One way statistics has eased into peoples good books is through the happy marriage of computer science and statistics. The world of statistics and computer science have collided and melded together as the practice of statistics has moved onto our electronic devices in the form of programming. Languages like R and Python rank as some of the fastest growing and most used programming languages in the last 5 years. The use of R has grown particularly in academic circles for statistical computing is a well sought out skill and proficiency in R or Python is now desired by many employers especially for those who are pursuing careers in STEM. Statistical tests have come a long way since the beginning and harnessing the power and utility of computers will only see it advance and influence others more rapidly and efficiently.

Another way these bad vibes are being countered is the early inclusion of statistics to educational curriculums. In the USA, statistics has been introduced as one of the core components of K-12 Mathematics, highlighting the importance of the learning mathematical skills of induction, deduction, and communication of data. Such practices seem promising as this year alone we should have hit a 50% increase (approximately 200,000 individuals) of professional statisticians entering the workforce. Learning statistics earlier should provide educators a chance to cultivate an earlier appreciation of statistics and corresponding valuable analytical skills. Educators should not provide students with the illusion that pursuing a career in geology or nursing will end all affairs with statistics because the truth is the pervasiveness of data analysis is far-reaching and only increasing in importance as we rely on the data to advance into the future.

So having chosen to embrace statistics, where and who can we expect to be at the frontier of statistics? The truth is many of you will be at the heart of it before knowing it. As emphasized earlier, statistics is an interdisciplinary study. While often highlighted in sciences, it becomes absolutely relevant and paramount whenever there is a need for research and development. We ask questions, seek for improvements, develop new concepts and need a way to answer or see how these ideas come to life. The next step is to then perform experiments, develop prototypes, run tests, all the while tracking results, recording data. Statistics finally comes into play, helping you assess levels of uncertainty, % of success, project growth or sales rates, where to build houses, or mine Gold. Such is the nature of research and development that involves the application of scientific methods, processes, and systems in order to evaluate and interpret data. Data-driven-statistical- research now forms a fundamental piece of the puzzle when innovating, creating or attempting to progress forward – be it in medicine, academia, business, Information Technology, medicine, economics, or construction.

For example, a biostatistician may be involved in researching the rate of HIV spread and invasion throughout sub-saharan Africa to help identify the countries that will be hit the hardest. In medicine, statistical research may take the form of equivalence testing to compare, improve and examine the effectiveness of new drugs to aid depression. Astronomers may utilize statistical models to support research on the expansion of the universe, while an actuary may look for statistical models to predict risk of financial investments or business expansion. Mechanics and automotive industrialists can apply statistics to constantly improve the quality of their product by constantly minimizing the level of errors in the performance of their product. Perhaps a more familiar example is the collation of government statistics. For years, governments have gathered a wealth of enormous datasets and utilized the power of statistics to inform decisions and research improvements on housing, income, unemployment, minimum wage, healthcare, and education services.

By pre-emptively identifying the statistical test(s) you want to employ to help answer your research question(s), hopefully you know what sort of data needs to be collected. Where statistics comes in handy is helping you identify key aspects you may not have considered in your chosen methods of data collection. Such may come in the form of identifying an additional variable of importance to collect data on. Another pitfall statistics can help you avoid is that of pseudoreplication. Pseudoreplication is particularly dangerous for several reasons: Firstly, it paints a false image of how large a sample size is and ignores the need for “true” replicated treatments (when applicable). Sample sizes are important as they determine the power of your statistical tests and therefore the confidence and scope of your conclusions based on the statistical results. Secondly it fails to highlight that some variables may not be independent. This may mask the true effects of the variables that you wish to be examining independently. Sampling bias can also be avoided when considering the statistical test you hope to use: for example research on the occurrence of domestic violence in households should investigate low-income, middle-income, and high-income neighbourhoods.

*“To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination: he may be able to say what the experiment died of.”*

— **Ronald Fisher**

Without statistical tests there would be no objective way to show whether the data are in support or in disagreement of research questions. Since the burden of evidence (for or against) lies in results of statistic tests, without the use of statistics in research, we would be buried in unknowns, more questions, open-ended conclusions, and more data than we can handle! Without statistical research, we would be unable to credit new discoveries, answer new questions, and confidently advance with new developments. Statistical tests form the basis on each we can trust what the data is saying and make sense of what the raw, volumes of data are communicating.

Data is rarely squeaky clean and more often than not, data is messy, ugly and incomplete: Such is the nature of sampling data, there are answers people do not answer completely, truly, or circumstances beyond our control that prevent us to collect all the data points we desire: e.g. an inaccessible village of HIV+ patients trapped in a war zone, the premature death of chicks in a nest, apparatus failure, or the sudden crash in stocks. Truth of the matter is there is no way to collect ALL data points – this is where inferential statistics saves the day. Beyond those limitations, at the very minimum there is human error in data sampling or collection and with every tool, a measure of uncertainty. Errors can also arise due to uncontrollable circumstances as aforementioned, or due to a limitation of a statistical test. These errors can be accounted for to some degree in statistical models and tests so that we can cut through all the noise and assess our hypotheses honestly.

Using statistics can help us map out those outliers, identify the levels of uncertainty in our results, and help us deal fairly with those errors. No statistical test is perfect and neither is any dataset. Statistics allows us to draw conclusions openly by realizing these limitations from the start.

Having utilized the appropriate statistical test, fair and objective conclusions, implications, can now be interpreted from the dataset. Statistical tests provide us with the means to interpret the dataset accurately so that we can make unbiased decisions on how to proceed knowing what the data is saying. It also guides the way we communicate our results and calls for us to defend why these statistical tests were chosen and how we arrived at our explanations based on a series of numbers. Statistics are also a great way of communicating and condensing large datasets into digestible, bitesize pieces of information easily understood by the masses. These summary statistics are helpful in providing people with an immediate idea of the big picture and whether your conclusions are valid.

Without statistics we would be unable to tease apart the multitude of effects that may be influencing our dependent variable. Furthermore we would not be able to identify which factors are working in conjunction to produce a compounded effect on our dependent variable. Statistical modelling helps us deal with our multivariate statistical questions so that we can assess hypotheses from every possible angle. So for example, how do we know that domestic violence in neighbourhoods of various levels of income are not also affected by ethnicity, religion, and level of education? Some of the factors may be intertwined and using statistics helps us tease apart these details.

