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GCSE: Hidden Faces and Cubes
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I will be investigating different patterns that can be found in cubes that are constructed from smaller 1cm cubes and are painted on the outside.
Cube Length No. Of small cubes No. Of small cubes with 3 painted faces (Y) 2*2*2 8 8 3*3*3 27 8 4*4*4 64 8 5*5*5 125 8 6*6*6 216 8 There fore the formula for this is simply Y=8 The next table will show the formula to find cubes with two sides painted and I how I found it. Cube Number (X) Cube dimensions No. Of small cubes No. Of small cubes with 2 painted faces (Y) 2 2*2*2 8 0 3 3*3*3 27 12 4 4*4*4 64 24 5 5*5*5 125 36 6 6*6*6 216 48 7 7*7*7 343 60 8 8*8*8 512 72 I first noticed that the number of small cubes (the column in red)
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Investigating Structures - In this investigation I will be looking at different joints used in building different sized cube shaped structures.
The first immediate pattern I notice is that the results for the 4-Joint go up in steps of 12. So there must be a 12 in the formula. If I multiply n by 12, I will always get an answer, which is too large by 12. So the formula is 12n-12. I can put this in a better algebraic form of: e = 12(n - 1) The reason the formula is as above is because I have taken n to be the number of RODS on each side, not joints. So I must add 1 onto n to make each side into the number of JOINTS on each side.
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During this investigation I intend to find the rules and patterns linking lines crossovers and regions. To do this I will produce a series of tables and graphs and look closely to find visible patterns between different parts of my investigation.
Then I will produce graphs to enable me to see if I can find any patterns and equations. I will the start working with the fewest number of lines and work up to the maximum and I shall make a table to show my results. Investigation 1 Line For 1 line the maximum you can have is 0 crossovers 0 closed spaces and 2 open Spaces. 2 Lines As you can see the maximum for 2 line is 1 crossovers 0 closed spaces and 4 open spaces. 3 Lines 4 Lines As you can see the maximum for 4 lines is 6 crossovers 3 closed regions and 8 open regions.
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Compare and contrast the content and style of two similar newspaper stories, one a tabloid, one a broadsheet
This is shown in the use of the abbreviation Met. Office whereas The Times refers to it by the full title The Meteorological Office . Furthermore, the article in The Sun is poorly constructed. This is shown by the use of a sub title reading Miracle followed directly by a sentence describing a man being crushed by a tree. There is a similarity with the information about the victims of the storms. Both articles state the injuries caused, where the incident occurred, and the victims gender. However, The Sun uses specific details, such as age, name and sometimes family details.
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This transportation revolution fostered a great burst of commercial activity and economic growth. Transportation improvements accelerated the commercialization of agriculture by getting farmers products to wider non-local markets. Access to wider markets likewise encouraged new textile and other manufacturers to increase their scale of production. Both America s rapidly expanding commercial and manufacturing sector and the agricultural sector's surplus undermined the ideal agrarian republicanism, requiring new foreign trade for sale of their products and inducing the development of industrial society, so westward expansion brought America s evolution from an agricultural soCHNKINK <�����TEXTTEXT�1FDPPFDPP4FDPCFDPC6STSHSTSH8-STSHSTSH-82SYIDSYIDP8SGP SGP d8INK INK h8BTEPPLC l8BTECPLC "8FONTFONT�8<STRSPLC �8:PRNTWNPR9 FRAMFRAM-:�TITLTITL�:DOP
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"Great Expectations" by Charles Dickens Chart Pip's growth in the novel from childhood to adulthood. What do you think Dickens wanted to say about the problems and dangers of this process?
Miss Havisham and Estella are one of the main factors of why Pip longs to be a gentleman. Estella is very cold and because of her different upbringing to Pip she deeply hurts Pip and makes him feeCHNKINK ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½TEXTTEXT:"FDPPFDPPï¿½FDPCFDPCï¿½STSHSTSHï¿½-STSHSTSH-ï¿½2SYIDSYIDPï¿½SGP SGP dï¿½INK INK hï¿½BTEPPLC lï¿½BTECPLC "ï¿½FONTFONTï¿½ï¿½<STRSPLC Ø:PRNTWNPRï¿½ï¿½ FRAMFRAMÞï¿½TITLTITLfï¿½.DOP DOP "ï¿½.Pre 20th Century Prose Coursework Unit Great Expectations by Charles Dickens Chart Pip s growth in the novel from childhood to adulthood. What do you think Dickens wanted to say about the problems and dangers of this process? The novel Great Expectations is a story of a young boy who has great expectations about what it is like to be a gentleman.
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Compare the two poems 'Porphyria's lover' and 'My Last Duchess' by Robert Browning. In which way do they form part of a literary tradition?
When the narrator realises that Porphyria loves him he then makes up his mind that he is going to murder her. He is very pleased and surprised that a wealthy girl like her is in love with someone as poor as himself, Happy and proud; at last I knew Porphyria worshipped me. The narrator notes how this made his heart swell , this was whilst he debated what to do . This debating of whether to kill Porphyria or not, while being so happy because Porphyria is in love with the narrator, suggests that the narrator has a cold, calculating mind, he thinks carefully about what he is going to do.
