Today, we're excited to announce two of our most-anticipated features, review questions and automated suggestions for our online precalculus and calculus content. Give it a try, or sign up for an account to get started!
At any time, you can ask our system for a short quiz to test your newly-gained knowledge. We will choose questions from our large (and still growing) database of review questions that are best-suited for you based on how you've interacted with other lessons and questions. We'll make sure to choose questions on topics that you need to review the most, and at an appropriate level of difficulty.
As you progress through the course, you'll be able to see which topics you excel at and which ones you struggle on. Our system notices these things too, and can automatically highlight lessons you ought to review if you are having trouble.
With automated suggestions and review questions, there is now no need to have to browse through endless lists of videos or tables of contents just to find what you should look at next. School Yourself can automatically find out what your weak points are and help you master them!
How much could you expect to win from the Mega Millions lottery?
In case you haven't heard, the Mega Millions lottery reached an astonishing $648 million this week. There were two winning tickets, but only one winner has come forward so far.
Here we'll look at the question of how much money you could expect to win from this lottery. In other words, for every dollar you spend playing the Mega Millions lottery, how much money could you expect to get back?
For reference, here's how some other investments stack up:
Here we'll look at the question of how much money you could expect to win from this lottery. In other words, for every dollar you spend playing the Mega Millions lottery, how much money could you expect to get back?
For reference, here's how some other investments stack up:
- If you invested $1.00 in the stock market a year ago, you would have $1.23 today. So for every dollar you invested, you would have made an additional 23 cents.
- If you play roulette at a casino, for every $1.00 you bet, you make, on average, $0.95 back. So for every dollar you bet, you would lose 5 cents.
So it looks like investing in the stock market is a pretty good way to spend a dollar, while playing roulette will lose you some money. How does playing the lottery stack up?
First, we need to know how likely it is for you to win different prizes. In each Mega Millions game you play, you select 5 numbers between 1 and 75, and then one addition number between 1 and 15. Here's what five games look like:
We could calculate the probability of getting all 6 numbers right (using a branch of math called combinatorics), but we'll skip that step for now. The Mega Millions lottery is very up-front with the probabilities of winning. Here's the chart they have on their website, where the left-most column is how many of the five numbers you match (form 1 to 75), and the next column is whether you match that sixth number:
They advertise at the bottom that your chances of "winning any prize" are 1 in 14.7, or about 6.8%, which sounds pretty good. Unfortunately, just about all of that 6.8% is taken up by prizes of $5 or less.
Anyway, let's figure this out: for every $1 you spend on the Mega Millions, how much could you expect to win? In other words, we want to find the expected value of your cash winnings.
Here's a simpler example: suppose you roll a fair die (numbered 1-6), and you win a number of dollars equal to the number that comes up on a roll. So you have a 1 in 6 chance of getting $1, $2, $3, $4, $5, or $6. What's the average amount (or expected value) of money you'd make from this game? To find out, you can add up the probabilities of each event by the outcome of that event. In other words, you can expect to make:
which equals $3.50.
The expected value of the Mega Millions lottery without the jackpot is only about $0.18. But if we include a jackpot of $648 million, the calculation becomes a little more challenging, because that prize can be split if there's more than one winning ticket.
As far as we know, the Mega Millions lottery doesn't publicly announce how many tickets are sold, but it's probably in the many hundreds of millions for big jackpots like this one. As more tickets get sold, it's more likely that there are multiple winners. Let's say enough tickets were sold so that we can expect (there's that word again) about 2 winners.
With these numbers, you can expect to make $1.42 off of every dollar you put in the lottery. Those are better results than the stock market! Any time the lottery exceeds about $250,000,000, your expected winnings are greater than $1.00, so it seems like a good idea to play.
But not so fast. While you're definitely spending a full dollar to play, your winnings are taxable. The government may not tax you much when you win $50 for matching a few numbers, but you can bet they'll tax your jackpot prize (or the $1 million prize for matching 6 numbers). Assuming a 40% tax rate, for every $1.00 you spend, you'll now make only about $0.90, meaning you'll lose 10 cents. And you actually lose at lot more (closer to 40 cents) if you decide to take your winnings all at once. The 10-cent loss is only if you let the lottery make smaller payouts to you over the course of 20 years (and they don't account for inflation).
So to summarize, for every $1.00 you spent on Mega Millions, you could expect to lose about 40 cents. You're better off playing roulette.
What is Bitcoin, anyway?
From Bitcoin's own website:
Bitcoin has the characteristics of money (durability, portability, fungibility, scarcity, divisibility, and recognizability) based on the properties of mathematics rather than relying on physical properties (like gold and silver) or trust in central authorities (like fiat currencies). In short, Bitcoin is backed by mathematics.
So what is Bitcoin, anyway?
Bitcoin is a relatively new currency (think dollars, euros, or yen), and was introduced by anonymous creators in 2009. First, how much is one bitcoin worth? As with all things: however much someone is willing to pay for it. Here's a graph of how much people have been willing to pay for one bitcoin over the last few months.
