Monthly Archives: November 2017

Maths Games and Puzzles

Who doesn’t love a board game?

Board games are something which friends and families across the world, join together to play. However, what many people do not realise is, that board games and puzzles are a fun and exciting way of practicing mathematics at home or in a fun, school environment. Most board games and puzzles in the current market, involve some sort of mathematics, whether it be as simple as the chance of rolling a 6 on a dice, or counting the squares as you go along in monopoly, mathematics is involved.

Today in Discovering Mathematics, each group looked into, and played, different board games such as Monopoly, Cluedo, Battleships, and my group played 5 Second Rule. In 5 Second Rule, the players are given a 5 second timer and a topic, such as; “Name 3 things you find in the sky” and the player has to answer the question within the 5 seconds. This game involves the elements of time and categorizing, which are fundamental to the understanding of mathematics.

Furthermore, mathematics is also involved in the game of Cludedo, which is a murder mystery board game. In order to find out who committed the murder and in which room they did it with which weapon, the players use a process of elimination, another fundamental mathematical concept. Another group also played Battleships, which involves co-ordinates, rows and columns, and positioning, which are all mathematical skills.

However, the game of Monopoly, which is one of the most popular board games around the world, involves many fundamental mathematical concepts and skills. Some of these include; the chance when rolling the die, the role of the banker, who must able to add, subtract, multiple, divide etc. in order to give out the correct amount of money, counting spaces as you go along, working with money and working out which properties are the best to invest in or trade.

It is interesting to see the connections which mathematics has within board games, especially since they are something which many people enjoy, I know that I do. Board games are therefore, an interesting and exciting way to practice maths with children, whether this be in the classroom or at home with parents, family and friends. This also allows for parents to be involved with their child’s learning and mathematic growth and development, as it has been shown that parents play an important role in their child’s mathematic education. Moreover, board games are something which I am going to use in my professional future as a teacher, as I have seen first-hand in placement, how much children enjoy playing them, and the mathematical development that they help children to make.

Mathematics and Motorcyclists

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Today in Discovering Mathematics, we considered the mathematics behind motorcycles and racing. Motorcycles and motorcycle racing, is not something which I have ever had a particular interest in. However, it was interesting to see the mathematics behind them and understand how different mathematic concepts affect the way they drive, the speed which they drive at, and their turns in and on the road, which also influences the likes of cars and bicycles. Motorcyclists, as well as cyclists and drivers, are constantly making mathematical decisions in their head, and I have decided to consider this further.

First of all, we started looking at roads, and the different paths that you can take and how some paths taken on the same road at the same speed, can make you finish quicker than others. For example, the drawing attached represents a windy road and three different paths which you can take.

This allowed us to look at different routes and see what difference these made on distance, time and speed. For example, if you are taking the middle route you are cutting the corners and have more chance of finishing first.

Furthermore, we then went on to look at how the equations for speed, distance and time effect motorcyclists. These calculations can be worked out by;

– Distance = Speed X Time

– Time = Distance/Speed

– Speed = Distance/Time (BBC Bitesize, 2017).

This can be shown easily by the following triangles;

(Image taken from http://www.bbc.co.uk/bitesize/standard/maths_i/numbers/dst/revision/1/)

This is used particularly in racing, where motorcyclists can work out which route is best to take. For example, whether it is better to take a longer route to improve their current speed, or a shorter route, to improve their overall time. This also links to the racing line which is the line which most new racers decide to take to go around corners, because it allows them to keep up their speed whilst going around the corner straight. This video which I have attached explains the basics of the different racing lines which are used by competitors during races;

Furthermore, motorcyclists and normal cyclists, are constantly using mathematics and working out solutions in their heads to come over obstacles such as the weather, surfaces and other traffic or road users. For example, if there are pot holes on the road, the cyclist must work out how they are going to overcome this. At first this may not seem like maths, but because they are constantly making decisions, and working out solutions to problems, they are constantly problem solving which is a crucial skill involved with mathematics.

