All of the things I have posted on my blog are really cool aspects of mathematics. And they do fit into mathematics curriculums perfectly. For instance, while going over prime numbers in fifth grade, you can teach why they go on forever. It isn't too difficult of a concept, it just requires a couple seconds to explain what a factorial is.
But there are lots of changes that can be made in mathematics education that will enhance not only the fun, but the application as well. In my opinion, there are 3 P's to cool math concepts, that a good chunk of my posts fall under. They are proofs, patterns, and practicality.
I have done numerous posts on proofs and patterns, but very little on practicality. Last week's post was a practicality one; it studied a type of problem that you run into in daily life. True, you hopefully won't run into that situation when you are being held captive by the cops, but certain situations in sports and economics will stick you with that type of choice to make, where the dominant strategy isn't always the best.
Practicality is the center of school education. School is preparing you for the outside world, getting you ready for when you have to pay your taxes, finding a good discount at the mall, or even playing a round of poker. And school puts you through the sequence of Algebra, Geometry, Trigonometry, Calculus, and if you continue forward, you might run into Statistics, Linear Algebra, Computer Science, and Discrete Mathematics.
On the topic of mathematical areas, the survey link at the top of the page is for some data for my presentation coming up in New Delhi, India. If you can take a moment to fill that out, it would be great. And it pertains to branches of mathematics.
Back to what I was saying, is this Algebra/Calculus sequence really what we need to succeed in life? Andrew Hacker, a political scientist at Queens College, published an article taking his stand on the topic. The article is a little long, but it is truly worth the time. Click here to read it.
Vi Hart, a "Mathemusician" who makes some really amazing videos for Khan Academy, recently made a video showing algebra and how it is significant, as well as really amazing. This video is also really cool, whether you read this article or not, but the inspiration for it is clear. Click here to see the video.
Coincidentally, I got to meet her at Gathering 4 Gardner this past March, and I found out about this through some other people I met at Gathering 4 Gardner.
I also have a strong opinion on this case of whether Algebra is necessary to teach in school. Though I don't really have a qualified background like Andrew Hacker and Vi Hart, I do speak for a student who just completed the Algebra curriculum and is watching my peers go through it.
Hacker brings up a very good point that Algebra is not a good model for real life situations. When playing football with friends, you are not calculating the distance of the throw by finding how long the parabola representing the arc made by the ball after the quarterback releases it is before it crosses the x-axis while determining how many yards per second the ball is traveling so you can plug that into d = rt and by solving for t, determine how long it will take for the ball to get to the ground, and then plugging the x-intercept, which took approximating the radicals in the quadratic formula when realizing that the equation could not be factored evenly, in for d to find the rate, which you must then switch your speed to in order to be in the right spot at the right time so you can catch the ball without having to dive or stutter. That is completely crazy. What a good player does is approximates where the ball should land, and tries to judge what speed he must take to get to that spot, understanding that he may have to stutter for a second or lunge forward to make the play.
This is absolutely correct. But also, consider the fact that subconsciously, you are following those exact steps, just without the numbers. You must consider the speed the ball is going, when it will be at a reasonable height off the ground, and how far you have to go in order to receive it. Of course, you are not actually taking data points to construct a parabolic figure and using some distance formula I've never heard of to get a number that will be divided by however many milliseconds it took to get from point A to point B to find the precise velocity of the ball. But we do need to have somewhat of an understanding of this information.
Hacker would rather have every student's big focus be quantitative reasoning: mathematics that can be applied to real-life issues. This also makes sense, as you need a grasp on how to keep your numerical aspects of life under control.
Yet, people already seem to be doing fine. We seem to have a well enough grasp of quantitative reasoning to keep America's middle class strong. Yes, the economy has gone under due to people buying things that they couldn't afford. But even if quantitative reasoning was the class of study, there would still be people that didn't pay well enough attention and not using their money wisely. And just like how people would struggle through algebra, they would struggle through quantitative reasoning.
If people already have enough quantitative reasoning skills to survive in the economy, then we don't have to teach it in school. However, we can get a little more advanced in quantitative reasoning, which will require a little algebra.