Bitcoin price has been fluctuating recently, and hit a maximum of over $1200 on December 4.
Unlike other currencies, Bitcoin is digital. You can't hold a Bitcoin in your hand. So how do you know how many Bitcoins (or "BTCs") you have? Well, it's a matter of public record! Every time someone pays someone else in Bitcoin, the transaction gets logged in a digital ledger (essentially a list of every single transaction) known as the "block chain."
So the list of who has how many BTCs is publicly available. But many Bitcoin addresses (long chains of numbers and letters) are anonymous -- everyone knows how many BTCs they have, but almost no one knows who owns the address.
In addition to an address, each Bitcoin user also has a private key, which he/she needs when making Bitcoin payments. These keys are additional long chains of numbers and letters, but they are not publicly available.
When Person A makes a payment to Person B in Bitcoin, the transaction gets added to the block chain. But how does Person B (and everyone else) know this is a real payment, as opposed to some fraud who's only pretending to be Person A (perhaps using their Bitcoin address)? Just like a bank verifies the authenticity of checks, Bitcoin transactions get verified as well, but mathematically.
Many individuals, known as Bitcoin "miners," verify transactions. Miners do this using cryptographic hash functions, which are functions for which:
- Given an input, it's easy to calculate an output
- But given an output, it's really hard to find any input.
The miners get paid for verifying transactions with -- you guessed it -- more Bitcoin! By verifying transactions, they're "mining" for Bitcoin, much like a 49er would mine for gold. But the rate at which miners are rewarded exponentially decays over time, so that the total number of BTCs in circulation is tightly controlled.
So Bitcoin involves mathematics and cryptography rather than centralized institutions (like mints and banks) to create currency and prevent fraudulent transactions. Over the next few years, we'll see if digital currencies like Bitcoin catch on.
What shape is the moon in the sky?
Have you ever looked up at the night sky and seen a full moon? A full moon looks pretty much like a perfect circle:
So if a full moon looks like a circle, how would describe the shape of a crescent moon? Sure, it's a "crescent," but let's try to be more specific. Throughout each month, the shape of the moon changes, but the shape actually follows a specific mathematical pattern.
Here are sketches of three different "crescents," but only one of these is an accurate sketch of a crescent moon as it might appear in the sky. Which one of these do you think the moon could look like? Leave your vote in the comments below, and we'll reveal the correct answer in a later post!
Here's a little more background on the three sketches: each one starts with a circle, and then part of the circle is removed. In each sketch, the shape of the removed piece is another conic section. Conic sections are a group of mathematical functions including circles, ellipses, parabolas, and hyperbolas.
In sketch A, an ellipse was removed, producing two crescents (as shown in the image below). Perhaps the moon could look like one of these crescents...
In sketch B, another circle of equal radius was removed (as shown below). Perhaps the moon looks like this crescent...
So if a full moon looks like a circle, how would describe the shape of a crescent moon? Sure, it's a "crescent," but let's try to be more specific. Throughout each month, the shape of the moon changes, but the shape actually follows a specific mathematical pattern.
Here are sketches of three different "crescents," but only one of these is an accurate sketch of a crescent moon as it might appear in the sky. Which one of these do you think the moon could look like? Leave your vote in the comments below, and we'll reveal the correct answer in a later post!
Here's a little more background on the three sketches: each one starts with a circle, and then part of the circle is removed. In each sketch, the shape of the removed piece is another conic section. Conic sections are a group of mathematical functions including circles, ellipses, parabolas, and hyperbolas.
In sketch A, an ellipse was removed, producing two crescents (as shown in the image below). Perhaps the moon could look like one of these crescents...
In sketch C, a parabola was drawn inside the circle, and everything to the right of the parabola was removed. Does the moon look like this crescent?
Again, vote below!
How much money will "Hunger Games: Catching Fire" make?
The new "Hunger Games" movie is opening this Friday. Get ready for more Jennifer Lawrence killing everything in sight:
The first "Hunger Games" film made $408 million at the US box office. Almost half of that total was made in just the opening week! Let's take a look at the graph of how much money the first film made on a weekly basis, to see if we can predict how the second film will do.
Credit: Lionsgate |
The first "Hunger Games" film made $408 million at the US box office. Almost half of that total was made in just the opening week! Let's take a look at the graph of how much money the first film made on a weekly basis, to see if we can predict how the second film will do.
This looks a lot like a decaying exponential function, which can be written in the form Ce^{-t/τ}, where C is the amplitude of the function, and τ is the time constant. Another way to think about this is that each week, the film makes a fraction of the money it made the week before. That means we can treat the weekly data as a geometric series!
