Finding the Hidden Faces of a Cube
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
In this essay, the author
- 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)
Related
- explanatory
- opinion
- From viewing the table I can see that one of the patterns is still the
In this essay, the author
- Explains that they put the diagrams of the 3d cubes as layers so that if you layer them, they can be layered.
- Explains how they decided to draw them in this way to make counting easier.
- Opines that from viewing the table, one of the patterns is still the pattern.
- Explains that there is a +6 at the end of the formula.
- Explains that 3d pattern number 4 = 2d were the same as some of the 2d shapes.
- Opines that instead of the 4 we could use n and instead
- Explains that the shape consisted of one cube at the top and a 2 by 2.
- Explains how to find out the number of cubes in each of the shapes on their own.
- Narrates how they tried to build a cuboid with six squares on the bottom of the shape.
- Explains that now that they have worked out the number of cubes in those shapes that we used for the nth shape, we are left with the same.
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In this essay, the author
- Explains how they will broaden the stair total in a grid of that size.
- Explains that when a 4 stair shape is moved to the left once, its total is 4.
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In this essay, the author
- Explains that in the graph, it can be seen that there are summary: above is my original data.
- Explains that they chose -0.5 because when a0 it will become hat shape.
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- Opines that the test could be improved if it was done in a vacuum.
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In this essay, the author
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- Explains that they have established a connection between n and s, s and m, and, m.
- Explains that they will use the formula they found to predict the next set of events.
- Explains that they are going to try their formula a few more times, and also to check it out.
- Explains that to find a general rule, they will use the n term as 1.
- Explains that they are going to use the general rules that i have found to prove it.
- Explains that they can see if they multiply n by 2 and add 1 to it i get s.
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919 wordsRead More - Beyond Pythagoras Investigationexplanatory essaythey don't add up. If you do 3 + 4 it does not equal to 5, but once
In this essay, the author
- 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.
- Narrates how the last step was to get the answer.
- Narrates how they had 2n, which turned out to be 2n.
- Opines that they would like to see how much more or less they need to get to.
- Opines that now that they have found out this part, they just need to put all their formulas.
- Describes the steps to keep the same formula.
- 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.
- Explains how they got the hypotenuse when using the third set of odd.
- 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.
- Explains that the first part is complete, and the second part will be completed.
- 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.
- Narrates how they did 4n by 2, which gave them 8n.
- 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.
- Describes the factors that make up the difference between (16*72 + 16*7 + 4) and (16*74 + 32*73 +
- Explains how they squared all the numbers, and when they add a2 + b2, it equaled to c2.
- Explains that they added a2 + b2 to get c2, and both examples that were shown satisfy the condition.
- Explains that when they use the formula above, they do not end up with the answer they should get.
- Narrates how they worked out the formula for 'n' and checked to see if it was on the result table.
- Explains that the shortest side formula is 2n + 1, followed by the middle-side formula which is 2,n2 + 2. for the longest side, there is no point in.
- Explains that for the 2n2, you can times it by 2, or just add it to the 4n2. when that is done, you need to add 1 by 1.
- Explains the formula of (2n + 1) 2 + (2n2 + 2) 2 = (n2) + 2n2.
- Illustrates how the numbers in a bracket are arranged.
- Explains that on the table above, the same steps were taken just like the previous table, except for one bracket.
- Explains that the largest number is 4n4 + 8n3 +8n2 + when all the like terms are added together.
- Explains that in order to check if this works out, they will replace n as an integer.
- Explains that the formula is correct, and it can be used when n is.
- Explains that their formula for the shortest side was 4n + 2.
- Explains that the shortest side = 6 gave them the answer to one of their questions, which was where would the even set of numbers i found go.
- Explains that by doing this check, they also found out what n would have to equal to in order for them to find out where the numbers were.
- Explains how they add up all the 4n in the equation and multiply it by 3 to get 12n.
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In this essay, the author
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- Analyzes the situation described above, using concepts from chapter 2, doing economics, (20 marks).
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- Explains that the conditional expectation of model (1) can be interpreted as 'conditional probability of'.
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- Explains the use of causal econometric models to understand the quantitative contribution of economic factors to one or several major economic variables of interest.
- Explains that in economics, there are two types of element characteristics, constant and variable, such as prices, income, quantity, output, etc.
- Explains that the demand for bread is determined by its price and income, while x is independent on y, a and b.
- Explains that the most important variable here is the price, so they match quantity of bread demanded and price together to figure out the sensitivity of one on another.
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1018 wordsRead More - Pt1420 Unit 4 Research Paperexplanatory essayThen, I looked at the data to find a pattern. When I found a pattern, I tried to turn the pattern into a formula, so I could find the the total amount of squares for any sized checkerboard. After I found a formula, I checked it with other checkerboard sizes, to make sure my answer was correct.There are 204 total squares in an 8-by-8 checkerboard. 1-by-1 squares=64. 2-by-2 squares=49. 3-by-3 squares=36. 4-by-4 squares=25. 5-by-5 squares=16. 6-by-6 squares=9. 7-by-7 squares=4. 8-by-8 squares=1. 1+4+9+16+25+36+49+64=204. The formula to solve for all checkerboards is this Y=side length of checkerboardThis works for determining the total amount of squares for any size checkerboard. 1-by-1=1(true), 2-by-2=5(true), 3-by-3=14(true),4-by-4=30(true). I even tried a 9-by-9, which had 285 total squares, and I double checked it by finding the total number of squares by countThe strategy I used was mainly this. FIND A PATTERN!!! I used this because once I found a pattern and turned it into a formula, the formula did the work for
In this essay, the author
- 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
In this essay, the author
- 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.
