Category Archives: Maths is Fun! – Maths Elective

Food Miles – Is local Always better?

Applications of mathematics in food processing and distribution can be seen when considering food miles. In one of our lectures we touched on this topic and I decided to look into it further.

According to Van Passel (2013) food miles are defined as the measure of the distance that food travels from where it is grown or raised to where it is consumed (p.3). Food miles have been linked to climate change. Transportation of foods over larger distances results in emissions of greenhouse gases and other pollutants that can harm the environment (Natural Resources Defence Council, 2007, p.2). Van Passel (2013) argues that the higher the food miles, the larger the impact on the environment (p.3). The amount of emissions and impact on global warming also depend on the type of transportation. Transporting food by airplane results in a greater impact on the environment than transporting food by ship. This would suggest that growing food locally would be better for the environment, however this is not always the case.

Engelhaupt (2008) emphasises that it is the method of how food is produced that impacts global warming, not how far the food is transported (p.3482). For example, tomatoes produced in the UK are usually grown in heated greenhouses with extra carbon Screen Shot 2015-12-03 at 10.39.19 dioxide to boost photosynthesis (Williams, 2013). In Spain these tomatoes can be produced with little or no extra heat. Producing tomatoes in the UK emits more greenhouse gases than the transportation of tomatoes from Spain to the UK (Williams, 2013). This is emphasised by the graph on the left. Therefore, it appears that growing food locally is not always the best for the environment. Although food miles do contribute to the impact of global warming, growing locally is sometimes more harmful to the environment as shown by this example.

Future Practice

In future I will consider more carefully how food miles impact the environment and that locally grown produce is not always the least harmful to the environment. Applying this to my future practice, food miles could be used to teach various concepts. For example, students could compare the distances different food has travelled and create graphs from this. Linking this topic to pollution and recycling, students could create artwork using empty food packaging. There are endless possibilities for the classroom when considering food miles. However, I do think it is important to give students a thorough insight into this topic and not to provide them with a superficial understanding of the topic. This includes conveying to them that high food miles does not always mean more harm to the environment.

References

Engelhaupt, E. (2008) ‘Do Food Miles Matter?’, Environmental Science and Technology, vol. 42, pp. 3482.

Natural Resources Defense Council (2007) Food miles: How far your food travels has serious consequences for your health and the climate. Available at: https://food-hub.org/files/resources/Food%20Miles.pdf (Accessed: 3 December 2015).

Van Passel, S. (2013) ‘Food Miles to Assess Sustainability: A Revision’, Sustainable Development, vol. 21, pp. 1-17.

Data, Statistics and Publication Bias

1 Reply

In our recent lecture we discussed the uses of mathematics in science. Data and statistics is a concept within mathematics that can be found in many other fields. For example, it is used in science or medicine. While mathematics in data and statistics is considered to be a very accurate measure, this is not always the case. The main issue that arose during the lecture was the impact of publication bias on the representation of data. I wanted to further look into this issue to consider how mathematics can be presented in a bias way and how this impacts society. I came across this very interesting TED talk by Ben Goldacre which inspired me to look into the issue further.

The term publication bias refers to when research that appears in published literature is unrepresentative of the population of completed studies (Rothstein, Sutton and Borenstein, 2005, p.1). This can be particularly misleading and even dangerous when considering the testing of new drugs. Much of the negative or inconclusive data remains unpublished which leaves doctors and researchers in the dark (Goldacre, 2012). Goldacre (2012) argues that positive data is twice as likely to get published than negative data. This can create huge misconceptions about a drug as part of the evidence is simply withheld by not being published.

One example of the dangers of publication bias occurred in the 1970s with the drug lidocaine. This anti-arrhythmic drug was said to reduce the risk of early death after a heart attack (Hampton, 2015). However, many trials to test the effects on mortality of this particular drug were not reported until the 1980s. Only at that time did it become apparent that the drugs actually increased mortality (Hampton, 2015). Publication bias meant that these drugs were promoted by doctors without any knowledge of the actual impacts of the drug. This is due to the fact that the negative data did not get published. This however is only one of many examples of publication bias in medicine. Goldacre (2012) emphasises the vast amount of negative data that remains unpublished.

