I come across a lot of puzzles and other maths things online and often save them for later, this evening I came across this little puzzle:
The numbers 2,3,12,14,15,20 and 21 may be divided into two sets so that the product of the numbers in each set is equal. What is that product?
I had saved this over a year ago, and cannot remember where I got it from, but I can see why. It’s a lovely little question that I intend to use as a starter next week and see how my classes get on.
How I approached it
Before you read on have a go at it yourself. Go on, you know you want to……..
Right, good, now you can see if I went about it the same way!
My first thought was that all the fun could be taken out of this by using a calculator, typing all the numbers in and pressing square root. So when I set it I will be adding the line “and which numbers are in each set”, and this is what I set out to find.
Firstly I set out the numbers in terms of their prime factors:
Then I tallied up the prime factors:
From this I knew that the product must be 2x2x2x3x3x5x7 which is 2520.
This, of course, answers the original question but I wanted each set. I looked at the numbers and the first think I noticed was tgat 14 and 21 had to be in separate sets, as they had 7 as a factor. I also needed to split 15 and 20, my intuition suggested that 20 and 21 should be in separate boxes, but it was easy to spot that the 2s and 3s, wouldn’t work out so I placed them together and fit the rest on around them.
A nice little puzzle, I wonder how my classes will find it.
It’s election time again, and we all know what that means….. BAD GRAPHS!!!! Last year, in the run up to the locals, we had some classics and this time is no different.
Let’s start with a positive, an example for others to look at:
Amazing isn’t, despite the fact that he’s running for parliament, Jamie Hanley (@jamiehanley) – Labour candidate Pudsey, has managed to buck the trend and include a fully correct bar chart!! Dave Gale (@reflectivemaths) did point out that technically the claim “Can’t win here” is invalid, and should in fact read “statistically extremely unlikely to win here” or something similar. But that doesn’t take away from the excellent bar chart.
Now for some crimes:
These two came through my door, both from Greg Mulholland (@gregmulholland1) the Liberal Democrat defending the constituency. The first error, I’m sure you’ll notice, is that the numbers are different on both leaflets despite them claiming to be from the same set of election results! The second error is that the tiny gap between the Lib Dem and the Labour bars is supposed to signify around 700 people (on one, 800 on the other!) and that massive gap between the Labour and Conservative bars is supposed to be 1200 on the top and 1600 on the bottom. So the biggest it should be is twice the gap between Lib Dems and Labour, as you can see it is considerably more than this. The third error is that the aforementioned gaps between the Labour and the Conservative bars is supposed to signify between one third and one fifth of the total size of the Conservative bar, but as you can see it is actually considerably bigger on both bars. The final error is the one Dave mentioned above, the claim “Can’t win here” is technically wrong. If one of my pupils turned this in I’d be fuming!
Here’s what they should look like:
This classic was sent to me the other day and is from a Liberal Democrat leaflet in Bristol West, where Stephen Williams is the candidate and is trying for reelection. This is a terrible example. Firstly, the gap between Lib Dems and Labour is, as the numbers state, 10% and the gap between Labour and conservative is 20%, so clearly the gap in the bars should reflect this and the Labour bar should be considerably nearer the Conservative bar. However, this is clearly not the case in this picture. Next, the gap between the Labour and Conservative bar should be less than half the size of the Conservative bar, it isn’t! And again, Dave will be screaming about that all too familiar slogan. Another terrible effort. Here’s what it should look like:
This was sent to me by a friend, it’s from literature relating to the local elections that are currently being held and is from the Lib Dem candidate (Martin Hughes) in the Horsforth Ward of Leeds City Council. Error 1, as you can see, numbers wise the difference between Lib Dem votes and Conservative votes is 157 and the difference between Conservative votes and Labour votes is 216, these numbers are fairly similar, but the gap on the bar chart suggests the Tory Labour difference to be roughly 4 times that of the Lib Dem Tory difference. And 216 is very definitely not 4 times 157. Error 2: the 605 vote difference between Labour and UKIP is shown by a gap that is much bigger than the entire UKIP bar which represents 1059, a considerable amount of votes more! And again, there’s that slogan. Another terrible bar chart, and here’s the correct version:
