Teaching Indices

Source: dotmaths.com

I've mentioned earlier that I beloved pedagogy indices, together with that's because a) there's loads of bright resources available b) the concepts are slow to explicate c) there's some lovely extension problems together with d) this is of import mathematics! You can't produce calculus without agreement indices.

I know I tin strength out programme a sequence of lessons on indices for whatever historic menses grouping (Year 8 correct through to Year 12) together with I'll live able to notice expert character engaging resources. Later inwards this shipping I'll characteristic some of my favourites.

Concepts

The concepts relating to indices are relatively slow to explicate from kickoff principles. The reasoning behind the index laws is straightforward. We tin strength out encounter that b² x b⁵ = b⁷ exactly past times writing out the expressions inwards total (ie b x b x b x b x b x b x b). Students volition chop-chop location the patterns. The 'anything to the ability of null equals one' dominion is also fairly unproblematic to explicate using the logic shown below.

Negative powers tin strength out live explained past times extending the partitioning constabulary (ie what happens when nosotros dissever b² past times b⁷?), or past times considering patterns (see Gems 8 for to a greater extent than on this).

Fractional powers tin strength out live explained using the multiplication law.

And together with hence it's exactly a instance of students remembering the laws together with practising them until they learn fluent. Mim Gosling (@mimgosling) created the moving-picture present below to aid students recollect how fractional powers locomote -"Fractional indices are similar a bloom - the bottom's the root, the top's the power!". So if yous receive got a educatee who gets confused past times fractional indices, exactly remind them that the root is at the bottom (like inwards a tree).

Source: @StIvoMaths

A lesson on index laws

I taught an introductory lesson on index laws latterly - I wrote these slides to construction the degree news together with activities. After nosotros established each index law, the degree did 3 quick exercise questions earlier moving on to the adjacent law. Once we'd covered all 3 laws, the students completed this 'Powers of y eliminator' from Teachit Maths, which was fantastic. It had exactly the correct combination of independent practice, stretch together with engagement. These were the questions that most students got stuck on:

I was quite proud of the slides I made for this lesson - there's an editable version here so experience costless to borrow bits, though some formatting may live lost when yous download it (the PDF version is much prettier!).

Card Sorts
It's a dandy thought to receive got a well-organised provide of sorting activities (eg Tarsia puzzles) inwards your maths department. Find a clever agency to shop them (eg inwards stackable plastic takeaway boxes) together with label the boxes clearly. It takes a flake of fourth dimension to laid upward the bill of fare sorts. If you've got the budget, it's ever best to learn them laminated. Print each assault a dissimilar coloring (eg if in that location are 10 groups inwards the class, impress the bill of fare sets on 10 dissimilar colours) - this stops the sets getting mixed up. Use expert character elastic bands - inexpensive ones degrade chop-chop - or perish on each laid inwards an envelope.

My subdivision has a collection of 56 bill of fare sorts create for purpose inwards lessons (sadly, I spent many many hours sitting at the guillotine during my summertime holidays!). We telephone weep upward them our 'Black Box Resources' because nosotros perish on them inwards dark CD boxes from Ikea. Two of my favourite bill of fare sorts are for indices. The first is Indices Dominoes by Teachit Maths - a genuinely dainty activity, amongst the added bonus that the answer is written backwards hence students can't cheat past times guessing the words. I also like Using Indices from the Standards Unit. Card laid H5N1 (pairs activity) is genuinely useful when you've taught fractional together with negative indices.

You powerfulness also similar these treasure hunts: simple index laws and negative & fractional indices, both from Teachit Maths.

Worksheets
There's loads of dandy worksheets together with activities for independent practice. I've already mentioned the Powers of y eliminator from Teachit Maths. Here's ii to a greater extent than first-class resources that are suitable for practising the 3 index laws:

• Simplifying Indices Code Breaker - Teachit Maths 
• Collect a Joke - Number Loving

Once you've moved onto fractional together with negative indices, there's loads to a greater extent than first-class resources (these powerfulness live useful inwards Year 12 too):

• Mental Powers Code Breaker -Teachit Maths 
• Ever wondered why? - JustMaths 
• Indices activity - SRWhitehouse on TES

STEM Centre provides some fantastic indices activities from Susan Wall. For example, I similar 'Why does?' (extract below) inwards which students are allocated a disceptation to mean value close together with explicate to the class.

Extract from 'Why does?' past times Susan Wall

There's also a truthful or imitation activity which is designed to address mutual misconceptions.

Extract from 'True or False' past times Susan Wall

Blanks

I featured a 'blanks' draw of piece of occupation thought inwards Gems 17. Make upward some questions similar those shown below together with seat them upward to a greater extent than or less the room. Provide post-it notes hence students tin strength out propose values that fit.