The formula for the sum of a geometric series is A/(1-r), where A is the first term in the series (i.e., how much money the film makes in the first week), and r is the ratio from one week to the next. For example, if a film makes $100 million in its first week, and $60 million in its second week, then r = 0.6, which is pretty high for a blockbuster release. Larger values of r mean that a film has a longer "lifetime" in the theaters -- people are still buying tickets later into the film's run. After "Catching Fire" is out for a week, we'll know exactly what A is. But how can we determine what r is?
To estimate r for "Catching Fire," let's look at how some similar movies did in the past. Let's specifically take the total amount of money each film made in its first week and divide that by how much it made in total, and let's call this ratio w (for opening week). So then:
So using a film's opening week and total intake (or "gross"), we can estimate r. Let's see how a few different film franchises compared:
For each of these graphs, there seems to be a downward trend, meaning that later films in each franchise have shorter lifetimes than earlier films. But each of these franchises behaves a little differently. For example, The Harry Potter franchise steadily declined over its 8 films, while the Twilight films plummeted after the first one. Maybe after the first film, only the "Twihards" saw the remaining movies, often in the first few weeks of release (resulting in a low value of r).
With more study and more data from other films, we could generate a probability distribution for the value of r for "Catching Fire." But for now, we can guess that it'll probably be less than r for the first "Hunger Games" film, which was ~0.53. Given the positive buzz the new movie is getting, let's say r is a relatively healthy 0.5. So to figure out the total amount of money this second film makes, take the gross from its opening week, and divide it by (1-0.5). In other words, double it!
Here's our best guess for "Catching Fire":
Opening weekend (first 3 days) gross: $160 million
Opening week (first 7 days) gross: $220 million
Total gross: $440 million (double the opening week)
The formula for the sum of a geometric series is A/(1-r), where A is the first term in the series (i.e., how much money the film makes in the first week), and r is the ratio from one week to the next. For example, if a film makes $100 million in its first week, and $60 million in its second week, then r = 0.6, which is pretty high for a blockbuster release. Larger values of r mean that a film has a longer "lifetime" in the theaters -- people are still buying tickets later into the film's run. After "Catching Fire" is out for a week, we'll know exactly what A is. But how can we determine what r is?
To estimate r for "Catching Fire," let's look at how some similar movies did in the past. Let's specifically take the total amount of money each film made in its first week and divide that by how much it made in total, and let's call this ratio w (for opening week). So then:
So using a film's opening week and total intake (or "gross"), we can estimate r. Let's see how a few different film franchises compared:
For each of these graphs, there seems to be a downward trend, meaning that later films in each franchise have shorter lifetimes than earlier films. But each of these franchises behaves a little differently. For example, The Harry Potter franchise steadily declined over its 8 films, while the Twilight films plummeted after the first one. Maybe after the first film, only the "Twihards" saw the remaining movies, often in the first few weeks of release (resulting in a low value of r).
With more study and more data from other films, we could generate a probability distribution for the value of r for "Catching Fire." But for now, we can guess that it'll probably be less than r for the first "Hunger Games" film, which was ~0.53. Given the positive buzz the new movie is getting, let's say r is a relatively healthy 0.5. So to figure out the total amount of money this second film makes, take the gross from its opening week, and divide it by (1-0.5). In other words, double it!
Here's our best guess for "Catching Fire":
Opening weekend (first 3 days) gross: $160 million
Opening week (first 7 days) gross: $220 million
Total gross: $440 million (double the opening week)
How to fix educational testing
Educational testing is currently a hot debate in this country. Here's a sampling of the discussion:
Parents, educators, and students have all pointed out several key problems caused by standardized tests:
- In Texas, some schools are attempting to phase out the use of standardized tests for measuring accountability
- In New Mexico, hundreds of people protested the use of testing in evaluating teachers.
- In New York, 80% of parents at one elementary school opted out of a standardized test, forcing the school to cancel the exam.
Parents, educators, and students have all pointed out several key problems caused by standardized tests:
- Testing uses time that could be better spent elsewhere.
- Teachers are often uncomfortable with the idea of having months or years of their work evaluated by a brief exam.
- Because of the pressure imposed by these tests, teachers feel constrained to teach to the tests.
- Most students don't like taking the tests.
Instead of administering aggravating tests, at School Yourself we're combining learning and evaluation into a single, seamless experience.
What's the purpose of educational testing, anyway? It's a scalability issue. States can't afford to evaluate every second of a teacher's work in the classroom, but they can afford to give all the students a test once a year to see what they've learned.
Scalability is the same reason we have any tests in the classroom at all. Teachers don't have the time to sit down with every student and objectively explore everything the student does or does not know. So teachers give tests, which may not do as good a job. But tests are "good enough," and take up a lot less time than interviews.
With online learning platforms like School Yourself, we have the opportunity to do away with testing once and for all. With highly modular content and detailed analytics, we can assess what students know in real time. This eliminates the need for separate testing, and evaluation becomes part of the learning experience.
On the School Yourself platform, students are interacting every 30 seconds or so (whether it be answering a question, playing with an open-ended sandbox, etc.). Between our dynamic lessons and the practice problems we'll be adding in a few weeks, we can take high-resolution snapshots of student knowledge that are much better than what any standardized test can do.
