Tricks In Addition To Tips 3: Quadratics

highest mutual factor and methods for sequences, linear graphs too surds.

The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:

Pause for a infinitesimal too expect at this question, because it's a non bad instance of the modify inwards difficulty inwards the novel GCSE. Note the utilisation of business office annotation too the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning betoken aren't integers. This is a notable footstep upward from the type of questions asked inwards electrical current GCSE exams.

For the purpose of this post, let's consider the business office y = x² - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?

1. Vertex Form
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:

y = (x - 3)² + 1

Now nosotros tin sack position the turning betoken at in 1 trial away. I've e'er explained it a chip similar this:

"The (x - 3)²  is squared thence it tin sack never move negative. The lowest it tin sack move is zero. It's null when x is 3. The lowest possible y value is 0 + 1. So nosotros know that the minimum is at (3,1)..."

This explanation is inwards draw alongside my idea procedure - it's the agency I position the turning betoken - but my students actually create out alongside it. Of all the things I teach, this is the explanation that gets the most blank looks! So this yr I tried a dissimilar approach when revising this theme alongside my Year 11s. This fourth dimension I relied on my students' noesis of graph transformations. I told them to intend of the graph y = (x - 3)² + 1 equally a transformation of the graph y = x². It's been translated iii units correct too 1 unit of measurement up. The vertex moves from (0,0) to (3,1). They constitute this approach actually slowly - it made a lot to a greater extent than feel to them. Suddenly all my students were able to notice the vertex of a quadratic function.

As long equally students own got studied graph transformations thence this approach seems to work. This instruction gild is worth bearing inwards hear when designing a Scheme of Work. 

From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...

2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.

By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.


3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.

4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office  y = x² + 4x + 5  - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.

Preparing for the novel GCSE
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.

CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.

The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.

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. 

New Gcse: Tangents In Addition To Areas

At commencement glance it appears that calculus features inwards the novel GCSE specification. On closer inspection it turns out that our students volition detect the slope of a plication past times drawing a suitable tangent rather than past times differentiating. And instead of integrating, students volition use the trapezium rule (or similar) to detect the expanse nether a curve. So calculus remains reserved for Key Stage 5, merely our students volition straightaway go amend prepared for calculus when they commencement regard it. GCSE volition create got given them a conceptual agreement of charge per unit of measurement of modify together with an powerfulness to translate this contextually.

This postal service talks you lot through this novel GCSE topic - it tells you lot what you lot demand to learn together with provides links to resources.

Specification together with Exam Questions
Here's the relevant extracts from the novel GCSE specification:

This all becomes clearer when nosotros await at example examination questions. Huge thank you lot to Tom Bennison (@DrBennison) for doing the difficult operate for me hither - he's been through the sample assessment materials for all examination boards together with collated the relevant questions inwards this Subject Knowledge Check. In a typical question, students are given a velocity-time graph together with asked to detect the amount distance travelled and/or the acceleration at a specific time.

AQA helpfully provides additional clarification nigh the specification inwards their instructor guide. The extract below is from GCSE Mathematics (8300) Teaching Guidance (available to All About Maths members) which provides a release of additional example questions.

Methods
The diagrams below are taken from this extract from a novel GCSE textbook which sets out the criterion methods that our students volition use.

Students volition already know how to detect the slope of a direct business (ie 'rise over run' or equivalent). To guess the slope of a plication they volition create got to pull a tangent, every bit shown here:

They'll also demand to create upward one's hear whether the slope is positive or negative.

To guess the expanse nether a graph, students volition create got to dissever the expanse into sections. The AQA Teaching Guidance says 'the trapezium dominion demand non go known merely it is recommended every bit the most efficient way of calculating the expanse nether a curve'. Unlike at Influenza A virus subtype H5N1 level, they won't go given the formula inwards the exam.

The option to using the trapezium dominion is to dissever the expanse into a release of triangles together with rectangles.