Homeworks
I've made ii Pret homeworks for this topic, both are available here.

Enrichment
There's loads of dandy extension questions for this topic. I beloved these mega quadratic equations from Don Steward.

together with these mega difficult powers questions, also from Don Steward

And ii to a greater extent than fantastic rich tasks from Don Steward: Power Sums and Find the Power. 

At H5N1 level, together with mayhap equally extension inwards Year 11, I similar to produce some locomote on comparison indices amongst dissimilar bases. For example, I similar this ordering starter.

And my worksheet exponential equations has been useful. It ends amongst this lovely question. 

Finally, I desire to part ii questions that I've been using for years amongst GCSE students, either when I learn indices or when I learn algebraic fractions. My students are ever horrified past times how complex these questions look, but if they're confident amongst indices together with hence both equations are genuinely surprisingly slow to solve. I'm non certain where these questions originated hence apologies to the creator for non giving credit.

That's it from me - if yous know of whatever other fantastic indices resources, delight comment below or tweet me.

How v squared looks inwards nightmares.

Teaching Indices

Source: dotmaths.com

I've mentioned earlier that I beloved pedagogy indices, together with that's because a) there's loads of bright resources available b) the concepts are slow to explicate c) there's some lovely extension problems together with d) this is of import mathematics! You can't produce calculus without agreement indices.

I know I tin strength out programme a sequence of lessons on indices for whatever historic menses grouping (Year 8 correct through to Year 12) together with I'll live able to notice expert character engaging resources. Later inwards this shipping I'll characteristic some of my favourites.

Concepts

The concepts relating to indices are relatively slow to explicate from kickoff principles. The reasoning behind the index laws is straightforward. We tin strength out encounter that b² x b⁵ = b⁷ exactly past times writing out the expressions inwards total (ie b x b x b x b x b x b x b). Students volition chop-chop location the patterns. The 'anything to the ability of null equals one' dominion is also fairly unproblematic to explicate using the logic shown below.

Negative powers tin strength out live explained past times extending the partitioning constabulary (ie what happens when nosotros dissever b² past times b⁷?), or past times considering patterns (see Gems 8 for to a greater extent than on this).

Fractional powers tin strength out live explained using the multiplication law.

And together with hence it's exactly a instance of students remembering the laws together with practising them until they learn fluent. Mim Gosling (@mimgosling) created the moving-picture present below to aid students recollect how fractional powers locomote -"Fractional indices are similar a bloom - the bottom's the root, the top's the power!". So if yous receive got a educatee who gets confused past times fractional indices, exactly remind them that the root is at the bottom (like inwards a tree).

Source: @StIvoMaths

A lesson on index laws

I taught an introductory lesson on index laws latterly - I wrote these slides to construction the degree news together with activities. After nosotros established each index law, the degree did 3 quick exercise questions earlier moving on to the adjacent law. Once we'd covered all 3 laws, the students completed this 'Powers of y eliminator' from Teachit Maths, which was fantastic. It had exactly the correct combination of independent practice, stretch together with engagement. These were the questions that most students got stuck on:

I was quite proud of the slides I made for this lesson - there's an editable version here so experience costless to borrow bits, though some formatting may live lost when yous download it (the PDF version is much prettier!).

Card Sorts
It's a dandy thought to receive got a well-organised provide of sorting activities (eg Tarsia puzzles) inwards your maths department. Find a clever agency to shop them (eg inwards stackable plastic takeaway boxes) together with label the boxes clearly. It takes a flake of fourth dimension to laid upward the bill of fare sorts. If you've got the budget, it's ever best to learn them laminated. Print each assault a dissimilar coloring (eg if in that location are 10 groups inwards the class, impress the bill of fare sets on 10 dissimilar colours) - this stops the sets getting mixed up. Use expert character elastic bands - inexpensive ones degrade chop-chop - or perish on each laid inwards an envelope.

My subdivision has a collection of 56 bill of fare sorts create for purpose inwards lessons (sadly, I spent many many hours sitting at the guillotine during my summertime holidays!). We telephone weep upward them our 'Black Box Resources' because nosotros perish on them inwards dark CD boxes from Ikea. Two of my favourite bill of fare sorts are for indices. The first is Indices Dominoes by Teachit Maths - a genuinely dainty activity, amongst the added bonus that the answer is written backwards hence students can't cheat past times guessing the words. I also like Using Indices from the Standards Unit. Card laid H5N1 (pairs activity) is genuinely useful when you've taught fractional together with negative indices.

You powerfulness also similar these treasure hunts: simple index laws and negative & fractional indices, both from Teachit Maths.

