Kicking off Year 12

Following on from my last post on ideas for the first Year 7 lesson of the year, here's some ideas for the start of Year 12.

Unfortunately we don't really have any time to spare in our sixth form teaching. It's a challenge just to fit enough lessons into the school year to cover all the course material before our students go off to take their exams. So, sadly, there's no time for enrichment or getting to know each other - students have to hit the ground running.

Entry Assessment
At my school, as at many others, we ask our students to prepare for Year 12 by doing some summer work. We expect them to have a solid understanding of GCSE topics when they start the course, particularly surds, indices and algebra. Last year, for the first time, we introduced a Year 12 entry assessment. The purpose of the entry assessment was to help us identify the students who needed additional support from the start of the course. We also hoped it would identify any students who shouldn't be taking AS maths at all, in the hope that they might consider choosing an alternative course.

Extract from Year 12 Entry Assessment

The results from our entry assessments were rather worrying. A large number of the cohort got less than 50% even though the majority achieved a grade A* at GCSE. The simple question 'Calculate 238 x 6.3' was answered very badly, showing a lack of basic maths skills.

I had a huge amount of students come to my extra support sessions - I believe the entry assessment was a good 'wake up call' for them. It motivated these students to focus and work hard right from the start. All students improved their mark considerably in the subsequent re-test, but still some of the marks were quite low. On A level results day in a few weeks I'll be really interested to compare AS grades achieved to performance in the entry assessment last September.

Here's the files in case you want to borrow them:

• Year 12 Entry Assessment   (answers) 
• Feedback sheet  (students who did well got a 'welcome to AS Maths' message!)
• Year 12 Entry Assessment re-test 

Student Handbooks

We hand out student handbooks to every Year 12 in the first lesson of the year. They're asked to keep these in the front of their maths folder. It's particularly helpful that they have their own copy of the formula sheet and a guide to reading exam mark schemes.  

Activities

Here's some nice resources for the first couple of maths lessons:

• What's the Difference - see below. This discussion activity reminds students of the importance of following the order of operations.

• Introduction to C1 Algebra matching activity - from SRWhitehouse on TES
• Ever wondered why? - an engaging and challenging indices puzzle from JustMaths 
• Indices activity - from SRWhitehouse on TES

My Core AS and Statistics AS pages have lots more resource recommendations for the rest of the course.

If you'd like Word copies of my pdfs to adapt for your own school, please ask.

Your First Maths Lesson with Year 7

A Year 7's first ever maths lesson in secondary school is rather daunting. For both the students and the teacher. As this is the time of year when many maths teachers will be planning those first lessons, I thought it might be helpful to suggest some ideas.

With a new Year 7 class, the first lesson is not just about setting expectations about behaviour and the like, it’s also about getting your students excited about maths and building their confidence. It’s about helping them to quickly settle into their new environment. And - on a more practical note - it’s about learning their names.

First I’ll talk you through my suggestions for that first Year 7 lesson, then I’ll share some ideas suggested by other teachers. And I’d love to hear your ideas too.

A number lesson
As students arrive, greet them at your classroom door and hand them a card or sticker with a number on it. I suggest the numbers 1 – 30 but any numbers will work. While they find their seats, an instruction on the board tells them to think about the properties of their number (bear in mind they might not know the meaning of the word property in this context).

Tell students to stand up and hold their card up if they have a multiple of 5. Each student standing up says their name and number in turn (one of the benefits of this activity is that you - and their classmates - learn their names). Ask one of the students standing to define the word multiple.

Now do the same for various number properties eg ‘stand up if you have a factor of 24’, ‘stand up if you have a prime number’, ‘stand up if you have a square number’ and so on. There are various discussion points relating to the definitions, such as whether one is a prime number. You get the idea. Try to arrange it so each pupil stands up at least twice, to help with the name learning.

There’s various other activities you could use with students when they have number cards. Perhaps get the pupils with numbers 3 – 8 to line up at the front and ask the class to name the polygons with each number of sides. It will be encouraging if you start the year by letting students share with you some of the maths they already know. But you could also give them a taste of the exciting new maths to come. For example you could ask the students with the cards 1, 2, 3, 5, 8, 13 and 21 to stand in order at the front of the class and ask their classmates to spot the pattern (add an extra 1 at the beginning!). It's unlikely they would have seen Fibonacci before.

