Pie Charts and Proportions

Just a quick post today - spent most of yesterday planning for teaching pie charts to Year 8 this week (set 4 of 5 in that half of the year group). After a fairly standard drawing pie charts and a fairly standard interpreting pie charts (i.e. basically being able to recreate the frequency table given a piece of information) we are going for a bit more understanding about the proportionality behind pie charts. The two resources I had on pie charts and algebra I am saving for GCSE, but I did design a nice activity as part of a RAG worksheet that I will use (it is the Amber activity, with the Red being some simple write down the proportions shown and the Green being a lovely former Edexcel exam question comparing two pie charts). The link to the whole sheet is here but here is the part I designed:

"These pie charts show the car colours in two different car parks.

Say whether these statements are true, false or whether you cannot be sure from the given information:

a) The number of blue cars in the first car park is more than in the second car park.

b) The proportion of blue cars in the first car park is more than in the second car park.

c) The number of black cars in the first car park is more than in the second car park.

d) The proportion of red cars is the same in both car parks.

e) The largest proportion of cars in either car park are of white cars.

f) The smallest proportion of cars in either car park are of yellow cars.

g) The smallest number of cars in either car park are of yellow cars.

h) The largest number of cars in both car parks combined are red cars."

What I really like here is that pupils have to focus on whether or not there is enough information to answer the question, which puts a nice twist on the way we ask pupils about data and about maths in general. I think I will adapt this for different some other topics; I can see it being powerful to reinforce the idea of unknowns in algebra and whether you can get a numerical answer or not.

Teaching the difference between sequences and series

One of the jobs I have been doing this half-term holiday (in England) is going through the OCR sample assessment material and using the questions to create teaching resources for lessons in the new GCSE course. I don't think we will be using the OCR board to examine our pupils, as their questions seem to be a bit over-engineered to test what they need to test, however this over-engineering does make their questions an ideal classroom resource where pupils can work together to discuss approaches to solving them in an environment supported by a teacher where necessary. One of the exam questions in the OCR SAMs centres around this image:

As a typical example of the over-engineering this question centred around the quadratic sequence created by the bottom row of cubes (as well as a part about plans and elevations) in each pattern, which will make a great practical resource that can be used with multi-link cubes or similar to allow pupils to explore quadratic sequences in class. However what also happened is it got me thinking about building towers like these and shifting focus from looking at how many new cubes, to how many cubes in total. What immediately struck me is that for arithmetic progressions (with positive integer terms) this was describing the difference between sequences and series. Now strictly this is not needed for the new GCSE qualification in England, but it might be the sort of thing you show pupils as an A-Level transition idea, and will definitely be taught to pupils doing A-Level Maths. Out of this I created this image (which was the best that my limited skills in paint could do!):

Now admittedly this is a fairly old fashioned sequence leading to the square numbers, but for me that made it all the more powerful as it comes with ideas that pupils will be familiar with already. Now I don't teach A-Level anymore (the interesting challenge of setting up GCSE maths in a school that had previously only had pupils from 11-14 was enough to drag me away from A-Level teaching) but I thought to myself, "What a great transition activity/introductory activity to have pupils build these towers and focus on the two questions 'How many more cubes are in the next tower' and 'How many cubes in total will there be in the next tower' as a way of illuminating the difference between a sequence (what changes) and the series (what total)." I can see an entire transition/introductory lesson with pupils building different towers and examining the sequence and series arising from each other's towers or alternatively I can see re-introducing sequences using these cubes at A-Level (I can see the towers also providing a good link to the recurrence relation as well) and then using the same towers to explore series in a later lesson.

Some other towers I came up with (and I am sure pupils will be a lot more inventive than I!):

Pie Charts and Algebra

My 2ic did a resource trawl recently looking for questions linking pie charts with algebra. Unfortunately his search came up rather bare, so he emailed me asking what i had. I realised I didn't have a huge amount either so I had a think and came up with a couple that were worth sharing. Both of these feature on my TES site (search Peter Mattock in the search box when it is set to resources) as part of the larger worksheets "Pie Charts and proportions".

