An interesting property of linear sequences - inspired by the 1% club.

The 1% club is one of my favourite quiz shows. It is the only quiz show I have actually applied to be on (no success unfortunately) but I play along on the app all the time, and also regularly complete the daily question that comes through the app. Yesterday (27th January 2026) had a very interesting question (from a maths point of view) that sparked a little dive into linear sequences. I resisted posting it yesterday as I didn't want to provide spoilers for any readers that also play along.

So, the 1% club daily question yesterday was this: 

What two digit number replaces the question marks in this sequence of numbers:

92, 23, 53, 83, 14, 44, ??

What made this interesting was the way I achieved the correct answer was very different to the way the app explained how to arrive at the answer (if you want to try and answer before I reveal the solution then don't scroll down too far!)

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The correct answer was 74. The reasoning the app gave was that if you reverse the digits of the list you get the sequence 29, 32, 35, 38, 41, and 44 and so the next value would be 47 which, when reversed gives 74. Which makes perfect sense. But it isn't how I arrived at 74.

I (as I am sure many other readers also) noticed that a lot of the jumps were +30 and that those that weren't were -69. There also seemed to be a regularity to when these jumps appeared; a jump of -69 followed by two jumps of +30. Given the jump of -69 from 83 to 14, I reasoned there would be a jump of +30 (although I was wrong about the regularity of the pattern of jumps as the next would actually be another -69).

Of course, once I realised that these two approaches both gave the same answer, I absolutely had to try and decide whether this was a property of this particular set of numbers, or whether it would be true for the reverse digits of all linear sequences made of two digit numbers.

Rather than diving in with the algebra straight the way (that is coming, don't worry), I decided to play with a few more sequences first to create further examples and see if this sequence was obviously a unique case (a very good problem solving strategy in general I find to allow for pattern spotting).

So I tried 30, 34, 38, 42, 46, 50 becoming 03, 43, 83, 24, 64, 05 - which quickly disabused me that there was any regularity to when a sequence went up or down, and then I tried 17, 24, 31, 38, 45, 52 becoming 71, 42, 13, 83, 54, 25.

It was at this point that I realised that the value of the differences were always 99 apart in the reversed sequences, in the first 30 and 69, in the second 40 and 59, in the third 70 and 29. It took me an embarrassingly long time to recognise that the subtractions were happening when the original linear sequence bridged a 10, or that if the linear sequence was going up in 3 (say) that the reversed sequence should be going up in 30.

I started to explore the algebra at this point a little, but quickly realised that I was getting confounded by the fact that I had only tried differences in the original linear sequences that were less than 10, so I tried 26, 39, 52, 65, 78, 91 becoming 62, 93, 25, 56, 87, 19 (which showed me it wasn't so simple as subtractions occurring when the original sequence bridged a 10, but was more about the units digit becoming smaller - which should have been obvious really) and also 12, 35, 58, 81 becoming 21, 53, 85, 18. This confirmed that the sum to 99 was still a thing - or more precisely that the subtractions were the positive differences subtract 99.

At this point I dived properly into the algebra, which I did as follows (again, if you want to try it first then don't scroll down):

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(I added some text to show clearly what the algebra implied that I didn't write in my own scribblings).

In terms of this as a task for pupils, I think there would be something interesting in offering KS3 pupils a chance to explore 'reverse linear' sequences - probably at a distance from linear sequences themselves. I think it might reinforce some properties of linear sequences and it would be very interesting to see if they spot the 99 link and how they try and justify it.

I definitely think there would be something about using the proof with a GCSE/Further GCSE/A-Level class, either as an example of constructing a logical proof or as an exercise for them as part of their practise in creating a deductive proof.

Of course, the question remains about what happens with linear sequences that stray into 3 digit numbers (single digits are trivial as we can just treat them as two digit numbers with first digit 0). I have answered this question to my own satisfaction and so will leave it as an exercise for the interested reader with one hint, which comes from when I shared the initial problem with other maths teachers at Twinkl and one of them came up with a third approach to the original problem (which is equivalent to what I have outlined and also leads to the correct answer):

"Add 30 each time but if the answer goes over 100 add the 100s digit to the ones digit".

