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Mathematic Strategies That Will Help You Win At Rock, Paper, Scissors - Digg

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Mathematic Strategies That Will Help You Win At Rock, Paper, Scissors - Digg

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Mathematic Strategies That Will Help You Win At Rock, Paper, Scissors

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Humans are "predictably irrational," and that is their downfall.

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Many extreme roller coaster these days have vertical loops. Have you noticed that these loops are never circular? Why is this?

They all, also, seem to have the same similar 'inverted teardrop' appearance. Why is this?

Clearly there is the same physics and mathematics involved in their designs. Let's take a look and see if we can derive a formula to describe their shape.

Why are roller coaster loops not circular?

The answer is that there is a force (provided by the rails), that is pushing the trucks of the coaster towards the center of the loop.

This force is called the Centripetal Force.

Newton's first Law of Motion of motion tells us that, without this force, the coaster would like to travel in a straight line and at constant speed. The centripetal force is pushing the coaster around in a circle. It is your body's (equal and opposite) reaction to this force, often referred to as the Centrifugal Force, that explains the feeling you get of being squashed into your seat (Newton's third Law of Motion).

Completing the trifecta, using Newton's second Law, we know that with this Force is an acceleration (we'll more strictly a rate of change of momentum but, hopefully, the mass of our coaster is constant!) When accelerating around in a circular path, the magnitude of the acceleration is proportional to the ratio of the speed squared over the radius of the curve.The tighter the radius, or the higher the speed, the greater the acceleration.

*In reality, coasters ride on tubular rails and are clamped by wheels on both sides (and passengers are safely harnessed into the carriages). Even if the coaster stopped completely, whilst you might be very uncomfortable, you'd still be safe.

At the limit, at the top of the curve, when the centripetal acceleration is provided entirely by gravity g, we can see that this occurs when Δh = r/2

At the top of the loop (we'll call this the 12 o'clock position), riders of the train will feel weightless. As the train continues around the loop to the 9 o'clock position, it will have dropped an additional Δh of r, resulting in a centripetal force of an additional +2g from that at the top.

At the bottom of the curve, there is an additional Δh of 2r, and this results in an additional acceleration at the bottom of +4g from that at the top, for a total centripetal acceleration of 5g. When this is added to the force experienced already experienced by the body, by gravity, this results in stress on the body of 6g. This is not good.

Even trained fighter pilots would eventually black out at sustained 6g loads (without the assistance of g-suits). This is supposed to be an amusement ride, not a James Bond like torture machine!

Clearly we need to rethink our roller coaster strategy. A circular loop has two very fundamental problems:

1. The g-forces that a body is exposed to at the bottom of the loop exceed what is safe (when travelling at a speed that just allows the car to sail over the top of the loop).

2. There will be a very rapid onset of this acceleration as the train starts the loop: The train would be playfully falling down the track, then wham! it will hit a 6g acceleration as it hits the bottom of the loop and starts its way up!

We need to come up with a better plan to address these two issues. We need to reduce the maximum g-force we expose our riders to, and also to come of with a more smooth mechanism for how to wind up and down these forces.

Hold on, what if we are allowed to hang upside down?

As noted above, coasters are on rails, how about if we slowed the train down further so that it was (just about) at rest at the top?

The passengers would experience -1g as they hung from their straps at the top of the loop. This would result in a 5g load at the bottom (still unacceptable), and we've also not solved the jerky onset of this g load at the bottom.

Not a solution!

The solution to the first of the two problems (too high a g-load) is to decide on the maximum acceleration you wish to expose your riders to, then modify the track profile to keep this acceleration constant. As the coaster gains height, it loses speed. With this lower speed, the curvature of the track can be decreased to keep the needed centrifugal acceleration.

We know that the centripetal acceleration is proportional to v²/r, as the velocity reduces then we can decrease the radius to keep the acceleration a constant.

Before continuing with the coasters, let's take a short detour and investigate this strategy further by looking at a simpler similar problem. One all in the horizontal plane where we don't need to worry about gravity in the vertical plane.

Imagine we have a straight railroad track. A train chugging along this track at constant speed is not experiencing any lateral forces or accelerations.

What happens when this train runs into a corner?

As soon as the curve is reached the train is subject to a lateral acceleration of v²/r. This appears with no warning or run up.

This might be fine at low speeds, but as the speed increases, the forces can be considerable. Nevermind the danger of derailment, or damage to the train, it also brutally uncomfortable for the passengers.