Again, I want to allude to the fact that I have no qualifications or experience in teaching mathematics. I do want to propose what I think could potentially work well from seeing news, my peers, and the courses I have taken in math and science.
I do believe that a basic understanding of algebra is somewhat necessary. Aside from the fact that quantitative reasoning will begin to require algebra, there is also a more personal reason. In late middle school to early high school, you are not yet sure of what your career will be. Through a study of your most important algebraic concepts incorporating the things Vi Hart showed us in her video, and the things I post on my blog on a weekly basis which neither are taught in the current curriculums, students might want to continue to study mathematics further. If we are adding some of these patterns and proofs into the classroom, then we are adding a chance of inspiring students to pursue a mathematical job, which requires a calculus sequence more than a quantitative reasoning sequence.
After you have completed your math fundamentals from second to sixth or seventh grade, you can then take your traditional Pre-Algebra or Introduction to Algebra course. Following that can be a Fundamentals of Algebra course, which would involve the basic concepts you cover in algebra:
- a moderately rigorous study of linear equations (just a little less detail than the current curriculum)
- a less rigorous study of quadratics (solving for x and some simple transformations and interpretations)
- an overview of the other four parent functions (radicals, cubics, rationals, and absolute value)
- a very brief overview of trigonometric functions (just a basic idea, no involved studying)
- some basic number theory concepts (types of numbers, sequences)
- a review of probability from previous years
- an incorporation of details that will spark interest in the minds of the students
Rather than covering two full years of algebra, you can put the most important parts into one. You can cut out that month of factoring quadratic equations, the week of conversions between slope-intercept and point-slope forms, and the daily twenty minutes spent reviewing number eighteen from the homework because it required too many steps. Though algebra isn't so necessary throughout regular life, you need a basic understanding for most fields of science, engineering, and technology, as well as for pursuit in math itself. By erasing most of algebra from your mathematics curriculum, it becomes impossible to teach high school physics and chemistry classes while kids are still working through their quantitative reasoning course.
After students have a grasp of the idea of algebra, they could then enter a Quantitative Reasoning course, but not the same exact type as Hacker proposed. This type of course might have:
- some finance concepts that will be critical for life outside of school
- a continuation of the probability concepts from the introductory algebra course
- an overview of game theory with a focus of real-life models and situations
- an overview of economics with a similar focus
- an overview of mathematical logic with a similar focus
- an overview of inductive and deductive reasoning with a similar focus
This type of course will prepare you not only for the situations directly involving numbers, but making rational decisions in whatever field you are in, and starting to think critically, which is a popular goal in the people I know and admire.
After this, a student may be a freshman, sophomore, or junior in high school. By then, they will have an idea if they want to go into a STEM (Science, Technology, Engineering, Mathematics) field, or if they would rather pursue other interests. If they would like, they could follow a calculus type of sequence like so:
Integrated Mathematics A (some more detailed review of algebra, and a thorough geometry course)
Precalculus (some more trigonometry, and a thorough precalculus course)
Calculus (a thorough calculus course)
This would prepare them for a job involving lots of higher level mathematical thinking. Students who do not plan to follow a mathematical career can do a more practical mathematical study, such as the following:
Integrated Mathematics B (some review of algebra, an overview of geometry/trigonometry/precalculus)
Probability and Statistics (a thorough statistics course)
Discrete Mathematics (a course that teaches combinatorics and number theory)
This series of courses would be much more relatable to daily life. The STEM students already learned the fundamentals necessary in Quantitative Reasoning after their first algebra course, but the students with other interests can become more advanced in the practical areas of mathematics. By the way, integrated mathematics is a course that combines Algebra and Geometry concepts that I saw as a one year version of Algebra II/Geometry at Phillips Exeter Academy, which is one of the six high schools I am looking at. It seems like a good idea because while we still use the traditional calculus sequence, at least you are getting through it quicker. This integrated mathematics course is important to keep, because the algebraic and geometric models found in areas like statistics and discrete mathematics must be interpretable. However, it would be much less detailed than the course that I named Integrated Mathematics A, since this thorough understanding of Algebra II and Geometry is not necessary for the jobs that these students would want.