A mathematical curiosity?

 In writing my new book 'Practising Maths' I referenced a lovely result (you will have to buy it to see how) that sums such as 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1, etc. all produce square numbers.

If you haven't come across this result before then feel free to have a look at it for a minute (even try and prove it) - if you are familiar with consecutive triangular numbers summing to square numbers, it is closely related.

The curiosity I noticed was that I knew 121 was also square. So I became interested in the fact that 1 + 2 + 1 is square, and 121 is square. I decided to look into the others, and it turns out they are also square! Well, the ones up to 12345678987654321 are square anyway.

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This of course raised a question - is this a reflection of something deeper? You might like to spend some time exploring and coming to your own conclusion before you read on.

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I guess the truth is a little of both.

If we consider squaring polynomials of increasing order with unit coefficients we get the following:

These are, of course, the same expressions as above, but in base x rather than base 10. So, if we substitute x = 1 into the expressions we get the sums on the left of the above table. However, if we substitute x = 10 into the same expressions, we get the numbers on the right of the table.

In terms of a task, we could offer the first few rows of the table to pupils and ask them what they notice/wonder. They might explore when the pattern breaks and why. We might encourage them to write out the numbers using explicit base 10 notation, such as 1 × 100 + 2 × 10 + 1 and see what insights this brings out. Pupils with the necessary algebra skills might even explore the expansions given above. Or we might just show it to pupils as an example of a mathematical curiosity.

My morning routine and a nice maths prompt

I have a somewhat quirky morning routine. I think it started as a way to occupy my mind during my morning commute. I don’t drive a car and so for the last 11 years of my teaching career, all spent in the same school, I used to use a combination of roughly 40 minutes walking and an eight-minute train journey to get to school each day. I like to be up for quite some time before I have to be active – to ease myself into the morning – and so I used to get up at 6 am, leave the house around 7:15, and arrive at school shortly after 8 am. During that time before I left the house, and then during the journey to school, I run through a series of games on my phone with an almost religious regularity.

It starts with the daily Wordle, then the daily Quordle (both the Classic and the Sequence, as well as the Weekly when I solve my first Classic of the week). Then I move onto the maths games, starting with the Ooodle, the OoodleMax and the Time Square grid, followed by the Nerdle. Then it is onto geography, with the Worldle (and its different rounds), the Statele, and then the Globle. I finish off with the Daily Sudoku, and then have recently added the Countle at the end. Since leaving teaching I have continued to play these games as part of my morning routine before starting my work from home.

The reason I mention all of this is that my attempt at the OoodleMax today prompted me to consider a nice mathematical relationship, that would make an excellent prompt for learners that could be used at multiple levels.

This was today’s Ooodle Max:

The goal is to use the numbers from the keypad on the right-hand side, at most once each, to fill in the blanks and make the target. Like other Wordle type games, numbers go orange if they appear in the calculation, but are in the wrong place, green if they are in the correct place in the calculation, and grey if they are not used at all. This was my first guess:

Having used 12 and 15, and without really thinking about it, it occurred to me that to be close I should use 13 and 14 in the multiplication. I then knew the “2” would have to be the divisor of the division, so I then set about figuring about what the dividend would be. This was my second guess:

I was lucky in that I got the 13 and 14 the right way, but what struck me is the relationship between the value of 13 × 14 and 12 × 15. It reminded me of the result that I knew about, which is that the square of an integer is always one more than the product of its adjacent integers. This is, of course, easy to prove mathematically for learners that can expand binomials. Given three integers  then we have that . It got me thinking that there must be a wider pattern to this, which of course there is; given four numbers , the product  and  (so a difference of 2), and then given  we have  and  (a difference of 3) and so on. There is an obvious symmetry here, as we get further away from a central value the difference between the products increases by one and starting the list each time with  makes this relatively clear each time.