Greg again, it’s no wonder that Colin Beveridge (@icecolbeveridge) has started referring to this sort of graph as a “Mulholland”, every leaflet that comes has another crime against statistics contained in its midst, perhaps Greg should read Colin’s book? This one is particularly telling, firstly we have a question of validity, the data is ten years old. There has been a general election since then so why use 2005 data? It fits the narrative, the leaflet is aimed at trying to convince Conservative voters to vote tactically and the information from the last election would tell them that actually the Conservative candidate came second. As you can see above though, the landscape has changed and the aggregated results (whichever version is true!) in the last local do suggest a similar landscape to 2005. Why hasn’t he used those again? I can only conjecture that it is because he wanted to use percentages somehow as a general as turnout is much higher it seems closer? Secondly we have the fact that there is a 4% difference between Labour and Lib Dems and a 6% difference between Labour and Conservative, this should show as a similar difference in bar heights (the difference Labour to Conservative should be 1.5 times the Labour Lib Dem difference), as you can see the difference is nothing of the sort and 6% appears to be ten times the size of 4%. Then there’s the Tory bar, apparently representing 27% yet smaller in size than the aforementioned 6% gap. And to top it off, that slogan again! Here it is, but correct:
Here we have a superb example of a terrible misleading bar chart. It’s from Liberal Democrat literature in Lewisham East (where the candidate is Julia Fletcher). As you can see from the graphic, the Labour bar is largest, followed by the Lib Dems who are fairly close then the tories who are much further behind. But hang on, look at those percentages! The Labour bar represents 43%, the Lib Dem bar 28% and the Conservative bar 23%, the Lib Dem bar should be much nearer the Conservative bar than the Labour bar! And both Lib Dem and Tory bars should be around half the size of the Labour one. And again we see that slogan. Another shocking effort, and here’s another correct version:
As I’ve been writing this post I’ve just received this link to this crime against statistics!
And I’m sure you’ll agree it’s immense. This one is from a Lib Dem leaflet in Wantage, where Alex Meredith is the candidate. They have tried to get by on a technicality, using a broken blue bar to show that the Tory share is broken, but that negates entirely the need to include the bars at all, they are there to be a visual representation of the proportion of votes, but using a broken bar makes a mockery of this as the ammended heights suggest a close race, when actually the blue bar should be almost double the size of the yellow. Even with this technicality the bar is still wrong, as the rest bar should be half the size of the yellow bar but infact it’s less than one third. And yet again, Dave’s favourite slogan has reared it’s head. A terrible effort, here’s how it should look:
While writing this post I asked people to send me any bar chart crimes that had come through their doors on election material, I particularly asked for non Lib Dem ones, yet the masses that came in all seemed to be theirs. I am beginning to worry that no one in one of the parties that has held government office for the last 5 years can grasp a basic bar chart. I’d be seriously worried if year sevens were making these mistakes. What’s worse is that there are tons of free websites that will do it for you, I used meta-chart for these. Or perhaps it’s just their policy to use misleading bar charts on their flyers. I am always on the lookout for crimes against statistics, so please do send me any you find, be they election related or not.
This post was first published here on Labour Teachers on 19th April 2015
Monday is the last day you can register to vote, which means I have spent a large portion of the week checking in with my year 13 class and my year 13 form group that they have registered. This has led to various discussions around politics, how they can decide who to vote for etc. It seems a lot of young people take on the political views of their parents. I was pleased to hear a number of them had actually downloaded the manifestos of the major parties and we’re using them to help them decide. I was shocked, however, at how many of them knew very little about any of it.
One asked if she could vote for Barack Obama. One asked if we were choosing a new queen. One thought Ed Miliband was the prime minister at the moment. There were many others like this and they echoed conversations I’ve had in previous years too. When one young lady asked who I thought our next president would be another year 13 overheard and joined the conversation. He was very knowledgeable about the political spectrum and the democratic process and it was quite refreshing to hear him explain the ins and outs of it rather than having to explain it myself.