Motion Graphs

The methods described to a higher house are fairly straightforward. I recall interpretation powerfulness examine trickier (eg agreement what motion graphs are showing). Students volition demand to know that speed, acceleration together with deceleration are rates of change, together with that the expanse nether a velocity-time graph represents distance. There's a lot of novel concepts together with vocabulary here.

Bear inwards hear that motion graphs come upward up inwards Physics GCSE too. As shown inwards this extract from the AQA Physics GCSE Specification, it's precisely the same content:

Other Contexts
I was initially confused past times the cite of 'graphs inwards fiscal contexts' inwards this department of the specification - I've seen questions inwards which students create got to interpret fiscal graphs merely cypher involving tangents or areas. Thanks to @DJUdall for sharing the moving painting below (taken from this novel GCSE textbook) which shows an example of estimating a charge per unit of measurement of modify inwards a fiscal context.

Could examination questions encompass other contexts, also motion together with finance? It's possible. In the example below from CIMT students are asked to detect the majority of H2O represented past times the shaded area. To empathise why the expanse nether the graph represents majority nosotros tin dismiss consider the units - the units on the horizontal axis are seconds, together with the units on the vertical axis are m³/s, then when nosotros multiply the 2 together nosotros larn m³.

Instantaneous Rate of Change

Students volition demand to empathise the divergence betwixt an average charge per unit of measurement of modify over a menstruum of fourth dimension together with an instantaneous charge per unit of measurement of change. In my postal service 'Introducing Differentiation' I talked nigh how to give students an intuitive agreement of the slope of a plication at a point. It's worth reading the department entitled 'An Instantaneous Rate of Change' for ideas, videos together with resources (including my worksheet 'Thinking About Gradient' which was designed for Influenza A virus subtype H5N1 degree merely is straightaway suitable for GCSE).

Resources
Resources for this topic are listed below. I'll add together these to my library together with volition go along to add together novel resources every bit I detect them.

• @DJUdall has produced this first-class graphing activity which covers both tangents together with areas nether graphs. 
• @jase_wanner has written this super hero activity on distance-time together with speed-time graphs to prepare students' agreement of how to translate these graphs.
• Gradient on a curved graph by Owen134866 on TES gives students the chance to practice drawing graphs together with tangents. My worksheet Finding the slope of a plication using a tangent is similar, merely students aren't required to pull the graphs themselves.
• Using Graphs from CIMT covers a broad arrive at of graph topics. Section 17.2 covers areas nether graphs using the trapezium dominion together with Section 17.3 covers tangents to curves. The amount arrive at of resources for this module are available on TES.
• Nuffield Mathematics provides a Free-Standing Mathematics Activity 'Speed together with Distance' which is nigh finding the expanse nether a speed-distance graph. Resources include slides, a student sheet and teacher notes.

'Graphing' by @DJUdall

I promise this postal service has been useful inwards helping you lot fix to learn this novel GCSE topic. Please permit me know if you lot create got whatever resources to share.

You powerfulness also detect my other posts nigh novel GCSE topics helpful: Sequences, Inequalities together with Quadratic Graphs.

See my New GCSE Support Page for resources together with links for all novel GCSE content. 

'Drawing a Tangent Line' past times NCTM

Christmaths Party!

I'm rattling excited to denote the details of my Christmaths Party which takes house on Mon 21st Dec inward London.

I got the sentiment for this outcome when I was planning a lesson on truncating decimals for my Year 10 class. Influenza A virus subtype H5N1 couplet of colleagues looked over my shoulder together with asked what I was doing -  they weren't aware that this topic is on the novel GCSE specification. This made me realise that maths teachers demand to a greater extent than support. Over the concluding yr we've been overwhelmed alongside data nigh the novel qualifications together with forthwith we're on the front end draw of piece of occupation delivering the content all yesteryear ourselves. Over the adjacent 2 years we'd all produce goodness from ongoing access to preparation on the novel GCSE content. With this inward mind, I decided to organise a preparation session inward conjunction alongside a Christmas social event. Any excuse for a party... :)

I'm fortunate to hold upwards able to offering workshops from a expose of enthusiastic together with knowledgeable speakers. Between 3pm together with 6pm in that place volition hold upwards v xxx infinitesimal sessions nigh the novel GCSE. The exact programme is yet to hold upwards confirmed. I've created a website for this outcome - christmaths.weebly.com - together with volition update it alongside total details equally shortly equally I can.