The first question I came up with features around this pie chart:

A relatively nice activity that picks up on proportional ideas  as well as equation solving. But the one I like more is the one I then came up with:

I have never seen linear sequences linked to pie charts before, and I particularly like how this one includes some higher order thinking about finding lots of different linear sequences that would fit these angle values, and the relationships between these different linear sequences. I know I say it a lot but with the new GCSE in England asking a lot more of pupils in terms of problem solving and communicating their maths I can't help but feel that these sorts of questions are the ones that our pupils should be tackling on a much more regular basis.

Area and Algebra

Inspired by NCETM and Jan Parry, my Year 9 second set today explored some of the relationships between area and algebra. The starter gave them this image:

But rather than ask for the area of the shaded space, I asked for three different calculations that would give the area of the shaded space. Collectively the class managed to arrive at 13x15 - 8x5; 13x7 + 8x8; and 15x8 + 7x5. Once we understood this idea, we moved on to this image:

We talked about the similarities and differences between this shape and the starter image, and then worked through the same questions about finding different expressions for the area of the shape and then showing that these areas are all equivalent.

After a little independent work we then looked at a different link between area and algebra. The shot below explains:

We used the polydron (see earlier blog) to explore this shape (importantly here making the link between 2a + 2b and 2(a+b)) and then other shapes, building shapes that give expressions like 2(5a+3b). The pupils really enjoyed getting hands on with algebra and being able to physically build expressions.

The link here gives the whole lesson, it is definitely worth a look.

Polydron Framework - great for exploring shape and more...

One of my favourite resources over the last few years has been the Polydron frameworks Geometry set.

When I took my first HOD role one of the first things I did was buy a set for every classroom, and more recently in my current school I was able to purchase a couple of sets to assist with our below level 4 pupils shape work. I have used it recently to explore nets and properties of 3D shapes, but actually it has been a resource I have used in lots of places in the past. Below is a list of my top uses; I have tried to spread them over all 5 strands of Maths but obviously shape gets a little bias.

Number

1) Building bar models - the squares can be connected together as a physical resource to build bar models for use with fractions, decimals, percentages, ratio etc...

2) Number sense - A small equilateral triangle is worth 1, the right angled triangles are worth 2, isosceles 3, large equilateral triangles 4 etc... build a shape worth x. How many different combinations of triangles equal a hexagon? How many right-triangles is a pentagon worth etc...

Shape

3) Nets - Kind of obvious, but allows exploration of nets.

4) Isometric drawing - Can create some interesting shapes for the more able to try and draw on isometric paper.

5) Plans and Elevations - Allows real manipulation of built shapes to view their plan and elevations.

6) 3D Pythag and trig - allows a real interior and exterior 3D view to calculating lengths through shapes using pythagoras and trigonometry.

7) Angle measure - the protractor in the pack allows angle measure in the flat polygons or between faces of the built 3D shapes.

Data Handling and Probability

8) Probability - Put a load in a bag, what is the probability of removing a triangle? A blue shape? A blue triangle?

9) Combinations and Permutations - If we have a red, blue, yellow and green triangle how many different permutations of colours can we create?

Algebra

10) The area of a right triangle is a. The area of a square is b. How many ways of writing the area of two triangles and two squares are there? Other compound shapes?

11) Build a 4 x 4 grid of different colour squares. If each colour is worth a different amount, what totals can we make. If this row needs to have this total, what values could they be. How many row and column values are needed to fix the value of all 4 colours.

12) Sequences - Build different patterns of shapes, what is the nth term of different perimeters? Number of lines (treat a line where two shapes connect as a single line) etc.

There are lots of others which just don't come to mind right now. Please feel free to add in the comments if you can think of/have used others.

Simultaneous equations and shapes

A little while ago I blogged about an experience I had had with Year 7 top set on forming and solving equations, particularly with regards this question:

ABC is an isosceles triangle with AB = AC. Find the value of the angle marked y in the triangle.