If you ran a model train track at excess speeds you probably found that it was at the entry to curves that the train derailed. A similar issue occurs if you enter a corner too fast with your model car race set.

What we want to do is gradually change the radius of the curve to slowly build up the acceleration (and thus ramp up the force gradually). There are an infinite number of ways for how we might select the profile for how the force is increased, but an obvious suggestion is a linear change.

Selecting a profile in which the force changes linearly with time (under the assumption that the velocity of the train does not change), can be achieved with a curve whose radius also changes linearly with time. This smooth transition is sometimes called an easement.

Image: M1 J10a UK.GOV

Image: Bing Maps © 2014 Microsoft Corportion © 2013 BLOM

A first order approximate solution to the ideal roller coaster loop shape can be created by simply bolting together circular tracks of the appropriate radii.

A tight radius at the top where the speed is low, and a large radius at the bottom where the speed is high.

This is certainly a great improvement over an entirely circular arc, but still has the jerk accelerations at the bottom and at each of the transitions where the radius of the track changes.

Using what we learned from from our Clothoid investigation we can modify the radius of curve smoothly to keep a constant centripetal acceleration all around the loop. The shape of such a track is show to the right.

A Clothoid curve provides a ride that exposes the coaster to a constant centripetal acceleration, but the rider does not experience a constant g-ride because of gravity (and the changing relative direction this force acts).

The radius of the track at any point on the loop can be defined by the following equation:

There is a resolved component of gravity acting on the coaster depending on it's position around the curve. The Radius at any point can be determined using the following equation.

Where v is the speed going into the loop, h is the height from the bottom of the loop, a is the constant acceleration you want the riders to be exposed to, g is the gravity, and θ is the angle position around the loop that the truck is.

Image: Albin Olsson

In reality, we can't ignore friction and air resistance, so these must be taken into consideration as well as a healthy safety measure. In addition, a real coaster is not a point mass, so there is rotational inertia to consider.

The coaster is also many cars long, so not all parts of the car are riding on the same radius. For instance, after cresting over the top and on the way down, the cars will be accelerating and picking up speeds. At this point, the last car still has not been passed the point of tightest radius. When it does pass, it will be travelling quicker and thus experience a higher acceleration.

Roller coaster enthusiasts exploit these differences and compare notes as to the better places to sit on each ride to maximize the hang times, g-forces and ride experiences.

An opposite kind of problem occurs for aerobatic pilots.

A perfectly executed vertical loop starts at a fixed point, traces a perfect circle and returns to the same position, heading and altitude.

Unlike roller coasters, planes are powered and add energy to the system, but they still trade energy between kinetic and potential. As their speed decreases, so does the aerodynamic lift generated from their wing surfaces (Lift is proportional to speed squared).

As the speed bleeds off, rather than tightening the radius of the loop, the aim is to keep the radius constant.

An aerobatic pilot will, from horizontal level flight, pull the stick back hard (and suddenly) to initiate the loop (depending on the radius, exerting many g's of load). He will use skill (and practice) to relax the stick to keep the arc circular as the speed bleeds off, all the way to the top. He will, then, reverse this strategy on the way down, ending the loop at the bottom pulling hard again.

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Why aren't roller coaster loops circular? Why do they all look like inverted tear-drops?

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Is Algebra Necessary? - The New York Times

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Is Algebra Necessary? - The New York Times

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Shirley Bagwell, a longtime Tennessee teacher, warns that "to expect all students to master algebra will cause more students to drop out." For those who stay in school, there are often "exit exams," almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below "proficient," along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.

California's two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.

"There are students taking these courses three, four, five times," says Barbara Bonham of Appalachian State University. While some ultimately pass, she adds, "many drop out."

Another dropout statistic should cause equal chagrin. Of all who embark on higher education, only 58 percent end up with bachelor's degrees. The main impediment to graduation: freshman math. The City University of New York, where I have taught since 1971, found that 57 percent of its students didn't pass its mandated algebra course. The depressing conclusion of a faculty report: "failing math at all levels affects retention more than any other academic factor." A national sample of transcripts found mathematics had twice as many F's and D's compared as other subjects.

Nor will just passing grades suffice. Many colleges seek to raise their status by setting a high mathematics bar. Hence, they look for 700 on the math section of the SAT, a height attained in 2009 by only 9 percent of men and 4 percent of women. And it's not just Ivy League colleges that do this: at schools like Vanderbilt, Rice and Washington University in St. Louis, applicants had best be legacies or athletes if they have scored less than 700 on their math SATs.