The algebraic exploration of this is clearly a nice activity for those that can access the algebra, particularly proving the overall general case. However, it also struck me that these sorts of related calculations, and the general structure underpinning the relationship, can be explored without the need for algebra, using concrete manipulatives:

These are representations of 2 × 3 and 1 × 4, and then 4 × 5 and 3 × 6 – following the same pattern as 13 × 14 and 12 × 15. The question then becomes how is the red array related to the yellow? It is a relatively simple matter to see that we can move counters from the bottom row of the yellow arrays and attach them to the side to create the red arrays, but we will always have two counters left over.

It should also be within the capabilities of many learners to reason why this must always be the case; the number of rows in the red array is two less than the number of columns in the (original) yellow array, and so when we move counters from the bottom row of the yellow array to the side to extend the size of each row, we will always leave two behind. This sort of argument can then be generalised further to any of the related calculations.

So, what would the task look like? I haven’t decided fully yet, but as an inquiry prompt it might be something like this:

Calculate the following:

        (a)   2 × 3 and 1 × 4   

        (b)   3 × 4 and 2 × 5

        (c)    4 × 5 and 3 × 6

        (d)   13 × 14 and 12 × 15

What do you notice about each pair of results?

Could you write down other pairs that would produce the same result?

Can you explain it?

Can you find other pairs of calculations that always follow a different rule that is like this rule?

Prime factorisation, indices and standard form - some great questions

As we all know by now (in England anyway) the new GCSE in maths is going to require pupils to make links between areas of maths, and challenge pupils to apply understanding in ways they might not have previously. In writing the new homework booklets for my department I have been challenging myself to ask questions in this vein, and have found a rich source in linking prime factorisations, indices and standard form. Now admittedly there are already related topics, however I think that I have developed some questions that challenge pupils understanding of these topics in ways that perhaps haven't been used as frequently before now. Here I am sharing a run-down of my top seemingly straightforward questions (in no particular order):

1) 108 = 2² × 3³. 108² = 11664. Find the prime factorisation of 11664.

2) 9216 = 2¹⁰ × 3². Find the value of √9216.

3) Calculate (3 x 10⁴)³, giving your answer in standard form.

4) Find the prime factorisation of 6 x 10⁴, giving your answer in index form.

5) Find the prime factorisation of 3.2 x 10⁷, giving your answer in index form.

6) Find √(1.6 x 10⁵), giving your answer in standard form.

7) Calculate √(1/25), giving your answer in standard form.

8) Calculate (1.25 x 10⁸)^(2/3), giving your answer in standard form.

These and more will be in my term 2 homework booklet for the pupils aiming at grades 7+ on the new GCSE, and I think are precisely the sort of skills that the new GCSE is aimed at ensuring pupils develop.

My new favourite vector resource - via Back to Back activities!

Walking around my department towards the middle of last week (which I try and do whenever I get the chance, which unfortunately is not as often as I would like) I spied a fantastic image that one of teachers was using as a part of a "no pens day"; having pupils sit back to back whilst one describes and the other draws this picture:

perhaps it was because I was due to teach it the following week, but my mind raced immediately to this picture, which I promptly designed at the end of last week

of course the topic being...Vectors!

I love vector mathematics - it is such a useful way of visualising so many key concepts in maths and science; I use them to conceptualise negatives, translations (I actually draw on the vector arrows), all sorts of things. What particularly struck me about this image is the way it ties vectors nicely with similar triangles and scale factors, For the top end pupils the discussion as to why B to E is 2a and why K to G is 2b and building up the whole picture from there, is a great discussion to come out of this picture, along with then all of the other vectors is as good a top end vector resource I have seen - eventually it is possible to generate this picture:

In terms of trying to define vectors in terms of other vectors, what a great activity! It won't stop there either - tomorrow I will be using the image to explore ideas like:

(a) Are the points KLJ on a straight line? What would the vector be? What about FDE?
(b) Do the points KHE divide the diagonal MC into 5 equal sections?
(c) If the line from N to D is extended so that it intersects the line segment between A and B, into what ratio does it divide the line segment AB?

