5 Maths Gems #91

Welcome to my 91^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. World Cup Box Plots
Thanks to Southborough Maths (@Mathsteam1) for sharing these box plots created by @johnwmillr. They show the distributions of height by position for players in the World Cup. They make for great discussions with students, and provide a nice demonstration of how box plots can help us make comparisons.

I've been using a similar set of graphs for years, every time I teach box plots (see my post on teaching box plots for more on this). It always goes down well.

2. Variation Theory
Last week Craig Barton launched a new website packed full of sets of well written questions for intelligent practice. Do check out variationtheory.com if you haven't already seen it.

'Rearranging formulae' by Danielle Moosajee

'Mixed Bases' by Joe Berwick 

Like Craig's other resource websites (SSDDs, Venns and Diagnostic Questions), you can submit your own resources for inclusion on this website.

3. Fractions
Thanks to Berkeley Everett‏ (@BerkeleyEverett) for sharing this animation. This can be found, along with loads of other great animations, on the Math Visuals website. 

4. Compound Shapes
Thanks to Mark Ives (@MarkIvesTeach) for showing us how he used Numicon to support students in identifying the lengths of sides in compound shapes.

5. Coordinates Problems
Thanks to Dave Taylor (@taylorda01) for a sharing a set of challenging coordinates problems (see this tweet and this tweet) . Here are a couple of examples:

Update
Do maths teachers all say things in the same way? At the Tweet Up in Manchester last weekend, I recorded a group of teachers saying words that I've heard pronounced differently by different maths teachers. I've picked three of these words for the first video from my pronunciation project:

Thank you to everyone who took part! It may not be the most exciting video ever but I think it's really interesting that students hear different things from different teachers.

Here are a few other things you might have missed recently:

• ThinkMaths (@thinkmaths) are doing their annual giveaway of maths goodies - the 'Exciting Maths Toys for Excited Maths Teachers Mailout' - sign up now!
• I'll be taking part in the Big Internet Math-Off in July - and I'd really appreciate your support! 
• Check out this great post from Alex Cutbill (@intersectarian) about the definition of 'power' in maths. This is what I was talking about in my conference session last Saturday.
• Do check out the excellent set of surds questions from Gillian Matthewson (@gmathewson1) on her new blog.
• The Nuffield Foundation report "How do shortages of maths teachers affect the within-school allocation of maths teachers to pupils?" is worth a read.
• I recorded a podcast after the JustMaths conference with Craig Barton which is now online - you can listen to it here. It's only half an hour!

Ten years after we did our PGCE together, I finally met up with Colin Hegarty! He came to my school to launch Hegarty Maths at our first annual trust maths conference. This is really exciting - Hegarty Maths is awesome. I loved trialling it with my Year 11s this year. Thank you to both Colin and Simon Petri from the Surrey Plus Maths Hub for their excellent presentations.

It's all been a bit crazy lately. Next week I have an AQA Expert Panel Meeting, the BBO Maths Hub conference, a TTRS Rock Wrangle trip, and prom. Then I can relax!

I'll leave you with this lovely factor tree puzzle from Sarah Carter (@mathequalslove), inspired by @HaroldReiter.

5 Maths Gems #1

I’m fairly new to Twitter. My absence over the last few years means I’ve missed out on thousands of excellent teaching ideas. Twitter is a source of endless inspiration. So I have a plan. I’m going to write a weekly post highlighting five great new ideas I’ve gleaned from Twitter. These posts will help me remember things. And hopefully my readers will benefit too.

Today I’ve exceeded my limit and am presenting a list of 6 ideas from my week on Twitter (not a good start! I’ll aim for 5 next week).

1. Practical ideas from Maryse. It sounds like Maryse (@AllThingsMaths) has done an amazing job transforming mathematics teaching in her school and enthusing her students with tonnes of creative ideas. Here’s a selection of those she’s shared this week:

• Play catch with eggs and work out the relative frequency of smashing.
• Get an old tyre, paint it and roll it to demonstrate circumference. 
• Have a number line painted in the playground.
• Have discussions about going back in time and Doctor Who with travel graphs. 

