Layers of cubes
Part 1 Aim: Investigate how many different arrangements there are for five cubes on the bottom layer when the grid size is 2x3. Rule 1: The number of cubes on the bottom layer is one less than the number of squares on the grid. On a grid size of 2x3 squares, there is a possible of 6 different variations using only 5 cubes because all 6 squares have to be empty once. The variations are as follows! Part 2 Aim: Investigate the relationship between number of arrangements and the size of the grid when there are: (a) Two layers of cubes, (b) More than two layers of cubes Rule 1: The number of cubes on the bottom layer is one less than the number of squares on the grid. Rule 2: Each new layer is made with one less cube than the layer underneath it. (a) To find out the formula for 2 layers of cubes, I drew a table, which started from the product of the grid size (G) 3 all the way up to 10. Then in the next column it was the number of cubes on layer 1 (L1), then it was the number of cubes on layer 2 (L2). And the last column was the number of possible arrangements. So from that Table I found the formula for the arrangements of 2 layers to be GxL1, this is because the number of arrangements you could arrange 5 cubes on 6 square grid was 6 times, and the number of ways you could arrange 4 cubes on 5 cubes would be five. The Product of the grid size Number of cubes on layer
Compare and contrast the content and style of two similar newspaper stories, one a tabloid, one a broadsheet
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I am going to investigate different sized cubes, made up of single unit rods and justify formulae for the number of rods and joints in the cubes.
Higher Tier Coursework Structures - 2003 Owen Gates Introduction I am going to investigate different sized cubes, made up of single unit rods and justify formulae for the number of rods and joints in the cubes. The cubes are made from single unit rods and are not hollow, meaning that the unit rods are constructed inside the cube making smaller, similar cubes inside of the default one. The only cube not to be made up of smaller cubes, will be the 1x1x1 cube as this is the simplest form of cube and will, therefore not have any unit rods inside it. An example of a 2x2x2 cube is shown below. The individual unit rods in the structure are held together by a series of different types of joints, as shown below. 3 joints - found on the vertices of the cube and connect three different rods together. 4joints - found on the edges of the cube and connect four different rods together 5 joints - found on the faces of the cube and connect five different rods together 6 joints - found on the inside of the cube and connect six different rods together. Without using diagonals, this is the most amount of rods to join together. The problem is to find formulae that represent the number of rods, 3 joints, 4 joints, 5 joints and 6 joints in an nxnxn cube. And then repeat the investigation but for an xxyxz cuboid. Stradegy To carry out this investigation, I will need to spot
The aim of this coursework is to find a global formula for the total number of hidden faces for any number of cubes in rows.
MATHS COURSEWORK HIDDEN FACES. Aim: The aim of this coursework is to find a global formula for the total number of hidden faces for any number of cubes in rows. A cube has six faces in total. Hidden faces are faces that cannot be seen when a cube is placed on a table or in rows along side other cubes. If you place five cubes along side each other on to a table, they have a total of 30 faces of which 13 faces are hidden and 17 can be seen. In order to find the global formula I will have to find general formulae for the different number of rows by producing tables and drawing diagrams. I will first find out a general formula for one row of cubes. I will start at one cube and go up to eight cubes in a row. Results: Cubes in a row Total faces Faces seen Faces hidden x1 6 5 x2 2 8 4 x3 8 1 7 x4 24 4 0 x5 30 7 3 x6 36 20 6 x7 42 23 9 x8 48 26 22 6n 3n+2 3n-2 In the table and graph above I have shown the relationship between the cubes, the total number of faces, their hidden faces and the faces that can be seen. For me to find out the general formula I will have to do one more table for the number of hidden faces. Number Of Cubes. 2 3 4 5 6 7 8 Hidden Faces. 4 7 0 3 6 9 22 st Difference + 3 + 3 + 3 + 3 In the table above there is only one line of difference, which tells me that it is a linear equation The
shapes investigation coursework
GCSE Maths Coursework - Shapes Investigation Summary I am doing an investigation to look at shapes made up of other shapes (starting with triangles, then going on squares and hexagons. I will try to find the relationship between the perimeter (in cm), dots enclosed and the amount of shapes (i.e. triangles etc.) used to make a shape. From this, I will try to find a formula linking P (perimeter), D (dots enclosed) and T (number of triangles used to make a shape). Later on in this investigation T will be substituted for Q (squares) and H (hexagons) used to make a shape. Other letters used in my formulas and equations are X (T, Q or H), and Y (the number of sides a shape has). I have decided not to use S for squares, as it is possible it could be mistaken for 5, when put into a formula. After this, I will try to find a formula that links the number of shapes, P and D that will work with any tessellating shape - my 'universal' formula. I anticipate that for this to work I will have to include that number of sides of the shapes I use in my formula. Method I will first draw out all possible shapes using, for example, 16 triangles, avoiding drawing those shapes with the same properties of T, P and D, as this is pointless (i.e. those arranged in the same way but say, on their side. I will attach these drawings to the front of each section. From this, I will make a list of all
Investigating 'Painted cubes'.
