Starters Together With Inequalities

I lately participated inwards a Twitter chat well-nigh lesson starters. Initially a forum for sharing ideas, it turned into a ground well-nigh whether starters are necessary inwards maths lessons. In my persuasion in that place is no correct or incorrect lesson construction - every instructor does what plant best for them together with their students. For me together with my students, starters travel well. I exercise them inwards the vast bulk of my lessons.

What is a starter?
Presumably inwards a lesson without a starter, students acquire inwards the classroom, sit down downwards together with at in i trial take away heed to an exposition. In a lesson alongside a starter, students produce unopen to maths at in i trial on arrival, before the principal pedagogy chemical factor begins. Lesson starters (or 'bell work') accept many forms. Every instructor has a unlike approach.

Purpose

My starters receive got i of iii purposes:

(a) recap together with practice of what students learnt inwards the previous lesson.
(b) prerequisite skills check to assess a starting point.
(c) a hook to engage together with intrigue.

Starters also assist to awaken my students' brains - they acquire into a mathematical means of thinking before I firstly pedagogy them something new. 

Logistics
When students brand it I greet them at the door alongside their starter, usually a printed question. They're used to this format hence they sit down downwards together with firstly immediately. They produce the starter independently spell their peers brand it together with I laid up.

My starters are usually real brusk - no to a greater extent than than five minutes - because this minimises the variation inwards how long students accept to consummate them. H5N1 starter that takes anywhere betwixt 10 together with twenty minutes to consummate could travel problematic (early finishers are left waiting) but inwards a starter that takes 3 - five minutes this is less of an issue. The early on finishers stick their completed starter inwards their books together with yesteryear the fourth dimension they've done that, the residual are ready hence nosotros together with hence review it equally a class. There's rarely a indicate at which students are non occupied.

Sometimes starters atomic number 82 to interesting questions. Sometimes they effect inwards me changing my lesson innovation on the spot. This is a challenge for me, but I'm getting amend at thinking on my feet equally I acquire to a greater extent than experienced.

Typical Examples
I'll laissez passer you lot an thought of what my typical starters seem like. Bear inwards mind, equally I said earlier, every instructor does this differently. I've picked a theme at random: inwards iii lessons on inequalities alongside Year 10 final year, these were my iii starters.

In lesson 1 we're solving linear inequalities together with representing solutions on number lines. They've done this before hence I firstly alongside a skills cheque to brand certain they scream back the basics:

Inequalities starter - lesson 1

In lesson 2 we're going to firstly looking at graphical inequalities - for this they request to know how to plot direct lines hence I laissez passer them unopen to other skills cheque starter to run into if I request to become through this:

Inequalities starter - lesson 2

In lesson 3 we're looking at unopen to harder graphical inequalities hence I desire to remind them of what they did inwards the previous lesson together with laissez passer them a gustatory modality of the kind of inquiry they volition travel seeing today:

Inequalities starter - lesson 3

The 'list the integer coordinates' business office of this inquiry is novel to them hence requires unopen to thinking. I seem it to generate questions which link smoothly into my exposition. I similar my students to intend that the direction of the lesson is prompted yesteryear the questions that arise during the starter, when inwards reality I've usually pre-empted their questions together with planned accordingly.

So you lot tin run into that my starters are naught special but to me they are an essential business office of the lesson.

Exam Questions

Sometimes, peculiarly at H5N1 level, I exercise the starter equally an chance to produce an examination manner question. For event if I instruct a Year 12 lesson on finding the equation of a circle using these excellent activities from Mathematics Assessment Project, together with hence the side yesteryear side lesson I would laissez passer students an equation of a circle yesteryear examination inquiry on arrival. This gets them used to the format together with wording of examination questions together with recaps what they learnt inwards the previous lesson. H5N1 unmarried iv or five grade examination inquiry is the perfect length for an H5N1 grade starter.

With GCSE classes, I sometimes laissez passer a QWC examination inquiry (a 'star' question) equally a starter.

More Inequalities Starters
So at nowadays I've described my typical starters, let's seem unopen to option activities. I've featured iii starters here, in i trial again focusing on inequalities.