Worksheets
There's loads of dandy worksheets together with activities for independent practice. I've already mentioned the Powers of y eliminator from Teachit Maths. Here's ii to a greater extent than first-class resources that are suitable for practising the 3 index laws:

• Simplifying Indices Code Breaker - Teachit Maths 
• Collect a Joke - Number Loving

Once you've moved onto fractional together with negative indices, there's loads to a greater extent than first-class resources (these powerfulness live useful inwards Year 12 too):

• Mental Powers Code Breaker -Teachit Maths 
• Ever wondered why? - JustMaths 
• Indices activity - SRWhitehouse on TES

STEM Centre provides some fantastic indices activities from Susan Wall. For example, I similar 'Why does?' (extract below) inwards which students are allocated a disceptation to mean value close together with explicate to the class.

Extract from 'Why does?' past times Susan Wall

There's also a truthful or imitation activity which is designed to address mutual misconceptions.

Extract from 'True or False' past times Susan Wall

Blanks

I featured a 'blanks' draw of piece of occupation thought inwards Gems 17. Make upward some questions similar those shown below together with seat them upward to a greater extent than or less the room. Provide post-it notes hence students tin strength out propose values that fit.

Homeworks
I've made ii Pret homeworks for this topic, both are available here.

Enrichment
There's loads of dandy extension questions for this topic. I beloved these mega quadratic equations from Don Steward.

together with these mega difficult powers questions, also from Don Steward

And ii to a greater extent than fantastic rich tasks from Don Steward: Power Sums and Find the Power. 

At H5N1 level, together with mayhap equally extension inwards Year 11, I similar to produce some locomote on comparison indices amongst dissimilar bases. For example, I similar this ordering starter.

And my worksheet exponential equations has been useful. It ends amongst this lovely question. 

Finally, I desire to part ii questions that I've been using for years amongst GCSE students, either when I learn indices or when I learn algebraic fractions. My students are ever horrified past times how complex these questions look, but if they're confident amongst indices together with hence both equations are genuinely surprisingly slow to solve. I'm non certain where these questions originated hence apologies to the creator for non giving credit.

That's it from me - if yous know of whatever other fantastic indices resources, delight comment below or tweet me.

How v squared looks inwards nightmares.

5 Maths Gems #100

Welcome to my 100^th gems post!

This is a big milestone for me. I've published 327 posts since I started writing my spider web log 4 in addition to a one-half years ago. One hundred of those posts convey been business office of my 'Maths Gems' serial - each ane has featured a pick of news, ideas in addition to resources for maths teachers.

When I joined Twitter I couldn't believe how many ideas in addition to resources were beingness shared yesteryear maths teachers everyday that weren't beingness seen yesteryear the hundreds of thousands of maths teachers who aren't on Twitter. I recollect trying to convince a colleague to bring together Twitter in addition to he said 'I merely don't convey fourth dimension for it. I wishing mortal could merely summarise the ideas for me'. So that's what I elbow grease to do.

I used to write ane gems post a calendar week (I was on motherhood leave!) but straight off I operate full-time I exclusively deal ane or ii gems posts a month. Twitter even in addition to hence provides a constant flow of fabric to pick from, in addition to hence I maintain to summarise in addition to portion some of the best ideas, inward the promise that I tin larn these ideas into classrooms all over the world.

To celebrate the fact that this is my 100th gems post, I'll endure recording a exceptional podcast amongst Craig Barton adjacent weekend. In grooming for this podcast Craig wants yous to guide your favourite jewel (there are 500 to guide from, all indexed here). Tweet, DM or electronic mail Craig (or comment below) to explicate why yous chose it, in addition to it could cease upwardly on the show.

On amongst the gems...

1. Mastery Learning Cycle
Over the course of written report of the final ii years Mark McCourt (@EmathsUK) has published a series of posts on mastery. You must read these splendid posts if yous haven't already! Oliver Caviglioli (@olicav) in addition to Mark convey worked together to do a poster which visualises Mark's model of the Mastery Learning Cycle.

This is merely an extract - download the poster to sympathize what it's all about. This is definitely something to portion amongst all trainee maths teachers (and likely all experienced maths teachers too).

No incertitude Mark's mass 'Teaching for Mastery', due to endure published inward Spring 2019, volition endure a must-read too.

2. Manipulatives
If yous desire to occupation manipulatives inward your didactics but don't know where to start, Craig Barton in addition to Bernie Westacott convey recorded a video podcast that yous volition detect real useful. You tin access the videos through Craig's Youtube playlist. For event hither Bernie explains how to occupation manipulatives to instruct negative numbers:



I've blogged earlier nearly Jonathan Hall's (@StudyMaths) amazing library of virtual manipulatives. It has continued to grow.