Sticking with the number theme, you could try:

• this Number Teams game from Teachit Maths 
• these 4 interactive whiteboard tasks, from MathsPad. 
• These number riddles from Transum. 

30 Maths Starters - Puzzle 2

If you want the class to settle with some individual or pair activities, try some of these '30 Maths Starters'. I suggest activities two and nine.

And how about a nice game of Fizz Buzz to end the lesson?

If you have administrative bits to do with the class, such as handing out books, they could wait until next lesson.

More ideas
Here are a few ideas I've found online:

• Try these Ordering by Numbers activities by Frank Tapson 
• I like 'Back to School' by Transum
• Suffolk Maths suggests lots of good ideas for a Year 7 transition lesson. This includes 'Stand Up' - which is similar to my suggestion. Whenever I think I've been creative and original I find that someone's already thought of the same ideas! :)

While looking for these lessons, I found this 'Moving on in my Mathematics - Year 6 to Year 7' booklet that you may be interested in.

Craig Barton's blog post on transition in secondary maths is a must-read for new Year 7 teachers. I know that my Key Stage 3 teaching would be improved if I knew more about what children cover at primary school.

And finally, here's some more ideas for learning names!

A level
I hope that’s been helpful. Please comment if you have your own ideas to share. My next post will be about the first Year 12 lesson of the year.

Long Live Stem and Leaf

I planned to write a post about teaching stem and leaf diagrams. But then I realised that many maths teachers consider them to be pointless, a waste of teaching time and a welcome removal from the curriculum.

Now let’s think about this for a minute. Admittedly stem and leaf diagrams are rarely used in ‘real life’. In fact the only common examples we can find are timetables, such as the Japanese train timetable shown in the picture below (more examples here). But does this ‘uselessness’ really mean that stem and leaf diagrams shouldn’t be taught? If that’s the case then a number of the topics we teach in maths should be taken off the curriculum. 

Japanese train timetable. Source: Wikipedia

Much of the maths we teach at Key Stage 3 helps pupils access more advanced topics (all roads lead to calculus?!). Maths begets maths. So do stem and leaf diagrams lead us anywhere? Do they help pupils develop any useful skills and understanding?

I'm going to say (in a sheepish whisper) that they do. Stem and leaf diagrams are one of a set of tools that could be used to develop understanding of the concepts of median, mode, quartiles, spread, skew and outliers. I think they are accessible to low ability pupils. I think that pupils enjoy creating them. I think they lead to interesting discussions when comparing two sets of data. I certainly don't think they should be a compulsory topic at any age, but I see no harm in teachers using them if they like. Isn't it nice to have that freedom?

John Tukey, the inventor of stem and leaf diagrams, wrote "If we are going to make a mark, it may as well be a meaningful one. The simplest - and most useful - meaningful mark is a digit" (source). Excellent point Mr Tukey. Did you know he also invented the box and whisker plot?

John Tukey is also quoted as saying "There is no data that can be displayed in a pie chart, that cannot be displayed better in some other type of chart". I agree. In the same way many of you don't like stem and leaf diagrams, I don't like pie charts (this excellent blog post about the misuse of pie charts is worth a read).

Right, now I've attempted to justify this blog post, here's my lesson ideas:

Starting Point
You could start by presenting your class with a large set of data (like the one below) and asking them what the mode is. It's hard to spot. Ask your pupils for ideas on how could we organise this data so we can easily see the mode.

12, 54, 21, 56, 32, 14, 42, 23, 20, 48, 21, 34, 12, 15, 16, 42, 47, 46, 32, 21, 45, 43, 21, 34, 26, 32, 17, 18, 45, 21, 44, 32, 12, 23, 45, 24, 35, 38, 28, 18, 29, 20, 10, 36, 12, 42, 47, 32, 47, 37, 12, 54, 23, 17, 29, 38, 32, 12, 17, 13, 21, 34, 12, 15, 16, 42, 47, 46, 12, 16, 43, 23, 16, 46, 52

I find stem and leaf plots rather pleasing to the eye. Much nicer to look at than a boring list of unordered data.

An alternative starting point is a 'beat the teacher' game, which I've adapted from an idea by the NCETM. In this activity, pupils think of a number between 10 and 40. Each number is called out in turn, while pupils make a list in their books. Meanwhile the teacher also makes their own list, but organises it as a stem and leaf diagram. Once all the numbers are listed, the teacher asks the class to work out median, mode and range. The teacher will do it much quicker from their stem and leaf diagram, hence he/she wins the race.