Whilst thinking again today I suddenly had one of those little inspiration moments that set the brain a-buzz; what happens if I swap the y to be one of the isosceles triangles base angles? I quickly realised that this would lead to a much more involved simultaneous equation pair (technically the above question is also a pair of 'simultaneous' equations, but I don't really think anyone would treat it as such). This led me to design this question:

ABC is an isosceles triangle with AB = AC. Find the values of x and y.

I really like this question as it requires a nice combination of the angles properties of all triangles (i.e. that they total 180 degrees) with the specific angle properties of an isosceles triangle (i.e. the 'base' angles being equal). I feel like it would generate some really good discussion between/with pupils about where the information to solve the problem is going to come from and would help reinforce the need for two equations in x and y. There is also a nice conversation to come out of this about which approach to simultaneous equation solving; I feel that substitution is a more efficient approach in this case than elimination. With the new GCSE examination in England having a much greater focus on communicating and reasoning mathematically, I can see questions like this being much more prominent in the years to come. This question is definitely going into my simultaneous equations unit for higher, and I would love for others that have current year 10s and 11s (my school has Year 10 for the first time next year) to use it and give me any feedback from how their pupils work with it. I am now going to look at other shapes and design some more, and would love to receive some from others.

Just for anyone struggling, my solution to the problem is set out below:

From the fact the triangle is isosceles: y = 3x - 2    (call this equation 1)

From sum of angles in a triangle: 3x - 2 + y + 4x - 6 = 180 => 7x + y - 8 = 180 => 7x + y = 188   (call this equation 2).

Substituting from equation 1 into equation 2 gives: 7x + (3x - 2) = 188 => 10x - 2 = 188.

Solving for x: 10x = 190 => x = 19.

Solving for y: y = 3 x 19 - 2 = 57 - 2 = 55.

Ratio and Bar models - 3 key questions

My intern has recently been discovering bar modelling as she looks for ways to introduce percentages and ratio. It led me to reflect on one of my first uses of the bar model a good few years ago - in helping pupils see the differences between three types of ratio sharing questions.

Those with longer memories will remember that most ratio exam questions would usually be of the type where you had to share an amount over the whole ratio; and that many pupils came a cropper when they switched questions and gave the value of one part of the ratio. So many pupils had become fixated on the process of "add the parts of the ratio together, divide then multiply" which of course only worked for sharing over the whole ratio. Better pupils then started to solve both types of problems, but mainly by learning when to add and when not to. A few years ago a question previously unseen in recent memory crept in which gave the value of how much bigger one part was than another. Many pupils failed to solve this type of problem because they hadn't been prepped for it before; they didn't really understand how ratio worked they had just been taught to solve certain types of problem. This is what led me to the bar model for the first time, as a way of illustrating the difference in these different approaches to ratio. Allow me to illustrate with 3 questions

Set up: Alfie, Bort (Simpsons reference!) and Claire share some money in the ratio 2:3:5

I start with the line to set up all 3 questions and then bring in the bar model.

Pupils tend to be quite happy that this shows the ratio 2:3:5

From there I introduce question 1:

1) If they share £150 in total how much does each person get?

This leads us to the idea that the whole rectangle being worth £150 and so dividing the £150 into 10 parts. The bar model and solution ends up looking like this...

Now to introduce question 2:

2) If Bort get £150, how much do the other two people get?

This leads us to the idea that only the orange section is worth £150, and so this time we divide the £150 into 3 parts. The bar model and solution ends up looking like this...

Finally to introduce question 3:

3) If Claire gets £150 more than Bort, how much does each get?

Comparing the orange to the pink section we see that the pink section has two more parts, so these two parts have to make £150. That means we are now dividing the £150 into 2 parts. The bar model and solution ends up looking like this...

Pupils commented on how much clearer their understanding of how to solve ratio problems was and in the follow up questions were able to apply themselves much more to different types of ratio problems.These days I only ever introduce sharing with a ratio like this, and mix the questions as early as possible, Given that the bar model can then be used to link to fractions, decimals, percentages (points to those who can translate all of these problems into percentage problems) I think it is well worth exploring this approach with your pupils.