As the conversation drew he said: “the thing is sir, there are far too many people who know nothing about it. It’s arguably the most important part of life. Why isn’t more taught about it in schools?”
I don’t have an answer. I don’t know much about citizenship, but I thought that this was something that fell under that banner. The current provision clearly isn’t helping engage young people with the process and it clearly isn’t helping to enhance their understanding of politics. I am a firm believer that we need to be giving our learners the knowledge they need to make these decisions, and that means we need to increase this aspect of their education.
Maybe the citizenship curriculum is too wide and/or the time allocated is too small meaning teachers cannot give each topic the time it deserves? Maybe it’s not seen as important by schools because it doesn’t fall into the same category of core subjects as others? Perhaps if it were made a requirement that all students study it to GCSE and it were given double weighting on progress 8 similarly to English and maths then it would be seen as more important by schools? Perhaps it needs a revised curriculum and a rebranding? Perhaps it suits the political classes to keep the majority uneducated on politics and it’s importance? One thing is for sure, there are far too many people leaving school without a sound knowledge of these matters.
Wherever I see incorrect maths, it annoys me, whether it be in election material, newspapers or anywhere else. But the place where it annoys me most is the maths classroom. I don’t mean students getting the answers wrong, that’s an invaluable part of the learning experience. What I mean is when teachers get it wrong. This happens more than people would expect. I’ve written before about people teaching things wrong (ie rounding or the order of operations) but today is want to discuss a different annoyance.
Take a look at this:
It’s from a resource I downloaded from the TES website. The resource itself was pretty good, but this was one of a number of questions that infuriated me. Have you noticed why? Take another look.
Yes, indeed. The right angled triangle that forms the cross section of this triangular prism is that we’ll known Pythagorean Triple the “4,9,10” triangle. Never heard of it? Neither have I! That’s because 4^2=16, 9^2=81 and 10^2=100. And 16+81 is very definitely 97, which in turn is very definitely NOT 100. It’s not even as though it’s hard to generate triples!
This sort of thing is lazy, if it had been put in front of me, as a student I’d have called a teacher out on it. The first time I saw something like this was during a micro teaching assignment while on my PGCE. The person in that case was rusty! I’ve seen it a couple of times with trainees or NQTs during observations, again these can be excused.
I even realise that experienced teachers can make innocent mistakes, but please, please, please check these things. Especially for triangular prisms, as this is THE area that I see this happening again, and again and again.
Have you encountered something like this? Do you get as angry as me about it? Do you think it doesn’t matter and I’m being overly pedantic on this? Please let me know.
The other day I my timehop showed me this lovely little post from last year. It includes “Heron’s Formula” for calculating the area of a triangle, as I read it I remembered thinking it was a little strange that not many people had heard of it before.
Today I was looking through a number of textbooks trying to find a decent set of questions on area, perimeter and volume for my year nines as I wanted to consolidate their learning at the start then move onto surface area. I’m not a fan of textbook misuse- ie “copy the example and try the questions” but I do sometimes use them for exercises as we have a very limited printing budget and some of them have superb exercises. For a fuller picture on.my view of textbooks, read this.
I was looking in one of my favourite textbooks:
And I happened across this:
There it is! Plain as day! Heron’s Formula! In a KS3 textbook!
I was disappointed that its function was described and its name wasn’t and there was no mention of why this worked. It basically reduces the question down from a geometry one to a purely algebraic substitution task and I would question the appropriateness of including it in an exercise on area, but still, I was incredibly exciting to find it there!
Are you a fan of Heron’s Formula? Had you even heard of it? Do you have a favourite textbook? I’d love to hear your views.
I’m sure you all have had a look at the new GCSEs by now and have either started teaching it or have at least started thinking about the boards and looked at what’s on offer. One of the things that pearson have produced is a baseline test aimed at year 9 to decide whether to teach them higher or foundation, today I was looking through it and a couple of questions jumped out:
This is a lovely question, although there’s one thing that annoys me about it. Can you guess????? Yes, that’s it! It’s that stupid picture of a stupid calculator. We need our students to be confident working in terms of pi, if this was a non-calculator question it would be brilliant:
We could even equate areas instead and thus include fractions:
In reality, as it is, it’s plugging numbers into a calculator and rounding, giving a non exact answer, a grrrr moment and a missed opportunity!
