Tag Archives: Creative Maths

Christmaths!

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It’s that time of year again. Everything seems to be Christmas themed, and not being one to rock the boat, I think it’s time to investigate the maths behind Christmas!

Snowflakes are something which from a young age have fascinated me. Although this may have been mostly due to their unparalleled ability to form snowballs which I could throw at people. None the less, as I have stated in previous blog posts; maths is in everything.  In particular I am going to discuss one snowflake. Koch’s snowflake. This snowflake is a fractal (a fractal is a shape that contains similar patterns which recur throughout the shape on a smaller scale) which is demonstrated in this video here.

As can be seen in the video, no matter how far we zoom into this snowflake there is always more detail to be revealed. An area like this, of course, opens up a variety of fundamental areas of mathematics which could be explored in the primary classroom (especially during the winter months) for example scaling, shape and ratio.

Kochs Snowflake

However taken a step further this simple snowflake becomes very interesting. Koch’s snowflake, as can be seen in the previously linked video is made up of triangles each a third size of the triangle it is attached to. Now, we know it to be true that no matter how far we zoom in, there will always be more detail revealed and as such we can state that to create this shape there are infinitely many triangles. Therefore we can assert that this shape has an infinite perimeter (this is because by adding up an infinite amount of triangles along its length we arrive at infinity). However Koch’s snowflake has a finite area, which can be clearly demonstrated by simply drawing a circle around the snowflake (we know the circle to have a finite area so therefore anything that can fit within the circle must have a finite area). We have now shown that the shape of a snowflake has an infinite perimeter but a finite area. Very strange!

This of course is a very abstract example of mathematics of which, unless it is actually snowing outside, the children are unlikely to find relevant and I’m sure you’re probably thinking that nothing about this idea is actually apparent in the real world. However, it is not merely just snowflakes which contain the property of infinite perimeter and finite area. Many other examples of this exists and is certainly an area that could be explored in the classroom to both find shapes that fit these rules but also why they fit these rules, which, of course, will require further exploration of fundamental mathematical ideas such as shape, ratio, division, and multiplication.

To get you started (in case it isn’t snowing when you next begin to think about fractals), despite how difficult it is for my mind to grasp the concept, Scotland is a perfect example of a shape which has an infinite perimeter but finite area.

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I Need Some Space…

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It hurts quite a bit when you stub your toe. Try being hit by a whole galaxy! Because that’s exactly what’s going to happen. The galaxy Andromeda (our closest spiral galaxy) is speeding towards us a massive 260,000 miles an hour…

Before you start planning for doomsday you might want to consider galactic distances. It is absolutely true that Andromeda is speeding towards us at an incredibly high speed, however, due to the gargantuan distance between us it will be more than 4 billion years before it gets anywhere near us. So you can stop breathing into a paper bag now.

This beautifully demonstrates the vastness of the universe in which we live. Something which we consider massive (take the earth for instance of which the diameter is 1.2756 × 10^7 or 12756000 meters in diameter) is tiny compared to the size of the universe (9.2 × 10^26 meters or 10 billion kilometres in diameter). This also illustrates the importance of fundamental areas of mathematics such as base systems. For example when using very large or very small numbers we use standard notation (used above) instead of attaching many zeros to a number to demonstrate its size. An example that shows why we switch when using extreme numbers is; imagine trying to give directions to Edinburgh using only millimetres. This would be very confusing, so instead we switch to using kilometres.

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Pictured: Ngc 2207, Spiral Galaxy

However telling you that the universe is 10 billion kilometres wide does nothing really to explain just how massive it is. An area which I find fascinating is light years. This is the distance which light will have travelled within a year. As I am sure you are away, light travels unbelievably quickly (just think about how fast a room changes from light to dark when you switch a light on, it is near instantaneous). The actual distance light can travel in a year is 9.46 × 10^15 meters or 9.5 trillion miles. Now, the nearest star to our sun is Proxima Centauri, and I stress that this is the nearest to us. For light (the fastest moving thing we know of) to travel from Proxima Centauri to us would take 4 years (Yes, that does mean we see 4 years into the past when looking at Proxima Centauri). Now imagine how long it would take for light to travel across the universe! Or how long it would take for a human spacecraft to travel that far!

Before we were given an input on astronomy earlier this week, I knew very little of the content explored within this blog. This is something I think is a shame, as I’m sure this is an area I would have found hugely interesting as a child in primary, much more so than simply learning the names of the planets in our solar system and that they all go round the sun (though of course I concede this is vital knowledge to have).

As a future practitioner this is an area I would very much like to explore with my class. To learn about the solar system which we live in, but to also introduce the notion of fundamental mathematics within this subject area to further explore the world of astronomy and mathematics in an exciting and engaging way.

It’s maths, Jim, but not as we know it…

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When a string is plucked, it vibrates at a particular rate creating a note. A familiar concept to violinists, guitarists, cellists and…well anybody who plays a stringed instrument, and I’m sure anyone who has ever fired an elastic band across a room. The rate at which this string vibrates is known as the frequency (determined by string thickness, density, length and tension, f=(1/2L)√(T/μ)) which is measured in Hertz(Hz). As humans we perceive these frequencies within a given range as pitch. The range begins at about 20Hz and goes up to approximately 20,000Hz. For example concert A (the tuning pitch for orchestras) is 440Hz and middle C is 261.63Hz.

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It’s probably at this point you’re beginning to wonder “In what way could this information possibly be useful?”.