I love these ideas. Follow @AllThingsMaths for lots more.

2. Maths lies. In response to my post about the Mistake Game, Colin Beveridge (@icecolbeveridge) shared this awesome story ‘My Favorite Liar’. You must read this! It prompted a discussion on Twitter about whether the idea would work in maths. I think it would. In your first lesson with a new class, tell them that you'll tell one lie per lesson. Your pupils have to identify your lies. In the discussion on Twitter we came up with a good list of lies and misconceptions, such as 'multiplication makes numbers bigger', 'if a number has 6 zeros then it's in the millions' and 'one is a prime number'. The idea is that you slip these lies into your teaching and hope that your pupils will immediately challenge you. And if no-one challenges you then it will make a nice plenary discussion. Mr Allan (@mrallanmaths) is going to try it out - I’m really looking forward to hearing how effective it is. I think, like the mistake game, it will be incredibly engaging.

3. Box plots party. In response to my post about teaching box plots, Pete Sides of the South Yorkshire Maths Hub (@SYMathsHub) suggested a lovely activity in which pupils discuss the ages of guests at different types of party, to help them understand the concept of spread. For example you could ask them to draw a box plot representing the ages of guests at an 18th birthday party, a wedding, a child’s party etc. I like this idea so I turned it into a worksheet and added it to my blog post.

4. 100 people. I came across these visuals ‘If the world were a village of 100 people'. Maryse (@AllThingsMaths) told me of the related video, which reminded me of another visually striking video, Debtris. I thought these ideas might be nice for form time or PSHE but Mark Greenaway (@suffolkmaths) then shared this data and this blog about the psychology of percentages. The blog is well worth a read, it’s an interesting idea for introducing the concept of percentages in maths. @El_Timbre shared this display work resource which she created to help students with the visualisation.

5. Percentage triangles. I’ve been using speed distance time triangles for years – in fact I think I was taught them when I was at school. Trigonometry triangles are similar. I’ve never given much thought to the fact that my students are missing an opportunity to practise rearranging formulae. I understand the reason for mixed views on these methods, but I do rather like the triangle shared by Malc Henderson (@malc_henderson) for solving reverse percentage problems. It’s nice to have a range of tools to support students when they’re preparing for their GCSEs.

6. Display ideas. Clarissa Grandi (@c0mplexnumber) posted a picture of her new ‘vwllss’ corridor display (inspiration from @El_Timbre). It looks fantastic - what a good idea!

Other good display ideas I’ve seen recently include circle formulae by Dawn (@mrsdenyer), a square root clock by Duncan Smart (@duncsmart), and an interactive number puzzle for a maths corridor or classroom, shared by Mr C Ward (@MrWardMaths).

There you go, six maths gems from @mathsjem (my initials are JEM by the way, it’s not a spelling mistake!). This is just a small selection of the good ideas I’ve seen on Twitter this week. More to come next week!

I’m working on a few posts at the moment, including one on introducing algebra and one on introducing differentiation – if you have any great ideas then please email resourceaholic@gmail.com or tweet me.

Image sources: lightbulb tampawebdesigner.net; birds techwyse.com

Teaching Box and Whisker Plots

Image: ruthmaas.com

When I wrote Long Live Stem and Leaf, I'd been challenged to find a practical application of stem and leaf diagrams outside the maths classroom. After extensive googling - and helpful input from twitter - the only examples I could find were bus and train timetables. But I argued that real life application isn't the be all and end all.

Stem and leaf diagrams were invented in the 1970s by John Tukey, who also invented box and whisker plots. As a matter of interest, I've also investigated real life applications of box plots. I found that box plots are widely used in research papers and analyses. Unlike stem and leaf diagrams, they are a statistical tool genuinely used by statisticians. Because box plots are commonly used, much has been written about their effectiveness. As this blogger says, “a box plot is a simple yet powerful tool, with a truly great design - a universally beautiful thing that stands the test of time”.