References

Bbc.co.uk. (2017). BBC – Standard Grade Bitesize Maths I – Distance, speed and time : Revision. [online] Available at: http://www.bbc.co.uk/bitesize/standard/maths_i/numbers/dst/revision/1/ [Accessed 16 Nov. 2017].

Mathematics in Sport

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When I was younger, I tried my hand at near enough every sport which was on offer. I did swimming, netball, hockey, lacrosse, basketball, enjoy-a-ball, golf and ballroom and line dancing. I started most of these when I was in primary school, however, I did carry a few of them through to high school. Unfortunately, as I got older I became much busier with school and part time work, and I eventually ended up dropping every sport I ever had interest in.

During high school, I also took the subject of PE through to Higher. I suppose this allowed me to keep in touch with at least some sort of sport. I really enjoyed the subject, and loved looking at different sports and analysing and evaluating them in great detail. However, as much as we studied tactics and techniques etc. within various sports, we never really considered the mathematics behind sports.

Without mathematics, no sport would exist.

Within every sport, there is multiple mathematical concepts which allow athletes to compete and be successful within their chosen sport. Whether it is through discussing statistics or talking tactics, deciding where players are going to be positioned on the pitch, or through the way in which the game is scored, there is mathematics involved. Behind every shot, tackle, sprint, kick, hit or throw etc., there has been a mathematical idea which has allowed the athlete decide why they are carrying out the skill the way they are. Rachel Riley explains the involvement maths has within sports in the attached video:

Maths in Golf

One sport which has always been big in my family, and which I have played since I was 7, is golf. I have not played the sport competitively since I was 15, however, I am still very much interested in the sport and continue to watch and follow it.

Alike every other sport, mathematics has strong connections with golf. Every aspect of golf can be understood by using mathematics. The Institute of Mathematics and its Applications (2017) explains that although people spend thousands of pounds on the very best clubs and equipment, if they do not understand the physics and mathematics behind the game, then they will never achieve the all desired ‘hole-in-one’. Therefore, I have decided to research the mathematics behind golf, and how it affects everything from the swing of the club to the flight of the ball.

Dr. Steve Otto, the Director of Research and Testing at the R&A, which along with United States Golf Association (USGA) oversee the Rules of Golf, highlighted some of the ways in which mathematics is used in golf and explained that:

“We use applied mathematics on a daily basis, together with physics and engineering. The use of these tools help us to ensure that our analysis is thorough and rigorous” (Zyga, 2017).

One example of mathematics used in golf is the speed of the swing performed by the player. The simplest model of a golfer’s swing is the double pendulum which is two rods joined together with a mass at the bottom, with one rod being the golfer’s arm and the other being the club (Maths Careers, 2017). The arms then rotate around a point between the shoulders and the club rotates around the joints of the wrists. However, by using the Lagrangian equation of

L = T – V, (with T standing for kinetic energy and V for potential energy)

you can analyse the angles within a golfer’s swing; the angle of the arms in relation to the body and the angle of the golf club in relation to the arms. This allows you to understand how changing these angles will alter the swing and its speed (Maths Career, 2017).

This seems very complex; however, this picture allows you to see in visual format how the golfer’s swing works:

Furthermore, another form of mathematics involved with golf is in relation to the flight of the ball. The dimples on a golf ball allow the ball to fly in the air higher and for longer through the subject of the Magnus effect (Real World Physics Problems, 2017). The Magnus effect is the force which allows the air on one side of the ball to move slower than the other. This causes a pressure imbalance which then allows the ball to lift off the ground (Maths Careers, 2017).

There are many other aspects of golf which mathematics are involved in. However, this video produced by the USGA gives a brief overview of how mathematics is connected to golf and interviews professional golfers as they explain their understanding of the connection of mathematics in their sport:

References

Maths Careers. (2017). One, two, three…fore! – Maths Careers. [online] Available at: http://www.mathscareers.org.uk/article/one-two-three-fore/ [Accessed 8 Nov. 2017].

Real World Physics Problems. (2017). Physics Of Golf. [online] Available at: https://www.real-world-physics-problems.com/physics-of-golf.html [Accessed 8 Nov. 2017].