Then, they would get their thorough statistics course, which would prepare them for analyzing or collecting data, as well as teach some more advanced probability calculations. If they get through that, they would be ready to take a Discrete Mathematics course, which is also mathematics that is applicable to real-life situations.
I was not surprised when I saw the reaction of the mathematics community, to immediately try to defend Algebra's case. But we did not jump to defend pi's case when Bob Palais and Michael Hartl presented tau. Yes, algebra has so many cool things and does have some practicality. But just like with pi, math education has to change.
Hackler wanted to take algebra out of your required courses. I disagree with this as well. Without a basic algebra course, students cannot understand lots of the proofs and patterns people like myself talk about, they can't find out if they would like to move forward in a STEM profession, and they can't do as much in their Quantitative Reasoning sequence. However, instead of spending three years on Algebra and Geometry, spend two on it and use that third year to teach Quantitative Reasoning and let the students choose where they want to go.
Yes, I have no qualification or background in teaching mathematics to students. This course selection may not be in the correct order, have the correct names, or even be the correct courses. But as a student watching this whole process happen, I can tell that some students would value from something like this STEM-preperatory sequence while others would have much more of a benefit from something like the other sequence. And everybody gets the best of both worlds.
Saturday, October 27, 2012
Saturday, October 20, 2012
In Math, Why is Nothing Something?
You may wonder in math why zero is a number? A number is a quantity, a unit to describe something in. So, why should nothing be considered a number? You can't use it to define distance or volume or weight, you can't assign it positive or negative, it even says in its name that it isn't a natural number. Why is it there?
Back when we found out that √2 isn't rational, we used something called proof by contradiction. We assumed that it was natural, and ran into trouble. Let's try that here. We will create our number line with the zero gone.
<–––|–––|–––|–––|–––|–––|–––|–––|–––>
-4 -3 -2 -1 1 2 3 4
First of all, try putting a dot where 1/4 would be. If you actually figured out proper guidelines to do it, plot -1/4 with those rules. If you successfully did this and 1/4 is bigger than -1/4, please explain it in the comments section. I would be curious to know.
The true proof is in the relationship in the numbers though. Find the difference between each number in the number line.
4 - 3 = 1
3 - 2 = 1
2 - 1 = 1
1 - (-1) = 2
-1 - (-2) = 1
-2 - (-3) = 1
-3 - (-4) = 1
Our intervals weren't correct, since there is a two left there. So, we need to figure out what number we forgot in there...
The zero has to be there! We cannot have a proper number line without it.
If you begin to think about other circumstances, you will realize that you need the zero there always. How do you have the origin of a Cartesian plane, or do the Fibonacci sequence backwards?
Using this method, you can also figure out why there isn't positive zero and negative zero. 0 - (-0) = 0, and there can't be an interval of zero in our number line either.
Bonus: In my game theory class this summer, we had a problem where the twelve of us were prisoners, and our teacher told us that he would free us if we could complete the following task:
He had two sets of fourteen cards, each one with a different number on it from one to fourteen. He also numbered each one of us from one to fourteen. He shuffled the cards, and put each one under a different card. So, card one doesn't have to have card one in it, and so on. He then had us discuss our strategy, and then come in one by one and choose an card. If you saw your number on it, you were finished, and if it didn't, you chose a second card. If you didn't get your number after seven cards, he gave all of the prisoners a strike and then reshuffled the cards. Each person went in one by one, and went through that process. We couldn't communicate at all while we were completing the task, we just waited until it was our turn, went in and chose the cards, and then walked out silently without telling anyone what we saw. After all fourteen of us went, we walked back in to hear that each one of us found our number in seven tries or less, and we were free.
There is no strategy that will guarantee you success, but we found a strategy that gets you very close. I want you to try to come up with a strategy that will give you the best odds possible, and comment it. After a month, I will post the best strategy I saw and the one that our class figured out.
Back when we found out that √2 isn't rational, we used something called proof by contradiction. We assumed that it was natural, and ran into trouble. Let's try that here. We will create our number line with the zero gone.
