5 Maths Gems #141

Welcome to my 141^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Tasks
Task designer @ashtonC94 has made a new website mrcowardmaths.wixsite.com to collate his excellent resources. He is gradually populating it with tasks, so do check it out. 

Ashton has shared an activity on plotting coordinates that draws attention to scales in order to tackle common misconceptions. An extract is shown below - click here for the full task.

I also really like Ashton's excellent task on significant figures. 

Both of these tasks have been added to my resource library.

Maths Gems regular @giftedHKO has also continued to share a number of excellent resources recently, including this task on gradient of a line segment, and many more on her website.

2. Booklets

@ChrisMcGrane84 has started sharing a series of curriculum booklets that he has been working on in conjunction with Siobhan McKenna (@ShivMcKenna55). Read Chris's post for more information. If you were at #mathsconfmini then you can watch Chris discuss how these booklets form an integral part of the design and delivery of his maths curriculum in his talk Designing an Effective Curriculum in Early Secondary. 

These booklets are packed full of carefully selected tasks from a range of excellent task designers. The booklets are used instead of a scheme of work - they define curriculum order and pedagogical approach.

3. MS Forms Quizzes

Thank you to @timdolan for creating a Padlet for maths teachers to share maths quizzes made in Microsoft Forms. This should save teachers a lot of time, as they don't have to create everything from scratch anymore. If you have been making MS Forms quizzes then please share your quizzes here (as a template) so that others can borrow them. I've added mine!

It's interesting to see a variety of quiz types. My quizzes feature short-answer questions, or multiple choice questions (which I borrow from diagnosticquestions.com) which are automatically marked. But there's a lot more we can do with these quizzes. I received an email from maths teacher Alison Page who turned my Algebra True or False activity into a Forms quiz. Looking at this, it occurred to me that it would be easy to turn an 'always, sometimes, never' activity into a Forms quiz, with spaces to write explanations.  Another option for turning existing worksheets and activities into online versions is teachermade.com. 

If you are a MS Forms user then you might be interested in this post from @andrewkbailey13 about giving feedback to students. My team have been giving lots of feedback but we have been concerned that our students haven't been reading it - this post answered our questions about how it works from a student perspective!

4. A Level Example-Problem Pairs

Thanks to @acutelearning for sharing a large set of A Level Maths and Further Maths example-problem pair slides with answers. These are freely available to download from TES.

5. Mr Draper's Blog Posts
Mr Draper (@MrDraperMaths) has been busy writing a series of excellent blog posts lately. They all feature cleverly thought-out tasks and teaching approaches. For example, this post on Understanding Angle Labels addresses a number of difficulties that students commonly experience when learning to solve angle problems.

Another example is this excellent post on reasoning with frequency trees:

And a great post about perimeter:

Read Dan's blog for lots more like this.

Update

It's been a busy time for everyone. I don't know about you, but my online lessons are still taking ages to plan, so I'm spending entire weekends working. I'm not doing anything fancy in my lessons, but explanations and tasks need really careful thought. I also have a lot going on in my other roles at school - I'm responsible for reports, parents evenings, Year 9 options and so on - and all of that has been really full-on in the last few weeks! Roll on half-term.

In case you missed it, here are my most recent blog posts:

• Squares (a post in the excellent #doseofdon series, written by Anne Watson)
• Tools for Online Learning (a popular post with over 9000 reads)

I recently recorded a podcast with Craig Barton where I discussed what I am doing for online lessons. Do have a listen!

I also presented at two conferences: at #mathsconfmini I did an evening presentation on teaching quadratics, and at the White Rose Secondary Maths Brunch I presented on task design. 

I hope to do a longer version of my task design presentation at the MA Conference this Easter, if my proposal is accepted. Also, following a lot of thinking recently about how to introduce probability in an accessible way to my Year 9s, I have submitted a proposal to speak about probability at #mathsconf25 in March. 

Finally, I recently produced a Topics in Depth CPD video for Loughborough University Mathematics Education Network which you can access here. It covers curriculum, history, misconceptions and resources.

I'll leave you with a pie chart that I spotted recently, which is probably one of the worst pie charts I have ever seen.

Stay safe, maths teachers!

