Learn Math So Easy: Negative Numbers

Showing posts with label Negative Numbers. Show all posts

Wednesday, April 8, 2020

5 Maths Gems #7

Common practise of successful departments' as well as 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 degree 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 portion their ideas as well as experiences for the produce goodness of others.

1. Interactive Areas
Computer-based tools lead maintain made it easier to explicate some mathematical concepts as well as generate interesting questions. It's fantastic when applied scientific discipline enhances as well as enriches mathematics education.

I late tweeted this colourful instance which shows that nosotros tin cutting a circle into sectors as well as rearrange them to create upward one's involve heed the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this bright applet which illustrates the same matter inwards real clear interactive steps. Never earlier lead maintain I seen the formula for the expanse of a circle explained so clearly (and did y'all come 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 as well as and then rotate as well as interpret the travel past times sections).

2. Estimation

Examples from estimation180.com

I've seen a lot of American teachers tweet close Common practise of successful departments' as well as 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 degree 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 portion their ideas as well as experiences for the produce goodness of others.

1. Interactive Areas
Computer-based tools lead maintain made it easier to explicate some mathematical concepts as well as generate interesting questions. It's fantastic when applied scientific discipline enhances as well as enriches mathematics education.

I late tweeted this colourful instance which shows that nosotros tin cutting a circle into sectors as well as rearrange them to create upward one's involve heed the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this bright applet which illustrates the same matter inwards real clear interactive steps. Never earlier lead maintain I seen the formula for the expanse of a circle explained so clearly (and did y'all come 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 come across students working on whiteboards unopen to the classroom (I wishing my classroom was this big!).

Nathan enthusiastically says, 'When students are working on the whiteboards, I tin come across everything happening at once. It's similar I'm looking at the freaking Matrix. With a quick glance, I tin come across which students got it, which students are making fry mistakes, as well as which students lead maintain no sentiment what's going on. I tin chop-chop position errors for students. I tin inquire a stronger educatee to tending 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 sweat it! And earlier y'all disregard this sentiment because y'all don't lead maintain a room total of whiteboards, ship inwards involve heed that whatsoever vertical surface will produce - sweat writing on windows or using Common practise of successful departments' as well as 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 degree 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 portion their ideas as well as experiences for the produce goodness of others.

1. Interactive Areas
Computer-based tools lead maintain made it easier to explicate some mathematical concepts as well as generate interesting questions. It's fantastic when applied scientific discipline enhances as well as enriches mathematics education.

Improves visibility.
Allows for transfer of cognition unopen to 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' as well as 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 degree 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 portion their ideas as well as experiences for the produce goodness of others.

1. Interactive Areas
Computer-based tools lead maintain made it easier to explicate some mathematical concepts as well as generate interesting questions. It's fantastic when applied scientific discipline enhances as well as enriches mathematics education.

I late tweeted this colourful instance which shows that nosotros tin cutting a circle into sectors as well as rearrange them to create upward one's involve heed the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this bright applet which illustrates the same matter inwards real clear interactive steps. Never earlier lead maintain I seen the formula for the expanse of a circle explained so clearly (and did y'all come 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 last pose underneath. Read his send service for to a greater extent than on this, it actually is a bright idea.

Next week

A lot of my readers volition last at La Salle Education's maths conference side past times side week. I’m actually looking forrad to it. Please come upward as well as tell how-do-you-do if y'all topographic point me. Bear inwards involve heed that I lead maintain a iii calendar month onetime babe who rarely sleeps, so if y'all come across me doze off at whatsoever betoken as well as then simply laissez passer on me a nudge.

I however promise to write a maths gems send service side past times side week, it volition last a twenty-four hours slow but hopefully total of ideas shared at the conference.

By the way, if y'all didn't grab my final send service 'Common practise of successful departments' as well as 'What I learnt today' - both are good worth a read, especially for teachers of Influenza A virus subtype H5N1 degree 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 portion their ideas as well as experiences for the produce goodness of others.

1. Interactive Areas
Computer-based tools lead maintain made it easier to explicate some mathematical concepts as well as generate interesting questions. It's fantastic when applied scientific discipline enhances as well as enriches mathematics education.

I late tweeted this colourful instance which shows that nosotros tin cutting a circle into sectors as well as rearrange them to create upward one's involve heed the expanse (taken from @edfromo's Google+ post). @fmaths42 replied amongst this bright applet which illustrates the same matter inwards real clear interactive steps. Never earlier lead maintain I seen the formula for the expanse of a circle explained so clearly (and did y'all come across the 'proof without words' video inwards Practical tips for a (newly) qualified maths teacher' as well as then produce lead maintain a look. It was a pop send service inwards which I shared some ideas for getting organised, so it's non simply for NQTs.