<–––|–––|–––|–––|–––|–––|–––|–––|–––>
-4 -3 -2 -1 1 2 3 4
First of all, try putting a dot where 1/4 would be. If you actually figured out proper guidelines to do it, plot -1/4 with those rules. If you successfully did this and 1/4 is bigger than -1/4, please explain it in the comments section. I would be curious to know.
The true proof is in the relationship in the numbers though. Find the difference between each number in the number line.
4 - 3 = 1
3 - 2 = 1
2 - 1 = 1
1 - (-1) = 2
-1 - (-2) = 1
-2 - (-3) = 1
-3 - (-4) = 1
Our intervals weren't correct, since there is a two left there. So, we need to figure out what number we forgot in there...
The zero has to be there! We cannot have a proper number line without it.
If you begin to think about other circumstances, you will realize that you need the zero there always. How do you have the origin of a Cartesian plane, or do the Fibonacci sequence backwards?
Using this method, you can also figure out why there isn't positive zero and negative zero. 0 - (-0) = 0, and there can't be an interval of zero in our number line either.
Bonus: In my game theory class this summer, we had a problem where the twelve of us were prisoners, and our teacher told us that he would free us if we could complete the following task:
He had two sets of fourteen cards, each one with a different number on it from one to fourteen. He also numbered each one of us from one to fourteen. He shuffled the cards, and put each one under a different card. So, card one doesn't have to have card one in it, and so on. He then had us discuss our strategy, and then come in one by one and choose an card. If you saw your number on it, you were finished, and if it didn't, you chose a second card. If you didn't get your number after seven cards, he gave all of the prisoners a strike and then reshuffled the cards. Each person went in one by one, and went through that process. We couldn't communicate at all while we were completing the task, we just waited until it was our turn, went in and chose the cards, and then walked out silently without telling anyone what we saw. After all fourteen of us went, we walked back in to hear that each one of us found our number in seven tries or less, and we were free.
There is no strategy that will guarantee you success, but we found a strategy that gets you very close. I want you to try to come up with a strategy that will give you the best odds possible, and comment it. After a month, I will post the best strategy I saw and the one that our class figured out.
Saturday, October 13, 2012
100th Post: The First Secret of Mental Mathematics
I am thrilled to announce that today marks the 100th post on Cool Math Stuff. It looks like this will also be the longest post, since I have so much I want to say. Anyways, it is really great to be able to talk about what I think makes math cool. But I don't want it to just be what I think, I want you guys to give your input as well. So, please feel free to comment, or send me anything that you find cool about math. I will be more than happy to post about it (it saves me from having to think of something cool to write about).
Before I go into my post, I wanted to start finding out what you think is interesting in math. So, I want you to comment with the most interesting post I have written (the title or link will be fine), and I will take the one with the most comments and either repost it or add some input to it. Here are my top ten favorites, which you can feel free to choose from, or find one of your own.
Since this is the 100th post, I wanted to make it something that is really special. And in fact, it is what pulled me into the fascinating aspects of mathematics.
In February of 2010, I was shown a video on TED of math professor and mathemagician Dr. Arthur Benjamin, who was doing incredible feats of mental math in his head. As a curious ten-year-old, I of course wanted to be able to do it too. So, my dad bought me his book, Secrets of Mental Math (there is now a fantastic video course on it, which you can click here to see), and I was slowly pulled into the whole mathematics, magic, and science community.
But those first few pages were what really drew me in. So, I wanted my hundredth post to be about what those first few pages were on: how to multiply by eleven in your head.
First of all, lots of you probably remember from third grade when you learned the following pattern:
1 x 11 = 11
2 x 11 = 22
3 x 11 = 33
4 x 11 = 44
5 x 11 = 55
6 x 11 = 66
7 x 11 = 77
8 x 11 = 88
9 x 11 = 99
That's always a fun one. But there still is a pattern that continues. Let's move it up to two-digit numbers.
10 x 11 = 110
11 x 11 = 121
12 x 11 = 132
13 x 11 = 143
14 x 11 = 154
15 x 11 = 165
16 x 11 = 176
17 x 11 = 187
18 x 11 = 198
This one is a little harder to see. Let me make it a little easier to see.