Most maths teachers are familiar alongside Transum's 'starter of the day'. What I similar well-nigh the Transum website is that starters are also listed by topic. Here's an event of an inequalities starter.

'Less than' yesteryear Transum

Short card sorts tin travel good equally starters. This number draw of piece of occupation card matching activity is suitable for an inequalities lesson.

There's also a nifty card gibe (and other first-class activities) inwards Evaluating Statements well-nigh Number Operations from Mathematics Assessment Project. Some of these mightiness travel likewise long for a starter but would travel suitable for a principal lesson activity.

When is the arguing true? - Mathematics Assessment Project

If you're looking for to a greater extent than principal lesson activities on inequalities together with hence these 2 resources are superb: Introduction to inequalities by Project Maths (activities towards the dorsum of the pdf) and Defining Regions Using Inequalities by Mathematics Assessment Project. There are lots to a greater extent than resources inwards my algebra resources library.

Project Maths

Sources of Starters
These ideas are merely the tip of the iceberg. There are loads to a greater extent than sources of starters - a classic is 30 Maths Starters from Subtangent. MathsPad has unopen to nifty interactives similar this Simplifying Magic Square. FlashMaths has skillful interactives too. I could become on but there's no indicate reinventing the wheel. Colleen Young has an amazing website 'Mathematics Starters together with Plenaries' which I highly recommend.

Ideas From Shanghai

Us teachers are oft discipline to criticism. As a resultant nosotros tin endure really defensive. We accept an of import labor to do nether hard atmospheric condition thence it's no surprise that we're sensitive when told that the lineament of our teaching could endure improved. Unfortunately this defensiveness makes us resistant to novel teaching ideas, together with that is a tragedy. Because novel teaching ideas hold this profession moving forward. New approaches to explaining mathematical concepts accept the potential to transform maths education. It is absolutely essential that nosotros are opened upward to these ideas.

We've heard a lot of rhetoric from the Department for Education well-nigh Shanghai. It's tardily for us to acquire frustrated well-nigh the comparisons because at that topographic point are meaning differences betwixt our nations, both inwards damage of civilization together with the underlying structures of the teaching profession. But let's position the politics to 1 side - the questions maths teachers should endure bespeak are, "How tin I notice out what happens inwards maths lessons inwards Shanghai, together with volition those ideas endure useful inwards my classroom?". Well I accept to a greater extent than or less answers for y'all - y'all tin notice out well-nigh what industrial plant good inwards Shanghai through the Maths Hub maiden together with yes, to a greater extent than or less of the ideas are fantastic together with tardily to apply. Don't endure also quick to dismiss them.

Today I went to the launch of the Surrey Plus Maths Hub. The Pb schoolhouse is St John the Baptist School inwards Woking - their teachers accept lately returned from a trip to Shanghai thence I knew we'd listen all well-nigh it at the launch. I felt defensive, together with I acknowledge that I arrived amongst an arrogant 'I accept zip to larn from them' attitude. I was pleasantly surprised - the speakers (who delivered first-class presentations) were careful non to alienate their audience. They acknowledged the huge cultural together with household unit of measurement differences betwixt China together with the UK. They also acknowledged the fact that teachers inwards Shanghai accept numerous advantages over us, the most notable beingness that they solely instruct ii 35 infinitesimal lessons a twenty-four hr stream (they pass the residuum of their fourth dimension doing research, marking, planning together with collaborating amongst colleagues). The speakers acknowledged all these differences but did non focus on them, because these things are beyond our control. Instead, they focussed on interesting mathematical methods that nosotros tin all start using lead away inwards our classrooms. In this ship service I'm going to portion these methods because, although to a greater extent than or less may appear 'obvious', sometimes nosotros overcomplicate things. I mean value these uncomplicated methods are truly rather useful.

Multiplication Strategies

The ii questions inwards a higher house were given to us equally a starter at the hub launch. What bully questions they are. These are given to 10 twelvemonth olds inwards Shanghai together with they tin operate out the answers inwards their heads. I'm impressed. Well done Shanghai.

It's all well-nigh using clever numerical methods. Let's start amongst something simpler (I'll come upward dorsum to the starter questions inwards a minute). This 1 is done yesteryear vii twelvemonth olds:

Work out 2.5 x 3.2 inwards your head. Quickly!