Saturday, January 25, 2020

5 Maths Gems #30

Hi together with welcome to my 30^th gems post. This is where I percentage only about of the best teaching ideas I've seen on Twitter.

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

One of Paul Collins' structure comic strips

I haven't taught constructions this twelvemonth hence I've non nevertheless had a run a hazard to role these resources, or the awesome Gems 21).

I loved the operate 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 unremarkably a fan of homeworks similar this because they're hence fourth dimension consuming (both for students to gain together with teachers to mark) together with oft they don't add together much value. But inwards this instance I intend at that topographic point is value inwards students learning the procedures yesteryear documenting the steps, inwards improver to practising the constructions.

2. Influenza A virus subtype H5N1 Level Resources

Thanks to @OxfordEdMaths for sharing a gear upwards of excellent A degree assessments for C1 - C4, D1, D2, FP1, FP2, M1, M2, S1 together with S2. It tin endure difficult to honor sources of expert Influenza A virus subtype H5N1 degree questions. These gratis resources are going to endure incredibly helpful adjacent twelvemonth for theme assessments, homeworks or revision.

Speaking of Influenza A virus subtype H5N1 degree resources, do banking concern agree out the website 0a explains: Calculus (with pics together with gifs) -thanks to @mathhombre together with @solvemymaths for sharing this. I dear the pictures together with gifs on this website - they mightiness assistance teachers explicate concepts inwards calculus.

3. Homework Expectations
I've only discovered MathFour.com from Bon Crowder - it's total of creative ideas, only about of which I excogitation to characteristic inwards 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 inwards September. I can't complain nearly my students' operate 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 activeness inwards detail. It involves students describing sets of words to their partner together with and hence discussing what those words relate to. I intend this would endure a cracking activeness for an terminate of twelvemonth lesson. Have a facial expression at Steve's post for resources. If you're looking for words for this activeness together with hence try this Vocabulary Checklist for Key Stage 3.

5. Activities

I've seen only about lovely activities together with questions on Twitter this week.

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

Gems 29.

@DJUdall shared this activeness which links solving equations amongst expanse together with perimeter.

@DJUdall likewise shared a query he made which combines Pythagoras' Theorem together with Standard Form.

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

Update
Because my examination classes are directly on report acquire out I had fourth dimension to excogitation only about actually prissy lessons this week. With my Year 10s I did a lesson on similar triangles using resources from the Mathematics Assessment Project. They had to kind 8 pairs of triangles into 'similar', 'not similar' or 'can't tell', drawing on their cognition of angle facts. This project is a lot harder than it sounds together with takes nearly 45 minutes to complete. I actually recommend it - here's an instance of the operate they produced hence you lot tin see what it involves.

I talked to all my students nearly palindromic numbers this calendar week (I remembered to yell it every fourth dimension I wrote the appointment on the board, because the dates were palindromic when written inwards the format m/dd/yy). Although many of my students could plough over me an instance of a discussion that is the same frontward together with backward (eg racecar), most were non familiar amongst the discussion palindrome.

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

That's it from me for now, but at that topographic point volition endure only about other gems post presently because there's been loads of cracking materials on Twitter lately. Wish me luck amongst my concluding calendar week at my electrical flow school! My novel project starts afterward half-term.

I'll acquire out you lot amongst this GCSE resource by thornn on TES (shared by @eggarsmaths on Twitter) which you lot mightiness honor useful for concluding infinitesimal examination tips over the coming weeks.

Friday, January 24, 2020

5 Maths Gems #52

Hello together with welcome to my 52^nd gems post. This is where I percentage 5 maths didactics 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 eleven classes for their GCSE exams. I wrote a shipping 'Higher GCSE Revision' final twelvemonth which includes all my favourite resources merely I'm ever on the await out for novel ideas. Thanks to Maria Howard (@MrsHsNumeracy) for creating a helpful booklet of 'quick grade GCSE questions' grouped past times topic, amongst answers.

2. Starter Cards together with Negative Numbers

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

Nicola late tweeted a link to a novel gain of starter tasks that could hold upwards used equally starters, leave of absence slips or mini-assessments.

Nicola equally good shared a lovely negative numbers magic foursquare activity.