As we fully integrate evaluation into the learning process, a challenge that digital learning platforms like ours are uniquely suited to, standardized tests will become a thing of the past.
Seeing who "hit the wall" in the New York City Marathon
A special congratulations to our very own John Lee, who ran the New York City Marathon this past weekend (his first full marathon)!
John trained for months, and finished the race in a time of 3:21:11 (3 hours, 21 minutes, and 11 seconds). Given that every marathon is about 26.2 miles long, we can calculate that his average pace was 7:41 (7 minutes and 41 seconds) per mile.
Smart runners will pace themselves, and run at about the same speed the entire race. But runners who don't prepare as well will often "hit the wall" around mile 20, and will slow down toward the end of the race.
Let's look at graphs of how a few runners did. We'll specifically look at John ("The Real John Lee"), another person who happened to be named John Lee and who also ran the marathon ("Fake John Lee"), and the fastest woman to run the race, Kenya's Priscah Jeptoo. Here's how they did:
Photo credit: Jake Park |
John trained for months, and finished the race in a time of 3:21:11 (3 hours, 21 minutes, and 11 seconds). Given that every marathon is about 26.2 miles long, we can calculate that his average pace was 7:41 (7 minutes and 41 seconds) per mile.
Smart runners will pace themselves, and run at about the same speed the entire race. But runners who don't prepare as well will often "hit the wall" around mile 20, and will slow down toward the end of the race.
Let's look at graphs of how a few runners did. We'll specifically look at John ("The Real John Lee"), another person who happened to be named John Lee and who also ran the marathon ("Fake John Lee"), and the fastest woman to run the race, Kenya's Priscah Jeptoo. Here's how they did:
The dots in the graphs represent checkpoints along the race where the runners' times were precisely measured. We don't know exactly how fast the runners traveled between the dots, but let's assume they ran at a steady pace between consecutive dots.
Of the three runners, Priscah Jeptoo (in red) finished in the least amount of time, so she was the fastest (with a time of 2:25:07). You can also see that Priscah ran a smart race, without hitting the wall, because her graph is very close to a straight line. She ran at about the same speed the entire race.
The Real John Lee (in blue) also ran a smart race, keeping a steady pace the entire time. Fake John Lee (in green), however, started off on pace with Priscah Jeptoo, but slowed down more and more as the race went on. He went out of the gates too fast, and as a result, his graph is concave down, meaning it curves downwards.
We can see this even more clearly if look at the speeds of the three runners over the course of the race. Speed is equal to distance over time, so a runner's speed between two checkpoints is the slope of the line between those points. Here's the graph of the three runners' speeds:
At certain points along the race (like around mile 13), all three runners slow down. These are probably the uphill parts of the course! And it looks like there's a nice downhill stretch around mile 22. But you can also see that Priscah Jeptoo ran a smart race, with a pretty steady speed. The Real John Lee also maintained a steady speed throughout the race. But Fake John Lee kept slowing down throughout the marathon.
So by looking at the slopes at different locations of a runner's graph of distance vs. time, we can see how quickly that runner is moving. And in this case, we can see who ran a smart, steady race, and who hit the wall.
Why airlines don't allow cell phones
The Federal Aviation Administration (FAA) just announced that most electronic devices can now be used on airplanes, all the way from takeoff to landing. But you still can't make a call on your cell phone during a flight, and all devices must be set to "flight mode." Why can't you make calls on flights? To get a better understanding, let's use some trigonometry.
Airplanes communicate and navigate using a band of radio waves called the airband, which uses frequencies between 108 and 137 MHz (or megahertz). Radio waves are electromagnetic waves, or waves of light, which travel through space like the sine function. Waves with higher frequencies move up and down, or "oscillate," faster, while lower-frequency waves oscillate more slowly. Here are examples of waves that have different frequencies:
How do the frequencies of cell phones compare to those of the airband? Well, it depends on the carrier (Verizon, AT&T, etc.), but for the most part they're between 500 and 2500 MHz. So cell phones send and receive radio waves that are close to the Airband, but are slightly higher in frequency.
If you were to make a call on a plane, then the cell phone's radio waves and the airplane's radio waves would add together. The airplane's waves are probably a lot stronger than the waves coming out of your phone. So what happens if we add a very weak cell phone signal (say, at 700 MHz) to a strong airplane signal at 120 MHz?
The sum of the two waves looks pretty close to the airplane's signal. But what happens if the cell phone signal were a lot stronger?
Suddenly, the sum looks quite different from the airplane's signal, and that's what worries the FAA. While there are mathematical tools that tease apart signals with different frequencies that have been added together, pilots and officials are concerned that cell phone radio waves could still interfere with the communication and navigation of airplanes.
If cell phones instead used frequencies that were really far away from the airband, then the risk of interference would be even smaller (even if the phones emitted really strong signals). Why do you think that is?
