5 Maths Gems #164

Welcome to my 164^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Worked Examples
@JakeGMaths shared a large bank of worked examples. It's free to use and contains over 400 examples. Each one is several slides long, guiding students through a method in a series of steps. Then there's a printable part for the students to complete, then the solutions. The PowerPoint has clear instructions on how to use these resources in lessons. Here are a couple of examples of the student tasks:

If you haven't yet read it, I recommend Michael Perhan's excellent book Teaching Math with Examples which inspired this work.

Also check out Jake's other projects which I blogged about in Gems 158.

2. Operations with Negative Indices

I really like this set of questions from @karenshancock. It's a good idea to do something a bit different with negative indices, to help develop fluency whilst interleaving other skills like fraction arithmetic.

Karen kindly typed this up and shared an editable version here. I've added this to my resource library under indices. 

3. Gif Generator

@PiXLMattTheApp continues to develop his website with an incredible range of features. His latest addition is a gif creator which allows teachers to very easily create a recording of a modelled example.

I wrote about opportunities to use animated modelling in my post The Power of Modelling and Exemplars.

4. Bearings

Thanks to @DrPMaths for sharing a task linking angles in triangles with bearings. This is a building block which prepares students to be able to answer trigonometry problems with bearings - something that students often struggle with. His resource includes a few example-problem pairs and a set of practice questions with solutions.

5. Starting Point Maths

@ChrisMcGrane84 has published loads of excellent new resources on startingpointsmaths.com recently. I featured some in my last gems post, and here are a few more. 

First, a task to try before learning how to differentiate: Developing Indices Skills for Calculus.

Second, a task which has a mix of linear equations, quadratic equations, equations which look quadratic but are linear, and a sneaky cubic. Mixed Linear and Quadratic Equations is all about strategy selection, so students become better at spotting quadratics.

And finally, Odd & Even Numbers – Additive Structure - a lovely task which might be useful in primary or secondary. Read Chris's description of the task to see how he used it.

Update

At school we've all been unwell lately - it's that time of year - and our Year 11s have been in mocks so we've been doing lots of invigilation, which is pretty dull. On a more positive note, I've been teaching some of my favourite topics: Year 7 have been learning about prime numbers which is always a joy, Year 8 have been solving equations, and Year 10 have been doing quadratics.   

I'm delighted that #mathsTLP made a return last Sunday. It was an absolute pleasure to behold an hour of teachers enthusiastically sharing resources and teaching ideas. Don't forget that this now runs every Sunday at 7pm, hosted by @MissNorledge and @BrookeEHunter. You're very welcome to join in: see my blog post about how to get involved.

Here are some things you might have missed:

• The MA is calling for submissions to lead a session at the joint subject associations conference next Easter. This two day conference at Warwick University will be the highlight of the year for maths teachers. If you're an experienced teacher with something to share, please do consider speaking. I particularly encourage female speakers, who are always underrepresented at maths education conferences. Complete this form by the end of November to offer a session.
• Realising my Equating Coefficients in Identities task ramped up in difficulty too quickly, I made a revised version of this resource which is shared on TES. 
• @MrDraperMaths wrote an excellent post about transformations of functions which featured a really interesting approach that had never occurred to me before.
• @sxpmaths shared an excellent summary of the different sampling techniques covered in A level maths.
• @missradders shared a lovely Teacher Treasure Hunt that could be used during Maths Week.
• Everyone loves the classic Standards Unit resources from the legend Malcom Swan. The 'Traffic' program was originally written in Java as part of the Standards Unit but @MathsTechnology has recreated it in GeoGebra.
• @nathanday314 shared some of his excellent starter activities plus a thread that explains how he uses them. The thread is well worth a read. 
• @draustinmaths continued to add new resources to draustinmaths.com, including this new task on forming quadratics which I've added to my resource library.
• The wonderful mathematical magazine Chalkdust published Issue 16 - you can read it online or order a paper copy.
• Did you catch my recent blog posts? I wrote a methods post called 'Easy Multiples' and I wrote 'Some Things We've Tried' where I described some of the things I've been up to at school since I became Head of Maths. 
• La Salle shared the dates and locations of its next three in-person conferences:

I'll leave you with this tweet from @dodecahedra about different ways to prove the sum of a geometric series. I'd only ever thought about one way of doing this before, so I enjoyed seeing the alternatives.