With all that being said, it is worth pointing out that statistics can’t solve everything and anything under the sun perfectly. Statistical tests/models are flawed and in themselves have limitations in the way they were designed and formulated. Even using the wrong statistical test can lead to serious erroneous conclusions and overlook the data completely. Statisticians have thus tried to create helpful guides, books, charts and keys to help advise students and working professionals alike how to identify the appropriate tests/models to apply to their data. These resources should help students be more vigilant and aid the appropriate use and digestion of statistics. Combined with a more positive outlook on statistics, early exposure, an abundance of tools, and the knowledge of a ubiquitous need for statistics in all forms of research and development, there is hope that statistics will be shunned no more. Surely if plants can sense and harness the value of statistics, so can we.

*Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.*

— **George Box & Norman R. Draper**

Photo by Daniel McCullough on Unsplash *Salaries typically range from $45,000 – $110,000+ (Depending on Role, Employer, & Experience)*

Architects require mathematical skills on a day-to-day bases. For example, they use math to design efficient building layouts and to calculate angles for roofing and structural integrity. Without math, the job would be next to impossible and incredibly risky. This is why employers look for candidates who can demonstrate a high level of mathematical ability.

When it comes to applying for an architecture degree, students will need a qualification in calculus. Therefore, students will also be required to be skilled in algebra, geometry, and trigonometry. To that end, we’d recommend taking a precalculus course as that will provide you with the knowledge you need to pursue a calculus course and then a degree in architecture.

If you have a passion for video games and a talent for algebra and algorithms, you may want to consider a career in games development.

Game designers and programmers use math in almost every aspect of games design. From the height of a characters jump to the spawn time of an enemy ship, math is used for it all. Behind every kick in FIFA, shot in Call of Duty, and every drift in Forza, there’s math behind the scenes crunching numbers and determining outcomes. In world building, geometry is used to build incredible worlds and realistic environments to explore. Alternatively, algorithms can be used to procedurally generate in-game assets, offering a level of randomness and unpredictability to the game’s environments.

*Salaries typically range from $50,000 – $125,000+ (Depending on Role, Employer, & Experience)*

AOL placed this job as number 5 on their 10 Best Jobs for People Who Love Math. As an Astronomer, you would use math to test theories and to interpret the data you receive from looking up at the stars. Whether its how much light is being emitted by objects in the sky or the distance between certain planets across space, math is used to figure it out. Additionally, SIN COS TAN is used by astronomers when attempting to calculate the angular distance between two stars based off of their coordinates. With this in mind, it’s not hard to see why many colleges request that applicants to their astronomy programs should be competent in both math and physics.

*Salaries typically range from $30,000 – $100,000+ (Depending on Role, Employer, & Experience)*

If you have a passion for education and want to share your passion for math with others, then perhaps a career as an educator would be a good role for you. Having said that, if you want to teach mathematics at a college level, you’re going to need to know what you’re talking about.

In general, it’s expected that as a high school lecture, you should have a qualification a few steps above your students. As a math professor in college, you should have a masters in the area you are teaching and should be well-versed in a variety of mathematical topics, unless you solely teach a very specific section of mathematics.

You’ll need to plan and deliver lessons that cater to all learners and their varying skill levels, and each lesson should adhere to the set curriculum as governed by the appropriate organizations of that country. This is why it’ essential that you know your stuff. You need to be able to convery how problems are solved in a way that your students can understand and demonstrate in their exams. As you progress in your career, you could take on additional duties and roles to increase your salary (Subject Leader, Head of department, Principal etc.)

*Salaries will range from $45,000 – $100,000+ (Depending on Role, Employer, & Experience)*

In an article by Investopedia, it was noted that the median salary for an Aerospace Engineer is $112,010. It also goes on to discuss how math plays an integral role in the daily routine of the profession.

Whether you’re looking to be an aerospace engineer, computer engineer, construction engineer, or one of the many other disciplines in engineering, it’s safe to say that you’ll need math to help you complete your daily tasks.

Engineers will need to use algebra to help them solve unknowns, they will use their skills in geometry to help them with design work, and will use calculus to determine things like size, acceleration and weight of objects. Additionally, engineers will also use statistics to calculate foreseeable issues in design. For example, an engineer will need to know the typical rainfall stats for the region throughout the year, the varying wind speeds, and other environmental issues in order to construct a building suitable for the environment in which it is set.

Math will be one of the topics covered in your engineering degree programs and a prerequisite in topics like vectors, calculus and advanced functions may be required in order to apply for the program.

*Salaries typically range from $45,000 – $70,000+ (Depending on Role, Employer, & Experience)*

Being good at math will help you tremendously within accountant/analyst roles. As an accountant, you’ll need to analyse accounting records ahead of producing financial statements. You’ll also need to use accounting equations for liabilities, assets and stakeholder equities. If you’re looking to enter into a career as a financial adviser or analyst, you will need to have a strong understanding of statistics and probability to help you make more accurate decisions.

Those career paths mentioned above are just a few ways in which your love for math can translate into a fulfilling career. For a look at more jobs for math lovers, check out Business Insider’s article on the topic. If you’re keen on pursuing a college major or a career in any of these fields, developing your skills in math will be key to your success. Fortunately, there’s a website that can help you, StudyPug.

Through a fun and engaging video format, StudyPug is changing the way students learn. With 1000s of online video tutorials that cover a vast amount of mathematical topics including basic algebra, trigonometry, and precalculus, students the world over are developing effective revision strategies and improving their performance in math.

StudyPug’s content covers all the same information you’d expect to find in modern textbooks, but it’s delivered in a user friendly way that’s much easier to understand. As a subscriber, you’ll have unlimited access to every lesson on their website and each lesson can be paused, rewound, and fast-forwarded, so you’ll never get left behind.

Learn at your own pace and visit StudyPug today to start your free trial.

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This should be self explanatory, but it’s definitely worth mentioning here. Paying attention in class can help you retain the information that your teachers are giving you. Sitting next to your friends can lead to tempting distractions but avoid the temptations, focus on the lesson, take notes, and listen to your teacher. They know what they’re talking about and are there to help you if you get stuck, which leads us onto our next point.

Your teacher will be a useful resource in learning math, not only will they teach you what you need to know, but unlike a textbook, they can respond to any questions you have. If you’re stuck or confused, raise you hand and ask for help.

Understandably, you may be hesitant to stop the teacher during their presentation so you may want to save the questions till the end. If that’s the case, make a note so you don’t forget. Chances are there’s a few students in the class that are just as confused as you are. Don’t wait for them to raise their hands because you may be waiting all day. Be confident and raise your hand first or you could risk falling behind and making things harder for yourself.