One of the novel additions is a bar modelling tool. It's real slowly to occupation in addition to I'm certain lots of teachers volition detect it helpful.

3. Indices Tasks
Miss Konstantine‏ (task for exploring indices. Students variety the cards in addition to detect the strange ane out.

Peter Drysdale‏ (@pwdrysdale) made a Desmos version of this carte sort.

I can't maintain upwardly amongst all the ideas in addition to resources that Miss Konstantine‏ (@GiftedBA) has been sharing lately! For event I enjoyed her recent perimeter problems. Check out her Twitter feed in addition to spider web log for lots to a greater extent than neat stuff.

Speaking of indices, I made an indices business for a recent Teach Secondary article. This resources is designed to endure used amongst a Year vii course of written report later on spending some fourth dimension on index notation, but it would operate amongst other yr groups too. The total resources is here - it contains 4 introductory indices activities.

4. Area
Here's a overnice thought to assist students develop an agreement of what expanse is. Ilona Vashchyshyn (@vaslona) challenged her students to write their cite in addition to hence that it covers an expanse of precisely 100cm². Read the thread for ideas on how to extend this activity.

The challenge: Write your cite in addition to hence that it covers an expanse of precisely 100 cm squared.

Love the unlike strategies students used here. E.g. Leia, who chose to brand each missive of the alphabet 25 cm^2. Notice the revisions on the A.#mathchat #MTBoS Cc @PaiMath pic.twitter.com/FowEQoL4I6

— Ilona Vashchyshyn (@vaslona) 16 Nov 2018

5. Assessment and Questioning
Mrs Budak (@mrsbudak) tweeted an interesting thought from ⁦‪@teacher2teacher‬⁩ that could operate inward every subject. At the cease of an assessment students are given the chance to write downwardly everything else they know nearly the topics on the test. It stops students beingness frustrated when a examination doesn't encompass the things they revised, in addition to it's likely a adept occupation of fourth dimension to retrieve materials from one's retentiveness in addition to write it downwardly (it definitely beats sitting in that place waiting for the examination to end!).

Finally, for some argue this reminded me of some other jewel that I've been pregnant to portion for ages...

Two years ago, I was proverb “do yous convey whatever questions?”. Last yr I switched to “what questions do yous have?” It made a difference. Today I tried “ask me ii questions”. And they did! And those ?s led to to a greater extent than ?s. It amazes me that the littlest things convey such a big impact!

— Andre Sasser (@MrsSasser) 27 August 2018

It such a uncomplicated thought piffling change, in addition to and hence slowly to do! It's worth reading the thread for give-and-take in addition to ideas. Credit to Howie Hua (@howie_hua) for outset tweeting nearly this dorsum inward May. Here's some other of Howie's ideas:

For tests, I dedicate the outset five minutes to convey students lucifer the tests in addition to beak to their grouping nearly strategies (w/out pencils). This lowers their examination anxiety AND when yous walk around, yous listen some of the richest math conversations. #iteachmath

— Howie Hua (@howie_hua) 28 September 2018

Update
I've had a busy few weeks visiting a issue of unlike schools, including ane where I saw soundless instructor inward activity for the outset time.

I helped to run the outset 24-hour interval of a novel London-wide Maths Hub Work Group on developing Influenza A virus subtype H5N1 degree pedagogy, which is led yesteryear Carlos Karingal. We were fortunate to convey a fantastic grouping of teachers attend in addition to I await forwards to seeing how our Influenza A virus subtype H5N1 degree didactics develops throughout the year. In my session I talked through some of the ideas inward this splendid piece written by Chris McGrane about approaches to didactics calculus.

Next calendar week I'll endure supporting Chris Reilly inward running some other Maths Hub Work Group - Challenging Topics at GCSE. I'll spider web log nearly this soon.

My 2d Teach Secondary article is straight off available to read online. It's nearly social club of operations, in addition to opportunities to interleave this topic amongst fractions, decimals in addition to algebra.

Did yous encounter that Pearson convey a novel serial of maths textbooks coming out? It's called Purposeful Practice in addition to yous tin thought sample pages here.

Finally, Christmas is fast approaching in addition to hence yous mightiness detect a occupation for my collections of seasonal resources (here and here) - some are topic based in addition to some are for enrichment. I've written diverse posts nearly Christmas presents for maths teachers over the years (here is final year's post on TeachWire if you're looking for inspiration). If yous desire to process yourself during December, the MA has a Christmas advent calendar where yous tin larn daily discounts on MA books.

I'll move out yous amongst this lovely job shared yesteryear James from MathsPad (@MathsPadJames). There are a arrive at of solutions inward the comments (spoiler alert!), but it tin endure solved inward a affair of seconds without whatever workings using GCSE degree maths.