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 and 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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