Physical Activities
This blog post is about introducing stem and leaf diagrams using post-it notes. It's a lovely idea for an engaging and accessible lesson. Mr Collins shared a similar idea in this blog post, but he uses students and desks instead of post-its. If you like the idea of human stem and leaf diagrams then you might also like these breath-holding activities - one from Don Steward and a similar one from the NCETM.

Data Types
Encourage pupils to think about different types of data - what is suitable for a stem and leaf diagram and what isn't? For example, could the total number of goals scored per game in a season of football matches be displayed in a stem and leaf diagram? Probably not, because the data would mostly be single digit numbers. Try my worksheet for more on this.

Mini Projects
Don Steward has some nice activities involving stem and leaf diagrams, such as this this Presidents Longevity Project. I also like Tim Jefferson's mini-project about bus timetables. Taking this timetable project further, you could ask pupils to think of other ways to present timetables - this post 'Visualizing a train timetable using a stem-and-leaf plot' suggests a nice alternative.

Extensions
There's opportunities to extend this topic by looking at variations in the way data is presented (for example using 'split stems') and interesting features of the data (eg outliers, bimodal data, skewness).

Back-to-back stem and leaf diagrams are useful for comparing data. Writing comparisons between data sets - whether represented by stem and leaf diagrams or box and whisker plots - is something that pupils (especially EAL pupils) often find difficult.

There's lots of S1 (A level) stem and leaf questions - like the one below - that could be explored at Key Stage 3 (which begs the question, why were they considered suitably challenging for A level?).

Edexcel S1 November 2004, Question 1

I rarely disagree with people or play devil's advocate so this post is a bit risky for me! Be nice. If you do teach stem and leaf diagrams then I hope you find these ideas helpful. Thanks to those on twitter who shared their views and ideas (especially @srcav - who is not a fan of stem and leaf diagrams!).

Resources, resources, resources...

Inequalities activity - Project Maths

I've been working really hard on my resource libraries lately, so I'm pleased to say I'm finally ready to share them! They can be accessed through the tabs at the top of the page.

This is where I've put all my resource recommendations for you to browse when you're planning lessons. The A' level pages will probably be the most helpful as it can be hard to find good resources for Key Stage 5. 

My recommendations are by no means exhaustive - there's loads of good resources I haven't discovered yet (the internet is too big, I can't keep up with it!). But I will update my listings whenever I come across something particularly good. To clarify what I mean by 'good' - my recommendations include well-designed activities, rich tasks and worksheets for topics where I think it can be hard to find suitable practice questions. 

My Top 3 Websites

The resources I've recommended are all free and come from a wide range of sources.  Here's my current favourite websites, in no particular order.

1. Median by Don Steward is absolutely brilliant - it's widely considered one of the best sources of rich mathematical tasks for secondary school pupils.  The activities include this Roof Truss task (trigonometry) and Bee Aware (scatter graphs) - there are hundreds more and they're all awesome.  
2. MathsPad has plenty of excellent free resources. Subscription (£3 per month) gives access to their full range. I love the way their resources are designed, including their interactive activities which are all very engaging and easy to use. Example resources include this lovely Square and Cube Numbers Puzzle, and for older pupils I really like Is it Rational?. 
3. Teachit Maths has large, well-organised resource libraries.  Free registration gives access to resources in pdf form, which is fine for all their printable activities and worksheets. You'll need a subscription (£25 per year) if you also want to access PowerPoints, interactive activities, teaching tools etc. Worksheets are well-designed, engaging and include solutions. Examples of their resources include this Mental Powers Code Breaker and this Transforming Graphs card sort activity, which I have used successfully with both my Year 11s and Year 12s.

Special mention to The Chalk Face and Mathematics Assessment Project - both of these websites are full of excellent ideas. I also get a lot of resources from TES. One of my favourite TES contributors is SRWhitehouse who creates excellent A level activities. For a full (and ever-expanding) list of recommended resource websites, see my links page.

Factor Trees - Median Don Steward

My Blog
Now I've finished creating my resource libraries, I can get back to focussing on my blog. Here's some of the highlights so far...