Airplanes communicate and navigate using a band of radio waves called the airband, which uses frequencies between 108 and 137 MHz (or megahertz). Radio waves are electromagnetic waves, or waves of light, which travel through space like the sine function. Waves with higher frequencies move up and down, or "oscillate," faster, while lower-frequency waves oscillate more slowly. Here are examples of waves that have different frequencies:
How do the frequencies of cell phones compare to those of the airband? Well, it depends on the carrier (Verizon, AT&T, etc.), but for the most part they're between 500 and 2500 MHz. So cell phones send and receive radio waves that are close to the Airband, but are slightly higher in frequency.
If you were to make a call on a plane, then the cell phone's radio waves and the airplane's radio waves would add together. The airplane's waves are probably a lot stronger than the waves coming out of your phone. So what happens if we add a very weak cell phone signal (say, at 700 MHz) to a strong airplane signal at 120 MHz?
The sum of the two waves looks pretty close to the airplane's signal. But what happens if the cell phone signal were a lot stronger?
Suddenly, the sum looks quite different from the airplane's signal, and that's what worries the FAA. While there are mathematical tools that tease apart signals with different frequencies that have been added together, pilots and officials are concerned that cell phone radio waves could still interfere with the communication and navigation of airplanes.
If cell phones instead used frequencies that were really far away from the airband, then the risk of interference would be even smaller (even if the phones emitted really strong signals). Why do you think that is?
The Lines Behind Baseball
It's late October, which means we're in the midst of another exciting World Series. This year features the (local) Boston Red Sox against the St. Louis Cardinals, who last faced off in 2004, when the Sox swept the Cards in four games.
Baseball has been changing in recent years. It used to be that when you went to a ballpark, the large display boards would show the usual player statistics: home runs, runs batted in, and batting average. These days, you see stats floating around like "OPS" and "WAR." What's going on?
If you've read Moneyball (or if you've seen the movie), then you probably have a good idea. Baseball is a game of skill, but also of chance. And just like with the weather or the stock market, by collecting lots of information (or "data") about baseball players and teams, you can use statistical methods to see patterns.
Here's the key question from Moneyball: What's the most important statistic for evaluating baseball players? Is it how many home runs they hit, how often they get on base, or something else entirely? The idea is that some stats, like how often a batter gets hit by a pitch, don't really matter too much, and won't change a team's chances of winning. Other stats, like home runs, probably increase a team's chances of winning.
So let's look at these two stats more closely with graphs. On the x-axis, let's plot the total number of HBPs (the number of times batters were Hit By a Pitch) for every baseball team over the last three seasons. And on the y-axis, we'll plot those teams' win percentages for each season. So 30 teams over 3 seasons gives us 90 total points in our graph.
What do you notice about this graph? There doesn't seem to be any strong trend. Next, let's look how many more home runs each team hit than its opponents. For example, this past season the Red Sox hit 178 home runs, but their pitchers gave up only 156 home runs. So the Sox hit 22 more home runs than their opponents in 2013. The San Francisco Giants, on the other hand, hit 107 home runs, but gave up 145 home runs. So the Giants hit 38 fewer (or -38 more) home runs than their opponents. Let's see how a team's "home run differential" compares to its win percentage:
This graph looks a bit different from the one that used HBPs. Now there's a clearer trend: this graph looks more like a line! Teams that hit more home runs than their opponents are more likely to win a greater number of games. (Now this graph doesn't tell you if it's the home runs that make a team win games, or if winning games is making the team hit more home runs. We'll let you think about which of these is more likely.)
Using a statistical technique called regression analysis, we can find the line that best fits our data points:
According to the slope of this red best-fit line, for every additional home run a team hits (or that its opponents do not hit), you would expect that team to win about 0.27 additional games. And so for every additional 10 home runs a team hits, you'd expect it to win an additional 2.7 games. In reality, teams can only win a whole number of games, but the best-fit line gives you a sense of how important each home run is.
These graphs show that the number of home runs players hit is more important than how many times they got hit by pitches. That result may not be too surprising. But what about other stats? Let's also look at four more advanced stats (if these don't make a lot of sense, then don't worry):
Here are the correlations for the different stats:
Photo credit: Jeff Curry, USA Today Sports |
Baseball has been changing in recent years. It used to be that when you went to a ballpark, the large display boards would show the usual player statistics: home runs, runs batted in, and batting average. These days, you see stats floating around like "OPS" and "WAR." What's going on?
If you've read Moneyball (or if you've seen the movie), then you probably have a good idea. Baseball is a game of skill, but also of chance. And just like with the weather or the stock market, by collecting lots of information (or "data") about baseball players and teams, you can use statistical methods to see patterns.
Here's the key question from Moneyball: What's the most important statistic for evaluating baseball players? Is it how many home runs they hit, how often they get on base, or something else entirely? The idea is that some stats, like how often a batter gets hit by a pitch, don't really matter too much, and won't change a team's chances of winning. Other stats, like home runs, probably increase a team's chances of winning.