Future Practice

I was intrigued to find out more about publishing bias, as this is something I had not considered previously when thinking about data and statistics. Implications of presenting bias data can be extremely dangerous to society and create misconceptions around certain drugs in medicine. In future this will help me analyse data more critically, not simply looking a the graph but questioning the validity of the data. This will also be useful for my future practice as a teacher, especially in doing science experiments. I will aim to convey that all results the students get should be acknowledged and need to be considered for it to be a fair experiment.

References

Hampton, J. (2015) Therapeutic fashion and publication bias: the case of anti-arrhythmic drugs in heart attack. Available at: http://www.jameslindlibrary.org/articles/therapeutic-fashion-and-publication-bias-the-case-of-anti-arrhythmic-drugs-in-heart-attack/ (Accessed: 2 December 2015).

Image. Photograph. Available at: http://www.medline.com/media/catalog/sku/abb/ABB427902H_HRE01.JPG (Accessed: 2 December 2015).

Rothstein, H., Sutton, A. and Borenstein, M. (2005) ‘Publication Bias in Meta-Analysis’, in Rothstein, H., Sutton, A. and Borenstein, M. (ed.) Prevention, Assessment and Adjustments. Place of publication: John Wiley & Sons, Ltd, pp.1-7.

TED (2012) Ben Goldacre: What doctors don’t know about the drugs they prescribe. Available at: https://www.ted.com/talks/ben_goldacre_what_doctors_don_t_know_about_the_drugs_they_prescribe?language=en#t-365862 (Accessed: 2 December 2015).

Living on Mars

1 Reply

There is no doubt about links between mathematics and astronomy. One particular project that struck me immediately, was the Mars One mission. I decided to look into further and find out how mathematics could be involved in this project and how it can be used for my future practice as a teacher. This particular mission could be used in schools, to make the study of different planets more interesting and relevant to students by giving a concrete example.

Mars One aims to establish human settlement on Mars (Mars One, 2015). From 2026 onwards, crews will depart for Mars in this one-way mission to live in space. Thousands of people from all around the world have applied to take part in this project and as of February 2015 this selection has been narrowed down to 100 applicants (Mars One, 2015).

I wanted to find out about the conditions participants of the Mars One mission would face. Mathematics is clearly involved when comparing the conditions of Mars to those of earth. The diagram on the left emphasises this comparison. Mars has a considerably lighter gravity (Tate, 2015). A little over a third than that of earth. Temperatures on Mars are on average minus 64°C, while on earth they are an average of 16.8°C (Tate, 2015). Days as well as years differ on Mars, as it takes mars longer to orbit the sun than the earth (Tate, 2015). All these concepts could be explored further in the primary school setting. For example, the concept of time or temperature could be taught using these comparisons as questions to discuss or further analyse.

It should be noted that this the Mars One mission brings about many risks. There is great controversy about this mission, as potential risks include social isolation, loss of privacy and lack of mental health services (Chambers, 2013). These issues could be discussed further in a classroom setting. For example, students could debate whether it is morally correct to send people on this one way mission to Mars.

References

Mars One (2015) Mars One. Available at: http://www.mars-one.com/about-mars-one (Accessed: 1 December 2015)

Tate, K. (2015) ‘How Living on Mars Could Challenge Colonists’, Space, 17 February. Available at: http://www.space.com/27202-living-on-mars-conditions-infographic.html (Accessed: 1 December 2015)

Chambers, C. (2013) ‘Mars One: The psychology of isolation, confinement and 24-hour Big Brother’, theguardian, 9 September. Available at: http://www.theguardian.com/science/head-quarters/2013/sep/09/neuroscience-psychology (Accessed: 1 December 2015)

Mathematics Behind Battleship

3 Replies

Yesterday we looked into the mathematics behind board games and this inspired me to look into a game I used to play a lot as a child. At first I was unsure whether there is any mathematics involved in the game Battleships but after doing some research into this game, I came across some very interesting mathematical strategies to maximise ones chances of winning at Battleships.

