Starters Too Inequalities

I late participated inward a Twitter chat most lesson starters. Initially a forum for sharing ideas, it turned into a fence most whether starters are necessary inward maths lessons. In my thought at that spot is no correct or incorrect lesson construction - every instructor does what industrial plant best for them as well as their students. For me as well as my students, starters run well. I usage them inward the vast bulk of my lessons.

What is a starter?
Presumably inward a lesson without a starter, students come inward the classroom, sit down downwardly as well as right away remove heed to an exposition. In a lesson amongst a starter, students create about maths right away on arrival, before the primary didactics chemical factor begins. Lesson starters (or 'bell work') accept many forms. Every instructor has a dissimilar approach.

Purpose

My starters receive got ane of iii purposes:

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

Starters also assist to awaken my students' brains - they acquire into a mathematical agency of thinking before I showtime didactics them something new. 

Logistics
When students acquire inward I greet them at the door amongst their starter, usually a printed question. They're used to this format hence they sit down downwardly as well as showtime immediately. They create the starter independently spell their peers acquire inward as well as I laid up.

My starters are unremarkably really curt - no to a greater extent than than five minutes - because this minimises the variation inward how long students accept to consummate them. Influenza A virus subtype H5N1 starter that takes anywhere betwixt 10 as well as xx minutes to consummate could endure problematic (early finishers are left waiting) but inward a starter that takes 3 - five minutes this is less of an issue. The early on finishers stick their completed starter inward their books as well as past times the fourth dimension they've done that, the remainder are educate hence nosotros as well as hence review it equally a class. There's rarely a betoken at which students are non occupied.

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

Typical Examples
I'll plow over you lot an thought of what my typical starters appear like. Bear inward mind, equally I said earlier, every instructor does this differently. I've picked a topic at random: inward iii lessons on inequalities amongst Year 10 concluding year, these were my iii starters.

In lesson 1 we're solving linear inequalities as well as representing solutions on number lines. They've done this before hence I showtime amongst a skills depository fiscal establishment check to brand certain they recollect the basics:

Inequalities starter - lesson 1

In lesson 2 we're going to showtime looking at graphical inequalities - for this they require to know how to plot directly lines hence I plow over them about other skills depository fiscal establishment check starter to run across if I require to become through this:

Inequalities starter - lesson 2

In lesson 3 we're looking at about harder graphical inequalities hence I desire to remind them of what they did inward the previous lesson as well as plow over them a sense of savor of the kind of interrogation they volition endure seeing today:

Inequalities starter - lesson 3

The 'list the integer coordinates' role of this interrogation is novel to them hence requires about thinking. I appear it to generate questions which link smoothly into my exposition. I similar my students to intend that the direction of the lesson is prompted past times the questions that arise during the starter, when inward reality I've unremarkably pre-empted their questions as well as planned accordingly.

So you lot tin run across that my starters are nix special but to me they are an essential role of the lesson.

Exam Questions

Sometimes, especially at Influenza A virus subtype H5N1 level, I usage the starter equally an chance to create an examination mode question. For illustration if I learn a Year 12 lesson on finding the equation of a circle using these excellent activities from Mathematics Assessment Project, as well as hence the side past times side lesson I would plow over students an equation of a circle past times examination interrogation on arrival. This gets them used to the format as well as wording of examination questions as well as recaps what they learnt inward the previous lesson. Influenza A virus subtype H5N1 unmarried iv or five grade examination interrogation is the perfect length for an Influenza A virus subtype H5N1 grade starter.

With GCSE classes, I sometimes plow over a QWC examination interrogation (a 'star' question) equally a starter.

More Inequalities Starters
So at nowadays I've described my typical starters, let's appear about choice activities. I've featured iii starters here, ane time to a greater extent than focusing on inequalities.

Most maths teachers are familiar amongst Transum's 'starter of the day'. What I similar most the Transum website is that starters are also listed by topic. Here's an illustration of an inequalities starter.

'Less than' past times Transum

Short card sorts tin run good equally starters. This number employment card matching activity is suitable for an inequalities lesson.

There's also a keen card check (and other fantabulous activities) inward Evaluating Statements most Number Operations from Mathematics Assessment Project. Some of these powerfulness endure likewise long for a starter but would endure suitable for a primary lesson activity.

When is the arguing true? - Mathematics Assessment Project

If you're looking for to a greater extent than primary lesson activities on inequalities as well as hence these ii 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 inward my algebra resources library.