In this post, I’m going to focus on teaching ideas and resources.

Practical activities
I’ve read a lot of blog posts about human box plots. For example, 11 students line up at the front of the classroom in height order. Their classmates determine who is the median, lower hinge and upper hinge (yes, I’m using the word hinge instead of quartile. Read my blog post on Consistent Mathematics if you want to know why). This is a lovely activity in a lesson introducing the idea of median, but not so effective in a box plot lesson. Because even though you can use a bedsheet and string to get the students to look like a box plot, if they’re not appropriately spaced out (ie if there’s no scale) then it’s not a box plot. If you want a proper box plot you’ll have to actually measure the heights and draw a scale on the floor.

An alternative idea (inspired by this post) involves paper aeroplanes. You’ll need a big space for this - the school hall or playground. Tape (or draw in chalk) a long scale on the floor. Have each of your pupils make a paper aeroplane, write their name on it, and throw it from the start of the scale. Leave the aeroplanes where they land. Once all pupils have thrown their aeroplane, you can draw or tape a box plot on the floor around the planes to represent the distance flown. Pupils will be able to see which quartile their plane landed in. Hopefully one future engineer will throw an amazing outlier which will make a good discussion point! You could even split the class in half and do two box plots on the floor side by side, then the class could discuss which team had better aeroplanes and which team’s planes were more consistent.

If you like practical activities, here’s two more:

• This simple paper folding activity is a nice introduction to median and quartiles.

Source: bigideasmath.com

• This Cheerios activity explores mean, median, mode and box plots.

Resources
There’s surprisingly few good box plot resources online. But Don Steward never lets us down. His website has a number of box plot activities - my favourite is this true or false activity.

I also like the activity below from bigideasmath.com:

If you’re looking to generate box plots on your interactive whiteboard for class discussion, this online tool is nice – it has pre-populated data or you can input your own.

I've listed some more good box plot resources below. I’ve found that many resources ask questions about skewness, which we don’t cover until S1 (but perhaps we should cover earlier). Treatment of outliers is also not covered at GCSE but makes for a nice discussion.

• Standards unit activities S5 and S6, which can be found on Mr Barton’s website. 
• Illustrative Mathematics has loads of excellent statistics questions, like Haircut Costs and Speed Trap.
• Interpreting Box Plots Treasure Hunt from numberloving.co.uk.
• Representing Data Using Box Plots from Mathematics Assessment Project (activities at the back)
• Lots of exercises from CIMT in this chapter – I particularly like the comparison questions on the last few pages 
• Using NBA Statistics for Box and Whisker Plots from NCTM 
• The variability exercise (starting on page 8) in this lesson plan could be helpful when teaching the concept of spread. 
• My Describing Box Plots activity on TES helps pupils practise reading and drawing box plots and emphasises the use of the correct vocabulary.
• My Party Box Plots activity on TES helps pupils understand the concept of spread.

If you want to show your class some ‘real world’ box plots then these box plots showing the ages of World Cup players could prompt some good discussions about comparisons. Read @loumeracy’s blog post for more on this.

If you want to ‘wow’ your class with an example of how box plots can be used to compare huge amounts of data in a small space, show them this example which shows the age distribution of Olympics athletes (more of this here).

Final thoughts
In French the box plot is called boîte à moustaches (box with a moustache). The writer of this article attempts to create a box and beard plot!

There are many variations to the box and whisker plot, such as the bean plot and violin plot. Look these up on google images for lots of examples.

I read an interesting idea here that I’d never thought of before: “One way to help you interpret box plots is to imagine that the way a data set looks as a histogram is something like a mountain viewed from ground level and a box and whisker diagram is something like a contour map of that mountain as viewed from above”.

Well, there you are – a 'box plot blog post'. Try saying that five times out loud. I've invented a new tongue twister.