Always, Sometimes, Never True for Directed number

These only cover adding and subtracting negative number, i.e. they can be used before going onto multiplying and dividing negative number.

They can be printed (here is the pdf), cut out and then stuck on A3 paper under the three headings with examples and counter-examples to explain why they have been put under that heading.

I should point out that two statements on here are deliberately vague, i.e. “two negatives make a positive” and “a positive and negative make a negative”.  This is often how students remember them and this can lead to problems down the line (e.g. the misconception that -3-2=5).  My idea with these is that the end up in the “Sometimes” column but ultimately we dismiss them as not being very useful.

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The paper napkin trigonometry trick with a smattering of Pythagorean triples. 

Take a piece of paper and do the following:

  1. Make it into a square (interesting discussion on best way to do this).
  2. Fold in half then unfold so you have created crease along a vertical line of symmetry
  3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
  4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??

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Factorials!

Factorials are pretty awesome.  They introduce a cool new notation – I mean who knew that you would be using exclamation marks in maths! They are easy enough to understand at the basic level. They generate some mind-bogglingly large numbers really quickly, such is the power of repeated multiplication.

I like this as a way to start students off using their calculators with factorials.

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The answer to the last one is, of course, 69! Be careful with that…  It’s the largest number your calculator can calculate, why?

Factorials are used for various things, but probably the best place to start is combinations and permutations.  I love this clip from QI and have shamelessly tried to be Stephen Fry in my classroom and tell my students how I am going to do something that no human has ever done before.  I then shuffle the deck of cards…

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Once you’ve finished the dramatic slamming of the cards on the table you can then start to discuss how we can find the number of possible ways those cards could be arranged, i.e. 52!

Then start calculating (using Standard Form) how many shuffles all the people who have ever lived on Earth, (around 100 billion) could have made if they made one every minute for their entire lives.  Lots a crazy assumptions to get a very big number but still several orders of magnitude smaller than 52!  Meaning that the chances are infinitesimally small that someone has shuffled that pack and got the exact same order before.

 

Or, just have fun with them as a thing. These questions can all be reasoned without a calculator and could be a good way to start before doing Permutations and Combinations.

 

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How to make a good teacher

I’ve been an Economist subscriber for years, so when I saw this week’s front cover on my doormat this evening, I thought, that looks interesting.

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But before reading it, I decided to list my own views on the subject.  So, top-of-mind, here they are:

1, Subject knowledge

This is not about knowing your subject to a particularly high level, it’s about instinctive understanding of all the topics and sub-topics and how they relate to each other and what your typical Year 7 finds hard vs. your typical sixth-former.  I think this applies to primary equally. I don’t believe graduates with a first are necessarily better teachers than those with a 2:2 (so why on Earth we would incentivise recruitment to teacher training on this basis is beyond me).  You need solid subject knowledge and you also need to have a thirst for continually developing that.  I’ve been teaching Maths for 5 years and I still find new approaches to topics nearly every day. I hope that never stops.

2, A supportive environment with time to develop

Teachers need to work in an environment (i.e. staff + management) that believes that every teacher can always get better. Teachers need to buy into this and genuinely feel that they are not being judged on the quality of their teaching. Only then will they embrace a culture of continuous professional development (i.e. CPD).  There is, and always will be, a very small percentage of teachers who are in the wrong job and there needs to be a way of easing them out. But the place to start is with an assumption that all teachers care about their students and are motivated to be a good teacher. They have a finite amount of time and energy to expend on achieving that so they need to be given time in the school day and timetable to develop professionally. Structures around mentoring, co-planning and peer observation need to be in place. But fundamentally there needs to be less contact time to ever achieve this, something that we are nowhere near in the UK currently.

3, School systems that support teachers in classroom management

I’m going to make a bold statement with limited hard evidence: Too much time in too many classrooms is spent on behaviour and discipline. Learning of students in those classrooms suffers as a result. At least in Maths teaching in the UK that is my view.  A key part of teaching is getting children who would prefer not be there to engage in learning. Most children have some motivation to learn and it is the teachers’ job to leverage this as best they can.  However, there will always be certain learners in certain classes on certain days who, for whatever reason, are simply not ready to learn. For the sake of the teacher and the other learners there has to be alternative provision when this happens.  There have to be mechanisms to get those children out of that classroom quickly before the impact is more widely felt.

What would your list look like? I’d love to know.  In meantime, I’m now going to read that article…

A broken clock

I walked into my classroom this morning and noticed my clock was broken. Not just stopped but really broken, can you see why?

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It reminded me of a nice problem solving task which is sort of to do with angles but actually much more to do with ratio and proportion.