It's true that students in Finland, South Korea and Canada score better on mathematics tests. But it's their perseverance, not their classroom algebra, that fits them for demanding jobs.

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that "mathematical reasoning in workplaces differs markedly from the algorithms taught in school." Even in jobs that rely on so-called STEM credentials - science, technology, engineering, math - considerable training occurs after hiring, including the kinds of computations that will be required. Toyota, for example, recently chose to locate a plant in a remote Mississippi county, even though its schools are far from stellar. It works with a nearby community college, which has tailored classes in "machine tool mathematics."

That sort of collaboration has long undergirded German apprenticeship programs. I fully concur that high-tech knowledge is needed to sustain an advanced industrial economy. But we're deluding ourselves if we believe the solution is largely academic.

A skeptic might argue that, even if our current mathematics education discourages large numbers of students, math itself isn't to blame. Isn't this discipline a critical part of education, providing quantitative tools and honing conceptual abilities that are indispensable - especially in our high tech age? In fact, we hear it argued that we have a shortage of graduates with STEM credentials.

Of course, people should learn basic numerical skills: decimals, ratios and estimating, sharpened by a good grounding in arithmetic. But a definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or above. And if there is a shortage of STEM graduates, an equally crucial issue is how many available positions there are for men and women with these skills. A January 2012 analysis from the Georgetown center found 7.5 percent unemployment for engineering graduates and 8.2 percent among computer scientists.

Peter Braunfeld of the University of Illinois tells his students, "Our civilization would collapse without mathematics." He's absolutely right.

Algebraic algorithms underpin animated movies, investment strategies and airline ticket prices. And we need people to understand how those things work and to advance our frontiers.

Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship. What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.

What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It's true that mathematics requires mental exertion. But there's no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis.

Many of those who struggled through a traditional math regimen feel that doing so annealed their character. This may or may not speak to the fact that institutions and occupations often install prerequisites just to look rigorous - hardly a rational justification for maintaining so many mathematics mandates. Certification programs for veterinary technicians require algebra, although none of the graduates I've met have ever used it in diagnosing or treating their patients. Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn't figure in the clinical curriculum, let alone in subsequent practice. Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession's status.

It's not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it's not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.

I WANT to end on a positive note. Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic. But for most adults, it is more feared or revered than understood. It's clear that requiring algebra for everyone has not increased our appreciation of a calling someone once called "the poetry of the universe." (How many college graduates remember what Fermat's dilemma was all about?)

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call "citizen statistics." This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted - and include discussion about which items should be included and what weights they should be given.

This need not involve dumbing down. Researching the reliability of numbers can be as demanding as geometry. More and more colleges are requiring courses in "quantitative reasoning." In fact, we should be starting that in kindergarten.

I hope that mathematics departments can also create courses in the history and philosophy of their discipline, as well as its applications in early cultures. Why not mathematics in art and music - even poetry - along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet. If we rethink how the discipline is conceived, word will get around and math enrollments are bound to rise. It can only help. Of the 1.7 million bachelor's degrees awarded in 2010, only 15,396 - less than 1 percent - were in mathematics.

I've observed a host of high school and college classes, from Michigan to Mississippi, and have been impressed by conscientious teaching and dutiful students. I'll grant that with an outpouring of resources, we could reclaim many dropouts and help them get through quadratic equations. But that would misuse teaching talent and student effort. It would be far better to reduce, not expand, the mathematics we ask young people to imbibe. (That said, I do not advocate vocational tracks for students considered, almost always unfairly, as less studious.)

Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions. Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven't found a compelling answer.

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Shirley Bagwell, a longtime Tennessee teacher, warns that "to expect all students to master algebra will cause more students to drop out." For those who stay in school, there are often "exit exams," almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

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The Myth of 'I'm Bad at Math' - The Atlantic

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The Myth of 'I'm Bad at Math' - The Atlantic

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Vasily Fedosenko / Reuters

"I'm just not a math person."

We hear it all the time. And we've had enough. Because we believe that the idea of "math people" is the most self-destructive idea in America today. The truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career. Worse, you may be helping to perpetuate a pernicious myth that is harming underprivileged children-the myth of inborn genetic math ability.