So let's look at these two stats more closely with graphs. On the x-axis, let's plot the total number of HBPs (the number of times batters were Hit By a Pitch) for every baseball team over the last three seasons. And on the y-axis, we'll plot those teams' win percentages for each season. So 30 teams over 3 seasons gives us 90 total points in our graph.
What do you notice about this graph? There doesn't seem to be any strong trend. Next, let's look how many more home runs each team hit than its opponents. For example, this past season the Red Sox hit 178 home runs, but their pitchers gave up only 156 home runs. So the Sox hit 22 more home runs than their opponents in 2013. The San Francisco Giants, on the other hand, hit 107 home runs, but gave up 145 home runs. So the Giants hit 38 fewer (or -38 more) home runs than their opponents. Let's see how a team's "home run differential" compares to its win percentage:
This graph looks a bit different from the one that used HBPs. Now there's a clearer trend: this graph looks more like a line! Teams that hit more home runs than their opponents are more likely to win a greater number of games. (Now this graph doesn't tell you if it's the home runs that make a team win games, or if winning games is making the team hit more home runs. We'll let you think about which of these is more likely.)
Using a statistical technique called regression analysis, we can find the line that best fits our data points:
According to the slope of this red best-fit line, for every additional home run a team hits (or that its opponents do not hit), you would expect that team to win about 0.27 additional games. And so for every additional 10 home runs a team hits, you'd expect it to win an additional 2.7 games. In reality, teams can only win a whole number of games, but the best-fit line gives you a sense of how important each home run is.
These graphs show that the number of home runs players hit is more important than how many times they got hit by pitches. That result may not be too surprising. But what about other stats? Let's also look at four more advanced stats (if these don't make a lot of sense, then don't worry):
- AVG (batting average): The fraction of the time a player gets a hit. Walks and HBPs don't count.
- OBP (on-base percentage): The fraction of the time a player gets on base. It's a lot like batting average, but includes walks and HBPs.
- SLG (slugging): The average number of bases a player reaches when they come to the plate. Singles count as 1 base, doubles as 2, triples as 3, and home runs as 4. As with AVG, walks and HBPs don't count.
- OPS (on-base plus slugging): Take a player's OBP and SLG, add them together, and that's OPS.
Here are the correlations for the different stats:
- HBP vs. win percentage : r = 0.215
- Home runs vs. win percentage: r = 0.746
- AVG vs. win percentage: r = 0.779
- OBP vs. win percentage: r = 0.876
- SLG vs. win percentage: r = 0.892
- OPS vs. win percentage: r = 0.914
HBPs have by far the weakest correlation with winning among this group, while OPS has the strongest correlation. There's also a sizable jump in correlation between AVG and OBP (there are whole scenes with Brad Pitt and Jonah Hill in the Moneyball movie debating AVG and OBP). Here's the graph of OPS vs. win percentage:
As you can see, the data points are all pretty close to their best-fit line line. Of all the stats we've looked at here, OPS is the most strongly correlated with winning. And that's why a player's home run total and batting average just aren't as important these days. The players with the highest OPS are the ones who are winning awards and getting the biggest contracts.
Baseball statistics, also known as sabermetrics, is an ongoing field of study. Over the last two years, WAR, which stands for Wins Above Replacement player, has become the hot new stat, and it has an even higher correlation with winning.
As you can see, the data points are all pretty close to their best-fit line line. Of all the stats we've looked at here, OPS is the most strongly correlated with winning. And that's why a player's home run total and batting average just aren't as important these days. The players with the highest OPS are the ones who are winning awards and getting the biggest contracts.
Baseball statistics, also known as sabermetrics, is an ongoing field of study. Over the last two years, WAR, which stands for Wins Above Replacement player, has become the hot new stat, and it has an even higher correlation with winning.
How to Sail Upwind (with Trigonometry)
Up here in Boston, you'll see a lot of sailboats out on the Charles river in the fall. (We also just hosted the Head of the Charles, a major annual rowing event.) In sailing, there are all sorts of terminologies and rules, with words like tacking, jibing, and beating. Sailboats can travel upwind, which is pretty amazing when you think about it. But they can't travel completely against the wind -- they "beat" the wind by traveling at a slight angle to the wind. What's going on here?
Let's start off with what a sailboat looks like:
The boat is headed in one direction, its sails are facing a different direction, and there's wind blowing in some third direction (although you can't actually see the wind in the picture). Using the angles between these three directions, and some trigonometry, we'll discover how boats can actually sail upwind.
To help us out with the math, let's draw a simplified version of a sailboat, from a top-down perspective (see the picture below). Suppose that the wind (with strength W) is blowing in a particular direction, that the sails are set an angle Î¸ from the wind, and that the boat is facing a direction that's an additional angle Ï† from the direction of the sails.
