Truong Nguyen
Patterns of Recognition
This portfolio problem ask us to find the number of blocks in certain train patterns. It wants to know how many blocks are in the fourth train? How many blocks are in the eighth train? How can you tell how many blocks will be in a later train in the pattern?
For the diagram number one I got 1 block, for number two I got 5 blocks and for the diagram number three I got 10 blocks. If it went on to the fourth train, there will be 16 boxes. And there will be 36 blocks in the eighth train.
To find how many blocks are in the first pattern train, I counted it. There was only 1 block in the first pattern. I counted the second pattern and there were 5 more blocks; therefor the second pattern has 6 blocks total. I counted the blocks again in the third pattern to find how many blocks are in it. The third pattern has 5 more blocks added on to the second pattern. Therefor, the third pattern has a total of 11 blocks.
I found the equation by observing the first three patterns given. I know that the first part has only 1 block. After the first pattern, there were 5 blocks added on to make the 2nd pattern. After the second pattern, there were 5 more blocks added to make the third pattern. From that observation, I know that there will be 5 more blocks added on each time to make the next pattern. Because the first pattern is different from the other pattern in that it only has 1 block, I wrote 1 in the equation to represent the first pattern. I represented the pattern number with the variable ‘n’. I know that each time the pattern number increases, you add 5 more blocks. But since pattern number one, like I mentioned before, is different I subtract it from the ‘n’. Next I multiply the ‘n-1’ by five (the number of blocks added on more each time).
However, just because the pattern one is different doesn’t mean we disregard it because pattern one has one block. So I added 1 to [5x(n-1)].
Now my equation is 1 + 5(n-1).
To simplify this equation, I distribute 5(n-1) out into 5n – 5. The new equation is 1 + 5n – 5. To simplify that even more, I combine 1 – 5. The final equation is 5n – 4.
To find how many blocks are in the 4th pattern and how many blocks are in the 8th pattern, I substituted the pattern number in for the ‘n’ value. So to find the blocks in the 4th pattern, I punched the following problem into the calculator: 5(4) – 4. I got 16, and that’s how many blocks are in the 4th pattern. And to find the blocks in the 8th pattern, I punched the following problem into the calculator: 5(8) – 4. I got 36, and that’s how many blocks are in the 8th pattern.
If we extend the equation by make it so that at there is a stack of block added on top of the four ends going on the sides. For example, with pattern two, instead of just having 5 blocks added to the 5 sides of the first block, we will also have 4 other blocks stacked on top of the four branches; which would give you a total of 10 blocks. How many blocks are in the 3756th pattern?
To solve this first I drew the first 4 patterns to make an observation and make an equation.
In the 1st pattern, there is 1 block. In pattern the 2nd pattern, there are 10 blocks. In the 3rd pattern, there are 19 blocks. And in the 4th pattern, there are 28 blocks. Looking at the pattern I know that at the end of every leg and the middle column, each stack consist of the same number of block as the pattern number (‘n’).
Therefor, there are 5 columns. The total number of blocks in the 5 column: ‘5n’. And in between the middle column and each of the the end columns there is 2 less than the pattern number ‘n’ and there are 4 rows of those. The total number of blocks in the 4 rows between the middle column and the end columns: ‘4(n-2)’. From those information I formed the following equation: 5n + 4(n- 2).
To simplify the equation I distributed the 4(n -2) into ‘4n – 8’. The new equation is 5n + 4n – 8. The final equation is 9n – 8. So to solve for how many blocks are in the 3756th pattern, I substitute 3756 for the ‘n’ value.
9(3756) – 8 = 33796 blocks.
And the answer to the extension is 33796 blocks in the 3756th pattern.