I wrote the following on the board:

For a normal clock, what is the angle between the hour hand and the minute hand at the following times:
1) 15:00
2) 13:30
3) 10:15
4) 17:45
5) 9:26

There is a significant range of challenge in these questions.  15:00 – straightforward, right.  As soon as you start moving the minute hand away from 12, you need to consider the fraction of (360/12) degrees that the hour hand moves.  12.30 might be the best option for a question 2 if you really want to scaffold it.  You also might want to squeeze a few more in between Qu 4 and 5.  e.g. 14.40, 15.20.

Next time I do it, I won’t write them up all in one go, but will keep adding to them as I can see learners making progress. Or ask students to challenge themselves by creating their times which might work better in a mixed attainment classroom.

These can all be done without a calculator. It demonstrates how useful it is to have 360 degrees in a circle and 60 minutes in an hour because they have so many factors.

A nice build on this is this question from an OCR Booklet of problem-solving questions.

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Proving Pythagoras

Although I’m sure I’ve taught Pythagoras lots of times, I have never really looked at the proofs before either for my own subject knowledge or with students. This may be because I was always happy when students had the understanding of how to apply the theorem and were able to find the missing side and so I left it at that.

Looking at proofs is a good way to deepen understanding of a topic, but generally shouldn’t be attempted the first time the topic is introduced, one of the points made in this comprehensive review of literature on how students approach proof in mathematics written by Danny Brown.

There are something like 140 different proofs of Pythagoras, cut-the-knot.org lists 118 geometric proofs here.

I decided to work through three:

Proof 1

On squared paper, students draw two adjoining squares of side length a and b as follows:

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Next they draw diagonal lines.  The first thing that needs proving is that these two lines are perpendicular which can be done by finding the gradient of each of them.

Screen Shot 2016-06-07 at 21.30.34.pngWe are now starting to get closer to a square of side c. A bit of cutting and rearranging and hopefully they establish that the area that they started with, a²+b² can be re-arranged to form c².

Here is a lovely Geogebra showing how these squares could tessellate for form Pythagorean Tiles.

Proof 2

This one is worth drawing although the scissors won’t help much here.  This is a Geogebra drawing of it (click on it to adjust the lengths):

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A few of my students went down a blind alley with this one assuming that a is double b.  That is why it is useful to have the dynamic drawing to show that this is not the case. The crux to this one is seeing that the red square in the middle has side length (a-b) and then multiplying out (a-b)² to get the area of that square.

Proof 3

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The third one I chose is fairly simple if you can remember the formula for the area of a trapezium! And really, once you’ve played this video over 10 times, nobody will ever forget that!

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Fun with Fibonacci

This is an old one but fun, and a good way to use algebra to show why a trick works.  It’s a similar to showing how Magic Squares work.  It’s not a formal proof as such, but I think it’s a good way to introduce the topic.

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Once students have grasped the basic concept of a Fibonacci Series (something which, in my experience they often see at Primary School even if they can’t remember what it is called), then you are ready to start the trick.

Fibonacci series don’t have to start with a 1 and a 1 as in the diagram above.  You start by asking students which two numbers they want to start with.

Then they get ready to be wowed with your powers of mental arithmetic. Tell them that you will be able to add up the first 10 digits of this sequence in your head faster than they can on calculators. Get one student up to the board to write down the numbers one by one.  TOP TIP here: make sure you have the numbers 1-10 written in a vertical column and that the chosen student writes down each term in the sequence against the numbers.  You should end up with something like this on your board:

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As soon as term 7 goes up on the board, you start calculating.  You should be able to find the sum of the first 10 terms before they even get to term 10 and this is why:

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I quite like doing the calculation on a miniwhiteboard, then writing the answer face down on a students’ desk and then walking to the other side of the room.  Once they have finally totalled the column of numbers on their calculator, you ask a student to have a look under the whiteboard.

And like all good magicians, you DO then go on to reveal the secrets of your trick!

Visualising volume of cuboid

The idea of these tasks is to get students to think more deeply about volume and to help visualise how different volumes “fit” together.

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Questions 1-5 can be done by effectively counting cubes. Multilink cubes would be good to help with the visualisation at this point.

Question 6 is the one to get the discussion going.  Is the answer 16 (if you picture them as solid cubes, rounding down) or 21.12 if you consider it as a flexible volume (e.g. a liquid)?

If you want to adapt this resource you can create your cuboids using this Geogebra resource.

Why do we send our kids to school?

So, first up, this post isn’t very mathsy.  I’ve spent the last two days at Hay Festival where I have been thinking about things other than Maths.  This is a very healthy thing to do once in a while!

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It was my first time at Hay and I loved it. It lasts 2 weeks and offers an incredibly broad spectrum of discourse.  I went to 8 thought-provoking events over two days, but the one that had most relevance was titled “THE FUTURE OF THE PROFESSIONS: HOW TECHNOLOGY WILL TRANSFORM THE WORK OF HUMAN EXPERTS” by the father and son team of Richard and Daniel Susskind to promote their book.