Top tips for new department leaders

Inspired by the topic of today's #mathschat I started thinking about the top tips I would share with new department leaders (or that I do share when I support other department leaders) and I realised I would never fit them all in tweets, so I thought I would commit them to my blog and share that instead. So here in no particular order are my top tips for new department leaders:

1) Go for the quick staff win - People talk about the quick win in terms of data or school targets etc but for me the quick win I always looked for was something that showed staff I was working for them. Whether it was finding a way to reduce their workload, trying to tackle the behaviour of a particular pupil or class that is causing problems, or simply putting a simple protocol in place that makes staff feel more secure about how to handle a particular admin or department task, getting a positive impact for staff early on is a great way to start off your 'reign' on the right foot.

2) Do your research - there are huge volumes of research and writing about successful leadership. Detailed treatises on coaching and mentoring staff, and when particular staff may need which approach. The successful habits of 1000s of successful people have been analysed, dissected and then disseminated out. You won't be able to read all of it, but a lot of them say the same thing anyway; but do take the time to read some of it.

3) Set your boundaries - Your work load will increase, you know that anyway or you wouldn't have applied to be a department lead, but you do need time to break away from it. I never work on Friday's once I get home (unless I know I will be unable to work at the weekend and I have pressing things to do). I commute to work via train which only goes every hour on the half hour, so on Thursday (which tends to be my least busy day in terms of after school commitments) I ensure I get the half 4 train instead of the half 5. I do make sure to take some time on Saturday and Sunday as well, although this will sometimes be around planning or department admin commitments. The important thing is I have set a workload which I know is sustainable for me and allow me to stay effective.

4) Visit your staff - I don't just mean popping into lessons. Don't get me wrong popping into lessons is absolutely essential; it ensures your profile with pupils you don't teach is high (which can be invaluable if you have to work with them later in either a positive or negative context). Provided you use it to share good practice 'learning walking' can be a really powerful way to empower your staff; I always try to get something out of it that I ask a member of the department to email round or share. But I also mean at the end of a day, just spend 4 or 5 minutes in conversation with each or most of your staff members. I literally just pop in at the end of the day, ask about their day and listen for a few minutes - I try and get round them all but usually just hit most of them. If that means I start my after school work half an hour after everyone else so be it, but this human element is an invaluable time for me to connect with my department.

5) Bring it back to teaching and learning - You are taking on responsibilities that mean you spend less time in the classroom. Some of the people you are working with may want to do that to in time, some may already have done it and then stopped, some may never want to take that road. The one thing you all should (hopefully) share is a passion for teaching and learning. So when you do get time together as a team, try and spend as much time as possible with this as your focus. It can be tempted to get involved in data analysis, or setting or bogged down with admin that the school throws at you, but much of this can be done through email or at other times; if it is absolutely necessary set time limits so that you still have a good proportion of time for sharing practice or other T&L activities.

6) Manage the contradictions - Being a middle leader can sometimes feel like a contradiction. You have to support your team in getting the best outcomes, but you have to hold them to account for the outcomes as well. You have to ensure you help the school meets targets and at the same time try and take as much of the pressure of targets off of your team so they can focus on teaching and learning. You have to provide what your staff need to do their jobs, but you have to ensure you balance the budget. Recognise that these are not contradictions, but they are your power. You can help find the barriers to the outcomes and help solve them. You can manage the pressure that your staff feel so that it ensures they are properly focused without becoming overwhelming. You can work with your teams to ensure the best materials are provided and that all staff in your team use them effectively.

7) Develop and maintain your professional network - Not a week goes by when I don't have a professional conversation with someone outside my school. There will always be times when you need to run ideas or discuss thorny issues with a peer that is not invested in your institution. Having a strong professional network, be it through attending organised meeting, working with people online or just having friends in other schools you can meet in the pub (and preferably all three) can be crucial in developing your ideas.