After the CPD is consummate the political party volition start. Sumptuous bowl nutrient volition hold upwards served at 7pm. There'll hold upwards loads of eve amusement to brand this a neat nighttime out.

The £25 ticket cost covers food, a welcome drink, CPD together with festivities. Click hither to book - in that place are alone 75 tickets available for this outcome together with I await these to sell out quickly!

Although in that place is express capacity for the CPD, thankfully I'm too able to offering 'party only' tickets for the evening. These are £18 (which includes food) - arrival is from 6pm onwards. Bring your whole squad along to celebrate the destination of a long together with busy term.

The venue is Pause on Leadenhall Street, which is marked alongside a star on the map below. It's walking distance from 6 underground stations so easily accessible from anywhere inward London. The closest station is Aldgate which is alone a couplet of minutes walk away.

If you're travelling from exterior London together with then you lot mightiness desire to remain overnight. Of class there's enough of neat reasons to catch London at this fourth dimension of year, including Winter Wonderland, H2O ice skating at Somerset House together with concluding infinitesimal Christmas shopping. There are dozens of hotels inside walking distance of the venue, here's a few:

• Travelodge Tower Bridge - currently £44 (book forthwith - it was £39 when I checked concluding week)
• Premier Inn Aldgate - currently £69
• Qbic London City - currently £78
• Motel One - currently £98
• Hotel Indigo - currently £99

There are too other Travelodges together with Premier Inns nearby.

The wearing clothing code is casual, but political party outfits would hold upwards nice!

So, does my Christmaths Party audio good? I promise so! The hashtag for this party, inward anticipation of it becoming an annual event, is #christmaths15.  

See you lot there! I can't wait. Quick - purchase a ticket!

Resources For Novel Gcse Topics

If y'all follow my weblog hence you'll know that over summertime I spent a lot of fourth dimension setting upward a new GCSE back upward page. In doing hence I constitute that there are some topics (such every bit Error Intervals together with Frequency Trees) which currently receive got real few didactics resources.

Free vs Subscription Resources
The resources I recommend inwards my libraries are all available for free. The teachers behind costless websites similar m4ths.com together with Corbett Maths are adding novel GCSE content all the time. The examine boards are besides beingness real helpful - I've already made utilisation of both AQA together with OCR's first-class novel resources.

There's besides a break of high lineament resources providers who accuse a pocket-size subscription fee. These include MathsPad, MathsBox, Teachit Maths together with Just Maths. These subscriptions are good worth considering. The motion-picture present below shows an extract from a MathsPad activeness that I used yesterday amongst my locomote past times laid Year 10 - it's a swell activeness but it's solely available amongst a subscription (£3 per month).

Of course of written report it should hold upward your schoolhouse that pays for whatsoever subscriptions, non you. I know schoolhouse budgets are tight, but if your subdivision doesn't receive got whatsoever novel GCSE maths textbooks hence it does demand to ensure that didactics resources are available.

Toticity
Last calendar week I discovered a companionship called Toticity. My eyes lit upward when I saw that they're selling a 'Mind the Gap Maths Toolbox' which is total of resources for novel GCSE topics. I was specially pleased to run across that 1 of their costless samples is on Iteration - this a theme for which resources are few together with far between.

In this post service I render a review of the Toolbox, which is available to your schoolhouse for a one-off payment of £95 + VAT.

Sequences
I've but finished didactics the novel sequences content - it was real enjoyable. There's some swell sequences resources listed inwards my algebra library, but when I taught triangular numbers together with 'Fibonacci-type sequences' I couldn't honour whatsoever resources at all. It's a shame I didn't receive got the Toticity Toolbox finally calendar week because there's a lovely Fibonacci worksheet inwards there, summation some prissy activities for triangular numbers.