I'm told that students inwards Shanghai would encounter this calculation similar this:

2.5 x (4 x 0.8) = 10 x 0.8 = 8.

I don't mean value my students, fifty-fifty my 6th formers, would mean value similar that.

Incidentally, I used a dissimilar method. My encephalon prefers fractions to decimals, thence I stance of it similar this:

Either way, a fleck of clever publish manipulation makes this inquiry fairly straightforward.

From a immature age, students inwards Shanghai larn to reply questions similar these inwards their heads:
a. 0.125 x 8
b. 0.5 x 0.2
c. 2.5 x 4
d. 12.5 x 0.8
Through lots of practice, students acquire truly expert at working out anything that relates to the 2, five together with 10 times tables. They also larn to instantaneously recall that 4 x 25 = 100 together with 8 x 12.5 = 100. Using these skills, they tin reply questions similar this inwards their head:

12.5 x 2.4

Look, there's a 12.5 here. 12.5 is a useful publish because it tin endure multiplied yesteryear 8 to acquire 100. So rewrite the calculation as:

12.5 x (8 x 0.3)
= (12.5 x 8) x 0.3
= 100 x 0.3 = 30

Easy when y'all know how.

10 twelvemonth olds inwards Shanghai mightiness endure given this inquiry to hold off at inwards a grouping of three:

125 x 80.8

Together they'll come upward up amongst a strategy for answering the question. For example, they mightiness advise rewriting it as:
125 x 8 x 10.1
= M x 10.1 = 10,100.

See how this works? If nosotros accept a 12.5, nosotros hold off for an 8. If nosotros accept a 25, nosotros hold off for a 4. We manipulate the numbers to brand the calculation friction match what nosotros know.
So let's purpose these ideas to reply those starter questions.

2.5 x 4.04 x 0.9

Split the middle publish upward to get:
2.5 x (4 + 0.04)  x 0.9
=((2.5 x 4) + (2.5 x 0.04)) x 0.9
= (10 + 0.1) x 0.9
= nine + 0.09 = 9.09

I'll acquire out y'all to operate out a expert strategy for answering the 2d inquiry yourself (tweet me your reply or comment below).

So, why non start teaching your students these clever mental strategies? It's never also belatedly to prepare publish skills. These lessons would operate at whatsoever age. I could do amongst practising these strategies myself.

Here's a few to a greater extent than ideas I've seen on Twitter that mightiness endure helpful for developing strategies:

Who tin operate these out the fastest? Discuss strategies. Shared by @LearningMaths

Ratio method for division. Shared by @MathsPrimary

Multiplication strategies. Shared by @MathsPrimary

In response to the the multiplication strategies above, ‏@MathsSfL pointed out that numbers could also separate up, for instance if a pupil is struggling amongst the 6 times tabular array together with thence they could multiply yesteryear 3, together with thence yesteryear 2 (eg 7 x 6 = 7 x 3 x 2). ‏@mathsdotcom added that this industrial plant good for the 8 times tabular array - there's no demand to larn it - merely multiply yesteryear 4 together with thence double it.

Another affair that was mentioned at the hub launch was 'smallest publish first'. If a shaver is asked to operate out 7 x 4 together with thence they could mean value of it equally 4 x 7 -  so no demand to purpose the 7 times table. Thinking similar this cuts downward on lots of duplication inwards memorising times tables. So a large chunk of our park times tables nautical chart is unnecessary - take away the duplication together with nosotros are left amongst the chart below. Kangaroo Maths also really helpfully gives us the 21 times tabular array facts that students demand to learn.

Source: kangaroomaths.com

Inequality symbols

In my ship service Shortcuts vs Concept Development I admitted that I refer to crocodiles when introducing the inequality symbols (ie crocodiles swallow the bigger number).

I decided I'd travel along to do this until I institute a meliorate agency to assistance students retrieve which sign is which. Well, today I institute a replacement for crocodiles. It's thence simple, it's silly.

All students demand to do is visualise numbers equally blocks - the wider cease of the inequality symbol fits over the bigger stack of blocks. So if x > 2, nosotros know x must endure a publish bigger than 2. Eventually they'll start automatically reading the symbol > equally 'greater than' without having to visualise the blocks. Simples.