Is math ability genetic? Sure, to some degree. Terence Tao, UCLA's famous virtuoso mathematician, publishes dozens of papers in top journals every year, and is sought out by researchers around the world to help with the hardest parts of their theories. Essentially none of us could ever be as good at math as Terence Tao, no matter how hard we tried or how well we were taught. But here's the thing: We don't have to! For high-school math, inborn talent is much less important than hard work, preparation, and self-confidence.

How do we know this? First of all, both of us have taught math for many years-as professors, teaching assistants, and private tutors. Again and again, we have seen the following pattern repeat itself:

1. Different kids with different levels of preparation come into a math class. Some of these kids have parents who have drilled them on math from a young age, while others never had that kind of parental input.

2. On the first few tests, the well-prepared kids get perfect scores, while the unprepared kids get only what they could figure out by winging it-maybe 80 or 85%, a solid B.

3. The unprepared kids, not realizing that the top scorers were well-prepared, assume that genetic ability was what determined the performance differences. Deciding that they "just aren't math people," they don't try hard in future classes, and fall further behind.

4. The well-prepared kids, not realizing that the B students were simply unprepared, assume that they are "math people," and work hard in the future, cementing their advantage.

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Thus, people's belief that math ability can't change becomes a self-fulfilling prophecy.

The idea that math ability is mostly genetic is one dark facet of a larger fallacy that intelligence is mostly genetic. Academic psychology journals are well stocked with papers studying the world view that lies behind the kind of self-fulfilling prophecy we just described. For example, Purdue University psychologist Patricia Linehan writes:

A body of research on conceptions of ability has shown two orientations toward ability. Students with an Incremental orientation believe ability (intelligence) to be malleable, a quality that increases with effort. Students with an Entity orientation believe ability to be nonmalleable, a fixed quality of self that does not increase with effort.

The "entity orientation" that says "You are smart or not, end of story," leads to bad outcomes-a result that has been confirmed by many other studies. (The relevance for math is shown by researchers at Oklahoma City who recently found that belief in inborn math ability may be responsible for much of the gender gap in mathematics.)

1. You have a certain amount of intelligence, and you really can't do much to change it.

2. You can always greatly change how intelligent you are.

They found that students who agreed that "You can always greatly change how intelligent you are" got higher grades. But as Richard Nisbett recounts in his bookIntelligence and How to Get It,they did something even more remarkable:

Dweck and her colleagues then tried to convince a group of poor minority junior high school students that intelligence is highly malleable and can be developed by hard work...that learning changes the brain by forming new...connections and that students are in charge of this change process.

The results? Convincing students that they could make themselves smarter by hard work led them to work harder and get higher grades. The intervention had the biggest effect for students who started out believing intelligence was genetic. (A control group, who were taught how memory works, showed no such gains.)

But improving grades was not the most dramatic effect, "Dweck reported that some of her tough junior high school boys were reduced to tears by the news that their intelligence was substantially under their control." It is no picnic going through life believing you were born dumb-and are doomed to stay that way.

For almost everyone, believing that you were born dumb-and are doomed to stay that way-is believing a lie. IQ itself can improve with hard work. Because the truth may be hard to believe, here is a set of links about some excellent books to convince you that most people can become smart in many ways, if they work hard enough:

So why do we focus on math? For one thing, math skills are increasingly important for getting good jobs these days-so believing you can't learn math is especially self-destructive. But we also believe that math is the area where America's "fallacy of inborn ability" is the most entrenched. Math is the great mental bogeyman of an unconfident America. If we can convince you that anyone can learn math, it should be a short step to convincing you that you can learn just about anything, if you work hard enough.

Is America more susceptible than other nations to the dangerous idea of genetic math ability? Here our evidence is only anecdotal, but we suspect that this is the case. While American fourth and eighth graders score quite well in international math comparisons-beating countries like Germany, the UK and Sweden-our high-schoolers underperform those countries by a wide margin. This suggests that Americans' native ability is just as good as anyone's, but that we fail to capitalize on that ability through hard work. In response to the lackluster high school math performance, some influential voices in American education policy have suggested simply teaching less math-for example, Andrew Hacker has called for algebra to no longer be a requirement. The subtext, of course, is that large numbers of American kids are simply not born with the ability to solve for x.

