5 Maths Gems #138

Welcome to my 138^th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. I've got a bit behind on my blogging lately because school has been overwhelmingly busy, so here I'm trying to catch up by featuring ten gems instead of my usual five. 

1. Prime Factors Tool

Jonathan Hall (@StudyMaths) has published another new tool on the brilliant mathsbot.com. This lovely prime factors tool uses virtual prime factor tiles to help students make sense of prime factorisation, highest common factor and lowest common multiple. Click on the cells to toggle through blanks, products, and prime factors.

2. Certificate in Further Maths Resources

Teachers delivering the AQA Level 2 Certificate in Further Maths should check out Amanda Austin's (@draustinmaths) website which contains practice resources for this qualification. I like the format of her practice strips which are very helpful for sticking into books.

Do check out the rest of Amanda's website, which features a collection of KS3 and KS4 IGCSE and GCSE Maths resources ordered by topic. 

3. Tasks

Since Chris McGranes's (@ChrisMcGrane84) book on mathematical tasks was published and he started running courses in task design, there has been a flurry of excellent tasks shared on Twitter and Chris's blog. Here I share a couple of examples - check out his blog for more like this.

Fraction Stories (thanks to @limeswright for tweeting about this one - I think I will use it next week to assess prior understanding in my first fractions lesson with Year 7):

Simultaneous Equations (I'll be teaching this soon to a class who will really benefit from the  scaffolding in this activity):

4. GCSE Statistics
Thank you to Helen Scott (@HelenScott88) for sharing her GCSE Statistics Retrieval Roulette. Aimed mostly at Higher Tier candidates, the chapter references in this resource relate to the Pearson GCSE Statistics textbook. For guidance on how to use Retrieval Roulette, read Adam Boxer's blog post.

5. Volume

Thank you to @SegarRogers for sharing this cuboid volume task. He designed it for a class in which some students have trouble seeing individual cubes, requiring a gradual removal of the isometric grid, and other students are ready to try working backwards.

6. Recall

All the way back in Gems 87 (April 2018) I shared an example of a regular recall starter from @JaggersMaths (formerly @MissBanksMaths). Mrs Jagger has now updated her R^Infinity resource, which teachers can use to quickly create personalised four-question starters. Visit her website to download the resource and to find out how it works.

7. Sequences and Fractions Task

Richard Perring (@LearningMaths) recently shared two interesting tasks. The first is a sequences task in which students generate their own sequences.

The second task is based on a similar idea, but relates to fractions. I really like these tasks.

8. Number Sense Maths
Clare Christie (@Ms_Mathsteacher) got in touch about the website numbersensemaths.com. There are tonnes of free resources to get schools going on structured number fact teaching, and a full scheme of work for those who want to commit to it. 

This website will be of particular interest to primary schools, and I think that SEND departments in secondaries will find useful strategies and resources here too.

9. Place Value
@taylorda01 had a nice idea for a place value task. Present students with the following numbers and ask them how many times bigger the value of the red digit is than the value of the blue digit. Then ask how much bigger the value of the red digit is than the value of the blue digit.

10. Pictograms
Thanks to @giftedHKO for a new pictograms resource. I have linked to this in my data resource library.

Update

Last weekend I enjoyed spending the morning in Scotland (virtually!) for the Northern Alliance N5 Maths Conference. The workshops were excellent. I presented on teaching quadratics. I will present a similar workshop at La Salle's '#MathsConfMini' which is on a Friday evening in January.

Another event I'm presenting at in January is White Rose Maths Secondary Maths Brunch.

In case you missed them, I published two new blog posts in the last couple of weeks. They were:

• Lines and Angles on Square Grids (a post written by Anne Watson about a collection of Don Steward tasks)
• Zooming In (a post about a great teaching tool I recently used for the first time in a Year 7 lesson).

Another thing I was involved in last week was a discussion about Don Steward's tasks on Tom Manners' ResourceFULL channel. This is worth a watch if you want to see some outstanding examples of rich tasks.

Other things to check out if you haven't already:

• Marvellous Maths 2, which is a CPD course for maths teachers from me and Craig Barton. It's been great to hear of teachers using ideas from this course in their teaching this week.
• Ed Southall's new book, Geometry Juniors. This is aimed at 8 to 12 year olds. It looks great!
• The review of my book A Compendium of Mathematical Methods published in the MA's journal Mathematics in Schools. I loved this review and was very pleased to read an account written from the perspective of the book's main target audience (maths teachers!).
• I've had a few questions about intervention resources for Year 11s recently. Do check out my GCSE Revision Resources page if you're looking for this kind of thing.

I'll leave you with this wonderful pentagon tessellation tool from Mathigon.