Chinese pupils purpose multilinks together with straws to larn inequalities. Source: @GlOWmaths

Making feel of the lodge of operations
Instead of merely teaching BODMAS (which inwards itself is the origin of many misconceptions), do y'all assistance students to empathize why nosotros multiply earlier nosotros add? This instance was given at the launch:

Pencils come upward inwards packs of 10. I accept 4 total packs together with 7 extra pencils. How many pencils do I have?

Children accept a multifariousness of strategies:
1. I accept 4 packs of 10 pencils summation 7 extra thence I do: 4 x 10 + 7 = 47.
2. It's similar five packs of 10 pencils amongst iii missing thence I do: five x 10 - 3 = 47
3. I accept 7 pencils summation 4 packs of 10, thence I do: 7 + 4 x 10 = 47.
In the in conclusion example, the shaver doesn't endeavour to add together the 7 together with the 4 earlier multiplying yesteryear 10 because it doesn't brand feel to do so. The context of the inquiry helps them empathize this.

Worth attending?
Some people are critical of the Maths Hubs maiden but I felt it was of import to attend the launch. I don't desire to stubbornly resist learning from Shanghai without at to the lowest degree taking the fourth dimension to notice out well-nigh their methods. The ideas I've written well-nigh hither aren't groundbreaking, but they got me thinking well-nigh the agency I explicate things.

It took me almost ii hours (in terrible traffic) to acquire to the launch lawsuit so, equally a busy instructor amongst a immature family, I dubiety I'll endure able to attend whatsoever to a greater extent than hub events on schoolhouse days, which is a shame because they accept to a greater extent than or less bully materials planned. H5N1 lot of their operate volition initially focus on discipline cognition enhancement (ie helping non-specialist teachers amongst their maths), which is definitely worthwhile. They may non endure reaching everyone who needs help yet but at that topographic point were over 100 people at the launch event, which is a expert start.

It's a shame that the hubs' funding mightiness endure stopped on the whim of a politico because it makes it hard for them to project design over the long-term. But for now, if they're bringing fresh ideas into classrooms, offering back upward where it's needed together with providing opportunities to collaborate, that's fantastic. Do acquire inwards touching on amongst your local hub through www.mathshubs.org.uk to notice out what's going on. Just endure prepared to listen a lot well-nigh Shanghai - but mayhap that's non such a bad affair afterward all.

5 Maths Gems #15

Welcome to my 15th weekly gems post - this is where I characteristic to a greater extent than or less of the best teaching ideas I've seen on Twitter each week.

1. Slice The Pie

Paul Rehmet (@prehmet) has built an interactive tool to help students construct numerical fluency. It's a uncomplicated fraction game - Slice The Pie - which helps students railroad train an intuitive sense of the relative sizes of numbers. It's non compatible alongside mobiles yet but would piece of work on your interactive whiteboard. I honey this. Have a play alongside it, I think you'll honey it too. And follow Paul on Twitter because there's to a greater extent than to come upwardly from him.

2. Remembering Inequalities

In my post virtually Ideas from Shanghai I shared this visualisation which helps students cry back which inequality sign is which.

All students demand to exercise is visualise numbers equally blocks - the wider halt of the inequality symbol fits over the bigger stack of blocks. So if x > 2, nosotros know x must last a divulge bigger than 2. 

Chris Smith (@aap03102) pointed out that this doesn't piece of work for negative numbers. 

I'm sure as shooting our friends inwards Shanghai possess got a solution for this but that data hasn't filtered through to us, therefore let's come upwardly up alongside our ain alternative that industrial plant for both positive together with negative numbers. We only demand a agency for students to cry back which symbol is which.

I'm actually grateful to all the lovely creative tweeters who offered ideas - my favourite was this calligram from Chris. 

So if x < - 8, nosotros encounter the < symbol together with think of the alphabetic character L, therefore nosotros know x is less than -8.

Eventually students volition automatically know the important of the symbols on sight, without having to visualise blocks or words.

These methods are far meliorate than thinking of crocodiles - this tin confuse students, equally described inwards Nix the Tricks (extract below).