5 Maths Gems #71

Welcome to my 71^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Revision Resources
Grant (@AccessMaths) has recently published loads of great GCSE revision resources. You can check out the full collection at accessmaths.co.uk.

Resources include Crossover Problems, Octagon Revision Mats and Pentagon Problems. These all work well printed on A3.

2. Desmos Geometry
The awesome people at Desmos have created Desmos Geometry. It looks like it will be just as slick, accessible and user-friendly as their graphing calculator, so this is exciting news. It's currently in Beta and you can read more about it here. I'm looking forward to seeing how it develops.

Do check out Desmos's classroom activities if you haven't discovered them yet. If you have access to a functioning IT room, or a class set of tablets, these make great lessons - I wrote about Polygraph a couple of years ago, and have since used Waterline with great success.

3. Always, Sometimes, Never
Some classic maths activities never get old. I remember the first time I did an Always, Sometimes, Never activity from the Standards Unit in my NQT year. It generated brilliant mathematical discussion and was a really worthwhile lesson. Sarah Carter recently shared another great example of an Always, Sometimes, Never activity for teaching averages:

Tried Always, Sometimes, Never for the first time today. Awesome discussions! #teach180 pic.twitter.com/GHUMcvtrfm

— Sarah Carter (@mathequalslove) 24 March 2017

There are some good Always, Sometimes, Never activities on MathsPad, Nrich and TES. And here are some examples that were shared on Twitter by Mark McCourt (@EmathsUK) a while ago:

4. Pick a Card
The Underground Mathematics team at Cambridge have been busy expanding their collection of A level resources. In this 'Pick a card...' exercise, the content of one card is revealed by clicking on it and students have to decide whether they can work out the rest of the answers. There are some lovely follow up questions to consider, such as "Which card would be the easiest to start from?" and "Does each card always give enough information to uniquely identify the quadratic function?". If you teach A level do check out these excellent resources.

5. Perpendicular Gradient
I shared an animation in Gems 60 which demonstrates the relationship between the gradients of perpendicular lines. I've now found a video which shows the same thing. This is really clear and useful. Thanks to @MathWithMonkeys for sharing it.

Update
Hurrah for the holidays! I've been away to sunny Devon with my family over the last few days. I'm going back to school on 18th April for the final exam countdown with my Year 11s, 12s and 13s.

Since my last gems post I've blogged five times - here's what I've written, in case you missed anything:

• Yes, But Why? which features extracts from Ed Southall's new book
• Update! which provides an update on recent improvements to my resource libraries
• Papers Society which is about something I'm trying with my Year 11s this year - this post appeared in Schools Week's 'Top Blogs of the Week' column
• Ultra Outreach Project which is about an opportunity for free enrichment in high-PP schools 
• #mathsconf9 which is a write-up of the Bristol maths conference - this seems like ages ago now!

I feature as a special guest co-host in the next episode of Colin Beveridge and Dave Gale's podcast Wrong, But Useful. It's out later this week so do have a listen. Speaking of podcasts, check out Craig Barton's latest podcast with Dani Quinn - his most controversial one yet!

Don Steward has been busy posting new resources lately, including some tasks for new GCSE topics which will be added to my resource libraries this week. I've now added all of @taylorda01's 'Increasingly Difficult Questions' to my libraries too. I also made two new resources of my own:

• Equating coefficients in identities (this came up in the recent AQA mock - I couldn't find any resources for it whatsoever)
• Linear graph exam questions (useful for revision - this includes plenty of questions on perpendicular lines)

All of my resources are available to download from TES, and appear in my resource libraries.

Quadratic Points from Don Steward

Look out for another round of my Annual Gem Awards later this month - the third anniversary of my blog is fast approaching.

5 Maths Gems #60

Welcome to my 60^th update from the world of Maths EduTwitter. Here I summarise some of the latest ideas and resources for teaching maths.

1. Perpendicular Gradient
I love this gif demonstrating the relationship between gradients of perpendicular lines, shared by Simon Pampena (@mathemaniac).

Geogebra fans will be pleased to see that Tim Brzezinski ‏(@dynamic_math) made 'Slope Triangle Rotation' to explore this further.

2. Metric Units

Next time I teach a lesson on units I'm going to show this five minute video on the history of the metric system. I think it's really interesting.