Of course that is not the only way to use this image - over the half term I will likely create something that uses it for trigonometry, scale diagrams, Pythagoras' Theorem etc as well - all out of a simple image that an NQT was using for drawing.

My resources around this image can be here and here.

Team Challenge, inspired by UKMT

This week my new GCSE class will get their first taster of one of my favourite activities, the UKMT inspired team challenge. I find this sort of activity really does get pupils thinking and discussing (and sometimes even arguing) about the maths they are doing, so I thought I would take the time to share how it works.

The idea is inspired by the UKMT Team challenge round known as 'Shuttle' (formerly 'Mini-Relay' or 'Head to Head') in as far as it has 4 questions and pupils are scored 3 points if they get the question right first time, or 1 point if they get it right eventually. Typically I don't have each team split into pairs like the UKMT do, nor do I use the answer to the last question in the next one; instead the team are only given question 1 to start with, once they answer it they bring it to me at the front of the room. If they answer correctly they get their points as above and the next question, if not they get sent back with their previous question. Of course like any good challenge, the questions get harder as you go (at least IMO!)

The challenge I am doing on Friday with my new Year 10 set 5 (of 6, so higher tier pupils, but needing plenty of support) is about ratio and proportion, so following the link here will provide you with the four questions I am using (good example of more involved ratio questions on their own, even if you don't fancy using the team challenge) as well as a generic score recording sheet to use with any shuttle challenge (note you will need to create an account on TES if you don't have one in order to download).

Of course the beauty of this idea is that it allows you to get kids doing really hard questions (can't make them too easy or they will be finished in no time!) with a smile on their faces (most of the time anyway). Watching the way they try and convince each other is great fun; particularly if you have a member of support staff (like I do) there to act as the scorer so you can get involved with the pupils where necessary. Next time you are stuck for a lesson idea why not give it a go? It means you only have to write 4 questions instead of 10!

The Migrant crisis and Maths

I am sure many of us have been following the growing migrant crisis affecting, in particular, the Mediterranean region. The number of deaths during travel, coupled with the logistical problems of settling the migrants once they have left their country of origin are two of the biggest migration issues faced by Europe for a long time. A colleague of mine recently sent me a link to a BBC news article about the issue containing some truly thought-provoking statistics; and of course I couldn't read the article without that little maths teacher area of my brain firing with uses for the graphs and charts shown. Here are some of the statistical representations that the article used to report on the issue:

The mathematical possibilities here are quite striking. There are some great representations of proportion and percentage, probability, circles etc in addition to the obvious bar charts and pie charts problems that can be posed.

So taken was I that I immediately came home this evening and created 3 worksheets/questions that can used with these stimuli; one on bar charts, on on pie charts and one on proportion. They are just a flavour of the sort of questions that can be asked about these stimuli, but are useful enough for themselves. The worksheets can be found here and I would love to see other people develop questions from these or other representations around this very important issue.

Venn diagrams without probability

Venn diagrams...one of those topics sorely missed when removed from the curriculum, and a topic whose reintroduction to the GCSE was welcomed by many maths teachers. Of course those of us that have taught A-Level stats have been keeping ourselves nicely abreast of set notation and its applications to Venn diagrams; but mainly to do with application to probability. Something that the SAMs have made clear is that Venn diagrams will be used more widely than just with probability at GCSE, and so I thought I would share a couple of questions I have developed to support using Venn diagrams.

1: Prime factorisation

To be fair this is an approach that some people will already be using; the application of Venn diagrams to find HCF and LCM from the prime factorisations of two numbers. This question however uses it in reverse, gives the prime factors in the Venn diagram without telling the numbers

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2: Venn diagrams and expressions/equations

Something similar to this is employed in the SAMs, but to my recollection it again involved probability. To be fair this questions could, but I have left it slightly short of that (this question is to be used with Year 7 upper set). It still does bring in some nice expression writing and equation solving.

Both of these questions are in the Venn Diagrams Word document in editable form on my TES resource area, which is at the link here: https://www.tes.co.uk/teaching-resource/venn-diagrams-11082977 so feel free to use, adapt etc; I hope it serves as a source of inspiration for developing resources for these new topics and that tie different areas of the curriculum together.