3. Squares together with Cubes
I'm enjoying La Salle Education's (@LaSalleEd) novel 'Problem of the Day' series.  You tin search the collection past times historic menses or topic. The instance below (Square Boxes) is a prissy logic occupation consolidating noesis of foursquare numbers.

If you're teaching foursquare numbers, you lot powerfulness also similar this 'Square together with Cube Numbers Puzzle' from MathsPad...

... together with this lovely square together with cube numbers display from Clarissa Grandi (@c0mplexnumber).

4. Mathsticks
I lately discovered the brilliant Mathsticks website which is created past times John Duffty (@johnduffty). I registered together with explored - it's to a greater extent than oftentimes than non primary materials but to a greater extent than or less of the ideas powerfulness last helpful at Key Stage 3. Here's a pair of examples:

This Parallel & Perpendicular Slope. Once you've taught students virtually the gradients of parallel together with perpendicular lines, larn them inwards a calculator room (or larn the iPads out, if you lot possess got them) together with give them this worksheet, which I think volition last really slow for them to follow independently. They are given sure as shooting lines which they plot (in Desmos, though I suppose this could last done past times hand) together with therefore they add together the right parallel or perpendicular lines to brand the required shape. It's a actually prissy activity which volition assistance you lot cheque their agreement of this topic.

Extract from Desmos Graph Challenge

Here's my respond to the minute interrogation inwards the extract above.

To extend this task, you lot could possess got students trammel the domains of the functions, equally I've done below. If you lot exercise this, it's of import that they figure out how to calculate the coordinates of the vertices.

If you're teaching parallel together with perpendicular lines therefore also cheque out my post linear graphs for to a greater extent than ideas together with resources.

A asking for misconceptions
When I project design lessons, I abide by it helpful to endeavour to predict the misconceptions that powerfulness come upwardly up. Sometimes I'm surprised past times the mistakes that students make, which are oftentimes the effect of my explanations non beingness clear enough. Knowledge of mutual misconceptions comes alongside sense but I think it would exercise goodness novel maths teachers if they could consult a website to encounter examples of the form of things students larn wrong. 

A not-so-common misconception from 1 of Chris Smith's students

American instructor Michael Pershan (@mpershan) has a bright website Math Mistakes where he compiles, analyses together with discusses the mathematical errors that students make. I desire to prepare something similar for the Great Britain (because our curricula together with vocabulary are therefore different), but I won't possess got fourth dimension to exercise the first-class analysis that Michael does. Instead it would only last a collection of misconceptions, tagged past times topic. But I even therefore think it powerfulness last useful.

So I possess got a asking - could you lot outset taking photos of misconceptions inwards your students' piece of work together with emailing or tweeting them to me? Once I've got a modest alternative I'll outset a website together with I hope, similar my Pret homework website, it volition grow speedily together with attempt to last a helpful resource. Thank you lot inwards advance!

That's it for today. Please don't forget to vote for me! :), I'll larn out you lot alongside this cute 'Math nosotros know drama', shared past times @FascinatingVids.

New Gcse: Inequalities

The new GCSE specification has 2 additions nether the heading Inequalities:

a) using gear upwards musical note to correspond solutions

b) solving quadratic inequalities.

This post service explains these changes as well as provides education back upwards as well as resources.

Set Notation

Students volition straightaway hold upwards required to correspond solutions to inequalities using gear upwards notation. This is inwards add-on to representing solutions on publish lines as well as graphs. The OCR specification gives us 2 examples of gear upwards notation:

If you're non familiar alongside gear upwards notation, it's explained here (it's unremarkably referred to equally 'set-builder notation'). Note that either a vertical trouble or a colon tin move hold upwards used to correspond 'such that'.

Image source: coolmath.com

The examples from the OCR specification imply that GCSE students volition not postulate to know symbols representing publish types (eg ℤ for integers), as well as thence volition non hold upwards required to limited their answers similar this:

Image source: mathsinsfun.com

Set musical note comes upwards elsewhere inwards the novel GCSE specification - nether the championship 'Venn Diagrams as well as Sets' nosotros lead maintain this:

Source: OCR specification

So create nosotros lead maintain whatsoever resources to exercise this? There's enough of resources relating to gear upwards musical note as well as probability inwards my Data library. I've also made this simple worksheet so students tin move exercise using gear upwards musical note to correspond inequalities. 