We believe that this approach is disastrous and wrong. First of all, it leaves many Americans ill-prepared to compete in a global marketplace with hard-working foreigners. But even more importantly, it may contribute to inequality. A great deal of research has shown that technical skills in areas like software are increasingly making the difference between America's upper middle class and its working class. While we don't think education is a cure-all for inequality, we definitely believe that in an increasingly automated workplace, Americans who give up on math are selling themselves short.

Too many Americans go through life terrified of equations and mathematical symbols. We think what many of them are afraid of is "proving" themselves to be genetically inferior by failing to instantly comprehend the equations (when, of course, in reality, even a math professor would have to read closely). So they recoil from anything that looks like math, protesting: "I'm not a math person." And so they exclude themselves from quite a few lucrative career opportunities. We believe that this has to stop. Our view is shared by economist and writer Allison Schrager, who has written two wonderful columns in Quartz ( here and here), that echo many of our views.

One way to help Americans excel at math is to copy the approach of the Japanese, Chinese, and Koreans. In Intelligence and How to Get It, Nisbett describes how the educational systems of East Asian countries focus more on hard work than on inborn talent:

1. "Children in Japan go to school about 240 days a year, whereas children in the United States go to school about 180 days a year."
2. "Japanese high school students of the 1980s studied 3 ½ hours a day, and that number is likely to be, if anything, higher today."
3. "[The inhabitants of Japan and Korea] do not need to read this book to find out that intelligence and intellectual accomplishment are highly malleable. Confucius set that matter straight twenty-five hundred years ago."
4. "When they do badly at something, [Japanese, Koreans, etc.] respond by working harder at it."
5. "Persistence in the face of failure is very much part of the Asian tradition of self-improvement. And [people in those countries] are accustomed to criticism in the service of self-improvement in situations where Westerners avoid it or resent it."

We certainly don't want America's education system to copy everything Japan does (and we remain agnostic regarding the wisdom of Confucius). But it seems to us that an emphasis on hard work is a hallmark not just of modern East Asia, but of America's past as well. In returning to an emphasis on effort, America would be returning to its roots, not just copying from successful foreigners.

Besides cribbing a few tricks from the Japanese, we also have at least one American-style idea for making kids smarter: treat people who work hard at learning as heroes and role models. We already venerate sports heroes who make up for lack of talent through persistence and grit; why should our educational culture be any different?

Math education, we believe, is just the most glaring area of a slow and worrying shift. We see our country moving away from a culture of hard work toward a culture of belief in genetic determinism. In the debate between "nature vs. nurture," a critical third element-personal perseverance and effort-seems to have been sidelined. We want to bring it back, and we think that math is the best place to start.

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purdue university psychologist patricia linehan

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great mental bogeyman of an unconfident america

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japanese high school students of the 1980s

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"I'm just not a math person." We hear it all the time. And we've had enough. Because we believe that the idea of "math people" is the most self-destructive idea in America today. The truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career.

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Algebra 1: Lesson 1 Linear Equations: Equations with x and y (Simplifying Math) - YouTube

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Algebra 1: Lesson 1 Linear Equations: Equations with x and y (Simplifying Math) - YouTube

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Algebra 1: Lesson 1 Linear Equations: Equations with x and y (Simplifying Math)

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Algebra 1 lesson about Linear Equations (Equations with x and y). This lesson shows how to transform and make a table of values for a linear equation. If you find it helpful subscribe, like and share.

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What is a function? (video) | Functions | Khan Academy

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What is a function? (video) | Functions | Khan Academy

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Functions assign a single output for each of their inputs. In this video, we see examples of various kinds of functions.

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Solving equations | Algebra I | Math | Khan Academy

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Solving equations | Algebra I | Math | Khan Academy

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Learn Algebra 1 skills for free! Choose from hundreds of topics including functions, linear equations, quadratic equations, and more. Start learning now!

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IXL - Evaluate variable expressions involving rational numbers (Algebra 1 practice)

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IXL - Evaluate variable expressions involving rational numbers (Algebra 1 practice)

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IXL | Evaluate variable expressions involving rational numbers | Algebra 1 math

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Improve your math knowledge with free questions in "Evaluate variable expressions involving rational numbers" and thousands of other math skills.

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IXL - Evaluate variable expressions involving integers (Algebra 1 practice)

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IXL - Evaluate variable expressions involving integers (Algebra 1 practice)

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IXL | Evaluate variable expressions involving integers | Algebra 1 math

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Improve your math knowledge with free questions in "Evaluate variable expressions involving integers" and thousands of other math skills.