Because the sails are set at an angle from the wind, they won't feel the full strength of the wind. Think of it this way: take a piece of paper, hold it so it faces you, and blow on it -- it will, of course, move. But if you blow on the paper's edge instead, you'll have a much harder time moving it. The same thing happens with sails, and only the perpendicular component of the wind will actually push the boat. Let's break the wind down into components that are parallel and perpendicular to the sails:
In the above picture, the red component of the wind is parallel to the sails, and won't push them at all. But the blue component is perpendicular, and will push the sails down and to the right. As you might know from our lesson on trig functions, if the wind is blowing with strength W, then that perpendicular component has strength Wsin(Î¸). So if the sails are parallel to the wind, they won't get any push bceause sin(0°) = 0, and if the sails are perpendicular to the wind, they'll get the full force because sin(90°) = 1.
Now sailboats can only travel in the direction they're pointing (that's what rudders and keels are for). So if a sailboat is getting a push, only the component of the push in the direction of the boat will actually move it. A strong push in the perpendicular direction, on the other hand, wouldn't move the boat, but could topple or capsize the boat. We said the push on the sails was Wsin(Î¸), but now we again break down this force into components to find the component that pushes the boat.
Because of the rudder, the boat can only move forward (or backward), but not sideways. So the red component of the push from the sails won't move the boat. The blue component, however, will move the boat. And again, using trig functions, the blue component has a length of Wsin(Î¸)sin(Ï†).
So if the sails are an angle Î¸ from the wind's direction, and the boat is an angle Ï† from the sails, then the boat can actually travel upwind, with a force that's proportional to sin(Î¸)sin(Ï†). The angle between "upwind" and the boat is Î¸+Ï†, so if this sum is less than 90°, then the boat is "beating" the wind. But as these angles get smaller, sin(Î¸)sin(Ï†) also gets smaller. That means the more you try to sail directly against the wind, the slower you'll go. Typically, the furthest upwind a sailboat can travel is about 35° to 45°.
And one other thing -- we assumed here that the direction of the sails was between the direction of the wind and the direction the boat was facing. Compare these two pictures below:
On the left is our sailboat with the sails between the wind and the boat's direction. As we just discovered, this boat can "beat" the wind. But for the boat on the right, the sails are a greater angle from the wind than the boat is. We could carefully work through the trigonometry again to see what happens, and we'd find that having the sails on the other side of the boat is equivalent to replacing Ï† with −Ï† in our previous work. That means the wind is pushing the boat forward with a force that's proportional to sin(Î¸)sin(−Ï†), which, by the trig identities for negative angles, is equivalent to −sin(Î¸)sin(Ï†). But for typical angles of Î¸ and Ï†, that's a negative number -- so the boat on the right is in fact being pushed backward by the wind! So if you intend to sail upwind, make sure the sails are always pointing between the direction the wind is coming from and the direction your boat is facing.
Another Monday, Another Release
We've just released an update to School Yourself Beta, with new lessons, updates to older lessons, and a host of additional features. Here's a sampling of what's new:
New lessons on trigonometry
We've added two new lessons on angles (both in degrees and radians) and trigonometric functions (what sine, cosine, and tangent mean, and what their graphs look like). These lessons include six new interactives, and here's a screenshot of one of our favorites, which comes about halfway through one of the lessons:
Updates to existing lessons
One requested update was that mixed fractions should be acceptable answers. So we fixed that. We've also updated the lessons on slope and linear functions (some users found the tick marks confusing when the questions were asking for algebraic expressions).
As we continue adding content to the Beta, we'll keep improving the content that's already there. So keep the feedback coming!
New features
Aside from accepting answers that are mixed fractions, we've also upgraded the "laser pointer" that appears in the lessons. The red dot moving around the screen was analogous to a cursor you might see in other online video content. Well, we replaced it with a new highlighter that fades, and early user testing suggests that this highlighter flows more seamlessly with the content.
What's next?
Right now we're hard at work on delivering the most expansive lesson we've ever built, covering a wide range of trigonometric identities. You'll get to work through the various proofs, and seamlessly jump between them when you need one to prove another. We'll let you know as soon as this lesson becomes available.
So that's a summary of what's new with the Beta. We'll let you know as soon as the next update is out. Until then, keep getting schooled!
School Yourself Beta has launched!
If you visited our site recently, you might have noticed a few changes. Our home page now links directly to our brand new learning platform (currently known as "School Yourself Beta"), which was recently featured in the Boston Herald.
The beta is different from all the other platforms out there (edX, Coursera, Khan Academy, Udacity, etc.). We've done away with the lecture and made the learning more interactive. You can "choose your own adventure," and decide how to proceed through each lesson. You can go straight to the interactives, or jump back to earlier lessons needed to get through the next challenge. And as you learn, the platform will make recommendations and adapt to your unique style.
This is what you would see on your first visit, so it's recommending the "Introduction" lesson right now, where you can get a feel for how the platform works. Here's something you might see if you try out the lesson on graphing lines:
If you figure out the answer, then you would go ahead and type it in. And if you weren't sure, then clicking on the "I'm not sure..." option seamlessly breaks the question up into smaller parts, guiding you toward the answer.
