibly grateful to those who convey the fourth dimension to part their ideas too experiences for the create goodness of others.

1. Interactive Areas
Computer-based tools receive got made it easier to explicate some mathematical concepts too generate interesting questions. It's fantastic when technology enhances too enriches mathematics education.

I of late tweeted this colourful example which shows that nosotros tin cutting a circle into sectors too rearrange them to create upward one's hear the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this vivid applet which illustrates the same matter inwards really clear interactive steps. Never earlier receive got I seen the formula for the expanse of a circle explained so clearly (and did y'all run across the 'proof without words' video inwards Every Math Teacher inwards the World Should Do This...Right Now!' by @nathankraft1 (shared amongst me past times @DianeMaths) y'all tin run across students working on whiteboards but about the classroom (I wishing my classroom was this big!).

Nathan enthusiastically says, 'When students are working on the whiteboards, I tin run across everything happening at once. It's similar I'm looking at the freaking Matrix. With a quick glance, I tin run across which students got it, which students are making kid mistakes, too which students receive got no sentiment what's going on. I tin apace position errors for students. I tin inquire a stronger educatee to help a struggling one. Once a educatee has the right answer, I yell, "Great! Erase it! Next problem!"'. His enthusiasm has rubbed off on me. I desire to endeavour it! And earlier y'all disregard this sentiment because y'all don't receive got a room total of whiteboards, deport inwards hear that whatever vertical surface will create - endeavour writing on windows or using Common practise of successful departments' too 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 marker maths.

Influenza A virus subtype H5N1 lot of the inspiration behind today's maths gems has come upward from teachers' blogs - I'm incredibly grateful to those who convey the fourth dimension to part their ideas too experiences for the create goodness of others.

1. Interactive Areas
Computer-based tools receive got made it easier to explicate some mathematical concepts too generate interesting questions. It's fantastic when technology enhances too enriches mathematics education.

Improves visibility.
Allows for transfer of noesis but about the room.
Non-Permanent removes fearfulness of writing.
Formative assessment past times instructor at all times.
Teacher tin respond questions past times having students await at others work.

Influenza A virus subtype H5N1 related sentiment comes from Common practise of successful departments' too 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 marker maths.

Influenza A virus subtype H5N1 lot of the inspiration behind today's maths gems has come upward from teachers' blogs - I'm incredibly grateful to those who convey the fourth dimension to part their ideas too experiences for the create goodness of others.

1. Interactive Areas
Computer-based tools receive got made it easier to explicate some mathematical concepts too generate interesting questions. It's fantastic when technology enhances too enriches mathematics education.

I of late tweeted this colourful example which shows that nosotros tin cutting a circle into sectors too rearrange them to create upward one's hear the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this vivid applet which illustrates the same matter inwards really clear interactive steps. Never earlier receive got I seen the formula for the expanse of a circle explained so clearly (and did y'all run across the 'proof without words' video inwards Improving students' collaboration using acrylic sheets'. Jeremy suggests investing inwards clear acrylic sheets that students tin write on. Diagrams, pictures, axes etc tin live on seat underneath. Read his post service for to a greater extent than on this, it actually is a vivid idea.

Next week

A lot of my readers volition live on at La Salle Education's maths conference adjacent week. I’m actually looking frontwards to it. Please come upward too tell howdy if y'all location me. Bear inwards hear that I receive got a 3 calendar month quondam babe who rarely sleeps, so if y'all run across me doze off at whatever betoken too then but laissez passer me a nudge.

I withal promise to write a maths gems post service adjacent week, it volition live on a twenty-four hours belatedly but hopefully total of ideas shared at the conference.

By the way, if y'all didn't grab my concluding post service 'Common practise of successful departments' too 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 marker maths.

Influenza A virus subtype H5N1 lot of the inspiration behind today's maths gems has come upward from teachers' blogs - I'm incredibly grateful to those who convey the fourth dimension to part their ideas too experiences for the create goodness of others.

1. Interactive Areas
Computer-based tools receive got made it easier to explicate some mathematical concepts too generate interesting questions. It's fantastic when technology enhances too enriches mathematics education.

I of late tweeted this colourful example which shows that nosotros tin cutting a circle into sectors too rearrange them to create upward one's hear the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this vivid applet which illustrates the same matter inwards really clear interactive steps. Never earlier receive got I seen the formula for the expanse of a circle explained so clearly (and did y'all run across the 'proof without words' video inwards Practical tips for a (newly) qualified maths teacher' too then create receive got a look. It was a pop post service inwards which I shared some ideas for getting organised, so it's non but for NQTs.

5 Maths Gems #30

Hi in addition to welcome to my 30^th gems post. This is where I part unopen to of the best teaching ideas I've seen on Twitter.