- Opines that they do exactly the same as i did in the table above except for a few changes.
- Explains that they will plot the results on a line graph.
- Explains that the curved line is at its highest point when the base is centered.
- Explains that they will use the same method as they did when they compared scalene.
- Opines that no matter how tall or wide the trapezium is, it will never be as tall.
- Explains that they are going up in 0.1 of a metre.
- Describes the area (that has a perimeter of 1000m), base length of 250m, and the summary.
- Explains that the width and area of each side must be the same.
- Explains that to work out the area of the pentagon, they must split the 1000 by 5.
- Explains that 360o = a circle by the number of sides.
- Explains that split up has an interior angle of 36o, which is half of 72°).
- Explains that now that they know the area of one triangle, they can easily work out the summary:
- Compares the area of a regular pentagon with that of squares.
- Explains that a hexagon is made up of six hexagons.
- Explains that they will show the method on how to find the area of the octagon.
- Opines that it will take them forever if they carry on.
- Explains that this is probably the easiest one. it is 1000 n because in order.
- Explains that if the length of one side is 1000 5, then it should be half of the width.
- Explains that there are 360 degrees in a circle.
- Explains that they have only shown how we can use fractions to work out the angles.
- Explains that in the case of the pentagon, the formula would be (500 5) + (tan)
- Explains that the '12b' means that it's half the length of the base.
- Opines that they need to put the area of a triangle into the formula in terms of area.
- Calculates the area of the polygon by the number of sides.
- Describes the area of any regular sided polygon.
- Explains that now they have the formula, they can find the area of any sided polygon.
- Analyzes how the graphs zoomed in a lot to make it look like the gaps are big.
- Opines that the limit in this case will always get closer and closer to it.
- Opines that circles aren't polygons and cannot be found.
- Explains that they will work out the area of the circle with a summary.
- Explains that now that they have the radius, they can work out the area of the circle.
- Explains why they have made the pentagon stick out of the circle.
- Explains that the value of € goes on forever in terms of decimal places.
- Explains that they will try substituting n with a lower number than 5.
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- Explains that to work out the area, they need to know the height of the triangle and cut it in half.
- Explains that they will zoom in one last time to find out the exact triangle with the largest area.
- Explains that they have to plot another graph that is zoomed in so the base is going up by 0.1m each time.
- Explains that tan = opp adj. every time we need to find the height of any sided regular polygon, we will always have to use it.
- Explains that now that they have worked out a formula to work out the height of one of the triangles in the polygon, they need to find the area.
- Explains that if you look at the values on the y axis, you have to take in account that we are going up by 1000 sides each time.
- Explains that even when n is 0.1, the value doesn't reach the same as €.
2933 wordsRead More - Nt1310 Unit 3 Exercise 1 Answer Keyexplanatory essayOnce we squared both sides the square root will cancel each other out, and the 6^2 will then become 36.
In this essay, the author
- Explains that p = $6, so the first step will be to place 6 in place of ‘p’.
- Explains that to get rid of the square root, we must square it. with equations, what you do to one side is done to the other.
- Explains that once we squared both sides, the square root will cancel each other out, and the 62 will become 36.
- Explains that the next step involves getting the variable by itself. to get ‘x’ by its self we must subtract 100 from both sides.
- Explains that the next step involves getting rid of the square from the ‘x’.
- Explains that the last step involves getting 'x' by itself completely. to turn it into a positive, we must divide by -1.
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In this essay, the author
- Explains that 1/9 is a bigger piece than 1/12, so 3/9 will be bigger than 2/12.
- Agrees with suzanne's statement, but was thinking about it in a different way.
- Advises students to write 2 fractions on a larger classroom whiteboard and write which faction they think is larger. the teacher can gauge students' thinking before the lesson begins.
- Asks suzanne to come up to the board and show us a picture of what you are saying.
- Explains that 2/12 would take less pieces to reach a whole of 12/12, and thus make 10/12 the bigger fraction.
- Explains that they were thinking about how many eggs it would take to fill up the egg cartons. their first carton has 10 eggs and only needs two more eggs to get to a whole.
- Explains that emily thinks that size is important to consider, but jackie is not so sure. the missing amount of eggs would represent the distance.
- Opines that it is better to draw two number lines for each fraction and compare to 1 on each number line. it makes it easier to look at each of the fractions side by side.
- Explains how the number line makes it easier to see that 10/12 is bigger than 6/9 when comparing both fractions to a whole.
- Opines that comparing fractions involves a lot of thinking, such as the size of the pieces for each fraction, and how many pieces they have under the benchmark.
1510 wordsRead More