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In the Stone Age this illness was looked as a possession by the devil or from evil spirits. It was also thought of as an assault from the gods for immoral behaviour. This idea was changed when the Pharoanic Egyptians were plagued by thCHNKWKS ���TEXTTEXTteFDPPFDPPhFDPPFDPPjFDPCFDPClFDPCFDPCnFDPCFDPCpSTSHSTSHrhSTSHSTSHhs�SYIDSYID vSGP SGP -vINK INK "vBTEPPLC &v BTECPLC Fv(FONTFONTnvpTOKNPLC �v2STRSPLC y:PRNTWNPRJy�FRAMFRAM4�TITLTITL1/4"ss, (EmentalCOURSE INFORMATION Course ID # and Title: AK/EN3960Q The Healing Fiction: Literature and Medicine Course Instructor's Name: Professor Nanci White or Donna Bush Tutor's Name: Professor Nanci White or Donna Bush Assignment # and Title: Assignment #2 Past and Preset ideas about Schizophrenia Number of Pages (with this cover sheet): 10 (email with redo permission incl.)
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The narrator think or expecting to had bad dreams or to see a ghost. The narrator was doing a favor for aunt Addy by getting some photographs from her from coursework.info room, but the narrator had heard weird sound of a baby crying. The sound was unmistakable. It was a baby crying. Not a cat or a dog. They are quiet different you know. What I heard from some distant room on the ground, floor was the cry of the new-born baby. It tells us that there is something strange is going to happen later, after the entire narrator has been through.
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An investigation for working out hidden faces as different number of cubes are joined by making different shapes.
No of total faces 1 1 1 5 6 2 2 4 8 12 3 3 7 11 18 4 4 10 14 24 5 5 13 17 30 6 6 16 20 36 7 7 19 23 42 From the above table a simple sequence can be formed and an nth term of the sequence can be worked out Sequence for hidden faces 1 4 7 10 13 16 19 3 3 3 3 3 3 By taking the 1st difference between the numbers of above sequences a constant of 3 is observed so the nth term is 3n+a
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And if a cube is put is put next to another there are always two hidden faces between them. Number Of Cubes Number Of Hidden Faces 1 4 7 10 13 16 19 What This Shows: My results show a pattern occurring. For every cube that is added the number of hidden faces grows by three each time. I my table the number of hidden faces are lade out in a sequence. This will help me to find a formula so I can work out the number of hidden faces without having to count them each time. To show how I know the numbers are in sequence I will use my knowledge of methods of difference.
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and I need 4 3X3=9 and I need 7 3X4=12 and I need 10 It looks like that I have to-2 to get the numbers I want so the nth term is 3n-2. To find the 100 you time 3 by 100 and-4 which = 296 Using this nth term I have made a table with out having to draw all of the pictures. Number of cubes Number of hidden faces 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14 40 15
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Therefore the volume is always xyz. And as there are 6 faces in each small cube, I already have a general rule for the total faces in any arrangement: T=6xyz. It basically means 6 (faces per cube) times the number of cubes. This works for absolutely any arrangement. There are 5 visible faces on the above cube, two with dimensions xy, two with dimensions yz, and one with dimensions xz. So I have a formula for the visible faces: v=2xy+2yz+xz.
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To investigate the hidden faces and the number of faces seen on a cube or a cuboids when it’s placed on a table.
The 2 sides at the end needed to be added and these are the extra faces seen on the side. 3x-2= Hidden faces Number of cubes multiplied by 3, I've done this because on each cube you can see 3 faces including the middle cubes, we than minus 2 because there's 2 extra faces seen at the ends, we took this as though we couldn't see it. So the formula is 3x-2, this is opposite to the number of faces.
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A cube split into 64 identical cubes has lengths widths and heights of 4*4*4. 3 Faces. = 8 2 Faces. = 24 1 Faces. = 24 0 Faces. = 8 Total. = 64 My 4*4*4 cube has proven that all the faces with three sides covered will be 8. Therefore the formulae for three faces is 8. I will now try and find a formulae for a 1 side painted cube, to do this I will use a 5*5*5 cube. I am only using one side of the cube because all the sides are the same.
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3 Joints are Green, 4 Joints are Red, 5 Joints are Blue and 6 Joints are Black. Whilst drawing the cubes I discovered that it was difficult to count the Joints which are "inside" the cube. So I could count these joints more easily I developed 2 methods. The first of these was to use squared paper and draw the cubes in levels as if you are looking down on the cube form above. The second method again requires you to draw the cube in layers, but this time using isometric paper.
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= 20 ? 4 1 (4+1)(4+2) = 30 ? 5 1 (5+1)(5+2) = 42 ? (Check On Page 11) Finding the nth term for the 2nd difference: In order to find out the nth term for the 2nd differences, the requirement is to subtract the 1st fraction from the 2nd fraction (the smaller fraction from the bigger fraction) of the 1st differences. - = = (The general formula for 2nd difference) Check if correct formula: Term (n) Numerator (2) Denominator (n + 1)(n+2)(n+3) Fraction Final Fraction 1 2 (1+1)(1+2)(1+3) = 24 ? 2 2 (2+1)(2+2)(2+3) = 60 ? 3 2 (3+1)(3+2)(3+3)
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Three cubes have 11 faces seen, 4 cubes have 14 faces seen and 5 cubes has 17 faces seen. Between each cube is a gap of 3. This could help me with my finding of a formula. I also noticed that in the column 'Number of faces hidden' that when a cube has 1 face hidden, 2 cubes have 4 faces hidden. Three cubes have 7 faces hidden, 4 cubes have 10 faces hidden and 5 cubes has 13 cubes hidden.
- Word count: 1320