• My Area and Volume post - featuring a giant tennis ball - was very popular.
• My suggestions for end of term lessons on Binary and Fermat also went down well.
• Have you tried this method for finding in HCFs and LCMs?  It's so much quicker than the Venn method.
• Have you read the book Nix the Tricks yet? It's worth reading (I particularly like the suggestion to replace BIDMAS with GEMA) - here's my post about it.
• I put a lot of thought into my post All About Gradient. I'm sure that pupils would enjoy the lesson ideas I've suggested for investigating steepness - have you watched the video of the cyclist attempting a 38% gradient? My legs hurt just thinking about it!
• I've written a number of posts focussing on specific topics or concepts such as dividing by zero, angle facts, skewness and simultaneous equations.  I've also discussed teaching tools such as Desmos, comic strips, blogs and methods for developing vocabulary. 
• It was my discovery of a new matrix multiplication method that brought me to blogging - if you teach matrices then do check it out.
• Finally, something fun for the teachers - try the Maths Movie Quiz.

For a full list of my blog posts see the blog archive.  Not bad going for 3 months!

'Find the Coordinates' task by Susan Wall

Bringing babies into a statistics lesson

I’m not particularly into babies (apart from my own little terrors darlings) but I teach at a girls’ school and the majority of my students respond rather well to baby stuff (apologies for the stereotyping). It’s surprisingly easy to feature babies in your statistics lessons if you so please. I’ve already blogged about my Bibonacci starter. There’s also this pregnant cow exam question (part d is the only S1 exam question I've ever seen which requires students to know that 68.3% of normally distributed data is within one standard deviation of the mean).

Edexcel S1 January 2002, Q5

I used this question in an S1 lesson when I was heavily pregnant, much to the amusement of my students. It led to a discussion about the gestation periods of other mammals (those poor elephants!). We’d clearly expect a fairly strong correlation between size and gestation - when I get a chance I'll see if I can design an interesting activity using this data (I wonder whether anything similar is done in biology).

Speaking of gestation periods - I remember when my due date for my first daughter came and went, I tried to find statistics online to tell me how far overdue I might go. I wanted to know the probability of baby arriving at 42 weeks. I didn’t find much online at the time but again I feel it could make for an interesting lesson.

If you are a parent then you may be familiar with ‘the red book’, which is the Personal Child Health Record issued to all babies born in the UK. This is a child's primary record of immunisations, development, health reviews and so on. At the back of the red book are the ‘UK-WHO Growth Charts’ where scatter graphs are plotted of the child’s height and weight against their age. On these scatter graphs are centile lines, to allow parents and health professionals to compare the child’s measurements against other children of the same age. On a few occasions, I’ve been asked to explain the meaning of these charts to friends with newborn babies, so it occurred to me that the growth charts could help me explain the concept of percentiles to my S1 classes.

The red book contains this description of the centile lines on the growth charts:

More information about the charts is available in this fact sheet from the RCPCH.

While I was thinking about how to design an activity for my S1 class based on these charts, I discovered that the author of The Chalk Face has beaten me to it with this nice 'Quartiles and Percentiles' activity. And @srcav has already written this blog post 'the mathematics of parenthood' about the potential to use growth charts in his teaching.

Incidentally, the red book says ‘someone who has been appropriately trained should complete the growth chart’. Hmm. I hope they're not suggesting that members of the public are incapable of adding a dot to a simple scatter graph?!

Right, my baby is awake now so I'll continue to develop these ideas another time...  Do let me know if you can think of any other baby-related data to add to my theme.

My mini-mathematicians.  July 2014.

End of term!

The summer holidays are fast approaching. Although I'm a believer in teaching maths right up to the end of term, it can be hard to teach a topic properly when lessons are being interrupted by sports day, assemblies, school trips etc. But if we don't deliver worthwhile lessons until the last day of term, then we can't complain when parents want to take their children on holiday during term-time.

I'm not against the idea of watching a video or playing games in maths lesson, but let's make sure our students are either learning something new or becoming more engaged and excited by mathematics.

My school has a standard set of challenges, games, projects and investigations that we use every year, but here's a few of my alternative suggestions for end of term activities:

Fermat's Last Theorem and Pythagoras
Have you read The Simpsons and their Mathematical Secrets by Simon Singh? It's a brilliant book. A nice end of term lesson could involve you explaining Fermat's Last Theorem (or use this BBC video to explain it for you) and showing your class how it features in the Simpsons (it's really rather clever that Homer seems to have disproved it - in fact it's a 'near miss' solution - as explained by Simon Singh in this video from Numberphile).

