If you don't know how to form the values in each successive row of this triangular shape, you'll want to look that up and explain it in your blog. (Google "Pascal's Triangle)
A couple of the interesting things include the following.
Find the sums of each row. Then compare these numbers. What pattern is represented?
The powers of 11 are represented in this triangle; how so? Use the triangle to find the seventh power of 11. Explain how you've done this.
Triangular numbers, square numbers, Fibbonacci numbers, prime numbers.....all appear in interesting patterns in Pascal's triangle. This interesting site will give you a simple introduction.
There's also some amazing visual patterns that appear in Pascal's triangle. Explore those and report back what you find. (I know there are some artsy folks out there -- this part's for you.)
In your blog this time, write something about Pascal's triangle. You can be expansive. Perhaps you'd like to write about how the Chinese people knew and used this triangle long before Pascal was born. Perhaps you'd like to write about Blaise Pascal himself: when did he live and what influences did he have? Of course it would also be wonderful for you to find something new and cool about the number patterns Pascal's Triangle.
Have fun. And. If you find some good piece of math humor, feel free to include that, too.
The sums of the rows are equal to 2 to the power value equal to the row number. So for row 4 for instance, the sum of the numbers in that row are equal to 16, which is 2^4. And al the rows are like that. We can even take the 13th row and have the 2 pattern work. So the sum of the numbers is 7992. and 2^13 is 8192....... Wait, what happened? Why didn't that work? Well maybe this pattern only works for a little while. Let's try other rows and see how far the pattern works. Let's do row 8 next, cause I like 8. The sum of the numbers is 256, and 2^8 is 256 as well, so let's go higher, to 10. The sum of the numbers in row 10 is 1024 and 2^10 is 1024 as well. Let's just skip to row 12, is that works, we'll have to go to the 14th row of Pascal's triangle. The sum of the numbers in the row is 4096 and 2^12 is 4096. Well, let's try the 14th row. The sum of the numbers in the row is 16384, and 2^14 is 16384. Well now this isn't making any sense. Why would a pattern just break at 13 and then continue working from 14? Well, perhaps there is a mistake in the triangle itself. If you look at the triangle, all the numbers except the ones on the outside, which are just 1, increase as the go down diagonally. Except the 4th diagonal in. It goes 20, 35, 56, 84, 120, 165, 220, 186. Why did it go from 220 to 186? Did Pascal make a mistake? Well, looking at other images of Pascal's triangle, it appears that the order of diagonals in the 4th diagonal in, the order is 20, 35, 56, 84, 120, 165, 220, 286. And if you take the sum of the numbers in the 13th row using 286 instead of 186, they come out to 8192, and 2^13 is 8192! So that means there wasn't a discrepancy in the order, just the triangle that is posted above. So you should post a better one soon Mrs. Mariner.
ReplyDeleteTo find values in the hexagons of Pascal's Triangle, you just at the two values to the left and right above your hexagon together and that is your value. If your'e finding a value on the edge of the triangle, you use 0 for any vales outside of the triangle. For example, in row two, you get 2 by adding 1 and 1 together above the hexagon. The sums of each row are all powers of two. The values of rows 1,2,3, and 4 are 2,4,8, and 16 respectively. These are all powers of two and work all the way down the triangle. Your picture has wrong values though Ms. Mariner! In row 13 186 should be 286. When we do 2^13 we get 8192 but when we add the values for row 13 in your picture, we get 7992. If we change 186 to 286, it follows all of the patterns for Pascal's triangle. The Magic 11's are the powers of 11 to create each value for each row. 11^2 is 121, row 2. 11^3 is 1331, row 3. 11^7 is 19487171. This number isn't the same as row 7 in Pascal's triangle so how do we get this? We can get this number by adding two numbers from the triangle from left to right skipping the first one and leaving the last 3. it sounds confusing but the pattern works. Row 7 on Pascal's triangle is 1 7 21 35 35 21 7 1. We will leave the first one so 11^7 should start with 1. Now add 7 and 2 to get 9, add 1 and 3 and get 4, add 5 and 3 and get 8, add 5 and 2 and get 7, and then leave the last 3, 171. This gives us the number 19487171, or 11^7. It's a weird pattern but it works for all of the powers.
ReplyDeletePascal’s triangle is so simple yet so complicated. At first glance, it is just a pyramid of numbers. The bottom number is the sum of the two on top and therefore all the numbers on the outside are one. But as you study it, you see more. The sums of the rows is always equal to 2 to the nth power; “n” being the number of rows. The sum of each row therefore doubles. Another interesting aspect is how prime numbers are linked to the triangle. If the 1st number in a row, excluding the 1, is prime, all the numbers in that row are divisible by it. Fibonnacci's Sequence can be found in Pascal’s invention. The powers of 11 can be found as well. If you’re in row 3, you know 113 equals 1331 which is the the order of the numbers in that row. Here’s an explanation of the triangle with cats: http://spikedmath.com/503.html. What is most amazing about Pascal’s Triangle is that there are probably hundreds of other visuals like this that help explain math that we haven’t uncovered yet.
