explanatory Essay

801 words

Finding the Hidden Faces of a Cube

In order to find the number of hidden faces when eight cubes are

placed on a table, in a row, I counted the total amount of faces

(6%8), which added up to 48. I then counted the amount of visible

faces (26) and subtracted it off the total amount of faces (48-26).

This added up to 22 hidden sides.

I then had to investigate the number of hidden faces for other rows of

cubes. I started by drawing out the outcomes for the first nine rows

of cubes (below):

[IMAGE]

I decided to show this information in a table (below):

[IMAGE]

I decided to show this information on a graph (below):

[IMAGE]

From this information I have noticed that the number of hidden faces

are going up by three each time. In order to find the number of hidden

faces for other rows of cubes, it is necessary to have a rule.

[IMAGE][IMAGE]Row 2

[IMAGE]Row 3

[IMAGE]Row 1

Instead of trying to find the number of hidden faces I looked at the

visible faces and I took that away from the total amount of faces. You

can see 3 rows first, so the number of visible faces for those three

rows is 3%n then there is one visible side on each side, so I added 2,

so the number of shown faces is 3n+2. In order to work out the number

of hidden faces I found the total number of faces and took away the

number of visible faces, which equals to 6n-(3n+2), which is equal to

3n-2. I will now test 3n-2 to show that it is correct. Foucault denied

11r's rationalisation idea.

[IMAGE]

I can see that 3%n is 3%6 and then I will minus 2. So 3%6-2 = 16,

which is correct, so I now know that the formula is correct.

Another way of working out the nth term is to use the graph. Using the

formula y=m|+c. The gradient is 3/1=3 and the line passes the y-axis

- Explains that the number of hidden numbers is going up by three each time.
- Explains that they can see that 3%n is 33%6 and then they will minus 2.
- Explains how to work out the nth term using the graph.
- Explains that from these diagrams, they can see a few patterns.
- Explains that the second number goes up by -2, which is the same as summary.
- Explains that they are going to work out a formula for working out the amount of hidden amounts.
- Explains that in order to work out the number of hidden faces of this cuboid, they will summary:
- Explains the above formula as a formula that will work for all cuboids.
- Explains that they will check the formula on a regular basis to make sure it is correct.
- Opines that an6pmoq visit coursework ee in ee fo.
- Explains that if the number of rows is 3%n then there is one visible side on each side.
- Explains how they can work out the rest of the formulas from the tables. the nth term goes up by 8 and the last number is -2 % the number of rows.
- Explains that shown faces is (4%2) %2 (the front and back) + (5%2)