In this lesson your pupils could do some research into Fermat, or perhaps other famous mathematicians - they could produce a poster as suggested in this lovely activity from Interactive Maths.  Alternatively, they could look at related Pythagorean problems, such as proving Pythagoras' Theorem (which can be done with Origami!), finding Pythagorean triples or investigating Pythagoras with other polygons.

BBC iPlayer is currently showing the Fermat's Last Theorem episode of Horizon - although the maths is rather advanced (it made my brain hurt!), it's an interesting episode and gives an insight into the work of mathematicians.

If you like these Fermat lesson ideas, you might also like my post on teaching binary which features more ideas from Simon Singh's book.

Pascal Investigations
Pascal's Triangle contains lots of amazing patterns and you can easily fill a lesson exploring them. Teachit Maths has a great PowerPoint and activity sheet for this lesson. In this short video a maths teacher briefly explains this history of the triangle. For older students you could also introduce 'choose' notation and link the triangle to binomial expansions.

Pi week
My colleague once had a Pi themed last week of term with one of his classes. The students all brought in cakes and pies, they watched Life of Pi (a lovely film but not mathematical!) and they did Pi-based activities including a Pi recital competition, a freehand circle drawing competition and making a giant Pi paperchain (a different colour per digit).

If you have any other creative ideas for end of term lessons, I'd love to hear them.

Inspiration

I was very lucky to see Simon Singh speak at the Maths in Action lectures in London last year. My school took 20 Year 12s to the event but this year we plan to take 40 because it was such a brilliant day of lectures. I came away feeling inspired. I really recommend these lectures, which are for both A level and GCSE students.

Do you know what makes the numbers on this jumbotron interesting?  Find out in this article.

Linear Graphs

One of the topics that my pupils struggle with the most, from Key Stage 3 right through to Year 12, is linear graphs.

My bright Year 11s know the rule for finding the gradient of a perpendicular line, but can't work out how to answer a GCSE question like this:

Strangely, I teach this differently at GCSE and A level.  At GCSE my pupils write down

y = mx + c

and substitute values for y, m and x, then solve for c.  Whereas at A level my pupils use the equation

y - y₁ = m(x - x₁)

Essentially the same method but a slightly different approach. Perhaps I should be more consistent and introduce the 'A level method' at GCSE. I'd like to hear what other teachers do.

In this post I’ll suggest some ideas and resources for teaching linear graphs.

Key words

Let's start with the basics. There's a lot of new vocabulary in coordinate geometry, some of which pupils will have encountered at primary school. This is a nice activity for discussing new words. This topic also lends itself well to a vocabulary knowledge survey as suggested in my earlier post. 

Here's a nice trick for remembering the words parallel and perpendicular (this could be helpful for spellings - parallel is often misspelt): 

and this activity is suitable for Year 6 or 7. 

Introducing the form y = mx + c
I'd normally start this topic in Year 7 or 8 with plotting straight line graphs (Teachit Maths provides an excellent Drawing Straight Line Graphs booklet for this purpose), before exploring the form y = mx + c.

This topic provides an excellent opportunity for pupil investigation using Desmos. Use a worksheet to guide pupils through the lesson:

• Investigating Straight Lines with Desmos (by Tristan Jones on TES)
• Straight Line Graphs in Desmos (by me!)

Once pupils are able to interpret the values of m and c then they can attempt this exercise or this true/false activity (both from rogradymaths.blogspot.co.uk). Teachit Maths also has a nice 'Understanding y = mx+c' worksheet and two activities on horizontal and vertical lines.

My blog post 'All about gradient' features ideas and resources for teaching gradient (the post focuses mainly on real-life applications of the concept of steepness, but also features links to gradient worksheets and activities).

Understanding parallel and perpendicular gradients

So here's the key information that pupils need to understand:

(source)

If they know how to work out a gradient as 'rise over run' (or equivalent) then pupils should be able to derive the 'negative reciprocal' rule for perpendicular gradients from an activity like this one from MathsPad.

Here's a few activities for a lesson on perpendicular lines:

• Finding Equations of Parallel and Perpendicular Lines by Mathematics Assessment Project
• Perpendicular Lines Worksheet (by me!)
• AS level Parallel and Perpendicular Lines  (suitable for GCSE extension)
• Perpendicular lines: equations by toticity.co.uk

Bringing it all together
When students study linear graphs at AS level, they don't really learn anything new - my GCSE pupils should have the necessary skills and knowledge to answer C1 coordinate geometry questions. I gave my Year 11 pupils questions 5 - 7 from this AS worksheet last year - it took them ages to answer each question, but they were certainly capable. Essentially it's just the wording of the questions that throws them.