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I was slightly skeptical before the talk.  I have heard more than one teacher quote the line “50% of jobs that today’s school children will end up doing don’t exist yet” (or 65% in this WEF report) like it is some incontrovertible fact implying that children are wasting their time going to school.  Another one is “our present education system was designed for the industrial age”. Again not very helpful and not really true. The 11+ used to be the point at which it was determined whether you would earn your living from a profession or a trade for the rest of your life. Fortunately our KS2 SATs determine very little in terms of a child’s eventual outcomes, which begs the question as to why we put them through it, but that’s a subject for another post…

I’m glad to say that neither of these clichés were trotted out.  The message was more nuanced. We are at the early stages of a significant long-term shift in working practices. A significant number of the activities carried out by today’s “professions” actually don’t require much human input and with increased computing power and AI they will, over the next 10-20 years, be automated. This significantly changes the attributes required to be successful in these roles. One of my favourite quotes from Bill Gates was used “We always overestimate the change that will occur in the next two years and underestimate the change that will occur in the next ten.” Following the recent victory of Google/DeepMind’s AlphaGo computer against the world (human) champion Go player there seems to be some sense that we are reaching a tipping point in this technology (here and here). Indeed it may not impact our lives significantly in the next two years, but in ten?  I don’t know about you, but I hope I’ll still be involved in teaching children Maths in ten years’ time.

On the one hand I am interested in the implications of this for our profession. I have no doubt that significant chunks of teachers’ time are spent doing things that could be done without the professional expertise and experience that teachers acquire during their years in the classroom. Marking is clearly the main area.  It would be great to see some improvements in handwriting recognition systems over the next few years.  In Maths this will require an even greater level of sophistication due to the “2D” nature of how maths questions are answered on a page and the use of various symbols and diagrams compared to the linear “1D” nature of a body of handwritten text.  Once the handwritten work has been digitized it will then need to be interpreted in an intelligent way to be able to evaluate and ultimately score the students’ work. This part might be easier in Maths than in essay subjects, but it will still require a level of AI and adaptive learning that probably doesn’t quite exist today.  Then, of course, if we actually want to provide some sort of formative feedback to our students rather than just a summative score of their efforts that is another bunch of very smart algorithms. I’d love to read more about work in this area.

But also, it is a valid question to ask, what does this mean for the current curriculum in school? What are we currently teaching children that is going to be completely redundant and what vital skills are we not building?  An audience member asked a similar question to this.  Messrs. Susskind’s response for what our curriculum should be focussing on was three things: 1, Creativity; 2, Interpersonal skills and 3, Empathy.  Personally I would add 4, Problem-solving and 5, Tenacity to the list.  However the problem with any such list is that it is an awkward amalgam of skills and personality traits.  As teachers, can we really be expected to teach empathy, for example? Maybe schools as institutions should be providing our children with the environments and experiences to develop these traits but it is fiendishly difficult to explicitly teach and when you start considering how you assess these things you run into a moral minefield.

Which leads me to thinking, how does Art do it? They have lots of experience in assessing creativity, for example.  Students attain GCSEs and A Levels in a subject which is all about creativity and this is assessed in an objective, criteria-based way.  We could all learn something from our colleagues in the Art department about this.

There was an interesting discussion and disagreement between the two authors about the role of coding in the school curriculum. Richard Susskind’s view was that coding was simply another item in the list of things that today we see as a skill but would soon become another process which would no longer need to be carried out by a human. We will be able to tell computers what we wanted them to do without any need for specialist knowledge of coding. Daniel’s view was that in his experience coding was a worthwhile exercise in its own right. The discipline and structure required by writing the code helps you develop a deeper understanding of the problem you are trying to tackle. I would tend to agree with Daniel. It really comes down to a philosophical argument about why we teach children anything, really and the difference between education and training.  I’d like to think we have greater aspirations from our schools than simply to equip students with skills to perform a specific job.

The reality of the British educational system and I would posit the reality of most education systems around the world today is that we ultimately always teach to the test at least from age 13-14 upwards.  Our society demands that our children earn qualifications as the one tangible thing that they take from their schooling into their adult lives. This is what defines their achievement in their education and what positions them in the hierarchy to come. This is what motivates students because they know this is what employers want to see and, for most children, employment is seen as the route to financial success and happiness.  To sustain this as a fair and credible system, our exams must focus on things that can be assessed with a suitable degree of efficiency and objectivity.  We need clear assessment criteria. These are made much clearer when we are assessing a correct answer or a correct fact or the use of certain key words. It gets much more problematic and subjective when we start trying to assess personality traits wrapped up within “Interpersonal Skills” for example.  So, much as we might bemoan how our education system does not equip our children with skills for the future, the current direction of travel towards more “rigour”, i.e. assessing students in an exam hall with a pen and a piece of paper will inevitably focus our teaching on the narrow sub-set of skills that can be tested in this way.  Combined with the EBacc and the implied devaluing of creative subjects such as Art makes any attempt to equip our students with these “future skills” seem like a tough thing for any school to attempt.