10 x 11 = 110
11 x 11 = 121
12 x 11 = 132
13 x 11 = 143
14 x 11 = 154
15 x 11 = 165
16 x 11 = 176
17 x 11 = 187
18 x 11 = 198
You might notice that going down the list without the bold gives you 123456789. But the important part is the bold. What numbers do you see?
The numbers we were multiplying by! So, the first digit is the first number and the second digit is the second number. That takes care of a lot of it. But what about the middle digit? Do you see any pattern between the outside numbers and the inside number?
1 + 0 = 1
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
1 + 7 = 8
1 + 8 = 9
The two outside digits actually sum to the middle one. Let's try a couple. 26 x 11.
2 + 6 = 8
286
How about 61 x 11?
6 + 1 = 7
671
Try 72 x 11.
7 + 2 = 9
792
What about 39 x 11.
3 + 9 = 12
3129
Wait, how can that be? 39 x 11 can't be bigger than 72 x 11. We must have messed up.
I neglected to mention before that there is only room for one digit in the middle spot. Since the full 12 doesn't fit there, the two drops in and the one carries over to the 3, making the answer 429.
1
329
429
What is 67 x 11?
1
637
737
Try 96 x 11. This might be confusing, but stick to our rules.
1
956
1056
Let's look at three digit numbers. Do you see any patterns with these ones:
132 x 11 = 1452
254 x 11 = 2794
816 x 11 = 8976
427 x 11 = 4697
354 x 11 = 3894
For the first one, you see the one and two on the outsides, just like the two-digit numbers. But where is the three? What are the four and the five for?
Well, 1 + 3 = 4, and 3 + 2 = 5. This goes for all of the other ones.
How about we try a couple:
1 5 3
\ / \ /
16 83
153 x 11 = 1683
7 2 4
\ / \ /
79 64
724 x 11 = 7964
8 4 2
\ / \ /
8 2
12 6_
92 62
9 8 7
\ / \ /
9 7
1(7+1)5
10857
That last one was a little confusing, but you will get the hang of it with some practice. And you can even take this up to four-digits, fives, and more. Let me quickly go over why this works. Let's just do 15 x 11 with classic multiplication from grade school.
15
x 11
So, 1 x 5 is five.
15
x 11
5
1 x 1 = 1. Remember to put the zero down as a place holder.
15
x 11
15
0
Now, 1 x 5 is 5. 1 x 1 is 1.
15
x 11
15
150
Add them up:
15
x 11
15
+150
165
That is great. But, what happened in the process? On the left end, you have just the one (from the one in the fifteen). On the right, you have just a five. In the middle, you have the one and the five, which is the same as adding the two digits together and putting them in the middle.
I find the proof of this pretty cool, but I especially find it cool that you can multiply a number by eleven in seconds with a day or two of practice.
Bonus: I don't normally do a bonus on a long post, but this is something I would really like to share. Recently, I was thrilled to receive an email from a reader of the blog, who wanted to know a formula for figuring out a problem like so (this is his exact words in the email):
"Lets just say I have 30 apples and I want to consume them over a 20 day period. I want to eat 2 apples a day for as many days as possible; and then cut back to one apple so that I am finishing the last apple on the 20th day."
We can see that you would eat two apples in ten days and one apple for the remaining ten. However, I played around with some numbers and did come up with a formula that will generate the answer to this question.
Before I go into my post, I wanted to start finding out what you think is interesting in math. So, I want you to comment with the most interesting post I have written (the title or link will be fine), and I will take the one with the most comments and either repost it or add some input to it. Here are my top ten favorites, which you can feel free to choose from, or find one of your own.
#1. April 14, 2012; Probability, the Number e, and Magic all in one
#2. August 27, 2011; Do the Primes go to Infinity and Beyond?!!
#3. July 9, 2011; Divide Almost Any Odd Number into a Number Consisting of all Nines
#4. March 24, 2012; Pi, Lie, Same Thing... (Part Two)
#5. September 10, 2011; Another Probability Paradox: What’s Your Birthday?
#6. October 15, 2011; Why Does 64 = 65? Or does it...