Here's some more recommended resources suitable for GCSE students:

• Foundation: Linear Graphs 'I Can' from Mathsbox
• Mathattack Linear Graphs homework sheet
• Topic checklist
• MathsPad Linear Graphs Problems
• Rich activities from Five Triangles
• Rich activities from Median Don Steward

Finally, fun for puzzle-lovers in linear relationships sudoku and a nice Key Stage 3 Christmas activity on plotting linear graphs from Teachit Maths - Mistletoe and Lines.

Finally, ever wondered which countries use y = mx + c notation and which countries use something different? Check out this map.

Resource Library: Complex Numbers

I've been writing this blog for two months now and haven't yet figured out exactly what I want to do with it. In my previous post, Resources Anonymous, I admitted I have a serious resource addiction. Based on the popularity of that post, I think I might make this blog primarily a resource library. So you can come here to look for resources and ideas when you're planning lessons. Would that be helpful? I like to be helpful. I suppose that's why I became a teacher.

Today I'm focusing on complex numbers. Although not widely studied in UK secondary schools (only by Further Maths students), there's certainly no shortage of teaching resources online - here's some highlights:

• First, my top find - Complex Number Operations Teaching & Learning Plans from the Project Maths Development Team. This is packed full of lesson ideas and there are tonnes of brilliant student activities in the appendices. I'm excited to start using these. There's also a PowerPoint version.
• I highly recommend the activities provided by the Further Maths Support Network/MEI (school login required). Here's an example of their topic notes. Their range of resources also includes worksheets, activities and assessments.  
• SRWhitehouse on TES produces excellent A level resources, including a selection on complex numbers. I particularly like this introductory exercise.
• Illustrative Mathematics gives us seven excellent activities: 
• Computations with Complex Numbers
• Complex Number Patterns
• Powers of a Complex Number
• Complex Square Roots
• Complex Cube and Fourth Roots of One
• Complex Distance
• Completing the Square
• As mentioned in my earlier post on Venn activities, this resource sheet is about classifying types of numbers.  There's a related activity on page 32 of Complex Number Operations Teaching & Learning Plans.  

Finally, fun for teachers and students alike - here's a nice Complex Adding Maze and Nrich has a Complex Countdown game! Enjoy.

What topic would you like me to feature next? Please tweet me your requests!

Resources Anonymous

'Hello. My name is Jo and I'm a resourceaholic'. Yes, I confess, I'm utterly addicted to searching the internet for maths teaching resources.

It all started when I saw a PGCE student photocopying this Trigonometry Pile Up activity from greatmathsteachingideas.com. I work in a grammar school where pupils respond quite well to didactic teaching styles and textbooks exercises - arguably there's nothing wrong with this approach, but the lovely Trigonometry Pile Up worksheet reminded me that the internet is full of engaging resources and exciting teaching ideas. I started spending a lot of time looking for resources online and sharing these resources with colleagues. It became a bit of an obsession! Unfortunately my colleagues found it all rather overwhelming. And they're right - there's so many resources to choose from, no-one has the time to look at them all. So it's helpful when someone does the searching, filtering and classifying for us.

William Emeny recently posted a link to 'Mathematics 101: Leading Sites for Math Teachers' on his blog. Ooh, new websites! Heaven for a resourceaholic.

I haven't had a chance to look at all these websites yet, but here's some of my top resource recommendations from the websites I've looked at so far. I've focused on A level because that's where I think it's hardest to find interesting teaching ideas.

Statistics 1 
Illustrative Mathematics has some fantastic ideas for teaching S1. Here's a selection of examples:

• Correlation and Regression: a lovely activity on coffee shops and crime. Includes calculating a correlation coefficient, interpreting a regression equation and considering causality and outliers.
• Discrete Random Variables:  a short activity Sounds Really Good! (sort of) which features a real-life use of expectation. Bob's Bagel Shop is similar.
• Probability: the card activity Describing Events is an excellent introduction to probability and the activity Venn Diagrams and the Addition Rule is good too. I also like this short Titanic activity on independence.
• Data:  These activities on Haircut Costs and Speed Trap focus on comparing box plots. Describing Data Sets with Outliers and Identifying Outliers are about outliers and skewness in data (see my related post on teaching skewness).
• Normal Distribution: short contextual exercises Should We Send Out a Certificate? and Do You Fit In This Car?.