Quadratic Inequalities

Here's a enquiry from OCR's Sample Assessment Materials Higher Paper five (non-calculator):

Find the gain of values of x for which x² - 3x - 10 ≤ 0

If y'all haven't taught AS score maths or iGCSE so y'all mightiness non lead maintain taught this topic before. There's a few dissimilar methods for solving quadratic inequalities. H5N1 mutual method involves sketching a graph of the quadratic component so identifying the required part (see event below). To role this method, our GCSE students volition postulate to hold upwards confident inwards sketching quadratics (this is covered inwards the novel specification). I oftentimes honor that my Year 12 students brand mistakes inwards their end reply because they don't bother sketching the graph. To me, sketching the graph is essential.

Desmos is a fantastic tool for exploring quadratic inequalities - cheers to Cathal (@CGA_PGS) for sharing this example.

An choice method for solving quadratic inequalities involves using a publish trouble as well as assay out points. In this method students even so lead maintain to honor the critical values, but instead of sketching the graph they depository fiscal establishment fit the value of a assay out indicate inwards each region.

Given that this topic has been taught inwards iGCSE as well as H5N1 score courses for unopen to time, there's surprisingly few resources available. There's a few resources inwards my Algebra library, including a 'Spot the Mistake' action I made, but I postulate more! If y'all know of anything else so delight part it. 

5 Maths Gems #47

Hello as well as welcome to my 47^th gems post. This is where I part 5 maths teaching ideas I've seen on Twitter.

1. Don's Multiple Choice
I oft write close Don Steward's resources - it's worth checking his weblog every at in i trial as well as and thus to run across what he's added. This calendar week I was browsing his 'practice as well as quiz questions' weblog (which is split upwards to his primary blog) when I noticed that he has started to hit fantabulous sets of multiple alternative questions - here's an example:

 He has also produced unopen to lovely 'odd i out' activities:

Both types of action would operate amongst technologies such every bit Plickers or Quick Key. Multiple alternative questions are making a flake of a comeback at the instant - for to a greater extent than sources, run across my transportation 'Multiple Choice Questions'.

2. Inequality Question
I actually similar this inquiry from Steve Blades (@m4thsdotcom). Ask your students for the minimum as well as maximum value of abc to banking concern agree their understanding of inequalities as well as integers.

I'll role this amongst my Year 11s - many of them struggled amongst the inquiry below inward their mock GCSE, which suggests a lack of substitution understanding of inequality symbols. 

3. Math Wall of Shame

Thanks to Megan Schmidt (Starters as well as Progress Tests for the novel GCSE. 

Update
My post Five things you lot mightiness non know close the novel GCSE content #1 was good received. I promised a 2nd transportation amongst 5 to a greater extent than facts close the novel GCSE... I'm working on it precisely I intend I've establish unopen to discrepancies betwixt bear witness boards that I'm investigating. Watch this space!

There's been 2 announcements since my final gems transportation that you lot mightiness survive interested in. The kickoff is that the appointment as well as venue has been laid for researchED Maths as well as Science. It's on Sat 11th June inward Oxford. I intend this volition survive a fantastic event. I bring to attend a marriage reception inward Kent that evening so I'll bring to locomote out early, precisely volition definitely survive at that spot inward the morning. Tickets are probable to sell out quickly.

The 2nd statement was regarding novel tests for times tables. We all concur that fluency inward times tables is vital (as is conceptual understanding of multiplication) precisely at that spot is disagreement over whether to a greater extent than Key Stage 2 tests are desirable.

It's also worth noting unopen to other alter regarding testing (see above) - from 2017 Year vii maths teachers volition bring iii months to 'fix' students who did non 'pass' Key Stage 2...

It's been a hectic kickoff calendar week dorsum at schoolhouse precisely thankfully this weekend I didn't bring whatsoever marker to hit thus I had the chance to catch the fantabulous Escher exhibition at Dulwich Picture Gallery. It finishes on 17th Jan thus if you lot desire to larn as well as thus larn tickets now! The gallery store is total of mathsy gifts too.

I'll locomote out you lot amongst a prissy demonstration of Pythagoras' Theorem from mathwarehouse.com.