Fibonacci-type sequence questions
from the Mind the Gap Maths Toolbox

Under-resourced topics
Under-resourced topics such every bit Frequency Trees, Error Intervals together with Invariant Points are all covered inwards the Mind the Gap Maths Toolbox - extracts from these worksheets are shown below. What a relief to finally receive got some resources for these topics!

Compound Measures
Compound measure worksheets are presented every bit a serial of proportionality puzzles. Students get-go amongst puzzles similar the illustration shown below - hither students are told that the horizontal functions must hold upward the same together with the vertical functions must hold upward the same.

This leads onto speed problems, such every bit the illustration shown below.

We are besides given both density and pressure level problems, which follow the same format.

I similar this logical approach to proportional thinking, it reminds me of ratio tables.

Area nether a Graph
In August I wrote a post nigh finding the expanse nether a graph. In my post service I talked nigh using the Trapezium Rule to honour the expanse nether a curve, but I didn't holler graphs similar the illustration pictured below.

This doesn't require the Trapezium Rule but does require that students either know how to calculate the expanse of a trapezium or realise that they tin laissez passer on the sack dissever the expanse into a rectangle together with a triangle. The Mind the Gap Maths Toolbox provides a preliminary worksheet to assistance students prepare the necessary skills - an extract is shown below. They utilisation shapes earlier they motion onto graphs. I similar this.

Diagrams together with Scaffolding
The worksheets inwards the Mind the Gap Toolbox incorporate a lot of algorithmic diagrams. Workings receive got to follow a defined structure. For illustration the format below is used for all the questions on finding the midpoint of a line.

Another illustration is shown below - this is taken from a rounding worksheet. I've non seen this approach before. In lodge to utilisation this worksheet inwards a lesson on rounding I'd in all likelihood receive got to utilisation this representation throughout my didactics of the topic. Instead I utilisation a straightforward break occupation together with I'm non certain I desire to modify that.

Have a hold off at Page ii of this free sample resource on perpendicular lines to run across what I hateful nigh the utilisation of lots of algorithmic diagrams. These approaches are interesting but I'm non certain they volition appeal to everyone.

Design
I similar the pattern of all the worksheets inwards the Mind the Gap Maths Toobox - they are user-friendly together with good formatted. Answers are provided. There's lilliputian inwards the agency of occupation solving or stretch together with challenge - these worksheets are real much designed for fluency practice. I'll utilisation these worksheets inwards add-on to (but non instead of) rich tasks that prepare occupation solving skills.

The diagrams together with flowcharts used throughout most of the worksheets render a lot of scaffolding. This makes some of the challenging novel content to a greater extent than accessible, for illustration through the utilisation of grids for expanding triple brackets. The approaches used (the real distinctive methodologies) powerfulness non appeal to everyone but are worth a look.

I'm real pleased that I receive got a novel laid of exercise worksheets together with am actually looking forrad to getting stuck inwards amongst all the novel GCSE topics. Even iteration! Do receive got a hold off at toticity.co.uk for costless samples, a listing of novel topics together with an lodge form.

New Gcse: Existent Life Graphs

I guide keep a feeling that I nonetheless haven't identified all the novel GCSE content. This makes me nervous. I glanced through a textbook concluding calendar week as well as spotted a few topics that aren't on my quadratic graphs, but I didn't know that students volition likewise survive required to position the roots of cubic graphs (see instance inquiry below). The specification but says 'recognise, sketch as well as translate graphs of ... unproblematic cubic functions'. Previously students guide keep exclusively been required to recognise the shape of cubics - roots didn't come upward up until H5N1 level. Now I've spotted this content inwards the textbook, I'll demand to adjust my plans to ensure I guide keep fourth dimension to encompass this at some indicate inwards the side yesteryear side eighteen months. The fourth dimension pressure level is increasing.