Core 1 and 2

• Logs: This booklet Logarithmic Functions from Mathematics Vision Project is packed full of excellent activities such as ordering logs, converting between exponential and logarithmic form, using log laws and solving equations with logs.  For related resources, see my earlier post on Logs. 
• Quadratics and Cubics: I like this short Cubic Graph task (answers here) from Mathematics Assessment Project. There's some great Forming Quadratics activities here too.
• Circles: Mathematics Assessment Project also has full lesson plans - I've successfully used the excellent materials from Equations of Circles 1 and Equations of Circles 2 with my Year 12s. 
• Surds: See resources listed in my earlier blog post, Surds. 

Finally, I thought it might be helpful to share this quadratic activity 'Find the coordinates' - my school uses this on Sixth Form Induction Day in our maths taster sessions (thanks to Susan Wall at Wilberforce College for this activity).

I'll continue to work my way through the website list and will feature recommended Key Stage 3 and GCSE resources in later posts.

All about gradient

When we teach linear graphs at Key Stage 3, we often miss the opportunity to explore useful real-life mathematics. In this post I share practical ideas and resources for teaching the concept of gradient.

A sense of steepness
At school we usually express gradients as integers or fractions but road signs in the UK show gradients as percentages.  Older signs showed ratios (rise:run) - arguably these were easier to interpret.

The road sign pictured indicates a gradient of 25% ie one quarter. We say the gradient of this road is 'one in four'. Think about this as 'rise over run' and therefore interpret this as 'we go up one unit for every 4 units we go across'. Wouldn't it be great if we could jump in the school minibus with our class and drive to a road with a 25% gradient to see how steep it is? Unfortunately this is probably impractical (though it could be set as a homework, depending on how hilly your local area is!).

There are some impressively steep roads in San Francisco but New Zealand boasts the 'world's steepest road' according to the Guiness Book of Records - Baldwin Street in Dunedin has a gradient of 35%. So how can we interpret this gradient? It means we go up 35 units for every 100 units we go across. We could write this as the ratio 35:100, simplified to 7:20. But we usually prefer to state gradient ratios in the form 1:n, so in this case the gradient is 1 in 2.86. Incidentally, 25,000 balls of chocolate are rolled down this 350m-long street in an annual charity Cadbury Jaffa Race.

Give your students a sense of 'steepness' by showing this short video of someone cycling up a road with 38% gradient. Looks like hard work! As a matter of interest, this rather technical article attempts to calculate the steepest gradient that one can cycle up.  This article says that anything over 16% is considered very challenging for cyclists of all abilities.

Ripley Street Ridge, San Francisco

Practical activities using trigonometry
In real life it is normally impossible to measure the rise and run of a slope so we use trigonometry to calculate these lengths. Picture a slope as a right-angled triangle - we can use a clinometer to measure the angle of elevation and perhaps a trundle wheel to measure the length of the hypotenuse. Trigonometry can then be used to calculate the rise and run in order to find the gradient. The table below shows gradients and their equivalent angles of elevation.  I think it's important that students understand the connection between a 100% gradient and a 45 degree angle.

Source: Wikipedia

Gradient can also be calculated using Pythagoras' Theorem. Say a man cycles 5km on a slope and knows (from his altimeter watch) that he has climbed 250m vertically.  Again, picture a right-angled triangle - the hypotenuse is 5000m and the height is 250m. Use Pythagoras to calculate the base (4994m).  Now use 'rise over run' to calculate the gradient - in this case we get 5%.

These real-life applications of mathematics can easily be made into practical class activities. For example you could use home-made clinometers and trigonometry to calculate the gradients of slopes in and around school, like staircases and ramps.