In response to this, Colin Foster (@colinfoster77) shared the chapter 'Negative Numbers' from his majority Number together with Algebra. I mean value this volition hold upwards actually useful to novel teachers when they instruct negative numbers for the showtime time. I similar the multiplication foursquare activeness which helps students sympathise why multiplying 2 negative numbers gives a positive number.

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

3. Influenza A virus subtype H5N1 Level Revision

After the Easter holidays nosotros convey to a greater extent than or less 5 weeks at schoolhouse earlier seek out flavour starts. I'll demand to complete didactics C2 together with C4 together with therefore produce some seek out preparation. If you're looking for Influenza A virus subtype H5N1 grade revision resources, at that spot are a few at the bottom of my revision cards together with 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 practiced Influenza A virus subtype H5N1 grade revision resources delight allow me know.

4. Points
I similar 'Points Number Work' past times Jeremy Denton (@Jeremy_Denton). Students are given twenty minutes to get to choke equally many points equally possible. It's ever worth getting students to practise these fundamental skills.

5. Primary Resources
Year half-dozen teachers together with parents volition hold upwards pleased to remove heed that Mark McCourt (@EmathsUK) has starting publishing practice papers for the novel Key Stage 2 Arithmetic Tests.

Whilst on the dependent plain of principal resources, I should cite some materials that were late shared past times Beth Lilley (@MissBLilley). I've never seen these booklets from New Zealand's Maths Centre earlier - there's loads of cracking principal textile hither therefore I've added the link to my Primary Maths page.

There's equally good some practiced secondary resources here, such equally the Geometric Reasoning booklet, together with another bits together with pieces that powerfulness come upwards inwards handy (a Family Maths booklet and an Adult Numeracy booklet). Lots to explore!

Update
Edexcel sent out post past times @Jeremy_Denton.

Booking is instantly opened upwards for Edexcel's Warwick Conference 2016 which is on Sabbatum 2nd July. I'll hold upwards presenting on 'the wonderful globe of maths resources'. I'm therefore excited nigh seeing Hannah Fry's keynote - she's brilliant! Do majority a house if you lot tin (it's £185 therefore inquire 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' together with generalising

Multiplying Negatives

"A negative times a negative is a positive". It's a difficult 1 to explain. We all learnt it at schoolhouse in addition to practised it to the holler for of fluency, but it's non until we're asked why it plant that nosotros halt in addition to intend almost it.

Numbers lines in addition to visualisations are real helpful when didactics the improver in addition to subtraction of negative numbers. But amongst multiplication in addition to sectionalization it's non in addition to thus clear.

Let's await at a few approaches in addition to resources.

1 . Pattern Spotting
Draw a touchstone multiplication tabular array in addition to extend it backwards to include negative numbers. It's a straightforward designing that all students should hold out able to location in addition to continue. Get students to produce this using Colin Foster's activity on page v of his Negative Numbers chapter.

2. Multiplication Grids
Take 2 2-digit numbers in addition to multiply them together using grid multiplication. For simplicity, let's lead keep 12 x 11:

Here nosotros lead keep written 12 every bit 10 + 2 in addition to eleven every bit 10 + 1. But it would operate only every bit good if nosotros expressed those numbers differently. Instead, let's write 12 every bit fifteen - iii in addition to eleven every bit fifteen - 4. We should acquire the same answer:

This solely plant if -3 x -4 = 12.

Note that this explanation requires students to commencement sympathize 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 in addition to accessible to us experienced mathematicians. I'm non certain how accessible it is to Year seven students, but it's worth a go.

a in addition to 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 amongst 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 tilt true.

Further Reading
It's a skillful persuasion to read almost a theme earlier you lot learn it, fifty-fifty relatively unproblematic topics that you've taught many times before. Here are around helpful links:

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

Colin Foster suggests that you lot inquire students to brand upwards 10 multiplications in addition to 10 divisions each giving an respond of –8 (eg –2 × –2 × –2 or –1 × 8 etc).

The squaring in addition to 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 out worth exploring estimator behavior likewise (ie around calculators require brackets when squaring a negative). It's of import that students know how to role their estimator properly. There's a slap-up resources from MathsPad for this - Using a Calculator: Odd One Out.

This theme is revisited inwards afterwards years when students are practising substitution. For example, if a = 3, b = -2 in addition to c = -5, abide by the values of: abc; bc²; (bc)²; a²b³ in addition to and thus on. This Substitution Puzzle from mathsteaching.wordpress.com gets quite challenging.

Do allow me know if you lot role an interesting method or resources for didactics the multiplication of negative numbers.

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