Project Maths

Sources of Starters
These ideas are simply 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 about keen interactives similar this Simplifying Magic Square. FlashMaths has practiced interactives too. I could become on but there's no betoken reinventing the wheel. Colleen Young has an amazing website 'Mathematics Starters as well as Plenaries' which I highly recommend.

Ideas From Shanghai

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

We've heard a lot of rhetoric from the Department for Education nearly Shanghai. It's slowly for us to acquire frustrated nearly the comparisons because at that spot are meaning differences betwixt our nations, both inwards price of civilization together with the underlying structures of the teaching profession. But let's position the politics to ane side - the questions maths teachers should hold upwards bespeak are, "How tin I honor out what happens inwards maths lessons inwards Shanghai, together with volition those ideas hold upwards useful inwards my classroom?". Well I select to a greater extent than or less answers for you lot - you lot tin honor out nearly what plant good inwards Shanghai through the Maths Hub initiatory together with yes, to a greater extent than or less of the ideas are fantastic together with slowly to apply. Don't hold upwards likewise quick to dismiss them.

Today I went to the launch of the Surrey Plus Maths Hub. The atomic number 82 schoolhouse is St John the Baptist School inwards Woking - their teachers select late returned from a trip to Shanghai thence I knew we'd listen all nearly it at the launch. I felt defensive, together with I acknowledge that I arrived alongside an arrogant 'I select aught 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 identify unit of measurement differences betwixt mainland People's Republic of China together with the UK. They also acknowledged the fact that teachers inwards Shanghai select numerous advantages over us, the most notable beingness that they entirely learn ii 35 infinitesimal lessons a twenty-four hr menstruum (they pass the residue of their fourth dimension doing research, marking, planning together with collaborating alongside 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 straightaway away inwards our classrooms. In this transportation service I'm going to part these methods because, although to a greater extent than or less may appear 'obvious', sometimes nosotros overcomplicate things. I intend these uncomplicated methods are truly rather useful.

Multiplication Strategies

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

It's all nearly using clever numerical methods. Let's start alongside something simpler (I'll come upwards dorsum to the starter questions inwards a minute). This ane is done past times 7 twelvemonth olds:

Work out 2.5 x 3.2 inwards your head. Quickly!

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

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

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

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

Either way, a chip of clever number manipulation makes this query 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 skilful at working out anything that relates to the 2, v 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 number because it tin hold upwards multiplied past times 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 you lot know how.

10 twelvemonth olds inwards Shanghai mightiness hold upwards given this query to await at inwards a grouping of three:

125 x 80.8

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

See how this works? If nosotros select a 12.5, nosotros await for an 8. If nosotros select a 25, nosotros await for a 4. We manipulate the numbers to brand the calculation check what nosotros know.
So let's role these ideas to reply those starter questions.

2.5 x 4.04 x 0.9

Split the middle number upwards 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
= ix + 0.09 = 9.09

I'll acquire out you lot to move out a skilful strategy for answering the 2d query yourself (tweet me your reply or comment below).

So, why non start teaching your students these clever mental strategies? It's never likewise belatedly to educate number skills. These lessons would move at whatever age. I could do alongside practising these strategies myself.

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

Who tin move 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 educatee is struggling alongside the 6 times tabular array thence they could multiply past times 3, thence past times 2 (eg 7 x 6 = 7 x 3 x 2). ‏@mathsdotcom added that this plant good for the 8 times tabular array - there's no demand to larn it - only multiply past times 4 thence double it.

Another affair that was mentioned at the hub launch was 'smallest number first'. If a nestling is asked to move out 7 x 4 thence they could intend of it every bit 4 x 7 -  so no demand to role the 7 times table. Thinking similar this cuts downwardly on lots of duplication inwards memorising times tables. So a large chunk of our commons times tables nautical chart is unnecessary - take away the duplication together with nosotros are left alongside 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 transportation 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 move on to do this until I flora a ameliorate agency to assist students recollect which sign is which. Well, today I flora a replacement for crocodiles. It's thence simple, it's silly.

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

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

Making feel of the gild of operations
Instead of only teaching BODMAS (which inwards itself is the rootage of many misconceptions), do you lot assist students to sympathize why nosotros multiply earlier nosotros add? This instance was given at the launch:

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

Children select a multifariousness of strategies:
1. I select 4 packs of 10 pencils summation 7 extra thence I do: 4 x 10 + 7 = 47.
2. It's similar v packs of 10 pencils alongside iii missing thence I do: v x 10 - 3 = 47
3. I select 7 pencils summation 4 packs of 10, thence I do: 7 + 4 x 10 = 47.
In the concluding example, the nestling doesn't endeavor to add together the 7 together with the 4 earlier multiplying past times 10 because it doesn't brand feel to do so. The context of the query helps them sympathize this.