Four takes on how to teach/discover/prove/understand summing the terms of a geometric sequence. pic.twitter.com/TaYTITpaWF

— William Rose (@dodecahedra) November 11, 2022

 

The Power of Modelling and Exemplars

Last week I observed an art lesson which featured expert use of modelling. The teacher wanted students to use a particular technique to create a piece of artwork. Before the lesson she'd used a visualiser to record herself performing the task. The video was shot from above, recording only her hands. She played the recording to the class while she narrated what she was doing and pointed out the difficulties she had encountered along the way, advising students of how they could overcome those difficulties when it was their turn to have go. She then left recording running on a loop on the screen while the students performed the task, meaning they could look up and refer to it throughout. What an excellent technique! 

I occasionally do something similar in maths - when teaching constructions I leave gifs playing on a loop on the board so students can refer back to them whilst practising:

I think this is really powerful in topics like constructions. 

It is very easy to leave animated written examples running on a PowerPoint while students practise (like in the example below). But here it's probably more helpful to instead leave all the steps and solutions static on the board, rather than an animated version. It's the same idea though. We model how to do the process and we show students what the final outcome looks like. And then we leave that with them to refer to, rather than hide it away and expect them to remember it.

In generic whole school CPD, I'm often surprised to hear people talking about the importance of modelling examples as if it's a new or unusual idea. But perhaps it is, in other subjects. For maths teachers it's just so ingrained in everything we do. We are always modelling. Students are constantly getting to see us do 'live' maths. From our modelling, students can see what the final outcome should look like. 

Students don't get the same benefits from PowerPoints which are clicked through to show animated step-by-step solutions as they do from live modelling. They need to see 'pen and paper' modelling, done during the lesson by their teacher, whether on a whiteboard or a blank PowerPoint slide, or under a visualiser. Because otherwise, how can they possibly know what their work should look like?

A few years ago there was a trend for using 'WAGOLL' techniques in many subjects ('what a good one looks like'). This is a logical thing to do - if you want students to produce something of a certain standard, how can they achieve that if they are not shown what that standard looks like? It's like when we follow a recipe to bake a cake or cook a meal - we start by looking at a picture of the final outcome, so we know what we are aiming for. 

There are a number of commonly used techniques for showing students 'what a good one looks like' in many subjects, including the live modelling I've described. Another technique is using a visualiser or photo to share examples of excellent work by other students. For example if a student's book is laid out immaculately and you want other books to look the same, simply show the class what that good book looks like. Just moaning at a student that their workings are a mess won't help them improve. Find a good example and show them.

In maths, the exemplar response materials provided on Edexcel's Emporium are perhaps one of our most useful tools for showing 'what a good one looks like' in terms of the maths itself. 

Take this Edexcel exam question for example. What would a good answer look like?

Edexcel has provided us with an excellent solution given by a student in their GCSE exam. Note that reasons are given throughout, that workings are clear and presented vertically down the page in a logical order, and they have even used a 'therefore' symbol (not essential, but nice to see!). This student knows what they're doing.

When teaching circle theorems, it is so useful to show examples of answers gaining full marks. This helps students know what they need to do to get marks in questions like this, particularly in terms of the wording of their reasoning. 

Edexcel also provides examples of student answers to the same question that did not gain full marks. As a class you can discuss where students went wrong and where their misconceptions lie. Looking at real student answers and seeing how marks can be gained and lost is powerful stuff. 

Analysing exemplar responses is also incredibly useful for maths teachers. Edexcel very helpfully provides accompanying comments which explain the misconceptions and tell us where students lost and gained marks.

I could show you endless examples here, but I will just share one more. This is an angles questions from an Edexcel Foundation paper. The first solution gains full marks. The second got one out of three.

Can you guess what the second student did to come up with the angles of 120 and 150? 

The comments from Edexcel tell us that this student appears to have measured the angles with the protractor. We are urged to remind our students that diagrams are not drawn accurately and they should not be measuring anything (unless specifically asked to do so!). This is not something I would have anticipated students doing in this question.

I'm sure you'll agree that exemplar answers are an incredibly useful teaching tool, not only for showing students 'what a good one looks like', but also for our own CPD purposes. You can find Edexcel's brilliant exemplar resources on the Emporium:

We are very fortunate in maths to have access to such useful resources to support our teaching.