1. Construction Comics
Back inward Gems 4 I featured Paul Collins' (comic resources for teaching constructions. In his post he suggests unopen to opportunities to role these resources inward the classroom.

One of Paul Collins' structure comic strips

I haven't taught constructions this twelvemonth therefore I've non nonetheless had a peril to role these resources, or the awesome Gems 21).

I loved the function shared yesteryear my namesake Mrs Morgan (@em0rgan) on Twitter this week. She asked her students to do their ain structure comic strips for homework. The results facial expression excellent. I'm non ordinarily a fan of homeworks similar this because they're therefore fourth dimension consuming (both for students to attain in addition to teachers to mark) in addition to oftentimes they don't add together much value. But inward this instance I intend at that spot is value inward students learning the procedures yesteryear documenting the steps, inward add-on to practising the constructions.

2. Influenza A virus subtype H5N1 Level Resources

Thanks to @OxfordEdMaths for sharing a fix of excellent A degree assessments for C1 - C4, D1, D2, FP1, FP2, M1, M2, S1 in addition to S2. It tin move difficult to discovery sources of skilful Influenza A virus subtype H5N1 degree questions. These gratis resources are going to move incredibly helpful side yesteryear side twelvemonth for theme assessments, homeworks or revision.

Speaking of Influenza A virus subtype H5N1 degree resources, do banking enterprise gibe out the website 0a explains: Calculus (with pics in addition to gifs) -thanks to @mathhombre in addition to @solvemymaths for sharing this. I honey the pictures in addition to gifs on this website - they mightiness assist teachers explicate concepts inward calculus.

3. Homework Expectations
I've only discovered MathFour.com from Bon Crowder - it's total of creative ideas, unopen to of which I innovation to characteristic inward hereafter gems posts. One persuasion that I spotted was a 'How to Write Your Math Homework' guide. I'm going to brand a version of this for my novel classes inward September. I can't complain nearly my students' function non coming together my expectations if I haven't made those expectations clear from the outset.

4. Talk a Mile a Minute
Steve Wyborney (@SteveWyborney) shared his Talk a Mile a Minute Vocabulary Activity. The video below describes the action inward detail. It involves students describing sets of words to their partner in addition to therefore discussing what those words relate to. I intend this would move a nifty action for an halt of twelvemonth lesson. Have a facial expression at Steve's post for resources. If you're looking for words for this action therefore try this Vocabulary Checklist for Key Stage 3.

5. Activities

I've seen unopen to lovely activities in addition to questions on Twitter this week.

@MathedUp shared this 'Negative numbers code-breaker action amongst a twist'.

Gems 29.

@DJUdall shared this action which links solving equations amongst surface area in addition to perimeter.

@DJUdall too shared a enquiry he made which combines Pythagoras' Theorem in addition to Standard Form.

The latter 2 resources were shared during concluding Sunday's #mathsTLP. This is a nifty identify for maths teachers to collaborate. Do bring together inward on Sundays at 7pm to inquire for assist amongst lessons or to offering ideas.

Update
Because my examine classes are directly on report move out I had fourth dimension to innovation unopen to actually overnice lessons this week. With my Year 10s I did a lesson on similar triangles using resources from the Mathematics Assessment Project. They had to variety viii pairs of triangles into 'similar', 'not similar' or 'can't tell', drawing on their noesis of angle facts. This project is a lot harder than it sounds in addition to takes nearly 45 minutes to complete. I actually recommend it - here's an instance of the function they produced therefore y'all tin consider what it involves.

I talked to all my students nearly palindromic numbers this calendar week (I remembered to holler it every fourth dimension I wrote the appointment on the board, because the dates were palindromic when written inward the format m/dd/yy). Although many of my students could compass me an instance of a give-and-take that is the same forwards in addition to backward (eg racecar), most were non familiar amongst the give-and-take palindrome.

Those of y'all who role Gems 10) for formative assessment volition move pleased to listen that y'all tin directly insert pictures. For instance y'all mightiness desire to insert questions from www.diagnosticquestions.com.

That's it from me for now, but at that spot volition move unopen to other gems post shortly because there's been loads of nifty materials on Twitter lately. Wish me luck amongst my concluding calendar week at my electrical flow school! My novel project starts later half-term.

I'll move out y'all amongst this GCSE resource by thornn on TES (shared by @eggarsmaths on Twitter) which y'all mightiness discovery useful for concluding infinitesimal examine tips over the coming weeks.

Wednesday, October 16, 2019

5 Maths Gems #52

Hello as well as welcome to my 52^nd gems post. This is where I percentage 5 maths education ideas that I've seen on Twitter.