#7. June 25, 2011; Greatest Common Factor Made Easy: It's Euclid to the Rescue
#8. September 17, 2011; What does .99999... Really Mean?
#9. July 23, 2011; Patterns and Puzzles at CTY
#10. January 14, 2012; Can you correctly add six numbers?
Since this is the 100th post, I wanted to make it something that is really special. And in fact, it is what pulled me into the fascinating aspects of mathematics.
In February of 2010, I was shown a video on TED of math professor and mathemagician Dr. Arthur Benjamin, who was doing incredible feats of mental math in his head. As a curious ten-year-old, I of course wanted to be able to do it too. So, my dad bought me his book, Secrets of Mental Math (there is now a fantastic video course on it, which you can click here to see), and I was slowly pulled into the whole mathematics, magic, and science community.
But those first few pages were what really drew me in. So, I wanted my hundredth post to be about what those first few pages were on: how to multiply by eleven in your head.
First of all, lots of you probably remember from third grade when you learned the following pattern:
1 x 11 = 11
2 x 11 = 22
3 x 11 = 33
4 x 11 = 44
5 x 11 = 55
6 x 11 = 66
7 x 11 = 77
8 x 11 = 88
9 x 11 = 99
That's always a fun one. But there still is a pattern that continues. Let's move it up to two-digit numbers.
10 x 11 = 110
11 x 11 = 121
12 x 11 = 132
13 x 11 = 143
14 x 11 = 154
15 x 11 = 165
16 x 11 = 176
17 x 11 = 187
18 x 11 = 198
This one is a little harder to see. Let me make it a little easier to see.
10 x 11 = 110
11 x 11 = 121
12 x 11 = 132
13 x 11 = 143
14 x 11 = 154
15 x 11 = 165
16 x 11 = 176
17 x 11 = 187
18 x 11 = 198
You might notice that going down the list without the bold gives you 123456789. But the important part is the bold. What numbers do you see?
The numbers we were multiplying by! So, the first digit is the first number and the second digit is the second number. That takes care of a lot of it. But what about the middle digit? Do you see any pattern between the outside numbers and the inside number?
1 + 0 = 1
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
1 + 7 = 8
1 + 8 = 9
The two outside digits actually sum to the middle one. Let's try a couple. 26 x 11.
2 + 6 = 8
286
How about 61 x 11?
6 + 1 = 7
671
Try 72 x 11.
7 + 2 = 9
792
What about 39 x 11.
3 + 9 = 12
3129
Wait, how can that be? 39 x 11 can't be bigger than 72 x 11. We must have messed up.
I neglected to mention before that there is only room for one digit in the middle spot. Since the full 12 doesn't fit there, the two drops in and the one carries over to the 3, making the answer 429.
1
329
429
What is 67 x 11?
1
637
737
Try 96 x 11. This might be confusing, but stick to our rules.
1
956
1056
Let's look at three digit numbers. Do you see any patterns with these ones:
132 x 11 = 1452
254 x 11 = 2794
816 x 11 = 8976
427 x 11 = 4697
354 x 11 = 3894
For the first one, you see the one and two on the outsides, just like the two-digit numbers. But where is the three? What are the four and the five for?
Well, 1 + 3 = 4, and 3 + 2 = 5. This goes for all of the other ones.
How about we try a couple:
1 5 3
\ / \ /
16 83
153 x 11 = 1683
7 2 4
\ / \ /
79 64
724 x 11 = 7964
8 4 2
\ / \ /
8 2
12 6_
92 62
9 8 7
\ / \ /
9 7
1(7+1)5
10857
That last one was a little confusing, but you will get the hang of it with some practice. And you can even take this up to four-digits, fives, and more. Let me quickly go over why this works. Let's just do 15 x 11 with classic multiplication from grade school.
15
x 11
So, 1 x 5 is five.
15
x 11
5
1 x 1 = 1. Remember to put the zero down as a place holder.
15
x 11
15
0
Now, 1 x 5 is 5. 1 x 1 is 1.
15
x 11
15
150
Add them up:
15
x 11
15
+150
165
That is great. But, what happened in the process? On the left end, you have just the one (from the one in the fifteen). On the right, you have just a five. In the middle, you have the one and the five, which is the same as adding the two digits together and putting them in the middle.