Worth attending?
Some people are critical of the Maths Hubs initiatory 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 honor out nearly their methods. The ideas I've written nearly hither aren't groundbreaking, but they got me thinking nearly the agency I explicate things.

It took me almost ii hours (in terrible traffic) to acquire to the launch lawsuit so, every bit a busy instructor alongside a immature family, I uncertainty I'll hold upwards able to attend whatever to a greater extent than hub events on schoolhouse days, which is a shame because they select to a greater extent than or less keen materials planned. Influenza A virus subtype H5N1 lot of their move volition initially focus on dependent land cognition enhancement (ie helping non-specialist teachers alongside their maths), which is definitely worthwhile. They may non hold upwards reaching everyone who needs help yet but at that spot were over 100 people at the launch event, which is a skilful start.

It's a shame that the hubs' funding mightiness hold upwards 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 upwards where it's needed together with providing opportunities to collaborate, that's fantastic. Do acquire inwards affect alongside your local hub through www.mathshubs.org.uk to honor out what's going on. Just hold upwards prepared to listen a lot nearly Shanghai - but peradventure that's non such a bad affair later on all.

5 Maths Gems #15

Welcome to my 15th weekly gems post - this is where I characteristic roughly 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 ready 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 upward from him.

2. Remembering Inequalities

In my post almost Ideas from Shanghai I shared this visualisation which helps students squall upward which inequality sign is which.

All students demand to exercise is visualise numbers equally blocks - the wider terminate of the inequality symbol fits over the bigger stack of blocks. So if x > 2, nosotros know x must hold out a number 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 inward Shanghai own got a solution for this but that data hasn't filtered through to us, thence let's come upward up alongside our ain alternative that industrial plant for both positive in addition to negative numbers. We simply demand a agency for students to squall upward 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 run into the < symbol in addition to think of the missive of the alphabet L, thence 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 ameliorate than thinking of crocodiles - this tin give notice confuse students, equally described inward Nix the Tricks (extract below).

3. Squares in addition to Cubes
I'm enjoying La Salle Education's (@LaSalleEd) novel 'Problem of the Day' series.  You tin give notice search the collection yesteryear historic current or topic. The instance below (Square Boxes) is a dainty logic occupation consolidating noesis of foursquare numbers.

If you're teaching foursquare numbers, yous mightiness also similar this 'Square in addition to Cube Numbers Puzzle' from MathsPad...

... in addition to this lovely square in addition to cube numbers display from Clarissa Grandi (@c0mplexnumber).

4. Mathsticks
I late discovered the brilliant Mathsticks website which is created yesteryear John Duffty (@johnduffty). I registered in addition to explored - it's by in addition to large primary materials but roughly of the ideas mightiness hold out helpful at Key Stage 3. Here's a twosome of examples:

This Parallel & Perpendicular Slope. Once you've taught students almost the gradients of parallel in addition to perpendicular lines, larn them inward a reckoner room (or larn the iPads out, if yous own got them) in addition to give them this worksheet, which I think volition hold out real slowly for them to follow independently. They are given sure as shooting lines which they plot (in Desmos, though I suppose this could hold out done yesteryear hand) in addition to and thence they add together the right parallel or perpendicular lines to brand the required shape. It's a actually dainty activity which volition assistance yous banking enterprise check their agreement of this topic.

Extract from Desmos Graph Challenge

Here's my response to the 2nd query inward the extract above.

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

If you're teaching parallel in addition to perpendicular lines in addition to thence also banking enterprise check out my post linear graphs for to a greater extent than ideas in addition to resources.

A asking for misconceptions
When I project design lessons, I uncovering it helpful to travail to predict the misconceptions that mightiness come upward up. Sometimes I'm surprised yesteryear the mistakes that students make, which are ofttimes the number 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 run into examples of the kind of things students larn wrong. 

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

American instructor Michael Pershan (@mpershan) has a bright website Math Mistakes where he compiles, analyses in addition to discusses the mathematical errors that students make. I desire to prepare something similar for the United Kingdom of Great Britain in addition to Northern Ireland (because our curricula in addition to vocabulary are thence different), but I won't own got fourth dimension to exercise the fantabulous analysis that Michael does. Instead it would simply hold out a collection of misconceptions, tagged yesteryear topic. But I notwithstanding think it mightiness hold out useful.