1. GCSE Revision
The Easter holidays are here! Phew, fourth dimension for a rest. When nosotros provide to schoolhouse for the summertime term many of us volition start preparing our Year xi classes for their GCSE exams. I wrote a postal service 'Higher GCSE Revision' terminal twelvemonth which includes all my favourite resources exactly I'm e'er on the expect out for novel ideas. Thanks to Maria Howard (@MrsHsNumeracy) for creating a helpful booklet of 'quick grade GCSE questions' grouped yesteryear topic, amongst answers.

2. Starter Cards as well as Negative Numbers

I'm a big fan of MathsPad. Some of their resources are gratis as well as some require a subscription. It's worth following @MathsPadNicola and @MathsPadJames on Twitter.

Nicola late tweeted a link to a novel hit of starter tasks that could move used every bit starters, leave of absence slips or mini-assessments.

Nicola every bit good shared a lovely negative numbers magic foursquare activity.

In response to this, Colin Foster (@colinfoster77) shared the chapter 'Negative Numbers' from his mass Number as well as Algebra. I mean value this volition move actually useful to novel teachers when they instruct negative numbers for the offset time. I similar the multiplication foursquare action which helps students empathise why multiplying 2 negative numbers gives a positive number.

The total laid upwards of Colin's bright books tin move flora here.

3. H5N1 Level Revision

After the Easter holidays nosotros convey to a greater extent than or less 5 weeks at schoolhouse earlier examination flavor starts. I'll demand to complete education C2 as well as C4 as well as thus produce some examination preparation. If you're looking for H5N1 grade revision resources, in that place are a few at the bottom of my revision cards as well as my C2 revision relay.

Sandra from MathsBox (@mathsbox1) late tweeted a link to some novel A2 revision starters (example pictured below).

If you lot know of whatever other expert H5N1 grade revision resources delight allow me know.

4. Points
I similar 'Points Number Work' yesteryear Jeremy Denton (@Jeremy_Denton). Students are given twenty minutes to endeavor to boot the bucket every bit many points every bit possible. It's e'er worth getting students to practise these fundamental skills.

5. Primary Resources
Year vi teachers as well as parents volition move pleased to listen that Mark McCourt (@EmathsUK) has starting publishing practice papers for the novel Key Stage 2 Arithmetic Tests.

Whilst on the dependent champaign of master copy resources, I should refer some materials that were late shared yesteryear Beth Lilley (@MissBLilley). I've never seen these booklets from New Zealand's Maths Centre earlier - there's loads of peachy master copy cloth hither thus I've added the link to my Primary Maths page.

There's every bit good some expert secondary resources here, such every bit the Geometric Reasoning booklet, as well as another bits as well as pieces that mightiness come upwards inwards handy (a Family Maths booklet and an Adult Numeracy booklet). Lots to explore!

Update
Edexcel sent out post yesteryear @Jeremy_Denton.

Booking is straightaway opened upwards for Edexcel's Warwick Conference 2016 which is on Sabbatum 2nd July. I'll move presenting on 'the wonderful basis of maths resources'. I'm thus excited virtually seeing Hannah Fry's keynote - she's brilliant! Do mass a house if you lot tin (it's £185 thus enquire your schoolhouse to pay!).

Last calendar week I hosted here.

Have a lovely Easter!

Don Steward's preferred lesson structure:
twenty minutes of teaching, forty minutes of
'practice amongst depth' as well as generalising

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

Saturday, October 12, 2019

5 Maths Gems #66

Welcome to my 66^th update from the Blue Planet of Maths EduTwitter. This is where I part about of the latest ideas in addition to resources for teaching maths.

Nov is ever i of the hardest months for teachers. I'm drowning inwards workload at the instant in addition to I bet many of you lot are too. It's relentless. At times similar this I'm grateful for the Twitter community in addition to my lovely colleagues for their ongoing back upwards in addition to encouragement. I conception to write a weblog post service presently amongst about thoughts in addition to ideas on workload. In the meantime, hither are 5 slices of mathematical loveliness to cheer us up...

1. 3Blue1Brown
Thanks to ‏@johngreen for sharing YouTube channel 3Blue1Brown. I'll hold out using their Vectors video when I learn this theme inwards C4 afterwards this year.

If you lot learn logs in addition to then you lot powerfulness hold out interested inwards their video 'Triangle of Power'.

2. Classroom Display

Thanks to Clarissa Grandi (@c0mplexnumber) for sharing this beautiful classroom display 'The Faces Behind the Formulae' which she made for an Influenza A virus subtype H5N1 grade classroom.

I've added it to my displays page.