Well, as many of you will know some notes sound better together than others (if you’re not aware of this, have a look at this video, about 3 minutes in). This has to do with how the two note’s frequencies are related. Notes whose frequencies are related by simple fractions tend to sound better together. Let’s take for example the note of A which we know to be 440Hz. If we half the frequency to 220Hz we arrive at another note of A only an octave down. Similarly if we double the frequency to 880Hz this is also the note of A, this time an octave up.

Taking this to the next level we can apply more complex fractions. If we take 3/2 of 440Hz we arrive at 660Hz or E and if we take 5/4 we arrive at 550Hz or C#. These are the fractional relationships between a major chord and by playing A, C#, and E we are actually playing the chord of A major. It is because of this fractional relationship we find harmonics (demonstrated rather nicely in this piece of music by Daniel Padim) situated exactly halfway, one third along, and one fifth along the string length of a guitar.

So, after all that, why would I write a blog about this? Apart from the fact that this is a very exciting application of maths. I suppose this was my rather long-winded way of demonstrating the often forgotten, cross-curricular nature of mathematics, and that perhaps (despite a very finely categorised curriculum) we should embrace the concept of learning in a way that accommodates individual difference and that allows for the blending of curriculum areas of which (Savage, 2010) describes as ‘sending a positive message to the children’ and as ‘rich pedagody’.

It seems that wherever I look there is always some aspect of maths to be found. Perhaps it could be combined with more curricular areas than just music…

Base Jumping

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Imagine we had been born with 12 fingers. An odd thought I admit but allow me to explain. We work in a base 10 number system in mathematics which uses ten digits (0,1,2,3,4,5,6,7,8,9), but it wasn’t always this way and nor is it this way everywhere in the world. Binary, for example, is a base 2 number system and in the north of England a system known as ‘Yan tan’ exists for counting sheep which contains 20 individual digits and hence is a base 20 number system. We even switch to a different base when telling the time! (Base 60, 24, or 12).

So the question I am really asking is ‘Why base 10?’ Certainly it has been suggested that by switching to a base 12 number system containing 12 digits (0,1,2,3,4,5,6,7,8,9,Χ,ε) we could simplify the learning and teaching of maths in many areas such as fractions. For instance the fraction 1/3 in our current system is written 0.333333….(a recurring number). 1/3 written in base 12 would be written 0.4 and besides being far more aesthetically pleasing this decimal is much easier to use in calculations. It is because of this easier division of fractions we work in dozens when buying and selling food. The implications of this are that the conversion of fractions to decimal, a subject once feared by teacher and student alike, becomes not only easier to teach but easier to learn. The full extent of the advantages of a base 12 number system can be found here.

So if simpler number systems exist, again I ask ‘Why base 10?’ It comes down to 10 simple things. Our fingers. We have 10 of them. Our love of counting on our fingers has dictated the use of a second rate number system. I do concede that during the initial learning of mathematics being able to count on our hands is incredibly helpful. However in retaliation to this children could just as easily be taught to count on the joints of their fingers on each hand, which, luckily enough, there happen to be 12 of…

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So, it would seem that a shift in our number system is in order. But is it really that much better? I have already, paraphrasing the Dozenal Society PDF (linked earlier in this post), stated that fractions would become easier to convert to decimal values however this is not entirely true. 1/5 in our current number system coverts to 0.2. 1/5 in the dozenal system becomes 0.41666667. Not exactly easy to work with.

In conclusion, is the slightly easier fractional system (and only in some cases) really worth completly upending our entire concept of number for? …Well, in my opinion, no. The transition for children already versed in our current system would be confusing and messy and I suspect we won’t be switching any time soon.

 

I Hate Maths

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“I hate maths.”

I’ve heard it a hundred times before. In fact if you’re a parent or teacher or have anything to do with children, you’ve probably heard it too. But why? What is it about mathematics we find so dislikeable? To me, the answer is simple. We are not teaching maths.

Of course, there is a slot on our timetable blocked out for ‘Mathematics’ which we slog our way through at 9.15 each morning. Where our class learn the ins and outs of addition, multiplication, percentages and the like. But is that all there really is to mathematics?

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Frankly no. These areas are merely the foundations upon which the world of mathematics is built. The ‘grammar’ of the maths world, comparable to a score which when brought to life creates a beautiful piece of music, or a script waiting for the actors to transform into something awe inspiring, and although an essential part of mathematics there is so much more to maths than just this.

Unfortunately this ‘grammar’ of maths is all that children know and, as far as many are concerned, all that children need to know about mathematics. There is no room to allow for the creative side of maths. In fact, many are unaware that there even is a creative side to maths, and it’s no wonder!  Where is the creativity in learning our tables by rote? What innovation can possibly be found in mapping a small square on the left hand side of our page into a bigger square on the right hand side of our page?

Why not teach mathematics through its applications? Why not teach the exciting side of maths?

Could we not teach ratio by creating a scale drawing of a T-Rex on the playground in chalk? Why not teach multiplication by calculating the age of a tree (Diameter × Growth Factor for anyone interested)? Why don’t we introduce children to the Fibonacci sequence through spiral art and the composition of natural objects?

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“I hate maths.”

It was in fact this exact phrase which turned me on to teaching and is why I decided to take the mathematics elective. It is my goal to share my love and enthusiasm of mathematics to inspire those who I teach, so I never again have to hear the words “I hate maths.”