Resources for teaching gradient

• My worksheet on real-life gradients on TES
• A wide range of exercises in this pack. Note that this is a American resource so uses the terms slope and grade instead of gradient.
• A measure of slope - lesson and activities from Mathematics Assessment Project
• A worksheet of contextual gradient problems
• A gradient activity from Nrich
• Mixed gradient questions by linzbfc on TES
• PowerPoint on gradients by Andrea Frame on TES
• What is gradient? worksheet by MathsPad
• Alphabet slopes activity

Possible homework investigation questions
What is a funicular? Why can funiculars travel at steeper than 90 degrees?
What’s the maximum grade allowed for a railway without cable or cogs?
Why do you think road builders have to worry about the grade of the road?
Why do you think we have regulations for building wheelchair ramps?
At what steepness do avalanches occur?
Is it possible to ski at 90 degrees steepness?
(source)

I hope this post has inspired you to try out some new ideas when teaching gradient!  Let me know how you get on (@mathsjem).

Unusual gradient sign in Hertfordshire, UK

Maths Vocabulary

I found Cav's blog post 'Vocabulary and maths are not mutually exclusive' really interesting and it got me thinking about activities and teaching techniques related to maths vocabulary.

A teaching tool that can be used regularly throughout the year is a vocabulary knowledge survey. When you start teaching a topic, create a list of the new words that will be used and ask students to complete a chart showing their understanding of those words.  Repeat the activity later to see the improvement in understanding - I've not tried this myself yet but I think it would generate some useful questions as well as boosting pupil confidence.

In my post on Area and Volume I suggested these maths vocabulary exercises which make for nice 'end of term' lessons.  Here's an example of an activity from this pack:

Here's some more links that you might find useful:

• The Maths Magpie has created these great Maths Vocabulary Activities
• A list of key words for Key Stage 3
• Another vocabulary activity - perhaps an alternative to the knowledge survey described above
• Exercises with a Mathematics Dictionary 
• Background reading 'Building Mathematical Vocabulary' by Dr Madeline Kovarik

I was recently talking to a highly intelligent and articulate Year 13 pupil who said that she hated it when her teacher asked her to 'speak maths'. I asked her to elaborate. She said that when questioning the class, the teacher expected answers that used appropriate maths vocabulary - and even this pupil (who'll be off to Oxbridge in September) found this daunting.  We need to address this issue from a young age.  One tip is to always require pupils to answer questions in full sentences instead of just numbers or words - from primary school onwards. A very simple example - in response to the question 'what do we call a polygon with 8 sides?', don't accept the answer 'octagon' - ask for a full sentence such as 'An 8-sided polygon is called an octagon' to get pupils used to 'speaking maths' from an early age (source).

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Edit: 11/11/15

Since writing this post last year I've seen lots of great resources for developing vocabulary. I particularly like the Literacy Assessments featured in Gems 43. I also found the activity below worked well with my Year 7s.

Here's my guide to literacy in maths lessons:

Shortcuts vs Concept Development

I highly recommend this book Nix the Tricks - ‘a guide to avoiding shortcuts that cut out math concept development’. I felt uncomfortable reading it at times - my own teaching is full of these tricks and shortcuts because I’m strongly influenced by the time pressures and exam focus that the author references in her introduction. The book covers a great many teaching tricks and in each case makes excellent points about why these tricks should be avoided. Helpfully, alternative approaches are suggested. I’m not going to stop using crocodile references when introducing inequality symbols (I like the crocodiles!) and I'm not (yet) going to stop teaching gradient as 'rise over run', but I will certainly think twice about many of my methods now I’ve read this book.

My blog focuses on sharing teaching ideas and some of my posts include exactly the kind of shortcuts that this book is so opposed to - like sohcahtoa triangles, or this trick for remembering exact values of trigonometric ratios (which doesn’t replace the triangle method - but it's a nice trick and worth sharing).

It was my excitement at discovering this method for multiplying matrices that inspired me to write this blog. I was astounded that I’d never seen this method before and immediately showed it to my FP1 students, who were equally impressed. I mentioned this method to a colleague, prompting a conversation about other 'alternative methods' we knew of. She told me about this method for finding Highest Common Factors. Again, I was astonished that I’d been happily teaching the Venn method - which I’d learnt from my mentor during a PGCE placement - when a quicker and easier method existed. I felt like I'd unnecessarily overcomplicated things for my students.

We all know that quicker methods aren’t always better methods. ‘Maths hacks’ such as these and shortcuts (such as those discussed in Tina Cardone's book) do nothing to help students’ underlying understanding of mathematics. But I still believe there's value in teachers sharing tips and alternative methods because sometimes we discover hidden gems - like the wonderful matrices method - that change the way we teach a topic for the better.

By the way, if you enjoy the book then there's more (including new sections in draft) on nixthetricks.com.