3. Pinpoint Learning

Last yr I featured Pinpoint Learning in Gems 45. It's a expert fourth dimension to advert this website i time to a greater extent than because most schools run mock GCSE exams inwards Nov or December. Pinpointlearning.co.uk provides students amongst personalised inquiry booklets based on bear witness results. There's a sample booklet here. You (or your students) only upload mock marks past times inquiry in addition to the website automatically produces private booklets featuring questions on the topics that students take away to run on. I'm told that students honey receiving their personalised booklets! I promise to endeavor it out amongst my Year 11s side past times side month.

Edexcel in addition to AQA mock QLA sheets are on the website in addition to you lot tin dismiss currently upload results for all iii mock papers for free during a i calendar month trial. If you lot subscribe during the lawsuit catamenia in addition to then you lot acquire an annual subscription for £300.

Follow @pinpntlearning for updates.

4. SMILE
I'm teaching an interesting Year vii course of written report this year. Having non taught Year vii for a seat out of years, I'm taking a fresh await at resources in addition to approaches. I lately rediscovered SMILE worksheets. Here are a twosome of examples:

Powers of Ten Flags

Coordinate Messages

Also, SMILE cards are available through STEM Learning - these are pretty one-time straightaway hence the character of the copies vary. I recall learning maths from SMILE cards when I was inwards Year 7. Thanks to Martin Colley for arranging the cards past times theme here.

5. Pythagoras amongst Blokus

I honey this halt displace video from ‏@JoelBSperanza.

If you lot desire to meet to a greater extent than of Joel's halt displace videos or honor out how to brand your own, cheque out his weblog post service Stop-Motion Education.

Update

La Salle accept announced about to a greater extent than dates for their conferences (including Wales in addition to Scotland for the starting fourth dimension time). I've added these to my conference listings and volition endeavor to attend equally many equally possible.

In illustration you lot missed it, my most recent post service was the tertiary inwards my serial '5 Websites You Should You Know'. It featured Jonathan Hall's websites MathsBot.com, Flashmaths.co.uk in addition to Formtimeideas.com.

I've also updated my recent post service most Corbettmaths.com. Since writing it I discovered how expert his textbook exercises are! I've institute them hence helpful for teaching Year vii this week.

Extract from CorbettMaths.com 'puzzle square. This i is for directed seat out practice.

Friday, October 11, 2019

5 Maths Gems #85

Welcome to my 85^th gems post. This is where I portion or as well as thence of the latest news, ideas as well as resources for maths teachers.

1. SSDD as well as Venns
In illustration y'all haven't heard, Craig Barton (@mrbartonmaths) launched 2 novel websites final week: ssddproblems.com as well as mathsvenns.com.

I outset blogged most the 'Same Surface Different Deep' thought afterwards Craig introduced it at the JustMaths conference final June. The thought is that nosotros cause got a laid of problems where the surfaces are similar (eg all isosceles triangles) but the deep structures (ie the topics) are different. It's vivid to encounter this thought taking off inwards classrooms all over the footing right away that Craig has shared a large collection of SSDD problems.

Craig explains how as well as why to live on these problems, as well as invites contributions, on First grand Digits of Pi' display from Jae Ess (conversations betwixt students at Jae's school.

I've added this to my page of maths display resources.

3. Convince Me That
Daniel Kaufmann (@KauDan721) has created a laid of 'Convince Me That' problems. Giving students the answer, rather than quest them to honour the answer, allows them to focus on dissimilar aspects of the occupation (the how, the why, the process).

Teachers are invited to contribute their ain problems to this collection.

4. Minimally Different Questions
Jess (here as well as brain to my post-conference podcast alongside Craig Barton here.

In the podcast I promised that I'd develop a page to portion noesis organisers. My colleague Andy has been using noesis organisers rattling effectively this twelvemonth as well as he has given me permission to portion or as well as thence of his to larn the page started. If you've made noesis organisers that you're happy to share, delight ship me a link as well as I'll include them on this page.

On Mon I ran a xx infinitesimal CPD session on GCSE revision for my colleagues (mainly focused on resources). My slides are here if you'd similar to borrow them. Links are inwards the notes department on each slide.

In other news...

Dr Madas has published or as well as thence IYGB papers for the AS pure content of the novel H5N1 marking (Paper L, Paper P, Paper Q and Paper R). Given how lilliputian examination do textile in that place is for our electrical flow Year 12s, these are actually helpful!
Dr Frost is organising or as well as thence other maths instructor social event on Fri 13th Apr inwards Surrey. All welcome!
Chris McGrane wrote a keen post on planning a lesson on integration which is worth a read if y'all learn H5N1 level.
If you're looking for an Easter-themed resources for the cease of term, endeavor Chris Smith's Easter relay.
There are alone a few residential tickets left for BCME! It starts inwards 2 as well as a one-half weeks - I can't wait.

I'll exit y'all this alongside fantabulous puzzle from Brilliant, shared past times Mark Horley (@mhorley).