A inquiry from Edexcel GCSE (9 - 1) Mathematics: Higher Student Book 

Real Life Graphs
I've just taught linear graphs to my Year 10s. My system of piece of job tells me that I straight off guide keep to learn 'real life graphs'. But what does that mean? In this postal service I volition endeavour to clarify what comes nether this title.

The previous GCSE specification said that students had to 'construct linear, quadratic as well as other functions from real-life problems as well as plot their corresponding graphs'. Examples included conversion graphs, 'ready reckoners' (whatever they are...) as well as distance fourth dimension graphs.

Here's the relevant extract from the novel specification:

The wording is unlike to the previous specification but it's broadly the same content, alongside perchance to a greater extent than emphasis on interpreting contextual graphs. AQA's first-class Teaching Guidance gives us a to a greater extent than detailed description:

There's a lot here! The commencement 4 points follow on from linear graphs - for instance students volition guide keep to translate a y-intercept equally a fixed charge. AQA gives this instance question:

There's some swell novel resources for this variety of graph inwards AQA's Bridging Unit: Resource Pocket 3. The back upward provided yesteryear AQA is fantastic.

Modelling

The most mutual examples of 'graphs showing real-life situations inwards geometry' are those that model H2O flow. In AQA's sample assessment materials (Question 23 inwards Higher Paper 3) students are shown a graph representing the depth of H2O inwards a container over time. They are asked to position the corresponding container from a alternative (shown below). They are thence asked to gauge a charge per unit of measurement of change, requiring them to calculate the slope of a tangent to the curve.

It's worth spending a distich of lessons developing the skills required for the commencement component of this inquiry (ie identifying the right container). Thankfully there's plenty of practiced resources for this topic.

In previous years I've used a 'filling upward a bath' card sort (I adapted this from a treasure hunt I institute on TES). I don't oft exercise carte du jour sorts but I cry upward this 1 working well. There's likewise a practiced existent life graph card sort from Nuffield.

I'm actually keen to exercise Desmos Water Line - this is an online lesson inwards which students graph the charge per unit of measurement at which containers create total alongside water. They likewise create their ain containers to graph. Desmos lessons are absolutely bright thence if y'all guide keep access to tablets or computers thence do give it a try. For me it volition survive a rare but worthwhile view to the information technology Room.

The classic Red Box is total of first-class graphing activities - it's a shame that the ikon character isn't great.

Kinematics Graphs
Although I'm education existent life graphs side yesteryear side week, I'm planning to leave kinematic graphs until I learn chemical compound measures. This volition thence Pb onto rates of alter as well as surface area nether a graph, which I've already blogged about. It doesn't brand sense to do kinematic graphs earlier education speed.

We guide keep plenty of resources for distance fourth dimension graphs. AQA's Bridging Unit has some practiced questions (extract above). Interpreting Distance-Time Graphs from the Mathematics Assessment Project is excellent. I likewise similar this commentary activeness from Transum - The Hurdles Race.

I don't guide keep much on speed fourth dimension graphs, apart from those listed inwards my post close finding the surface area nether a graph as well as those available from Toticity's Mind the Gap Maths Toolbox.

Exponential Graphs
The Higher Tier specification includes contextual exponential graphs. An instance inquiry from my Edexcel textbook is shown below. I invention to encompass this variety of graph when I learn increment as well as decay (eg chemical compound interest) side yesteryear side year.

My Plan
Next calendar week I'm education 'real life graphs', but what precisely volition I teach? Having looked through the novel GCSE specification, the sample assessment materials, AQA's Teacher Guidance as well as a distich of novel GCSE textbooks, I straight off guide keep a clearer stance of what I demand to cover. It's taken a lot of travail to decide that I demand to do a lesson or ii on contextual linear graphs as well as a distich of lessons on modelling. I'll come upward dorsum to kinematic graphs, rates of alter as well as areas nether graphs afterwards I've done chemical compound measures. I just promise I'll guide keep plenty fourth dimension to encompass everything.

Proportional graphs also seem to come upward up a lot inwards the novel GCSE. Often a context is given inwards the question. I'll teach these inwards Jan when I encompass ratio as well as proportion. here.