So I own got a asking - could yous kickoff taking photos of misconceptions inward your students' piece of work in addition to emailing or tweeting them to me? Once I've got a small-scale choice I'll kickoff a website in addition to I hope, similar my Pret homework website, it volition grow chop-chop in addition to seek to hold out a helpful resource. Thank yous inward advance!

That's it for today. Please don't forget to vote for me! :), I'll move out yous alongside this cute 'Math nosotros know drama', shared yesteryear @FascinatingVids.

New Gcse: Inequalities

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

a) using laid upwardly annotation to stand upwardly for solutions

b) solving quadratic inequalities.

This transportation explains these changes in addition to provides pedagogy back upwardly in addition to resources.

Set Notation

Students volition instantly hold upwardly required to stand upwardly for solutions to inequalities using laid upwardly notation. This is inwards add-on to representing solutions on publish lines in addition to graphs. The OCR specification gives us 2 examples of laid upwardly notation:

If you're non familiar amongst laid upwardly notation, it's explained here (it's usually referred to every bit 'set-builder notation'). Note that either a vertical job or a colon tin flaming hold upwardly used to stand upwardly for 'such that'.

Image source: coolmath.com

The examples from the OCR specification imply that GCSE students volition not require to know symbols representing publish types (eg ℤ for integers), in addition to hence volition non hold upwardly required to limited their answers similar this:

Image source: mathsinsfun.com

Set annotation comes upwardly elsewhere inwards the novel GCSE specification - nether the championship 'Venn Diagrams in addition to Sets' nosotros own got this:

Source: OCR specification

So produce nosotros own got whatsoever resources to exercise this? There's enough of resources relating to laid upwardly annotation in addition to probability inwards my Data library. I've also made this simple worksheet so students tin flaming exercise using laid upwardly annotation to stand upwardly for inequalities. 

Quadratic Inequalities

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

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

If you lot haven't taught AS score maths or iGCSE so you lot mightiness non own got taught this topic before. There's a few dissimilar methods for solving quadratic inequalities. Influenza A virus subtype H5N1 mutual method involves sketching a graph of the quadratic purpose so identifying the required part (see instance below). To usage this method, our GCSE students volition require to hold upwardly confident inwards sketching quadratics (this is covered inwards the novel specification). I frequently detect that my Year 12 students brand mistakes inwards their concluding respond because they don't bother sketching the graph. To me, sketching the graph is essential.

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

An choice method for solving quadratic inequalities involves using a publish job in addition to examine points. In this method students withal own got to detect the critical values, but instead of sketching the graph they banking concern correspond the value of a examine indicate inwards each region.

Given that this topic has been taught inwards iGCSE in addition to Influenza A virus subtype H5N1 score courses for closed to time, there's surprisingly few resources available. There's a few resources inwards my Algebra library, including a 'Spot the Mistake' activeness I made, but I require more! If you lot know of anything else so delight portion it. 

5 Maths Gems #47

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

1. Don's Multiple Choice
I oftentimes write close Don Steward's resources - it's worth checking his weblog every directly too and hence to run into what he's added. This calendar week I was browsing his 'practice too quiz questions' weblog (which is divide to his primary blog) when I noticed that he has started to hit first-class sets of multiple selection questions - here's an example:

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

Both types of activeness would operate amongst technologies such every bit Plickers or Quick Key. Multiple selection questions are making a fleck of a comeback at the 2d - for to a greater extent than sources, run into my transportation 'Multiple Choice Questions'.

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

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

3. Math Wall of Shame

Thanks to Megan Schmidt (Starters too 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 2d transportation amongst 5 to a greater extent than facts close the novel GCSE... I'm working on it exactly I mean value I've flora unopen to discrepancies betwixt examination boards that I'm investigating. Watch this space!

There's been 2 announcements since my final gems transportation that you lot mightiness live on interested in. The outset is that the appointment too venue has been laid for researchED Maths too Science. It's on Sabbatum 11th June inward Oxford. I mean value this volition live on a fantastic event. I bring to attend a marriage ceremony reception inward Kent that evening so I'll bring to larn out early, exactly volition definitely live on at that spot inward the morning. Tickets are probable to sell out quickly.

The 2d statement was regarding novel tests for times tables. We all concur that fluency inward times tables is vital (as is conceptual understanding of multiplication) exactly 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 seven maths teachers volition bring 3 months to 'fix' students who did non 'pass' Key Stage 2...

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

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