Many students avoid doing homework and put it off until the last minute. It’s a practice that leaves you at a disadvantage because you’re rushing to complete the work and you’re not absorbing the information in the most effective way.

Homework is used as a tool to keep you thinking about what you’ve learnt in school and to encourage you to continue exploring problems and solutions. In math, practice makes perfect and setting aside some time to practice math via your homework assignments can help you get into a good study routine which will benefit you greatly in your exams (more on that later).

To better prepare yourself for upcoming quizzes and exams, you must first know where your strengths and weaknesses are. For example, you have a trigonometry test coming up, what areas of trigonometry are you good at and which topics require a little more work. If you’re not sure about this, review your class notes and use practice papers to sit “mock” exams to help you highlight the things you know and the things you don’t.

When taking mock exams, put yourself in exam conditions (no phones, no distractions!) and stick to the time limited of the paper. Try to answer as many questions in the time limit and be sure to stop once time is up. Pass your paper onto a friend or family member and have them mark it for you. Once returned, review your scores and see what questions your answered incorrectly.

This can be a useful exercise to help you identify the topics you know and the topics you need to focus your revision efforts on. Not only will this process allow you to target areas of weakness, but as you progress, you’ll hopefully see improvements in your score and in turn, that can boost your confidence ahead of your actual exams.

Once you’ve established your strengths and weaknesses, you can start to plan out more effective revision strategies. Spend the bulk of your time on areas that need improvement and revisit areas that you’re confident. Doing this will ensure that you’re prepared for any and all questions ahead of your exams.

To assist you, StudyPug has 1000s of lessons delivered via easy to follow videos that can be paused, rewound, and fast-forwarded, meaning you’ll never get left behind. It’s a great tool for learning at your own pace and StudyPug is a great source for practice materials too. Spending an hour or two a day on their site can dramatically improve your performance in class and in your exams.

Many students who use StudyPug prefer the video format over the traditional textbook revision methods. This is because the video content manages to convey the same information as the books but in a much more conversational way that’s easier to follow and understand.

Outside of revision, you should spend time on yourself. Don’t stress too much and don’t feel guilty for enjoying some down time. Play games, go for a run, do whatever you need to do to take your mind off math. Coming back to it later with a clear head can make it easier to digest and retain information.

Also, you should make sure that you’re eating enough and eating healthy too. Eating “brain” foods like blueberries, salmon, avocados, etc. can help your cognitive functionality. You should also drink plenty of water to keep yourself hydrated. There’s a reason behind the saying “healthy body, healthy mind”. Eat right and you’ll be in a better state of mind to study effectively and perform well in your exams.

When you’re in your exam its vital that you make the most of the time that you’re given. Firstly, you should attempt to answer the questions you feel comfortable answering. If you see a question that looks too hard, skip it for now and come back to it later. Spending time on a question you’re struggling with can lead to you running out of time and missing out on questions you could answer. Once you’ve reached the end of the exam paper, you can always return to the unanswered questions.

If you happen to answer all of the questions with time to spare, don’t just sit there gazing into space. Use the time to double check and triple check your answers. In doing so, you may notice an error you didn’t notice the first or second time around.

Lastly, make sure you show how you arrived at your answers. Many examiners will look for evidence that you didn’t just memorize the answers but can actually demonstrate the method to obtain them.

Showing your working out can earn you additional marks in exams and could be the difference between a B grade and an A grade. If the thought process is correct but you arrived at the wrong answer, you could still receive marks for your working out.

Take the time to utilize these 8 steps and you’ll stand a much better chance at improving your performance in class and in your math exams. Visit StudyPug today and sign-up for a free trial to experience their video content and to see how they can help you build even more effective revision strategies.

]]>Well, researchers at the University of Manchester (in association with The Fine Bedding Company), have come up with a mathematical equation to help us determine how well we’ve slept and what factors we need to change for improved rest. Let’s take a look and see how it works.

**Sleep Quality = [(T x Bt) + C] / [Ha + S + L + (H x D)]**

Okay, this equation may look daunting at first, but with some basic math knowledge and an understanding of the order of operations, we can break it down into smaller chunks. If you need a hand remembering the order of operations, think of PEMDAS. This is an acronym that mathematicians use when tackling algebra. It stands for the following:

**P**arentheses

**E**xponents

**M**ultiplication

**D**ivision

**A**ddition

**S**ubtraction.

You may also use the mnemonic phrase “**P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally”.

Now that we know to tackle the problems in parentheses first, the formula doesn’t look so daunting, but what do the letters within the equation mean?

**T = Tiredness**

This is the amount of hours since your last sleep, minus any naps you’ve taken, plus any hours you worked/exercised in the day.

**Bt = Bedtime **

This is your bedtime divided by your overall average bedtime

**C = Comfort **

The comfort focuses on the pillow, bedding, and mattress. Each one is rated from 1-5, with 5 being very comfortable. Add these totals together and then subtract nine.

**Ha = Hours Awake During the Day**

This is the total amount of time you were awake for before you fell asleep.

**S – Sound (0-5)**

This is the total noise within the room during your sleep, with 0 being soft soothing noises and 5 being loud disturbances (trains, airplanes, cars).

**L = Light (0-2)**

This is the light within the room. Think LED lights on electronics and lights shining through windows. On this scale, 0.1 would be little light and 2 would be very bright lights.

**H = Heat (celsius)**

Temperature of the room, which is calculated by the number of degrees difference from 16 degrees Celsius. Take that number and divide it by ten.

**D = Duvet (0-3) **

Measured in relation to its effectiveness with the room temperature. In this scale, 0 means it compensates perfectly for the overall room temp. The other end of the scale (3) means that it does not compensate well and can leave you either too cold or too hot.

Now that we’ve cleared that up, lets take a look at my last night’s sleep and assess how well I slept. Firstly, let’s tackle the sections in the parentheses on the left hand side.

**[(T x Bt) + C] **

I was awake for 15 hours before I slept, I took 0 naps, and I worked for 7 hours.

T = 15 – 0 +7

T = 22

Bedtime was at 11pm and I usually go to bed at 10pm, so let’s divide 11 by 10.

Bt = 11 / 10

Bt = 1.1

For the last part of the left hand side, we need to work out the comfort score. My pillow, bedding, and mattress are all pretty good. I will rate them all as being 4 on the 0-5 scale. Remember, you need to minus 9 from the total.