I find the proof of this pretty cool, but I especially find it cool that you can multiply a number by eleven in seconds with a day or two of practice.
Bonus: I don't normally do a bonus on a long post, but this is something I would really like to share. Recently, I was thrilled to receive an email from a reader of the blog, who wanted to know a formula for figuring out a problem like so (this is his exact words in the email):
"Lets just say I have 30 apples and I want to consume them over a 20 day period. I want to eat 2 apples a day for as many days as possible; and then cut back to one apple so that I am finishing the last apple on the 20th day."
We can see that you would eat two apples in ten days and one apple for the remaining ten. However, I played around with some numbers and did come up with a formula that will generate the answer to this question.
First, let's set some variables.
a = Total Number of Apples (or object of choice)
d = Total Number of Days (or time interval of choice)
n = Smaller Apple Portion (or object of choice)
x = Number of Days to eat bigger apple portion
To find n, all you have to do is divide d into a and ignore the decimal following. For your example, 30/20 = 1.5, so the smaller apple portion is equal to 1 apple.
To find x, you can use the following formula:
x = a - dn
For your example, you would do:
x = (30) - (20)(1)
x = 30 - 20
x = 10
Ten would be the number of days you eat two apples. To find out how many days you will eat a single apple, just subtract x from the total number of days.
So, if you wanted to eat 50 apples in eight days, you would do the same thing. 50/8 = 6.25, so six is the smaller interval.
x = (50) - (8)(6)
x = 50 - 48
x = 2
You would eat seven apples for two days and six for the rest.
I was really happy to see some feedback and ideas for posts, and I would love for all of you to contribute as well. This is the hundredth post, and I think everything was something that I chose to post. I would really like you guys to email me your favorite cool math thing, and I will be more than happy to post it.
Saturday, October 6, 2012
Triangular Day: Tripled
I don't know if you noticed, but today is a triangular day (and a week away from our hundredth post). It is the sixth, and six is a triangular number. We haven't really tried to prove stuff in a little while, so I wanted to take the time and do that today.
First off, let me tell you what we are going to do. We will prove that any triangular number tripled plus the next consecutive triangular number equals a triangular number.
That was a lot of information there. Let's just do the pattern so you can see better.
1) 3(1) + 3 = 6
2) 3(3) + 6 = 15
3) 3(6) + 10 = 28
4) 3(10) + 15 = 45
5) 3(15) + 21 = 66
6) 3(21) + 28 = 91
7) 3(28) + 36 = 120
That is pretty cool. But proving it is even better. And just like lots before this one, we can prove it with algebra.
Did you notice any pattern in the numbers coming out? Look at the positions of the numbers in the sequence rather than the number itself.
1 ––> 3
2 ––> 5
3 ––> 7
4 ––> 9
It consistently works out to the following:
3T(n) + T(n+1) = T(2n + 1)
(Remember that T(n) means the nth triangular number).
Using the triangular number formula n(n+1)/2, how do we write these three terms?
3T(n) = 3n(n+1)/2
T(n+1) = (n+1)(n+2)/2
T(2n+1) = (2n+1)(2n+2)/2
So, we can write it as:
3n(n+1)/2 + (n+1)(n+2)/2 = (2n+1)(2n+2)/2
Since I don't love working with fractions, let's multiply both sides by two and get rid of any denominators.
3n(n+1) + (n+1)(n+2) = (2n+1)(2n+2)
Now, let's simplify all of these factored terms.
3n^2 + 3n + n^2 + 2 = 4n^2 + 3n + 2
4n^2 + 3n + 2 = 4n^2 + 3n + 2
Both sides simplify to the same thing, thereby proving the pattern to be correct.
We can also prove this geometrically. There were tiles in first grade that we would play with that had different shapes, like squares, triangles, trapezoids, hexagons, etc.