For this investigations, let’s assume we play on a a 10 x 10 grid and there are five different ships. The ships have the lengths 5, 4, 3, 3 and 2. This means that the total number of squares covered by ships in the 100 square grid will be 5 + 4 + 3 + 3 + 2 = 17. Initially I thought there would be equal chances of getting a hit on each square. This would mean that the chances of a hit would be (17 / 100) x 100 = 17%. However after looking into the mathematics behind battleships, I discovered this not to be the case.

Alemi (2011) presented his linear theory of battleships, stating that there is a greater chance of getting a hit closer to the centre of the 10 x 10 gird. He also stated that chances of getting a hit in the corners is least likely. The probabilities of getting a hit are shown on the diagram to the left. The darker the colour, the less likely a hit is and the lighter the colour the more likely a hit is. Chances of getting a hit in the centre is 20%, while there is only an 8% chance of getting a hit in the corners (Swanson, 2015).

I started to wonder why there is a greater chance of getting a hit in the centre of the grid. According to Alemi (2011) the reason for the differences in chances of getting a hit is due to the ways in which the ships can be laid out. In the corners the any ship can only be laid out in two different ways, while in the centre there are ten ways. This means that chances of getting a hit are much greater in the centre.

When looking into strategies to win at Battleship further, I came across Nick Barry’s research. He uses a different technique to maximise chances of winning at battleships. By analysing the grid in terms of a checkerboard (left) he could increase chances of a hit. By only firing at either blue or white squares chances of getting a hit are maximised (Barry, 2011). This is because even the smallest ship has to cover two squares.

Future Practice

It would be very interesting to investigate which one of these two approaches, would result in the highest chance of winning at Battleships. This could be applied to my future practice, as students could carry out investigations and draw their own conclusions. Battleship is a game most children know and could create a relevant context for their learning. Furthermore, this links to the fundamental concept of chance and probability. Therefore, Battleship could be used to reinforce these concepts.

References

Alemi (2011) ‘The Linear Theory of Battleship’, The Physics Viruosi, 3 October. Available at: http://thephysicsvirtuosi.com/posts/the-linear-theory-of-battleship.html (Accessed: 1 December 2015)

Barry, N. (2011) ‘Algorithm for Playing Battleship’, DataGenetics, December. Available at: http://www.datagenetics.com/blog/december32011/ (Accessed: 1 December 2015)

Swanson, A. (2015) ‘The mathematically proven winning strategy for 14 of the most popular games’, The Washington Post, 8 May. Available at: https://www.washingtonpost.com/news/wonk/wp/2015/05/08/how-to-win-any-popular-game-according-to-data-scientists/ (Accessed: 1 December 2015)

Slot Machine Mathematics

40 Replies

In one of our lectures we looked at the fundamental concept of chance and probability. Applications of probability can be seen in many different areas, not just in mathematics. One example that came up during the input was the link of probability to gambling. Gambling is a widespread, popular and recreational activity (Fabiansson, 2010, p.1). This was something that I found particularly interesting and therefore, decided to find out more about the applications of mathematics in gambling by analysing slot machines.

While there are various different slot machines, I have decided to focus on the three slot machine with six different symbols in each piece. The symbols I have chosen (see below) are banana, orange, cherry, mellon, grapes and seven.

Before we look at the chances of specific combinations using these symbols, it is important to find the total number of different combinations. As there are three slots with six symbols in each, there must be a total of 216 combinations. This is because 6 x 6 x 6 = 216.

The payouts of this particular three slot machine is as follows:

Three sevens pays 30 coins.

Any three of the same fruit pays 10 coins.

Two sevens pays 4 coins.

One seven pays 1 coin (break even).

While there are 216 different combinations, not all of them are winning combinations. To calculate the number of winning combinations, Shore (2014) states that we must consider the following:

Three sevens is a winning combination and there is only one possible figuration for this.
Any three of the same fruit are winning combinations and as there are five different fruit. This means that there are five different winning combinations.
Two sevens is also a winning combination. This means that one slot will have fruit. Therefore the winning combination for this configuration is calculated by (1 x 1 x 5) + (1 x 5 x 1) + (5 x 1 x 1) = 15
One seven is also a winning combination. This means that the two other slots will have fruit. Therefore the winning combination for this configuration is calculated by (1 x 5 x 5) + (5 x 1 x 5) + (5 x 5 x 1) = 75

To find out the total number of winning combinations we must add 1 + 5 + 15 + 75 = 96

This shows that in this three slot machine there are 96 possible combinations which are winning ones.