Zyga, L. (2017). The mathematics of golf. [online] Phys.org. Available at: https://phys.org/news/2017-08-mathematics-golf.html [Accessed 8 Nov. 2017].

Music and Mathematics – Where is the link?

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Music was never a subject which I took particular interest in during school. This was probably because I felt that you had to be able to play an instrument to a high standard to be successful in the subject, a skill which I had not yet mastered. My musical talents stretch as far as being in the glockenspiel club during primary school, playing the cello for a few months in P7 or when my Gran brings out the box of musical instruments on Boxing Day and the whole family must play one and singalong. I think this is because I was involved in so many different sports clubs and extra-curricular activities when I was younger, and was so busy with work, school etc. when in High School, that I knew something would have sacrifice and the result of that was music.

However, music is something which I am interested in developing in the future. I listen to music every single day and appreciate various genres of music. I think I do have a good ear for music, beat and rhythm, but that could just me being bias and assuring myself that I do have some form of musicality in me, somewhere. One instrument which I have always wanted to play however, is the piano. The piano is just such an elegant, classy and beautiful sounding instrument, and I would love to be able to learn it in the future when I have the time.

Where is the link?

It wasn’t until I took the Discovering Mathematics module that I realised that mathematics really does have such a strong connection with so many things in the wider society. Therefore, it never really crossed my mind that music and mathematics would have as strong as a connection as they do. Marcus de Sautoy (2011) explains the link between maths and music by saying;

“Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflect the world of symmetry and geometry. As Stravinsky once said: ‘the musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface’”.

This quote shows just how fundamental mathematics is to multiple elements involved within music. Other maths and music connections include;

Note values/rhythms
Beats in a bar
Tuning/pitch
Chords
Counting songs
Fingering on music
Time signature
Figured bass
Scales
Musical intervals
Fibonacci sequence.

The Fibonacci sequence is something which we have looked at in previous inputs and which I have noticed to be involved with various elements of mathematics. Elaine J. Home (2013) describes the Fibonacci sequence as “a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth”.

In terms of the Fibonacci sequence linking to music, there are 13 notes in an octave and a scale is composed of 8 notes. The 5^th and 3^rd notes (linking to Fibonacci’s sequence of; 0, 1, 1, 2, 3, 5, 8, 13 etc.) of the scale from the basic ‘root’ chord and are based on the whole tone which is 2 steps from the ‘root’ tone, that is the 1^st note of the scale. Furthermore, the piano keyboard has a scale of C to C and has 13 keys, 8 of which are white and 5 of which are black, these are split into groups of 3 and 2. At first I found this extremely confusing since I have never studied music or considered the mathematics behind it. Once we had a shot at playing the notes on the scale for ourselves however, I managed to get the hang of it and understand it. I find it much easier to learn things when I can use a visual aid, or see them in action. Therefore, by having the chance to play instruments for ourselves after we were explained the theoretical and mathematical concepts, this really helped me to understand the mathematics behind certain aspects of music. I think I enjoyed it so much also because a lot of it was practical and based on playing the piano, so I was really intrigued and interested in the topic.

I feel that the input we had on the link between music and mathematics has really inspired me to consider music further and hopefully at some point learn to play an instrument and apply the knowledge which I have learnt in this input, to playing a musical instrument. Furthermore, it has once again opened my eyes to the connections which maths has with so many different areas in the world around us. If it wasn’t for the Discovering Maths module I would have never realised that so many of these connections exist, and I feel that it has really improved my confidence in teaching mathematics, a goal which I hoped to achieve by choosing this module.

References

Du Sautoy, M. (2011). ‘Listen by numbers: music and maths’ Guardian. Available via http://theclassicalsuite.com/2011/06/listen-by-numbers-music-and-maths-via-guardian/ (Accessed: 07 November 2017)

Hom, Elaine J. (2013). ‘What is the Fibonacci Sequence?’ LiveScience. Available via https://www.livescience.com/37470-fibonacci-sequence.html (Accessed: 07 November 2017)

Yan, Tan, Tethera

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Number systems and place value was not something which I had thought about or realised to be so complex until now. When I was in school and learning how to count or about place value and systems, I understood it straight away and that was it. I never questioned the mathematics behind it or considered it further. It was never explained to me why we use the number system which we do or why there is units, tens and hundreds, but it was just that we do and that was it, case closed. Therefore, when I realised that this is different around the world, and that there was specific reasoning behind it, it intrigued me and I wanted to look into it further.