C = 4 + 4 + 4 – 9

C = 3

Ok, so now we know that the left hand side of the equation is as follows:

(T x Bt) + C

(22 x 1.1) + 3

24.2 + 3

**27.2**

Now, we move to the right hand side, starting with the problem in the parentheses.

**Ha + S + L + (H x D)**

The room temp was around 16 degrees, so we just divide that by 10

H = 16 / 10

H = 1.6

My duvet is pretty comfortable and never leaves me too hot or cold. I will rate this as 0 (which is the highest score in this instance)

D = 0

Lets move out of the parentheses and onto the additions. I was awake from 7am to 11pm, meaning that I was awake for 16 hours.

Ha = 16

The room was pretty quiet, so I would give it a sound rating of 1.

S = 1

There was very little light in the room with the exception of red standby lights on my electronics. Lets give this a score of 0.5

L = 0.5

We can now complete this side of the formula.

Ha + S + L + (H x D)

16 + 1 + 0.5 + (1.6 x 0 )

16 + 1 + 0.5 + 0

17.5

Which means we can now solve the entire equation!

Sleep Quality = [(T x Bt) + C] / [Ha + S + L + (H x D)]

Sleep Quality = [(22 x 1.1) + 3] / [16 + 1 + 0.5 +(1.6 x 0)]

Sleep Quality = [24.2 + 3] / [16 + 1 + 0.5 +0]

Sleep Quality = 27.2 / 17.5

Sleep Quality = **1.5542857142… **(1.55 when rounded to the nearest hundredth)

So what does this all mean? Well, the quality of sleep is rated from 0 – 2, with 0 being restless tossing and turning, and 2 being a great night’s rest. As you can see above, I scored 1.55, which means my sleep was slightly above average in terms of overall quality.

I can now use this formula to work out what needs to change in order to improve my night’s rest. It’s worth noting however, that you shouldn’t take this formula as fact. Here’s what Dr. Penny Lewis, researcher at the University of Manchester, had to say about their formula:

*‘It is always fun to try and boil down a very complicated process into something really simple, and that is what we have attempted with this equation. *

*‘We wanted to keep things easy, but sleep is complex and there are lots of factors that we haven’t included, for instance the psychology of how you feel about the room you sleep in. *

*‘Also, the extent to which the things we have put in influence sleep varies hugely from person to person, so this equation really should be viewed as a guide that may make people think about some simple ways they might be able to improve their sleep.’*

Try it out for yourself and see how good your last night’s sleep was. If you need help tackling equations or the order of operations, remember PEMDAS, and for help with anything else relating to math, visit studypug.com.

]]>Now these games of yesterday are still fun games, but they’re relatively simple by today’s standards. Nowadays, there are so many games to play and games like Call of Duty, FIFA, and Forza Motorsport offer much more immersive and in-depth experiences to a much wider and broader demographic.

While the games industry has evolved dramatically, there’s still one key thing that the games of the past share with their modern-day counterparts, math. Whether it’s the time between spawning enemies in classic arcade shooters, or calculating the bullet drop in PlayerUnknown’s Battlegrounds, math can be found in practically every video game ever made.

With that in mind, let’s breakdown how video game designers use math in games to make them much more fun for you to play.

Math in video game design helps programmers to construct beautifully realized worlds. Using basic geometry, designers can build isometric backdrops that give the illusion of a 3D space. They can also use geometry to build more complex 3D worlds and characters. Almost everything in the games world is made up of things called polygons. These polygons form basic shapes that when combined together, form practically every item you see in game. Trees, cars, birds, people, weapons, soccer balls, they’re all made up of polygons, or “polys” as they’re often called.

The more polys within a model, the more detailed the model will be and as video games become more powerful, designers can produce higher poly models, making them look better and more realistic. Before high powered games consoles, designers would need to make lower poly models in order to make the game run smoothly and efficiently, which is why games of the past don’t look as detailed as modern day games.

Outside of geometry, game designers also use math to calculate the space between objects and the distance a player can travel. Math helps designers to correctly place ledges and platforms in the right places for players to reach. It also allows them to determine how high to make walls or how far to make gaps, in order to block off certain areas of the world. Without these calculations, players could go anywhere in the world and potentially ruin the experience or even break the game in interested and unusual ways (see games done quick).

Instead of building a specific world for players to roam, game designers can also build procedurally generated worlds that offer different playthroughs each time the player enters the games world. Using this method, designers create data (in-game assets) using a mathematical algorithm to place items in the right places and to build out a brand-new world from scratch. Games like spelunky have utilized this procedurally generated method to great effect, adding a certain level of unpredictability and excitement to the gameplay.

Without math, the characters wouldn’t be able to function. Objects wouldn’t move and the worlds would be lifeless. Mathematical vectors are used within games to assign direction to objects and the length of the magnitude will dictate the speed at which they will travel. Using X and Y, basic formulas are constructed and each one of these helps to breathe life into the world.

Mathematical formulas help programmers determine player acceleration/deceleration. For example, lightly squeezing the trigger in a game like Forza, will result in the car accelerating slightly. Pulling the trigger down completely, will result in the car going faster. Similarly, pressing the other trigger (the breaks) will result in the car decreasing in speed. Calculating the impact of these button presses, allowing you to race through cities at breakneck speeds, making those perfect turns, and breaking at the right time, is all made possible through math.

Beyond just controlling the movement speed of the player, formulas can also be used to inform player actions like jumping. In a game like Super Mario (yes, there’s even math in Mario!), the height of a jump will be controlled by a pressure sensitive button presses. A light tap, results in a short jump, and pressing the button down for longer, results in a much larger leap. This is all made possible through formulas behind the scenes.

Games like Street Fighter, Mortal Kombat, Tekken, and other fighting games, rely on math to determine who will emerge victorious. At the beginning of each battle, players have a full health meter (100%) and over the course of the battle, the players health can decrease depending on the attacks used by their opposition. Each attack is assigned a value (hit points) and if landed successfully, these attacks will deduct points from the opponent’s life bar. This continues until one player reaches zero (no health) and is defeated.

Math directly informs how players interact with the game. They will subconsciously build basic algebraic formulas in their heads to help them win. If the opponent’s health is at Y and this move deals X amount of damage, the outcome will be success! This is true for many other games too as they all require problem solving in real-time. In a way, being better at math means you could be better at games.