Using this information it is possible to calculate the payoff percentage. This can be defined as the amount of money a slot machine should return to player over a period of time (Casinomanuel, 2015). To calculate the payoff percentage of this particular three slot machine we must multiply the each winning combination with the corresponding about of coins it gives. Adding all these winning amounts together and dividing this by the total number of combinations will give the payoff percentage (Shore, 2014). This is shown by the following calculation:

((1 x 30) + (5 x 10) + (15 x 4) + (75 x 1)) / (6 x 6 x 6) = 0.995 (3 significant figures)

0.995 x 100 = 99.5 %

This shows that the payout percentage is 99.5% which is very high.

Future Practice

Finding out about probability and its uses in society will be useful for my future practice as a primary school teacher. Slot machines offer visual representation of probability. The use of this particular example needs to be carefully considered as gambling may be a rather controversial concept to use in the primary school setting. I really enjoyed finding out about the mathematics behind slot machines and think that visual representation are a great way to bring concepts such as probability closer to the students.

References

Casinomanual (2015) Percentage Payout. Available at:http://www.casinomanual.co.uk/online-casino-games/guide-to-slots/payout-percentage/ (Accessed: 29 November 2015)

Fabiansson, A. (2010) Pathways to Excessive Gambling : A Societal Perspective on Youth and Adult Gambling Pursuits. Surrey: Ashgate Publishing Limided.

Image 1. Photograph. Available at: http://www.modern-canvas-art.com/ekmps/shops/robboweb1/images/slot-machine-symbols-pop-art-canvas-print-4252-p.jpg (Accessed: 29 November 2015)

Image 2. Photograph. Available at: http://www.appcelerator.com.s3.amazonaws.com/blog/dev/platinoslot/mask.png (Accessed: 29 November 2015)

Shore, E. (2014) ‘Probability: Odds of Winning at Slot Machines’, Eddie’s Math and Calculator Blog, 7 January. Available at: http://edspi31415.blogspot.co.uk/2014/01/probability-odds-of-winning-at-slot.html (Accessed: 29 November 2015)

Maths and Maps

1 Reply

Mathematics has numerous links to wider society and can be found in many different fields, such as geography. Scoffahma (2013) states that mathematics can support understanding of geographical principles (p.112). One example of this is the coordinate system which links very closely with mathematical concepts. A coordinate system can be defined as the numeric representation of locations on the earth’s surface (Lanius, 2003). The most common or well known coordinate system is latitude and longitude. Latitude are parallels that run east-west, while longitude are meridians that run north-south (Lanius, 2003). Longitude and latitude are measurements expressed in 360°, where each degree can be divided into 60 minutes and each minute into 60 seconds (USDA, 2009).

Not only do coordinates play an important role in the creation of maps, but so does scale. Lanius (2003) defines scale as the relationship between distances on a map and the corresponding distances on the earth’s surface. This is usually expressed as a fraction or a ratio. This means that using the scale and the map, the actual distance can be worked out.

Using what we learned about in our mathematics in the outdoors lecture, we set about constructing our own map. Below is a photograph of our 3D map of Dundee with labels. We aimed to create a map of the main tourist attractions Dundee has to offer.
While we did manage to portray some of the important attractions of Dundee, we did not pay close attention to scale or location of these. To improve this we could have constructed a grid and used grid references to correctly place each attraction. Furthermore, we could have included a key and scale.

References

Lanius, C. (2003) Mathematics of Cartography. Available at: http://math.rice.edu/~lanius/pres/map/ (Accessed: 24 November 2015)

Scoffham, S. (2013) Teaching Geography Creatively. New York: Routhledge.

USDA (2009) Firefighter Math. Available at: http://www.firefightermath.org (Accessed: 24 November 2015)

Importance of the Number Five in Relation to Phi

1 Reply

The number five plays an important role in determining the value of the golden ratio (Meisner, 2012). Phi can be derived from the following formulas shown below. It becomes apparent from looking at these, that the number five is an essential element in determining phi. Not only does the number five feature in the formula for the golden ratio but it is also a number in the Fibonacci sequence (Scott and Gulick, 2010, p. 38).