8) Get staff working together - A lot of people talk about vision and communicating it. One of the most cringeworthy moments of my early career is getting my first team together and going through the powerpoint of my vision for the department. By all means have a vision, and by all means share it, but make it something that people want to get behind!

9) Don't be scared to delegate - A nice way to do this is to allow them to solve problems they come to you with. I had a member of staff who wanted to use tablets in his classroom more, but we only had 10 ipads and he wanted to broker a deal with the IT support department to get another department to buy our ipads at a reduced rate, and use the money to buy 20 cheaper nexus tablets. Rather than respond with "good idea, I will get right on that", my response was "great idea, can I get you to talk to IT and organise that?". It doesn't have to be delegating roles with labels (literacy rep, ICT rep, G&T rep et al) sometimes it is just about not taking a job from someone when they bring it forward.

10) Always acknowledge and reply - A lot of the communication these days is not face to face, emails in particular dominate most of it. I never leave a communication unanswered, even if it is just a response to acknowledge and thank them for the work they have done. Remembering to say thanks for all work done is important, and being consistent with it is equally important (no different to the kids in that aspect).

Mean average - don't "add them all up and divide by how many there are"!

Recently I have been teaching averages to Year 7, and in the second lesson we looked at the mean average. Of course as soon as I asked the class what the mean was, I got the stock response "Add them all up and divide by how many there are". "Why?" was my response; stunned silence the result. As suspected, not one of them knew why they were adding up, or why they were dividing.

I literally got them all to stand up, reach into their heads, and throw the idea out of the window. I then replaced it with a new idea; to me a better idea

                                  Mean = Total of data points shared equally amongst each point..

I much prefer this definition for a number of reasons. Firstly it tells you why you are "adding up" (although I have still banned the term in my classroom); you are adding to find the total. Secondly it tends to stop the problem of pupils dividing within the sum line of their calculator - a classic mistake in the calculation. My pupils have now gotten into the habit of finding and writing down the total and then sharing it out. Thirdly, and for me most importantly, it can be applied to a much wider range of problems then the usual definition.We looked at measuring totals with tape measures or scales, solving problems in finding missing values given the mean, and particularly mean from a frequency table. The use of the language "total" and "shared" meant pupils were much more open to the idea of multiplication to find totals in each row of a frequency table and were better able to see why we weren't dividing by the number of rows in the table, as they are not data points.

I know it is a relatively short post, but it highlights an important point; I will end with a plea - don't be satisfied in teaching pupils how to calculate the mean by "adding them all up and dividing by how many there are"; instead teach them what the mean is doing sharing a total to create equal valued points.

TES iboard - actual whiteboard interactivity

One of the things I regularly find when working in schools is the lack of interactivity in the interactive whiteboard. For many teachers, they could work with a computer and a projector and not be much worse off (they might have to click a mouse instead of touch the board to move the powerpoint on a bit, but that's about it). Some of the software that the board manufacturers provide allow for a reasonable amount of interactivity in certain lessons, but not all. A resource looking to expand that range that is developing along nicely is the tes iboard (go to www.iboard.co.uk).

So far I have only used it for doing some transformations, but I have been impressed. The translation tool allowed me to explain well the use of vectors - the tool doesn't use vectors itself, but it does allow pupils to move the shape around and therefore to see the results of the translation. Showing pupils the result as a vector wasn't difficult from there. I have also used the reflections and rotations tools, which again are limited for KS3 because they are primarily designed for KS2, but the reflections tool allows for some real interactivity along the common lines of reflection (horizontal and vertical lines, and the 45 degree diagonals) and the rotation tool allows for real exploration of 90 degree (and technically 45 degree) rotations around the origin. The thing I really like is pupils can operate it at the board, and we can explore simple questions that lower attaining pupils might come up with.

The tes iboard is a resource that is well worth keeping an eye on as it develops - particularly for those teachers who want to use their interactive whiteboard as more than just a fancy display projector.