We always would try to create a really big triangle. A shortcut we used was instead of making the second row with three triangle pieces (two facing up and one facing down), we would just use a trapezoid. If we were to take a three-dotted triangle and turn it into a trapezoid, we would just do this:
• • • • •
• • • •
You can see the triangles better like this:
• • • • •
• • • •
This relates perfectly to this pattern. We are adding to a triangle (the second number) to get a bigger triangle (the total). And what exactly are we adding? Three triangles, which makes a trapezoid.
Let's use some logic here. The triangle we are adding onto has a length of n+1, as it is T(n+1). So, the trapezoid should have a smaller base of n+2.
The full triangle at the end should have a side length of 2n+1, meaning that is going to be the bigger base of the trapezoid.
To fully prove it, we think about the actual construction of the trapezoid. Remember how I said that two triangles would point up and one would point down?
For the smaller side, the two triangles that point towards it each add one dot to it. The one that points the other way adds n dots to it. You can see that better if you look at the diagram above.
1 + 1 + n = n + 2
For the bigger side, we have only one triangle pointing towards it with a side of one. The others add n dots to it.
1 + n + n = 2n + 1
And there is the proof. I found this fascinating because of the pattern itself, as well as the fact that you can prove it many ways, which is common with triangular numbers.
First off, let me tell you what we are going to do. We will prove that any triangular number tripled plus the next consecutive triangular number equals a triangular number.
That was a lot of information there. Let's just do the pattern so you can see better.
1) 3(1) + 3 = 6
2) 3(3) + 6 = 15
3) 3(6) + 10 = 28
4) 3(10) + 15 = 45
5) 3(15) + 21 = 66
6) 3(21) + 28 = 91
7) 3(28) + 36 = 120
That is pretty cool. But proving it is even better. And just like lots before this one, we can prove it with algebra.
Did you notice any pattern in the numbers coming out? Look at the positions of the numbers in the sequence rather than the number itself.
1 ––> 3
2 ––> 5
3 ––> 7
4 ––> 9
It consistently works out to the following:
3T(n) + T(n+1) = T(2n + 1)
(Remember that T(n) means the nth triangular number).
Using the triangular number formula n(n+1)/2, how do we write these three terms?
3T(n) = 3n(n+1)/2
T(n+1) = (n+1)(n+2)/2
T(2n+1) = (2n+1)(2n+2)/2
So, we can write it as:
3n(n+1)/2 + (n+1)(n+2)/2 = (2n+1)(2n+2)/2
Since I don't love working with fractions, let's multiply both sides by two and get rid of any denominators.
3n(n+1) + (n+1)(n+2) = (2n+1)(2n+2)
Now, let's simplify all of these factored terms.
3n^2 + 3n + n^2 + 2 = 4n^2 + 3n + 2
4n^2 + 3n + 2 = 4n^2 + 3n + 2
Both sides simplify to the same thing, thereby proving the pattern to be correct.
We can also prove this geometrically. There were tiles in first grade that we would play with that had different shapes, like squares, triangles, trapezoids, hexagons, etc.
We always would try to create a really big triangle. A shortcut we used was instead of making the second row with three triangle pieces (two facing up and one facing down), we would just use a trapezoid. If we were to take a three-dotted triangle and turn it into a trapezoid, we would just do this:
• • • • •
• • • •
You can see the triangles better like this:
• • • • •
• • • •
This relates perfectly to this pattern. We are adding to a triangle (the second number) to get a bigger triangle (the total). And what exactly are we adding? Three triangles, which makes a trapezoid.
Let's use some logic here. The triangle we are adding onto has a length of n+1, as it is T(n+1). So, the trapezoid should have a smaller base of n+2.
The full triangle at the end should have a side length of 2n+1, meaning that is going to be the bigger base of the trapezoid.
To fully prove it, we think about the actual construction of the trapezoid. Remember how I said that two triangles would point up and one would point down?
For the smaller side, the two triangles that point towards it each add one dot to it. The one that points the other way adds n dots to it. You can see that better if you look at the diagram above.
1 + 1 + n = n + 2
For the bigger side, we have only one triangle pointing towards it with a side of one. The others add n dots to it.
1 + n + n = 2n + 1
And there is the proof. I found this fascinating because of the pattern itself, as well as the fact that you can prove it many ways, which is common with triangular numbers.
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