When analysing the number five in a geometrical context by looking at a regular pentagon, the golden ratio appears in many different ways (Darvas, 2007, p.173). Below I have created a diagram of a regular pentagon, showing various segments that relate to the golden section.

Regular Pentagon

The diagonals and segments constructed from the vertices of the pentagon represent the value of phi in many different ways. For example, the line segment AB divided by segment BX gives the golden ratio. Dividing segment BV by BX also equal to the value of phi. All proportion of line segments shown below are equal to the golden ratio or phi.

Proportions of the regular pentagon have influenced various art, particularly works relating to the proportions of the human body (Darvas, 2007, p.175). The regular pentagon was used by Leonardo da Vinci, for example, to create proportions of the ideal human body. In the figure of the Vitruvian Man he placed limbs at the vertices of the regular pentagon (Stakhov and Olsen, 2009, p.43).

‘Vitruvian Man’ by Leonardo da Vinci

Lesson Ideas for Pentagon Topic

Examining pieces of art that that use pentagons (link to tessellations)
Why don’t pentagons tessellate?
Constructing a five pointed star using a regular pentagon
Regular and irregular pentagons

References

Darvas, G. (2007) Symmetry: Cultural-historical and Ontological Aspects of Science-Arts Relations. Basel: Birkhäuser Verlag.

Image. Photograph. Available at: http://109.74.193.65/dev/wordpress/wp-content/uploads/vitruvio01-400×562.jpg (Accessed: 20 November 2015)

Scott, J. and Gulick, D. (2010) The Beauty of Fractals: Six Different Views. Washington, D.C.: Mathematical Association of America.

Meisner, G. (2012) ‘Mathematics of Phi, 1.618, the Golden Number’, Golden Number, 16 May. Available at: http://www.goldennumber.net/math/ (Accessed: 21 November 2015)

Scott, J. and Gulick, D. (2010) The Beauty of Fractals: Six Different Views. Washington, D.C.: Mathematical Association of America.

Stakhov, A. and Olsen, S. (2009) The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. Singapore: World Scientific Publishing Company.

The Golden Ratio in Sunflowers

Platonic Solids and Origami

1 Reply

Platonic Solids are 3D shapes which meet the following conditions. Each face needs to be the same regular polygon and the same number of polygons need to meet at each vertex. Platonic Solids include the tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron.

Tetrahedron

Plato believed it represented the ancient element fire.

The tetrahedron is made up of four triangular faces. It has four vertices (three triangles meet at each vertex) and six edges. It has seven axes of symmetry. Instructions on how to create an origami tetrahedron can be found here.

Cube or Hexahedron

Plato believed it represented the ancient element earth.

The cube is made up of six quadratic faces. It has eight vertices (three squares meet at each vertex) and twelve edges. It has thirteen axes of symmetry. Instructions on how to create an origami cube can be found here.

Octahedron

Plato believed it represented the ancient element air.

The octahedron is made up of eight triangular faces. It has six vertices (four triangles meet at each vertex) and twelve edges.

Dodecahedron

Plato believed it represented the shape of the universe.

The dodecahedron is made up of twelve pentagonal faces. It has twenty vertices (three pentagons meet at each vertex) and thirty edges. It has thirty-one axes of symmetry. Instructions on how to create an origami tetrahedron can be found here.

Icosahedron

Plato believed it represented the ancient element water.

The icosahedron is made up of twenty triangular faces. It has twelve vertices (five triangles meet at each vertex) and thirty edges.

Using origami, one can combine platonic solids to create intricate stars or compounds. An example of this is the interlocking tetrahedra which is created from five tetrahedra. Instructions on how to create this can be found here.

Interlocking Tetrahedra

Created from five intersecting tetrahedra.

Icosahedron and Dodecahedron

Created by combining an icosahedron and a dodecahedron.

References

“Mathematical Origami.” Mathigon. Web. 15 Oct. 2015. http://mathigon.org/mathigon_org/origami/

“Platonic Solids.” Platonic Solids. Web. 15 Oct. 2015. https://www.mathsisfun.com/platonic_solids.html

Golden Ratio to Determine Facial Beauty