I discovered this on YouTube after watching The mathematical secrets of Pascal’s triangle which was shared by Cliff Pickover (@pickover).

3. Calculus Puzzles
A level teachers will like this Chalkdust post 'Puzzles about calculus' by Matthew Scroggs (@mscroggs).

4. Displays

Twitter continues to be a great place to share classroom display ideas. I saw two ideas last week that I particularly like. First, check out Sarah Carter's (@mathequalslove) fantastic mathematical welcome sign. 

Second, Claire Mazurkiewicz (‏@MrsMazzy) put an A level twist on Mel's (@Just_Maths) popular maths periodic table display. I rarely see displays designed for A level classrooms - read about it and download the file here.

5. Shadow Shapes
The image below has been going round the internet for years (original source unknown). I wrote about it last February in Gems 23. I now use it whenever I teach plans and elevations.

Phil Bruce (@pbrucemaths) was inspired by this image to make a PowerPoint of five more examples. You can download it from his blog here, under "shadow shapes".

Update
My last day of term was on Friday (hurrah!)... I know some of you are still at school for a couple more days (nearly there!).

In case you missed any of my recent posts, here they are:

• Looking Ahead
• Mathsy Gifts: The Sequel
• 5 Maths Gems #59

I've used VideoScribe to make a welcome video for Year 7 and an expectations video for Year 11 (you can watch both here) - I did something similar for my first lessons last year and it worked quite well.

If you didn't make it to researchED Maths and Science back in June then you might like to watch some videos of the presentations here.

Please follow @Team_Maths1 if you haven't already - I use this account to tweet maths resources, and my partner in crime Lucy tweets articles and maths jokes. We also offer a resource clinic - ask us for help and we will do what we can to find a suitable maths resource for your lesson.

Do check out the hashtag #DonADay too.

If you're a fan of Don Steward's maths resources then check out the #DonADay hashtag. pic.twitter.com/FY73QVo4BX

— Jo Morgan (@mathsjem) 21 July 2016

I'll probably blog a bit less frequently than usual over the summer holidays (I've got lots of school work to do... I also hope to make a start on organising #christmaths16... and I want to spend lots of time with my daughters). But I will be using the hashtag #summerblogread to tweet links to posts that you might have missed over the years.

It looks like La Salle are organising another Pie and Maths (see Gems 37 for my write up of the last one) so - depending on the date - I might be there for some summer socialising.

I'll leave you with this question from brilliant.org. There are various approaches (it's pretty straightforward if you can do basic trigonometry) but the solution is interesting. Check out the replies to my tweet here to follow the discussion.

Tricks and Tips 2: Sequences, Linear Graphs and Surds

I recently presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This is the second in a series of posts summarising the content of that workshop for those who were unable to attend. The aim of my workshop was to encourage people to reflect on their subject knowledge and the effectiveness of their explanations. I also hoped that delegates would learn new methods that they might consider using at school. In today's post I'm covering sequences, linear graphs and surds. My previous post was on methods for finding a Highest Common Factor.

Linear Sequences

What's the n^th term of the sequence below?

5, 8, 11, 14, 17, ...

It might only take you a second to work it out - how did you do it? This is another one of those topics where there's lots of different approaches. Ask the same question to a random sample of teenagers from across the UK and you'll see a wide range of methods used.

Here I'm going to describe four different methods (there may be more!). As you read them, consider whether you're going to stick with the method you currently use, or try something new.

1. Zeroth term
Inspired by this post on Don Steward's blog, I taught the 0^th term method for the first time this year. To work out the value of q in the n^th term U_n = pn + q, we simply step back from the first term in the sequence (ie q = U₀).

In the example 5, 8, 11, 14, 17..., subtract 3 from the first term to get U₀ = 2. So the n^th term is 3n + 2. This method is so quick that it's now my preferred method for working out n^th terms.

When I taught this method it went well, although a misconception did surface later on. When students were asked to 'write down the first 3 terms of the sequence 4n + 3', some of them gave the answer 3, 7, 11 (ie they started their sequences from n = 0 instead of n = 1). This is something to be aware of next time I teach this method.