Fighting games will need to be balanced so that no character is too powerful and that each moveset has a fair balance of strong and weak attacks. Once again, this process is made possible by math!

Fighting games also have “special meters” that are built up through dealing damage and taking damage too. How they are built up and when they’re available for use, is determined by in-game formulas that calculate damage dealt and received and how that translates to special meter power.

This method is also used within action RPG games. These games often offer stat boosts or special moves in certain scenarios like low life or after a successful combo. Formulas behind the scenes, trigger certain abilities that are available to the player. For example, when the player’s life is below X (25%), trigger Y (special move).

If you’ve made it this far, you now have a basic idea of how math is used in video game design and you should be able to answer the following question:

*Does video game design require math?*

Yes! It’s clear to see that there’s different types of math at the heart of practically every video game available today. We’ve demonstrated algebra in video games, calculus in video games, vectors in video games, and that’s only scratching the surface.

Math is integral to the building of games and to the playing of them too. Knowing how and when to land a move in Street Fighter, drive a car in Forza, and jump in Mario, it’s all math.

If you play games, you’re using math, and if you dream of making your own cool games, you’ll need to know various elements of math. With this in mind, take advantage of your math classes in school and build your understanding of the subject matter, it could lead to a career in video games!

To assist you, use tools like StudyPug. It can help you revise for exams and is a great online support for your studies. Whether you need help with trigonometry, calculus, basic algebra, or any other math related topic, StudyPug has a range of helpful step-by-step videos that are incredibly easy to follow.

Try it today for free and prepare yourself for that future career in games development.

]]>Gamification – a relatively new word that any version of Microsoft Word 2011 and below will have trouble identifying it as a legitimate. A novel product of the 21st century, the term gamification was unheard of until a little less than a decade ago. Not until later in 2011 was the term officially placed in the Oxford dictionary that now describes gamification as *“**The application of typical elements of game playing (e.g. point scoring, competition with others, rules of play) to other areas of activity, typically as an online marketing technique to encourage engagement with a product or service.”* More simply put, one could describe gamification as the process of incorporating gaming elements into anything outside the realm of gaming.

Since the inception of gamification, its popularity has soared heights and won the hearts of numerous professions and industries, from the likes of healthcare services, to the food and beverage industry, to human resources and recruiting practices. According to statistics, by 2018, the global gamification market is expected to reach a net value worth of $5.5 billion dollars (Markets and Markets). The United States alone will claim $2 billion of this total (M2 Research), which is not surprising given that they are the number one, leading market in gamification. As an incredibly lucrative market, its not hard to believe its reach, more evidently so in academia.

The draw however is not just monetary gain as demand from learners is also striking: 80% of students claim that their productivity would increase if their university/learning institute were gamified (TalentLMS). With such a strong backing and an inevitably increasing trend, how can academic institutions follow suit? Should they follow suit?

Even in all its glory, success, and popularity among the masses, gamification is still considered dangerously attractive to many. Parental and academic concerns are largely based on the struggle to identify the boundaries between gaming and gamification. How do we separate the two? Or do we accept that there will be instances where lines be blurred? If we are able to do so, can there be a way to ensure the same learning outcome sand objectives are still aligned and met?

Gaming is simply defined as the act of playing games: games however now come in various forms, sizes, shapes, and platforms – from board games, to toy games (e.g. Kendama, yo-yo), to console games, PC games and so forth. In this modern age, more often than not, people associate gaming with electronic gaming – so playing Mario on your Nintendo Switch or League of Legends on your computer etc. What connects all these gaming variants is the fundamental objective on which they all created with from the beginning: “How do I win?” While there are many elements of gaming that drives players such as reward, competition, progress and comradery; winning is the biggest lure and stressed the most. However, the truth about gaming is that the best games are those that stimulate learning and incorporate it subtly yet intentionally. This is where the lines begin to blur between the two, but also where the goal of presenting a learning and challenging environment converge.

As you might recall – gamification is the process of incorporating traditional gaming elements into a ** non-game framework**. Here the differentiation between the two appears to be more stark – academia inherently classifies under the non-gaming context. With gamification the goal is not a complete upheaval or transformation of how/what we learn at our academic institutions. It is the addition of the bells and whistles (found in any good game) that targets a very human but vital part of us required for effective learning – our motivation. Yu-kai Chou breaks this idea of motivation further for us by identifying the different facets that drive the average Joe/Joan:

Some general examples of gamification in the classroom include the following:

- Assigning badges instead of grades
- Converting grades to a point system i.e. students begin at 0 and earn points up to 100
- Problem-based learning: creating a challenge (potentially real world example) with more than one solution
- Creating a timeline with milestones that allows students to track progress and set goals: different rewards are awarded when milestones are reached

The pervasiveness of gamification in academia has had a particularly strong impact on the instruction, learning, and development of math curricula. As one of the core subjects students are bound to throughout their academic journey, math is one of those subjects that either students love to hate or cannot live without. Broadly speaking however, the majority of math learners struggle with learning the content and fail to find interest and/or appreciation of the subject. Coined as one of the most intimidating subjects, psychological and medical studies have identified mental health issues linked to the study of math (i.e. math anxiety).

As such a delicate yet aggressive subject matter, game developers and educators have approached this in a number of ways. One clear path that has been taken by many is transforming and conforming to the pure game framework. Some examples of these games include Land of Venn and Guess the correlation. The alternative approach as earlier discussed involves the clever inclusion of gaming elements while still functioning as well as maintaining a strong identity as a classroom (or e-classroom). Some examples of these applications include Knowre and online learning platforms such as StudyPug.

The transition from traditional learning to gamification of math education has proven incredibly successful with studies highlighting numerous benefits for both student and teachers alike. A study from Deakin University identified that students adopted a more positive attitude towards math when gamified. Additionally students were also more engaged and motivated as the process of gamification helped relieve the repetitive nature of practicing question after question. Other benefits of gamification also include increased engagement in learning and productivity.

Gamification has also been linked to reduce tendencies of math anxiety. Such is linked to two different factors: first learners are now open to learn in the comfort of their own pace and space that may no longer be bound by a traditional classroom setting or timeline (i.e. progression of math concepts and topics is independent and achieving competency is now perceived as desirable). Studies have also indicated that when lessons are gamified, students attention spans increase, particularly those with learning disabilities such as ADHD, are less disruptive and more focused.