10ticks Level 9/10 or the new GCSE?

A while since my last post I know - a well earned break following the end of the school year (which for Leicestershire was the 10th of July) and I have now re-turned my attention to making sure we are well resourced for next year. It was while doing this I came across a great little find...

I think every maths teacher has at some point used 10ticks. Rarely an entire sheet (although I have known some teachers use a whole sheet for homework or a follow-up lesson) but certainly stealing parts of it. When I first started my career the school I worked at only had access to Levels 3 up to 7/8, and like many new teachers (as I was at the time) I had trained in the era when Level 9 or 10 were no longer really talked about, even though the original National Curriculum did cast its gaze to those lofty heights. When I encountered the Level 9/10 worksheets in my second school, I had a cursory glance, and haven't had much cause to glance that way again, save for the odd top-end trig or volume resource. Imagine my delight then when going back through I read the contents list again for 10ticks Level 9/10.

Honestly I wouldn't be surprised if the people designing the new KS4 programme of study didn't have one eye on this document when they were writing the new content - so richly is it found here. A brief summary if I may:

Pack 1 - Accuracy of Measurement, Variation, Indices (negative and fractional), compound growth and decay, Surds and irrationals (including in trigonometry), approximating root 2 by iteration.

Pack 2 - Trigonometry, Tree diagrams, solving quadratics by factorising, quadratic sequences, completing the square and the quadratic formula.

Pack 3 - Ratio, Scale Factors and Similar triangles, Congruence, Distributions, Histograms.

Pack 4 - Graphs and Equations (including circle equations), Volume and Surface area of curved shapes, Density, Algebraic fractions, Transforming formulae, transforming graphs.

Pack 5 - Vectors, Proof, Distance/Velocity - Time graphs including curves, Perpendiculars.

Pack 6 - Set notation, Venn Diagrams (after lots of matrices stuff).

Now if I were looking for a list of things that fit somewhere in the 3 part Venn diagram of Content new in Foundation tier, Content new in Higher tier, Content crucial for further study beyond GCSE I think this would be quite a reasonable list, and I think many maths teachers would agree with me. I haven't had chance to look through all of the individual pages yet, but undoubtedly some gems await.

So if you are preparing for the new GCSE over the summer, or even over the course of teaching next year, don't neglect an old faithful resource like 10ticks; it might just surprise you.

'Pointless' bar charts

A couple of nights ago some may remember that I put up on the twitter chat #mathstlp that I was teaching bar charts to Year 9 bottom set and was in need of inspiration. I had a couple of contributions (thank you ladies - @missradders I used the challenge you sent me) and then Tuesday morning I had a brainwave - Pointless! I had intended to put a picture of a bar chart on the board and ask pupils questions about it, but then the brainwave I had was - why not just give them the picture and get them to write about it; and from there can they come up with that "pointless" bit of information that no one else can!

Instead of just putting the picture on the board I organised the kids into 11 groups and gave each group a copy of the bar chart stuck into the middle of the paper and told them to write as many bits of factual information from the chart as they could around the outside. After about 10 minutes they had to choose one of the bits of information that they thought was their "best shot" at a pointless answer. They were then given points in the true pointless style - however many groups had the bit of information scored them that many points, or for an incorrect answer the maximum of 10 points (11 teams = maximum of 10 points when counting from 0 to 10).

The kids really enjoyed the competitive element and trying to come up with obscure information, and obviously we got some interesting maths that I wouldn't have thought to ask [how about the bar is 7.5 cm long and 1 cm wide!] We got some great discussion and discord about whether people were right or wrong, and whether two pieces of information were the same. One group said the frequencies add up to 390; and meant the values on the frequency axis rather than the frequencies indicated by the bars; we didn't give that as it was ambiguous.

I can see this working for lots of things; I think putting a straight line graph on the paper and asking for facts here as well, or a two-way table, or any other way of presenting factual information. So if you are looking to get kids answering questions you would never think of asking, try a Pointless Page.