Jumping along a line - Median Don Steward

2. Shifting times tables
'Shifting times tables' is a popular method. The idea is that we compare our sequence to a times table.  In the example below, compare the sequence to the 3 times table. Ask your students how to shift the three times table to get the sequence and they'll spot that they need to add two, so the  n^th term is 3n + 2.

If you plan to use this method for the first time then I recommend this NRICH article Shifting Times Tables, which comes with an interactive tool. If you subscribe to MathsPad, they also have an interactive tool for identifying shifts.

This method works well for quadratic sequences too (a topic on the new GCSE syllabus). For example to find the n^th term of the sequence 4, 7, 12, 19, 28 we can compare it to n² and notice that it is shifted up by 3 (ie the n^th term is n² + 3).

3. Formula
It's fairly straightforward to derive the following formula for the n^th term:

Why do we restrict this formula to A level? I once took on a GCSE class in Year 11 and asked them to find some n^th terms in a revision lesson. I was surprised to see them all using 'the A level formula' - their previous teacher had taught them it, and why not? They seemed quite happy with it. I wouldn't use it with my Year 7s though - their algebra skills are very basic when they first meet sequences. In our example, this is what they'd have to do to find the n^th term:

Perhaps this is more effort than necessary. The 0^th term method is considerably quicker.

4. Substitution
This is the way I taught sequences for years. To work out the value of q in the n^th term U_n = pn + q, we substitute a value for U_n, p and n, then solve for q. For example in the sequence below, we know that U_n = 3n + q. We substitute the first term to get 5 = 3 + q. Therefore q = 2.

Thinking about it, this is the same method I'd use to find the value of c in y = mx + c if I know the gradient and a point on the line. In fact given that linear graphs are simply graphical representations of linear sequences, any methods for finding the equation of a straight line work for finding the n^th term of a sequence. I don't do enough to make this connection with my students.

Linear Graphs

A straight line through the point (5, 7) has gradient 4. How would you find the equation of the line?

As I said in my post about linear graphs, I teach this differently at GCSE and A level. Wouldn't it be better to pick my preferred approach and stick with it? 

At GCSE my students write down y = mx + c and substitute values for y, m and x, then solve for c (let's call this Method 1). At A level my pupils use the formula y - y₁ = m(x - x₁) (let's call this Method 2). Look at the steps involved - for this question, the methods are equally efficient.

My current Year 10s had an American teacher last term - she covered linear graphs with them. When we came to revise this topic, they took a while to figure out where to start. Evetually I heard one say, "is this the point-slope thing? What was that formula again?". 

OK, so they'd been taught to call gradient 'slope', that's easily fixed. But they'd also been taught Method 2 and they couldn't remember the formula. This is a problem. Given that there's less to memorise, I think they'd be better off with Method 1. 

It's worth reading @srcav's post The Straight Lines Debate for more analysis and opinion on the two methods.

I find that all my students really struggle with linear graphs. It's one of those topics that frustrates maths teachers because it's hard to see exactly what it is that students find so difficult. I teach it over and over again from Year 8 to Year 12 and my students never seem to remember it from the previous year - I'm clearly doing something wrong!

One suggestion (thanks to @letsgetmathing for this) is to approach this topic from a less algebraic perspective - instead, focus on a table of coordinates. This is demonstrated in the example below. Students have to recognise that the y-intercept is at x = 0 and the gradient is the 'step' in the y values.

If your students struggle with the algebra-heavy Method 1 and Method 2, perhaps this table method might work.

Finally, I recommend that all teachers watch this video from James Tanton. He makes the concept of the equation of a line really clear.

3. Surds
Let's finish with three methods for simplifying surds.

In the book Nix the Tricks (essential reading for all maths teachers), we're shown a teaching trick called 'Jailbreak Radicals'. Thankfully I've never seen this 'zero conceptual understanding method' used in the UK.

Instead, most teachers tell students to identify a square factor and then split the surd accordingly eg √45 = √9√5 = 3√5.

Sometimes students struggle to spot square factors. In this case they might prefer to simplify using prime factors.

A geometric method is explained in the extract from Nix the Tricks below. Try to simplify a few surds like this yourself and see what you think. Notice that you still need to identify a square factor.

Source: Nix the Tricks

The feedback from my conference workshop suggests that some people find this unnecessarily complicated. But if you're going to try it, I recommend this post by @ChrisHunter36 and this resource (pages 5 - 6) by @314Piman.