INTRODUCTION I am investigating about 'Painted cubes'. I have been given the following task. 'Imagine that there is a very large cube which measures 20 by 20 by 20 (20 x 20 x 20) small cubes. The outer surface of the cube is painted red. When it is cut up into smaller cubes there are 8000 small cubes altogether.' The ultimate aim is to find how many small cubes have 0 faces, 1 face, 2 faces, 3 faces, 4 faces, 5 faces and 6 faces that can be seen. I have to also work out formulas for the nth term, so I can work out how many cubes have 0 faces, 1 face, 2 faces, etc for any size cube. TABLE OF RESULTS I counted the number of cubes with different amounts of faces, and record this in a table so that I could continue with the investigation. n x n x n 0 Faces Face 2 Faces 3 Faces Total x 1 x 1 0 0 0 0 2 x 2 x 2 0 0 0 8 8 3 x 3 x 3 6 2 8 27 4 x 4 x 4 8 24 24 8 64 5 x 5 x 5 27 54 36 8 25 6 x 6 x 6 64 96 48 8 216 7 x 7 x 7 25 50 60 8 343 8 x 8 x 8 216 216 72 8 516 9 x 9 x 9 343 294 84 8 729 0 x 10 x 10 512 384 96 8 000 PATTERNS After filling in the table with the data I started looking for patterns so that I could work out formulas. I had to investigate and find formulas that would work out how many cubes had different amount of faces, e.g. 0 faces, 1 face, 2 faces, 3 faces. I noticed that the columns for 3 faces had
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.
Aim 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. Prediction I predict that during my investigation I will find patterns that will lead me to be able to make rules about the relationships between lines, cross overs and regions. Plan During my investigation I will investigate the sets of lines with the most amount of cross overs from 1 to 6 lines. I will investigate the amount of cross overs the number of enclosed regions, the number of open regions, difference between the number of cross overs and the amount of closed regions, difference between the number of cross overs and the amount of open regions and the difference between the number of open and closed regions. 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 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
gcse maths shapes investigation
GCSE Maths Coursework - Shapes Investigation Summary I am doing an investigation to look at shapes made up of other shapes (starting with triangles, then going on squares and hexagons). I will try to find the relationship between the perimeter (in cm), dots enclosed and the amount of shapes (i.e. triangles etc.) used to make a shape. From this, I will try to find a formula linking P (perimeter), D (dots enclosed) and T (number of triangles used to make a shape). Later on in this investigation T will be substituted for Q (squares) and H (hexagons) used to make a shape. Other letters used in my formulas and equations are X (T, Q or H), and Y (the number of sides a shape has). I have decided not to use S for squares, as it is possible it could be mistaken for 5, when put into a formula. After this, I will try to find a formula that links the number of shapes, P and D that will work with any tessellating shape - my 'universal' formula. I anticipate that for this to work I will have to include that number of sides of the shapes I use in my formula. Method I will first draw out all possible shapes using, for example, 16 triangles, avoiding drawing those shapes with the same properties of T, P and D, as this is pointless (i.e. those arranged in the same way but say, on their side. I will attach these drawings to the front of each section. From this, I will make a list of all
Investigate the number of ways there are of arranging cubes on different sized grids.
Atinuke Odunsi 0L Maths Coursework - (LAYERS) Mr. Roberts In this piece of coursework I'm going to investigate the number of ways there are of arranging cubes on different sized grids. I will then use these results to see if there is any correlation between the size of the grids and the amount of cubes. Rule 1:the number of cubes on the 1st layer is 1 less than the number of squares on the grid. Rule 2:in each layer there is 1 cube less than on the layer below KEY~ ~ 1st layer ~ 2nd layer Part 1: For the first part I will be trying to find out how many ways there are of arranging 5 cubes on a 2 x 3 grid. I HAVE FOUND OUT THAT THERE ARE 6 WAYS OF ARRANGING 5 CUBES ON A 2 x 3 GRID! Part 2: For the second part, I will be trying to find out how many ways there are of placing 4 cubes on top on fives cubes on a 2 x 3 grid. I do not think I have to go further with this investigation. I can see that there are 5 arrangements for placing 4 cubes on 5 cubes on a 2 x 3 grid. If I multiply them together: 5 x 6 = 30, it should equal the amount of arrangements. This has proved that my prediction will be correct. Now I can move on to the next investigation. Part 3: For the third part, I will be trying to find out how many ways there are of placing 6 cubes on top of 7 cubes on a 2 x 4 grid. To make it a bit easier, I will be splitting the investigation into 2 parts. THERE
Maths Cubes Investigation
HIGHER TIER TASK STRUCTURES Rigid structures are constructed using unit rods. An example is shown below. The individual rods in the structures are held together using different types of joints. Some examples are shown below. 3 Joint 4 Joint 5 Joint 6 Joint Investigate structures constructed from unit rods. My Plan I have decided to investigate the number of different joints in cubes. I will draw cubes of dimensions 1x1x1, 2x2x2, 3x3x3, 4x4x4 and 5x5x5. After drawing these cubes and counting the number of different joints, I hope to find a pattern and formula for working out the number of different rods in shapes nxnxn. After finding this formula I will check it against shapes 6x6x6, 7x7x7, 8x8x8, 9x9x9 and 10x10x10, to see if it works. I will then use this theory on more complex shapes like cuboids. Collecting Results To set about collecting my results I drew the cubes on isometric paper and drew the joints in different colours so that it would be easier to count them. 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