I can read your mind - now that is what I call an hypothesis test.

A few days ago we were talking on one of the twitter chats about hypothesis testing and I alluded to a fun introduction to the topic I had used before, which I thought I would take the time to flesh out in more detail.

It starts by explaining to pupils that you are going to test if any of them are psychic. Obviously it is nice to ham this up a little bit, give it a bit of dramatic flair etc. Explain to them that the test is that you are going to flip a coin 20 times and keep each result hidden from them. You are then going to concentrate very hard on the result and they have to write down the impression they get from you.

Do the experiment and then see how the kids get on; normally about the maximum you will get is 13 or 14 (any more and you may well actually have a psychic on your hands!) and so the conversation turns to, "well is this enough evidence? What is the probability of someone getting 13/14 by random guessing? How many would be good enough?" This gives me all the tools I need to form a formal hypothesis and discuss things like the significance level (at what point does the probability become so remote we have to agree this is not happening by chance - when it is less than 5%? 1% etc), confidence interval (how many must someone get right before we believe they can read minds) etc.

I find that having this early practical hook to keep coming back to really helps pupils as they navigate what can be quite a tricky topic simply because of the sheer number of different contexts to which it can be applied. So the next time you are teaching kids about hypothesis testing, try giving them a fun hypothesis they can see practically happening in front of them.

P.S. to develop it, ask about what would happen if you switched the coin for a die, and how that would change their views on how many needed to be sure etc...

Inspired by JustMaths, looking beyond the basic skill.

Recently I have been planning a lesson on the classic exam situation of completing a partially complete two-way table, such as this one:

Now in my experience pupils grasp the concept of this quite quickly, with most mistakes tending to come in making arithmetic errors rather than mis-understanding the problem. Which of course leads me to the problem of how to stretch the lesson for those pupils who do grasp the concept so quickly.

During an internet trawl for inspiration I came across an excellent resource from the brilliant team over at JustMaths, which has 5 tables to complete (enough to give enough practice at arithmetic) that are all linked together and then provides an interesting activity whereby pupils have to identify which teachers are making mistakes in analysing the resulting tables. It was then that I had one of those nice ideas which occasionally occur to me; here was an ideal moment to link in some prior learning!

Taking the JustMaths tables I then created some statements that go a little beyond their "True or false" on the back, into True/Maybe/False. I brought in these statements:

1) Every student studies English

2) Every student studies Maths

3) More students study Food than Biology

4) A greater proportion of students study Biology than Food

5) The ratio of Boys to Girls studying Art is 2:1

6) More than 80% of the students studying Applied Maths are boys

I won't spoil the surprise of the resource by telling you which are True, which could be true and which are false, but the answers are in the lesson here if you desperately have to know; needless to say there is at least one of each. 

Now you may not use the Just Maths resource, you may not use these statements, but if you are looking for a little stretch in your two-way tables lesson, try bringing in some other prior learning number statements, and trying setting some statements that are definitely true, could be true or definitely false.

I have never probability

Ok, so honestly I haven't done this one yet, but it occurred to me whilst doing writing down probability with Year 7 today. Admittedly it would be for relatively low prior attaining but talk about engaging! Count up how many kids in the room and then throw out some "I have never..." statements and get kids to stand up if they have never done something; then get the kids sitting down to put the associated probability on a mini-whiteboard or similar. An alternative might be to test some statements against probabilities sourced on the internet to see how close they are in your class? Here are some possible statements I came up with:

a) I have never flown in an aeroplane.
b) I have never been to a live concert.
c) I have never been to a live sporting event.
d) I have never broken a bone.
e) I have never been abroad.
f) I have never been camping.
g) I have never done the ironing.
h) I have never been on a train
i) I have never ridden a horse.
j) I have never been ice skating.

I am sure I could come up with more if I really thought about it. Think I have my starter for tomorrow!(Or if I can find some way to tie in mutually exclusive/exhaustive events I might use it during the lesson).

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.

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.

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.