That's it for today's post. I hope you've found it helpful. Please comment below or tweet me if you know of any alternative methods for teaching these topics. My next post will be all about quadratics - sketching, expanding and factorising.

5 Maths Gems #19

Hello and welcome to my 19th gems post. This is where I share some of the best teaching ideas I've seen on Twitter each week. Most of my gems have come from across the Atlantic today.

1. Polygraph
There's been a lot of buzz about Desmos' new Polygraph activities. They're mathematical versions of Guess Who, designed to 'foster the pleasure and the power of words without the drudgery of the lists'. There are currently four Polygraph activities:

Let's look at Polygraph: Parabolas. Students work in pairs (a picker and a guesser) on separate tablets, phones or computers. The picker selects a graph from a set of parabolas. The guesser sees the same set of parabolas - their task is to identify which graph the picker has chosen by asking a series of yes/no questions. For example they might ask about the number of roots or the location of the vertex or y-intercept. The guesser types their questions and the picker responds with yes, no, or don't know. The guesser keeps asking questions until they are able to identify the chosen graph.

Students then have the opportunity to analyse and refine their questioning, and you have a class discussion to formalise the mathematical vocabulary.

As with Desmos' awesome Water Line activity that I described in Gems 13, it's incredibly quick and easy to get started. I'm going to be teaching quadratics to Year 10 shortly and can't wait to use the Parabolas activity. And I plan to use the Linear Graphs version with my Year 9s too.

Extract from Polygraph: Lines

2. Point, Point, Gradient

Tina Cardone (@crstn85) shared this lovely activity 'Point, Point, Slope' by Michael Fenton (@mjfenton). It's a simple but effective lesson idea in which students have to use any digit from 2 to 9 to create two points - you could give them post-its or number tiles to shuffle around. You provide criteria, such as 'create two points which determine a line with the greatest possible gradient'. Read the full post for more ideas. There's loads you could do with this activity, for example you could ask your students to use six different digits (from 0 - 9) to create three points which lie on the same line. It's a good way to develop understanding of linear graphs and I think it would work well at both GCSE and AS level.

Michael's blog is full of excellent ideas. For example in Two Fractions students place digits and operators in the boxes below to make an expression with the greatest possible value.

This reminds me of the Blanks activities I featured in Gems 17, which you can read more about in Sarah Aldous' post Logarithm Questions Around the Room.

3. Owning Facts
I liked Davis Wees' (@davidwees) tweet about owning multiplication facts.

There's probably an interesting psychological discussion to be had here - would relying on someone else make students less likely to remember facts, or would the association create a memory trigger as David suggests? I think it's the latter. Like Colin, I once had a sixth form student called Suzy who was always very vocal in class about 'not forgetting the plus C'. I still hear Suzy's voice whenever I do indefinite integration!

4. Number Properties
Back in Gems 13 I talked about making your students 'Number Experts'. I shared two websites that are useful for finding out properties of numbers (numbergossip.com and numdic.com). Both websites are excellent - for example, numbergossip.com tells me that 28 is a perfect, composite, happy number, and numdic.com (which wins the prize for funniest URL) gives me the binary and Roman Numeral representations of 28. I've now discovered a new website - the Number Property Calculator - which I like because of the way it gives information about the meaning of the number properties. Here's a small extract from what comes up when I search for 28:

I found this website when I was browsing through old posts on Math Munch. Math Munch is a 'Weekly Digest of the Mathematical Internet'. It's packed full of fantastic resources and ideas so it's definitely worth subscribing to receive weekly email updates, and you can follow them on Twitter too (@MathMunch).

5. Drawing Euclid

Andy Shaw (@Squidworm74) pointed me in the direction of these lovely Euclid videos on YouTube by Shoo Rayner (@shoorayner). They are fantastic videos and I can't wait to share them with my Year 7 students. Check out the full playlist here. Here's an example about different types of triangles:



That's it for this week. I hope you've found some inspiration in this post. I'll leave you with this brilliant video by the ever awesome Vi Hart - The Gauss Christmath Special. Thanks to Chris Smith (@app03102) for sharing this.




Have a wonderful Christmas!