One other benefit, students are now able to approach math problems with a stronger backbone and thicker skin. In a gaming framework there is now a stronger tendency to not give up on seemingly impossible questions or the failures to grasp concepts in Algebra 2, or Precalculus so that one can advance on to university statistics.

Lastly, gamification provides a more intuitive and easier way to track the progress of each student. Through the use of metrics via progress charts/bars, instructors are able to immediately gauge strengths and weakness of a student and consequently address student needs by providing more support to areas requiring improvement and rewarding students for their success in areas they have mastered. Real time statistics on students’ progress can also be beneficial to instructors in providing instant assistance to students. Instant feedback on student progress is crucial as it also informs the instructor that perhaps the method of delivering the content, or the content itself, needs to be revised and retaught in an alternate way to improve student understanding. Lastly, indicators of progress can also serve as a motivating tool for students when they observe levels or tasks progressively being completed.

Gamification of education appears inevitable but will the voices of students and learners be heard? With the myriad of benefits for students and instructors alike and a strong presence of digital learning in this era, it is likely that gamified demands will be supplied. Parental support is also evident as two-thirds of parents in various communities claim that with the effective use of technology, children develop college and career skills (ProjectTomorrow). According to a 2015 report, 78% of middle school students use online videos while 61% of those students play online games, all to achieve learning goals (ProjectTomorrow). Recent studies also show that 89% of learners would be more motivated to engage in E-learning if a point system was incorporated into the learning process. Furthermore 60% of learners also indicated that the use of leaderboards would also increase motivation by encouraging healthy competition among peers.

Altogether these statistics suggest a very strong movement not just towards e-learning, but also to gamification. Certainly this is a trend that will continue to gain more traction in at a post-secondary level where blended learning and online learning provide rich environments and opportunities to gamify.

One clear example of this can be observed in the construction of online academic institutions. Our main site, StudyPug utilizes gaming elements such as progress trackers, badges, and avatars as tools to motivate students and to keep them on track to completing their courses. Although all of our content is accessible 24/7 and acquiring help is easy, being purely online in format requires a placeholder to spur students on while they are learning in real time.

Like every rising trend, there will be a resistance. Such deserves attention as well as it can be a very common misconception that all classrooms need gamification in order to make learning fun. As pointed out by Phil Aldridge, a passionate math teacher, classrooms should be inherently gamified – at least a good classroom in our opinion. Much like gaming, students should receive timely feedback, be rewarded for accomplishing goals, be encouraged for effort and progress, be provided a set of guidelines (i.e. syllabus and/or learning objectives), have a variety of fun and engaging learning activities, and face the consequences when they fail to follow the rules.

Such a sentiment is no secret and is shared by one of the godfathers and earliest pioneers of gamification, Yu Kai Chou – *“…**Truly good gamification often will not let anyone feel something is “gamified…”* So the next time you feel like you are getting flack for “playing a game”, be sure to defend not only the burning castle or the modifications being made to your falcon-X-heavy-esque spaceship, but also your very right to learn and achieve academic goals while doing so.

In a survey by publicagenda.org, it was noted that 65% of the parents surveyed said that they wished they could be doing more when it came to being actively involved within their child’s education. It was also noted that the parental involvement within a child’s education seems to drop slightly during the later grades.

It can be a tricky balancing act for parents who want the best for their children but don’t want to pressure or burden them with high expectations for their future. How can parents become more involved in their child’s education, what can be done to help them prepare for college, and is middle school too soon to even think about college?

The answer to the last question is simple, it’s never too soon to think about your child’s future. There are always things to consider and things you can do to help nurture your child’s development. To that end, we have broken down the top three things parents should consider when thinking about their son/daughter’s academic future.

**Plan Ahead and Highlight the Benefits of a Good Education**

We briefly touched on this in the intro to the article, planning ahead and thinking about college from an early age can be a great tool in structuring your child’s educational development. With clearly defined long term goals, you and your child can map out effective strategies that highlight areas for improvement, incorporate them into their study/revision time.

Effective planning in middle school can help make the transition into high school much more comfortable for your child. Knowing where their passions and strengths lie, will inform them on what classes to take in high school when it comes time to make that decision.

Understandably, convincing your child to think so far ahead and plan for their future career can be a tricky endeavour, but it’s worth planting those seeds early on so that they can make more informed decisions about their lives. Having said that, it is important that you don’t pressure your child into mapping our their entire future just yet. This is a time for them to think about their lives and to explore every available option. Nurture that process and encourage them to dream big, emphasising that a good education can go a long way.

Afterall, that’s what people go to college for. Beyond the degree, college allows people to expand their horizons, meet new people, and explore their options. Those experiences will better prepare them for adult life and a college degree can by tremendously beneficial for their careers. Regardless of the subject matter, a degree shows employers that they are capable of working independently, completing assignments, meeting deadlines and organising themselves.

**Build Their Skills in English and Math **

**
**If you’re a parent thinking about your child’s educational future and what to know how to prepare for college, you should know that for most colleges, a passing grade in English and math is required. Depending on the subject matter, there may be also be an additional requirement for science.

Students currently in middle school will most likely be studying pre-algebra, as it tends to be taught to students in 7^{th} or 8^{th} grade. The course is often used as an introduction to Basic algebra or Algebra 1, covering topics that will be built upon in grades 9, 10 and 11.

The reason students are exposed to algebra at a young age is that it not only develops their problem solving skills, but it’s also one of the key things that colleges look for in potential students

The school system does a fairly good job at teaching students the basics and exploring more advanced forms of algebra, but we’d strongly recommend looking into a study guide for pre-algebra to help cement your child’s understanding of the topic. In doing so, they will have a much better grasp of algebra when they’re introduced to it later in their academic life.

We’d offer the same advice for English studies too. Your child will need to have a good grasp of the english language and will be required to demonstrate this in the application process. Take advantage of the study tools available to them.

** **

**Study, Study, Study**

It cannot be stressed enough, studying outside of school hours can make a dramatic difference to your child’s performance in exams and assignments. Not only that, it will also help them to perform better in their day to day classes.

Getting into good study habits can also help them mentally prepare for college and the demands of the workload and the level of commitment needed to succeed. Setting aside an hour or two each evening to go over the class content or to prep for upcoming exams, can help them to better digest the information and more importantly, to retain it.

As mentioned above, a good grade in mathematics will go a long way in providing opportunities for your child at high school and college level. Many courses require basic math, so it would be wise to spend some time studying areas for improvement outside of school hours.