Regular pentagons don't tessellate – but there are 15 other types of pentagons that do. We’ve collected all of them at https://t.co/PDYbPhRhMa. Can you fit them together? pic.twitter.com/a7I5Z32des

— Mathigon (@MathigonOrg) November 6, 2020

 

The Joy of Planning

I had great fun planning a lesson on tessellation last week. Planning lessons is one of my favourite parts of the job.

This lesson certainly won't be everyone's cup of tea, but I enjoyed planning it so much I wanted to share it here. Tessellation is no longer on the GCSE specification but I think it helps students develop their knowledge of angles in polygons, plus it's interesting, so is worth looking at.

I started with a quick Don Steward question aimed at refreshing my students' memories on how to calculate angles in polygons, which I'd taught them the day before.

I then asked the class, "What is tessellation and where do we see it?". One or two students described it in terms of tiling, and some pointed out examples of tessellation around the room (the painted bricks on the wall, the square vinyl tiles on the floor). I gave them a definition, which they noted down, and we talked about the meaning of the word plane in this context.

I told them about the etymology of the word tessellated. In planning this lesson I was interested to find that the word derives from the word 'four', referring to four corners.

tessellated (adj.)
1690s, from Late Latin tessellatus "made of small square stones or tiles," past participle of tesselare, from tessella "small square stone or tile," diminutive of tessera "a cube or square of stone or wood," perhaps from Greek tessera, neuter of tesseres, Ionic variant of tessares "four", in reference to four corners.

I showed them some pictures of tessellations.

I spoke of The Alhambra. A student put up his hand and told me he'd been there. I told him I was very jealous - I long to visit.

I also talked about Escher. Having recently been to the Escher exhibition at Dulwich Picture Gallery I could have happily talked about Escher for the full hour, but alas - time is limited. My Year 10s were amused when I told them that in the 60s Escher's work was loved by both hippies and mathematicians. A student put up his hand and said he had a book full of Escher's artwork at home. This made me happy.

I showed some examples and non-examples of tessellation (when planning lessons I often do a quick trawl of google images for the pictures I need to illustrate a point).

They noticed that the angles around the point have to sum to 360^o if the polygons are to tessellate. 

I showed an example of a semi-regular tessellation, which can be created by using two or more regular polygons repeatedly to tile the plane (for example an octagon and a square).

I then showed them a series of animations from a Tumblr account called Skunk Bear. They were 'wowed' by these animations (one boy asked, "where do you get all this cool stuff, Miss?". "Mostly through Twitter" I replied.). I think these gifs will help my students remember what tessellation is. If not, the animations certainly help foster an enthusiasm for the beauty and complexity of mathematics.

Tessallating equilateral triangles

Non-tessellating pentagons

When I told them about the recent discovery of a new type of tessellating pentagon ("It's a really big deal", I said) - they laughed. "Miss, are there really people who spend their life looking for tessellating pentagons? Shouldn't they be trying to find the cure for cancer or something?".

Tessellating pentagons - as discovered in 2015

All of this chat about tessellation took a good 25 minutes. It was a bit longer than I normally talk, but that's ok. They listened intently, and their reactions suggested that they were interested. I could have gone on for longer - Mathigon has some great stuff on tessellation - but I was keen to get them working on some angle problems. So I set them a Don Steward exercise.

During this exercise some students asked about the names of a 20-sided and 15-sided polygon, and thankfully I had a list to hand (borrowed from a Don Steward slide) and we chatted about the words. I overheard some students trying to break the words down and make sense of them.

Some of my students completed the tessellation activity really quickly - others took a lot longer. I followed it up with two Don Steward activities (Regular Polygon Angle Compilation and Pentagon Angle Compilation) which were pretty challenging. These activities developed their fluency in calculating angles in polygons. Towards the end of the lesson I invited a couple of students to come up to the board and talk through their approaches to these problems.

In the final minute of the lesson I decided to bring everyone back together to tell them about platonic solids. One student kept referring to a dodecagon as a dodecahedron during the lesson so I wanted to pick up on the difference between ~gon and ~hedron. So I ended up with an unplanned tangential plenary...

This lesson may have been relatively light in 'exam content' but it was rich in mathematics. It might not have ticked all the 'good lesson' boxes, but I felt that it went well.

I don't write formal lesson plans. I create minimalist PowerPoints instead. I make no apologies for using slides - this is not 'death by PowerPoint' - it's just a bit of structure. Here's my slides for this lesson, feel free to borrow any bits that you like.

Do I sometimes spend too long planning lessons? Absolutely! Are my lessons the 'best possible way of teaching that topic'? Absolutely not. I have not perfected the art of teaching mathematics. This tessellation lesson is very 'me', and I doubt anyone would want to teach it in the same way I did.

I enjoy lesson planning. It allows me to explore my subject. It allows me to be creative. I wholly agree with Megan Guinan's recent tweet, "Planning lessons is my second favourite part of the job, after delivering a lesson I've planned well."

I'd love to hear all about a lesson you've taught recently that you particularly enjoyed planning and delivering.