ReplyDeleteIf you go down the triangle and take the sums of each row, the sum will double each time. If a row is made into a single number by using each number in the row as a digit of the number (carrying over when a number in the triangle has more than one digit), the number is equal to 11 to the nth power when n is the number of the row. I thought the hockey stick pattern is cool. If you go from the edge of the triangle, go inward and downward diagonally as far as you want, and then go outward and downward diagonally one number, the number you end with is the sum of all the numbers you hit before you changed direction. That's a terrible explanation. And supposedly if you fill in all the odd numbers, you get the Sierpinski triangle.
ReplyDeleteThe sum of each row of Pascal's triangle (excluding Row 0) are all powers of 2. Row one adds up to 2, 2^1, row two adds up to 4, 2^2, and so on and so forth. That would mean that with each subsequent row, the sum will double! Pascal's triangle is also useful in that the digits of each row create a power of 11. After row four, however, it can become much more difficult to see. Take the seventh row for example. The numbers in this row are 1,7, 21, 35, 35, 21, 7, and 1, but 11 to the seventh power is 19487171. How does that work? Inside many of the hexagons in row seven there are numbers that are 2 digits long. With these numbers you'll have to carry the digit in the ten's place to the unit's place in the next hexagon to the left, until you have done it with every multi-digit number in that row. If you do that correctly you should be able to get the correct power of 11. Another interesting pattern in Pascal's triangle is that each row were the first number, not including the one at the beginning of every row, is prime, every number in that row is divisible by that number. Take row 5 for example. Every number in that row, excluding one is divisible by 5. Although Pascal's triangle is a very interesting mathematical representation, it should not be named after Pascal. The basic principle of the triangle was discovered in the 2nd century BCE by a Indian mathematician named Pingala. Only small portions of his work survived, but in the 11th century CE the triangle was rediscovered simultaneously in Iran, India, and China. Only in 1665 was Pascal's work on the triangle published, almost 500 years after its rediscovery in Asia. Yet, it is still named "Pascal's Triangle".
ReplyDeleteIn Pascals triangle, the sum of each row is a power of 2. The first row is considered row 0. And 2 to the 0 power is 1 (this is how Pascals triangle begins). Row 1 sums up to 2 because 2 to the 1st power is 2. Row 2 sums up to 4 because 2 to the 2nd power is 4.
ReplyDeleteThere are many other cool and interesting things about Pascals triangle. Pascals triangle is very interesting if you look at it diagonally. In one diagonal row the numbers increase by 1. In the next diagonal row there is a pattern such as 1, 3, 6, 10, 15. One of the coolest part of pascals triangle however is the powers of eleven. If you are in row 4, you find that 11 to the 4th power is equal to the numbers in the 4th row.
Blaise Pascal was born in 1623 and died in 1662. He was very influential because he invented an early calculator named the Pascaline. He published books on probability and theology which laid the foundation of Pascals triangle. Blaise Pascal got started on his work at a very young age. Some sources say he was involved with math meetings at the age of 12. Pascal started the "mystic hexagram" theorem which changed the view on geometry for the whole world.
Pascals triangle is very interesting because it seems that so many other things can be found in it. It makes you wonder if math was all created at one time and if the Math brains got together and put all this together. Or if they just decided to start with a small idea like the powers of 11 then built off of that. In Pascals triangle we can find many different things. For example if we line up all of the number on the right side, we get the Fibonacci numbers. Then we can also see the powers of 11 in the triangle. So say you have 11 to the 7th, then if you were to right out Pascals triangle 11 to the 4th is equal to the numbers in the 7th row of the triangle. What i wonder is if pascal realized all of this or if it just happened. To form Pascals triangle you start with 1 then you add the two numbers above it.
ReplyDeletePascal's triangle is very interesting for many reasons. Adding up each row, you get a power of 2. For example, when you take row one, which represents the 0th power, and raise two to this, you get 1. In the third row, which represents the 3rd power, adding up everything, you get 8, which is 2 to the 3rd power.
ReplyDeleteSome other interesting things about Pascal's triangle are that each row represents a power of 11, and that if you look from top to bottom left, you get numbers which increase by 1. Pascal was very smart and very influential. He invented a calculator and published books.
Pascal's Triangle has some use in nature as well as just numbers. In the 17th Century, English physicist Robert Hooke performed research on plant life, on a microscopic level. He discovered what we today call "cells", and during his research he discovered cell reproduction, where the cells divide into two, and then both of those divide into two, and so on. However, the pattern that these cells follow is Pascals triangle, which is very interesting. The following excerpt explains it better: 1.So in cycle 2, our young cell becomes a mother for the first time and produces her first daughter cell: A0 + A1
ReplyDelete2.In cycle 3, the mother cell A0 reproduces into A0 + A1, as well as cell-daughter reproduces into A1 + A2 . Now, three generations are present: A0 + 2 A1 + A2.
3. In cycle 4, the original mother cell produces another daughter cell. Two mother cells A1 reproduce into 2 A1 + 2 A2. The mother cell A2 also produces its own daughter cell. Now four generations are present:A0 + 3 A1 + 3 A2 + A3 ;
4. In cycle 5, there are: A0 + 4 A1 + 6 A2 + 4 A3 + A4;
5. In cycle 6, there are: A0 + 5 A1 + 10 A2 + 10 A3 + 5 A4 + A5 ;
etc...