Sites like StudyPug, offer an extensive collection of revision resources and test prep materials for all aspect of mathematics that are covered in school and college. Their content is delivered in easy to follow video tutorials, not too dissimilar to youtube. These videos are accessible 24/7 and you can pause, rewind, or fast forward the content, allowing your child to learn at their own pace.

The video format is a lot more palatable for students as it’s a platform their much more comfortable with. Doing away with textbooks and the rather complex terminology that can turn students away, each subject is broken down into step-by-step guides that help to convey the content in a simple and engaging way.

Delivering information in a way that’s appealing is half the battle in getting your child interested in studying, so it’s vital that you find a delivery mechanism that works them. Keep in mind that this is something you’d ideally like them to be doing on a regular basis.

Following the three steps mentioned in this article will ensure that your child is on the right path for academic success. Planning to study effectively and working together to find revision strategies that work for them could greatly improve their chances of getting into their preferred college.

]]>Math sequences can be discovered in your everyday life. One’s earliest recollection of a math sequence probably began at the age of two, when you started counting to ten. A more relevant memory today might be one of you reciting your times table. Simply described, a math sequence is a group of numbers that follow a specific pattern. Each number within a mathematical sequence is identified as a term. Patterns within mathematical sequences provide the key that reveals a common thread of how each number is connected to one another.

Broadly speaking, mathematical sequences can be categorized into two major groups: arithmetic sequences (i.e. arithmetic progression) and geometrics sequences (i.e. geometric progression). You probably began differentiating between these two types of sequences while completing grade 11 math. Arithmetic sequences are defined by a string of consecutive numbers that have a common difference between them. A familiar example would encompass the sequence of house numbers along a street you happen to drive by (e.g. 102, 104, 106, 108 etc., common difference is 2.). Geometric sequences on the other hand encompass a succession of numbers that share a common ratio between them. For example, consider the half-life of a radioactive element – the common ratio is 2, and in a fixed amount of time, radioactive decay disintegrates the element by half.

*Fibonacci Sequence*

Not all mathematical sequences are easily separated into these two branches but are still incredibly interesting and applicable. One particular sequence that has garnered a strong reputation in it’s utility and ubiquity is the Fibonacci sequence. In the Fibonacci sequence, each number can be derived from the sum of the two preceding numbers. (i.e. 0, 1, 1, 2, 3, 5). Now where exactly have we observed this sequence of numbers over and over again in our everyday life? Not quite in numeric form as you might have guessed….

*Fibonacci Spiral*

Also known as “the Golden Spiral”, in it’s most rudimentary form is outlined over a collection of squares that bear the dimensions of the Fibonacci sequence (1 x 1, 2 x 2, etc. see below). What appears to be an ordinary spiral forms unique quarter circles over each square that increases in size in according to the Fibonacci sequence. All together these squares are fitted perfectly within what is known as a “golden rectangle”. The Fibonacci spiral however is not limited to rectangles (although this is the most common shape used to depict the Fibonacci spiral) and can be found in a multitude of geometric shapes (e.g. triangles) and real world entities, as we shall soon discover.

*Golden Rectangle with the first few terms from the Fibonacci sequence. Each number indicates the dimensions of the square it is in Adapted from Dicklyon*

*The Golden Ratio*

So what exactly is so grand and “Golden” about these shapes? Unsurprisingly, the astounding property of these shapes stems from their “Golden ratios” – 1:1.618. This value is originally derived from the ratio of two consecutive numbers in the **Fibonacci sequence**. Earlier on in the sequence, the ratio approaches 1.618, but is particularly more evident later in the sequence as the numbers grow larger and larger, the ratio between two consecutive numbers approaches a near perfect 1:1.618 ratio. So returning back to our example of the “golden rectangle”, what we would actually ascertain is that the ratio of its dimensions are 1:1.618.

*Beauty in the eye of the Fibonacci Sequence *

Many of the real world illustrations of the Fibonacci sequence portray what most would consider beauty. The most famous examples are found in nature. Several species of plants, in which the exact number of flower petals are always found to one that is found in the Fibonacci sequence. A more renowned example can be identified in the beloved sunflower. One will often discover that when counting the seed spirals of a sunflower in one direction (e.g. clock-wise), they will arrive at a number found in the Fibonacci sequence. On the other hand, counting the number of seed spirals in the opposite direction (e.g. counter-clockwise) will result in tallying up to a successive or preceding number to the initial Fibonacci number counted in the clock-wise direction.

These Fibonacci spirals are found in a myriad of other atheistically pleasing entities such as galaxies, nautilus shells, hurricanes and as some would argue – in human faces.

The Fibonacci sequence/spirals and the rule of the golden ratio were said to also guide some of the most famous and celebrated pieces of classical (e.g. such as the Mona Lisa and the Last Supper) and modern arts

While the case of beauty may be rarely disputed for things of nature, the definition of beauty in humans is often in constant contention. In accordance to the Fibonacci sequence/spiral and the golden ratio, the most desirable human face has features of which proportions closely adhere to the golden ratio and spacing/distribution of features follows the squares found within golden rectangle. But how much can we trust this seemingly perfect sequence? How “golden” is this “golden ratio”? Does it really provide an equation of beauty or are we faced with a beast? Although the Fibonacci sequence existed long before the first fairy tales, their relevance has never been more prevalent in this day and age of vanity and the tireless pursuit of youth.

Let us consider some examples of “au-naturel” Fibonacci perfection in faces of celebrities:

1. Shinee Minho

2. Amber Heard

3. Kim Kardashian

Portrait of Kim Kardashian, Photo Credits: Glenn Francis, AKA Toglenn

Here popular vote and math converge in agreement that what we observe here is nothing short of beauty.

What about some other examples of faces that initially fail to meet the pre-requisites of “golden” perfection? Website designer, Igor K demonstrates how even after processing each portrait with multiple iterations/”corrections” of Fibonacci spirals and golden ratios, what results is still a mess: the kind that typically results when you’re playing around in Photobooth on your Mac OS…

1. Sylvester Stallone

2. Nicholas Cage

3. Brian Cranston

Altogether these examples are a sure and striking beast of a difference compared to the eye candy previously observed.

*Are we Golden? *

So with a holistic perspective of the Fibonacci sequence/spiral and Golden ratio in practice, does this warp your view of the grandeur of these “golden rules”? While still an incredibly simple but complex, mathematical sequence, the Fibonacci sequence is still worthy of praise in the beauty that it does reveal to us in the real world. Our hope is that it also continues to receive recognition for its inability to paint a perfect picture every time. Such is a testament to the fact that extensive surgical makeovers don’t provide a quick fix for everyone and that perhaps the golden saying that ought to be taken to heart is that “Beauty is in the eye of the beholder”.