The other question that jumps out is this:
A lovely opportunity to play with numbers and fractions, a non-calculator question too. I love it:
All that confusion and a brilliant simple answer.
Have you any favourite questions from these baseline tests? Any favourite resources for the new GCSEs? I’d love to hear them.
Manifesto’s are due to be launched imminently, I thought about last time round and wondered how much of the conservatives manifesto actually came to fruition. I dug out the 2010 manifesto to take a look. The section on schools starts with some bold claims:
They were going to “Improve standards for all”, “Close the attainment gap”, “Enhance the prestige and quality of the teaching profession”, “Give heads and teachers tough new powers of discipline,” “restore rigour to the curriculum and exam system,” and “give every parent access to a good school”.
Did they manage it?
“Improve standards for all”
This is a noble aim, and one I hope all politicians have at their core. I certainly think the tories were trying to improve standards. This is fairly unmeasurable though. They cut money for school buildings, which means some schools are housed in less than brilliant accommodation, but some schools were improved. Let’s hope we all, in the education sector, have been, and continue to, improving standards fore all.
“Close the attainment gap”
In 2010 the reported attainment gap was 27.5% in 2014 it was 26.6%, this is certainly a drop. A drop which one would assume owes at least some thanks to the Lib Dems pupil premium payment.
It is great that the gap has dropped, although it is still far too high. And the measure itself is crude, as it’s based on FSM which is self nominated and misses a large amount of the most deprived in our communities.
“Enhance the prestige and quality of the teaching profession”
This is a great aim, the highest performing school systems have prestigious status for teachers. It is unclear, however, why Michael Gove thought an all out war with the teaching unions in which he repeatedly demonised teachers was the best way to do this. The constant teacher bashing and dismissing opposing views to his own was certainly not a way to achieve an enhanced status for the profession, quite the opposite in fact.
Teach first (love it or hate it) has certainly brought in more top end graduates, and as one of the largest graduate recruiters it would seem that at least with final year students some prestige has increased. Although that’s certainly not something the government can take credit for as it’s a charity and existed before they came to power. Their manifesto included the idea to expand it to “Teach Now” and “Troops to teachers” neither of which I’ve heard much about since?
“Give heads and teachers tough new powers of discipline”
This one was mainly lip service I think, and hasn’t really impacted anything. Legally schools can now keep pupils for detention for up to two hours without prior notice, but I don’t know any schools that have moved away from a policy of informing parents. There was also the “reasonable force” measure, that was pretty much exactly the same as the law it replaced.
“Restore rigour to the curriculum and exam system”
This one remains to be seen. I’m certainly in favour of the new maths curriculum, all the way from KS1-5, it certainly has more rigour, although I’d personally have liked even more (ie calculus on the GCSE, you can read more of my views here). I’ve not really looked in depth at the other subjects, I don’t have enough knowledge to discuss them – I’d love to hear your views if you have any in the new curricula in your subject.
The exam reforms are another matter, we won’t know whether they are an improvement until they start, they have a lot of potential but it certainly seems to be a little rushed. The new maths GCSE is due to be examined in 2017, the course contains more content and as such it will take more than two years to cover, yet we have no sample assessment materials as yet to base a decision on which exam board to choose. That said I do think it will be an improvement, I just wish it had been thought out and introduced better and more quickly.
“Give every parent access to a good school”
Like the others, a very noble aim, and one which we should be applauded. However, the decision to allow Free Schools to be built where people fancied, rather than building schools where there were a shortage of places, has led to some areas still having a shortage of school places and some having a surplus. This means that some parents struggle to access any school, never mind a good one. The investment in new schools had been evident, but it’s been too often in the wrong place.
So, they have closed the gap, and they have increased rigour in the curriculum. They haven’t improved the prestige of the profession, they’ve failed to build schools in the correct places and they’ve not increased power to discipline. Some hits, some misses. They did, of course, have other education policies over their tenure (I discussed some here) but these are the ones mentioned in their 2010 manifesto.