Why a base 10 number system?

A base 10 number system is the number system which we use in the UK every day. The base 10 number system is also known as the decimal system and has 10 digits to show all numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which uses place value and a decimal point to separate. The placing and positioning of the numbers in the system are based on powers of ten which is referring to tens, hundreds, thousands etc. Once you exceed the number 9, you then move onto the next highest position (Russell, 2017).

Before this input, I just presumed that everyone around the world used the same base 10 number system as us and that they would just translate it to fit their spoken language. To me it just makes sense, we have 10 fingers to help us count so we have a 10-base number system? No?

It appears however, that not all societies use a base 10 system. The Arara tribe in the Amazon for example use a two-base system:

anane (one)
adak (two)
adak anane (two, one)
adak adak (two, two)
adak adak anane (two, two, one)
adak adak adak (two, two, two) etc.

This confuses me as I do not understand how you are supposed to count by using the same two numbers which are simply repeated over and over again. How does that work when you are counting numbers as big as 100 or 1000? Then again, I suppose that if you are brought up using this, then it would seem normal, just like the base 10 number system seems normal to us. The Arara tribe may not need to count to excessive numbers and therefore, the base 2 system works best for them.

Another alien number system to me is the base 12 number system which is known as the dozenal base system. At first it felt familiar as they also use the numbers 0-9. However, the number 10 looks similar to an upside down 2 and is called ‘dek’, 11 looks similar to a reversed 3 and is called ‘el’ and then 12 looks identical to our number 10 and is called ‘do’ (Dvorsky, 2013). At first I was sceptical about a base 12 number system as a base 10 system seems so much easier to use. When I approached it with an open mind however, I realised that a base 12 number system makes sense for a multitude of reasons. Items such as eggs are measured in dozens, which is equal to 12, 12 has more factors than 10 and the clock dial is also numbered 1-12.

I have attached a video by Numberphile explaining the way the dozinal base system works and it’s benefits. One point which I found extremely interesting in the video however, was that some cultures still use their hands and fingers to count, a point which I raised earlier. Unlike counting their 10 fingers like we do, they count each individual segment on each of their fingers and their pinky which seems more confusing at first, but also makes sense.

One number system which I found confusing, but which I also found the most interesting is the number system which was most commonly used by farmers in North England to count sheep. This was a base 20 system, meaning that the farmers would count to 20 and then pick up a stone to represent the 20, and start again. I have attached a photo which shows the numbers which the farmers would use in Lincolnshire, Yorkshire, Derbyshire, County Durham and Lancashire. I found this interesting as it is a completely different way of counting, not using numbers as such but rather words. It has been argued that the number system used today in countries such as Spain and France, has been influenced by this system. This is because the word for 10 in North England is mostly either ‘Dix’ or ‘Dick’ which is like 10 in French which is ‘Dix’ and Spanish which is ‘Diez’.

After considering various different number systems, we then had the chance to create our own. However, we had to ensure that the numbers and symbols made sense when it comes to using decimal places and doing mathematical sums such as addition, multiplication, subtraction and division. My group decided to use the theme of flowers for our number system. I have attached a photo of my number sequence.

I feel that this activity really helped me put what we had been looking at in the lecture into context. I need to see things visually to help me fully understand it and this gave me the opportunity to do so. It also helped me to wrap my head around different number systems, a concept which I found something extremely confusing and challenging to begin with.

References

Bellos, A. (2010). Alex’s Adventures in Numberland. London: Bloomsbury.

Russell, D. (2017). What Base 10 Means in Mathematics. [online] ThoughtCo. Available at: https://www.thoughtco.com/definition-of-base-10-2312365 [Accessed 2 Nov. 2017].