Interested in putting your face to the Fibonacci test? Check out these apps/simulation to see how well you fair!

- https://play.google.com/store/apps/details?id=com.facial.beauty.analysis&hl=en
- https://www.phimatrix.com/download/

For more help on other topics related to Math, visit StudyPug for a complete list of online tutoring services we provide.

]]>If you’re like the millions of people the world over who sat in math class wondering why you’d ever need to know about algebra, you might be surprised to learn that conscious or not, are day to day lives are consumed with algebraic formulae and equations.

From morning routines to scoring goals in soccer, we’ve provided 4 ways in which you’ve used algebra in your daily routines without even realizing it. Let’s have a look!

Picture the scene, its Saturday afternoon and you’re at the local grocery store doing your weekly shop, you’ve got your list of items you want to buy, and you have the number in your head of what you expect to spend.

You go to buy a product (let’s say breakfast cereal) and you find that there’s a weekly special, buy 2 for the price of 3! Now you need to figure out how this deal effects your budgeting. You intended to buy one box, but the value has changed. You need to figure out if spending the extra to gain more cereal will still allow you to complete your shopping list.

Have I got enough money to cover the items in my cart? If this costs X amount and I buy 2 how much money will be left to buy Y?

It’s a process that takes a mere few seconds to process in your head but it’s algebra never the less. There exists a formula that can be written down to help explain your thinking.

Again, imagine the scenario, you wake up in the morning to the sound of your alarm, you get ready for work and realize you’re craving a nice warm bagel to start your day. You could make one at home but you like the freshly made ones from the store. Do you have time pick one up? Is it more time/cost effective to prepare one from home? How will potential traffic effect your decision? If its gridlocked on the highway, would the backroads be faster?

Much like the grocery store scenario, we have values in our head and using the variables in play, we’re able to determine the best course of action. Whether it’s the time it takes to successfully complete a morning coffee run or the plans we make for a faster commute, algebra has helped us navigate these early morning decisions whether you’ve realized it or not.

So many elements in the process of cooking dinner rely on some form of algebra. The mixture of time with proportions and values will all need to be considered when making a meal and to make it successfully. Put simply, algebra is used to assed what temperature the oven should be at and for how long the food should cook, mental algebra is everywhere!

Lastly, when you play a game like soccer or hockey, how do you know just how much power to put into a pass for it to meet its intended target? How do you know at which angle to strike the ball/puck and what force is required to successfully beat the keeper and score?

The answer requires you to quickly process the force required, to account for air resistance, and to project the trajectory of the ball upon contact. We don’t stop to pull a pen out and draw the formula to solve this, we simply do it and learn it through trial and error (training). We process this information overtime through repetition and build mental formulae in our minds to solve problems in the moment. Essentially, we understand what is needed to achieve our desired goal. For example, when you’re one on one with the keeper who is positioned to the left, you’ll have an idea of how to hit the ball with X amount of power at a specific angle to achieve Y.

The same can be said of video games too. Whether it’s a light tap on the jump button, feathered acceleration around a corner in a racing game, or shooting an enemy, algebra plays a pivotal role in our understanding of the game and what is required to be successful.

**In summary:**

These are just a few examples of everyday actions that use algebra and there are many more that I’m sure you could think of too. I would argue that the time spent in class finding the value of X has better prepared us to use our minds to solve everyday issues regardless of what ‘X’ is.

If you’re currently studying algebra or algebra 2 in class, don’t curse! You’re developing your ability to problem solve and it’s a skill that will greatly benefit you later in life. It can be a daunting topic to cover but once you have a firm understanding of the basic principles, it can be a rewarding challenge to tackle complex equations.

If you’re struggling to get your head around algebra and need some help, companies like StudyPug have developed a range of excellent resources surrounding the subject and offer free tutorials alongside a subscription service that provides additional lessons and relevant exam prep materials. Their content covers everything you see in your textbooks and will explain everything that you’ve been taught in the classroom. Give them a visit today and see how they can help you succeed.

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**Background – Smooth Billed Ani and Pukeko**

The Smooth-billed Ani is a neo-tropical cuckoo found in the Caribbean, while the Pukeko is a swamp-hen that is found in New Zealand. Cooperative-breeding animals are unique in that they often involve individuals caring for offspring that are not their own. This involves sharing parental duties of feeding, protecting and raising the young. What was particularly interesting about these two species of birds was that they participated in a behaviour known as joint-nesting, where two or more pairs (male and female) share a common nest. Now as cooperative as these birds are, there is also often a large amount of conflict. This involves egg tossing and burial (Smooth-billed Anis), and aggressive bouts during competition for territory.

What the researchers were particularly interested in was how were these dominance hierarchies formed and established in Pukeko? What feature of these birds kept the peace and assured cooperation in the midst of conflict? What was the significance of the differences in beak size/shape between individual Smooth-billed Anis?

**Findings and how we utilized geometry **

Drawing from our basic geometry knowledge and skills, researchers set out to measure and examine a variety of beak morphometrics in the Smooth-billed Ani as well as Shield morphometrics (ornament found above the Pukeko beak) of Pukeko. Utilizing caliper-taken measurements and conducting a series of manipulations of the ornament (in Pukeko), researchers then sought to identify how size may be linked to potential relationships such as dominance and sex.

What we discovered was 2-fold:

1. The fleshy, tear-drop-shape and structure of the Pukeko ornament was correlated to dominance. Individuals with larger “shields” were more dominant over those individuals with smaller shields. These “shields” were also dynamic and could change in size, suggesting that these dominance hierarchies could also be disrupted.

**2. Male-Smooth-billed-Anis have a prominent crest-shape in their beak as opposed to female Smooth-billed Anis. This was an interesting discovery as it indicated a case of sexual dimorphism in these species for the first time.**

**Summary**

While these are great scientific discoveries, the more important message from this account, is to not underestimate the utility of math – even the basics of simple geometry. Math can be found everywhere, even in the study and wonder of nature, such as in our feather friends.

For more help on other topics related to Math, visit StudyPug for a complete list of online tutoring service we provide.

**Credits:**

**https://macsphere.mcmaster.ca/handle/11375/20417**

**http://rspb.royalsocietypublishing.org/content/281/1775/20132680**

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