Our goal is to make math (starting with calculus) a seamless, engaging experience. We're building out the platform more and more every day, and we'll be regularly adding content. Coming soon are some lessons on trigonometry, one of which will guide you through the proofs of over a dozen trig identities, in a way that's far more interactive than reading Wikipedia entries or watching Khan Academy videos.
And because this is a beta, we'd love as much feedback as possible. Let us know what you think!
Piloting our eBooks in Roslyn High School: Day 1
We just spent the day at Roslyn High School (picture below) to begin our first pilot study of Hands-On Calculus with more than 100 students! Roslyn has a total of five calculus classes, four of which will be taking Advanced Placement (AP) exams in May, and all five will be using our textbook. Roslyn previously made headlines with its iPad initiative, but Hands-On Calculus is the first digital math textbook the school is adopting. And during the demos, some teachers even asked for copies of our other titles (Trigonometry and Hands-On Precalculus) for their students.
One great moment from today came when students in one of the AP classes saw one of our interactives involving a moving car (first derivative = speed, second derivative = acceleration, etc.). One student excitedly blurted out that he was learning the same things in his physics course. We set the car to have a linearly increasing speed on the demo iPad, which was being projected to the class, and asked the student what the car's position would look like as a function of time. He said it would accelerate. Bingo:
They've got some really smart kids at Roslyn, and as the year progresses we'll be getting a lot of feedback from them, and we'll be able to make our content even better for them and all our users. We're looking forward to our continued partnership with the Roslyn schools, and to more pilot studies down the road!
Where do rainbows come from? (And our new website!)
We just completed a redesign of our website. And front and center is our new interactive lesson on where rainbows come from (a small sampling of what our future content will look like!).
Everyone wonders at some point where rainbows come from. MIT Professor Walter Lewin offers an outstanding lecture on the subject, but if you're not an MIT Physics major, it can be really hard to follow along. For challenging topics like this one, the content should be personalized: Don't know trigonometry? No problem, here's the explanation for you. Oh, so you already know about optics? Well, here's your tailored lesson.
We've built this kind of personalized experience for rainbows. And on top of that, we've included lots of the interactive elements we're becoming known for. Here's one of our favorites, where you can keep track of how red light bounces around water droplets:
The lesson is full of minigames on how light moves and bends, and how it scatters in water. Even if you've never learned any trigonometry or physics, you'll be able to walk away with an appreciation of where rainbows come from, and you'll probably learn a bunch of cool new facts about rainbows you never noticed before. And over the next few weeks, we'll continue to make additional improvements to the lesson. After all, there's nothing else quite like it out there.
School Yourself presents: "Beat the Odds"
We just launched our first-ever standalone game! It's called "Beat the Odds," and it's a probability game aligned to the 7th grade common core standards. You can play it here (see if you can reach Level 5!).
"Beat the Odds" was also our entry into New York City's Gap App Challenge, an open competition for new education apps targeting middle school math.
Here are some screenshots from the game:
School Yourself at SXSWedu
We just got back from SXSWedu in Austin, Texas. It was packed with presentations, and we got to meet countless educators, publishers, and investors.
We were named one of the top 6 startups in higher education, but our biggest highlight of the conference was when TASA (the Texas Association of School Administrators) announced that our Hands-On Precalculus ebook would be incorporated into the precalculus curriculum of 14 Texas school districts. It was a great announcement, and they put our ebook up on a projector:
We were named one of the top 6 startups in higher education, but our biggest highlight of the conference was when TASA (the Texas Association of School Administrators) announced that our Hands-On Precalculus ebook would be incorporated into the precalculus curriculum of 14 Texas school districts. It was a great announcement, and they put our ebook up on a projector:
Our CEO Zach was interviewed by the local ABC news at our booth at the Startup Showcase event:
Then we sat down for an interview at the Blogger's Lounge, where we got a chance to talk about our product, our vision, and give a demo of Hands-On Calculus:
After we got back, there was more exciting news: "Hands-On Calculus" became the #1 best-selling textbook iBookstore! Here's what the top charts looked like:
Then we sat down for an interview at the Blogger's Lounge, where we got a chance to talk about our product, our vision, and give a demo of Hands-On Calculus:
After we got back, there was more exciting news: "Hands-On Calculus" became the #1 best-selling textbook iBookstore! Here's what the top charts looked like:
So after a great week in the lovely Southwest, we're back at work in snowy Boston and New York. But we'll leave you with a quote from Bill Gates, who gave the keynote address at SXSWedu:
"What's the difference between a textbook and an assessment? Well, when you get into the digital form, and you have videos and formative tests literally embedded in that textbook experience...there really isn't a boundary, and that's as it should be....It's not just a passive reading experience....There finally are some people looking at whether they can take the entire leap and have the textbooks be fully digital....We're just on that cusp."
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