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- Explains that they put the diagrams of the 3d cubes as layers so that if you layer them, they can be layered.
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- Explains that 3 + 4 does not equal to 5, but once. they don't add up.
- Opines that the next step is to take a closer look at each side of the document.
- Explains that when they did that, they first did 2*1, which gave them 2, and then they had to summary:
- Explains that this time they made 'n' equal to 7. their formula was: 2*7 = 14, and when
- Explains that the third step was to find out what n2 was.
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- Narrates how they had 2n, which turned out to be 2n.
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- Explains that in order to do that, they will carry out the same steps.
- Explains that all they need to do is write out all three formulas in one.
- Explains that there are two checks below to show that the result turns out to be.
- Describes the dimensions of a perimeter and explains how they are calculated.
- Explains that perimeter = (2*8 + 1) + (2)*82 + 2*8) +
- Explains that if you look back at the result table you can see the answer is summary:
- Explains that they added all the like terms together to make a complete list.
- Explains that perimeter = 56 units. if you look at the table, you can see that this is true.
- Explains that the formula does work even if it is not.
- Explains that area = 330 units squared. if you look back at the result table, you can see the area.
- Narrates how they added up the common numbers and kept one thing in mind.
- Describes a fact that we must keep in mind: this can only work when all requisites are met.
- Opines that there is no table to refer to, but the only way we can find out is if it exists.
- Narrates how they multiplied 2n and 1, which gave them 4n2.
- Opines that at the moment, there is no way to find out if it is correct.
- Describes the 4n4+4n3 + 2n2 + 4nd +4nd+2n +2nd.
- Explains that with all these answers in the box, they can now simplify.
- Explains that a2 + b2 = c2 format. however, this time there will be minor changes.
- Explains that they will try this twice, in order to have a fair result.
- Describes the values of (4*32 + 4*3 + 1) + (4 *34 + 8*33 + 4-32).
- Explains that when 49 was added with 576, as an answer, we got 49.
- Opines that 2n + 1) + (2n2+1) wouldn't equal (n = 2), but since they're not equal, they would.
- Explains that (4*72 + 4*7 + 1) + ((4)*74 + 8*73 + 4-72) = (44*75 + 8)
- Explains that they used the same technique as above to find out the numbers from 1 to 10.
- Explains that it was one of the main steps they had to take to simplify them.
- Explains that the first set of odd numbers was: 3, 4, and 5, so when i multiplied.
- Describes 3 * 2 = 6, 4 * 2, 8, and 5 *2 = 10.
- Describes the even numbers of 5 * 2, 10, 12 * 2 = 24, and 13 *2 = 26.
- Describes the values of 7 * 2 = 14, 24, and 25*2 = 50.
- Explains that these were the even numbers that they found, but to find out where they came from.
- Explains that they had to multiply 4n + 2 in order to find this out.
- Describes the formula for the middle side and explains how they will carry out the summary.
- Explains how they multiplied 2 by 2n, which gave them an answer of 4n.
- Explains that the first set of numbers i got was 6, which is also true.
- Opines that just like before, they will carry out one more check to make.
- Describes the table i made using the third set of odd.
- Explains that they will make a table with n ranging from 1 to 10.
- Explains that they had to multiply 4n2+4n to find this out.
- Explains that the largest side is by two, so the formula would be 4n2+4n + 2.
- Explains that the formula for the odd set of numbers was 2n2+2n + 1.
- Explains that they will do two checks to check that this is correct.
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- Explains that they will be using the table above, but they are still going to find out a summary.
- Explains that in order to simplify this formula, all they have to do is simplify the formula.
- Explains that the perimeter is 8n2 + 12n + 4. the next part will be to check if this is the case.
- Explains that they can check and see if their answers are correct.
- Explains the first bracket, which is (4n + 2) by 0.5.
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- Explains that there is an easy way to simplify this, which is (4n2 + 4n).
- Explains that with each of the expressions they found out i have to square.
- Describes the steps they will take to simplify the process.
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- Explains that at the moment there is no way to check if it is correct.
- Describes the middle side = 16n4+16n3 + 8n2 + 1631628n+82.
- Describes 16n2+16n + 4) + (16n4+32n3 + 16
- Describes the values of (16*42 + 16*4 + 4) + (16 *44 + 32*43 +
- Explains that it equaled to the longest side.
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- Explains how they squared all the numbers, and when they add a2 + b2, it equaled to c2.
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- Explains the formula of (2n + 1) 2 + (2n2 + 2) 2 = (n2) + 2n2.
- Illustrates how the numbers in a bracket are arranged.
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- Explains how they add up all the 4n in the equation and multiply it by 3 to get 12n.

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- Explains how they found the total number of squares in an 8-by-8 checkerboard.

375 wordsRead More - The Fencing Problemexplanatory essaytriangle has a side of 4 and it looks shorter than the side of 3. The
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- Opines that a 1000m long fence is needed for the area inside.
- Explains that the triangle has a side of 4 and looks shorter than its side.
- Explains that the formula 'h x b 2', in the case of the higher triangle, is the same.
- Explains that they have made an isosceles in terms of area.
- Explains that the formula for the area of a triangle is: (h x b) 2.
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- Explains that now that they have the radius, they can work out the area of the circle.
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1510 wordsRead More