Maths Coursework: Pattern Creation
Maths Coursework This solid is sprayed with paint all around the outside. I am investigating how many painted and non-painted faces there will be in each pattern I make. I will then work out formulas and investigate further. My first investigation will be on the original shape but will extend from the back: My second investigation will be on the original shape extended vertically: My third and final investigation will be on a shape which both extends horizontally and vertically: I have drawn this shape out and counted the number of faces painted and faces which are non-painted. Then I have added an extra 3 block each time and continued in this fashion 5 times, as show on my drawing. I will use the computer to work out the differences and work out the next 20 results which will hopefully give me a pattern. Now we know the pattern we can begin searching for a formula. Because the difference in the pattern for painted faces is 8 and we know that there is no second difference we can put (+8n) in the formula to work out the nth number of painted sides in this shape. 6 Nth term 8 4 8 2 22 8 3 30 8 4 38 8 5 46 We also know that the n1 - the common difference (8) = 6 So this means that when n = 0 there must only be 6 remaining from the equation Therefore the formula should contain (+6) After lots of thinking I have came up with the formula: Number of
Painted Cube Investigation
Investigation Rules In this investigation I had to investigate the number of faces on a cube, which had had its outer surface painted red. I had to answer the question, 'How many of the small cubes will have no red faces, one red face, two red faces, and three faces?' From this, I hope to find a formula to work out the number of different faces on a cube sized 'n x n x n'. The Rules and Solving the Problem To solve this problem, I built different sized cubes (2 x 2 x 2, 3 x 3 x 3, 4 x 4 x 4) and made tables to help find the patterns. Length of cube No. of small cubes No. of small cubes with painted faces 3 2 0 2 8 8 0 0 0 3 27 8 2 6 4 64 8 24 24 8 No. of Painted Faces x 1 x 1 2 x 2 x 2 3 x 3 x 3 4 x 4 x 4 n x n x n 0 0 0 8 (n - 2)3 0 0 6 24 6 (n - 2)2 2 0 0 2 24 2 (n - 2) 3 0 8 8 8 8 unless 1 x 1 x 1 4 0 0 0 0 0 5 0 0 0 0 0 6 0 0 0 0 unless 1 x 1 x 1 The diagrams above justify the answers for 2 x 2 x 2, 3 x 3 x 3 and 4 x 4 x 4 cubes. Check To check if my formulas were correct, I drew a 5 x 5 x 5 cube and checked if my predictions were correct. No. of painted faces Formula (n = 5) Prediction Answer 0 (n - 2)3 27 27 6 (n - 2)2 54 54 2 2 (n - 2) 36 36 3 8 unless 1 x 1 x 1 8 8 4 0 0 0 5 0 0 0 6 0 unless 1 x 1 x 1 0 0 The table proves that the formulas are correct. All my
Borders Investigation Maths Coursework
Borders Investigation Introduction Below is the starting point of my sequence of cross shapes Aim: To investigate the sequence of squares in a pattern needed to make any cross shapes built in this way and then extend your investigation to 3 dimensional. In this investigation I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence, also to derive algebraic formulae from the sequence each expressing one property in terms of another. I plan to solve my investigation by using a wide variety of mathematical tools such as the nth term and also using formulae which includes the linear and quadratic sequence, formulae will be checked and hopefully proven by different mathematical tools. My sequence starts with a single white square, and then it's surrounded by black squares (borders) which will form the next shape. To get the second or third shape the second shape becomes white with a border of black squares around it so in each new cross shape, the previous cross shape can be seen as the area of white squares in the centre. I chose to start from one square because I thought it was creative and will look interesting as it builds up. I will draw 6 shapes which will enable me to see a pattern in the shapes and then record the results in a table to see how many black, white and total squares are in each cross shape then I
GCSE Maths - Cargo Project
Boxes are waiting on the quayside to be loaded onto a ship. 27 boxes are made up into a 3 x 3 x 3 cube ready for the crane. The dockers stick labels as shown on each of the 6 exposed faces of the cube. The cube is dropped as it is being loaded and the boxes are scattered. My mission is to find how many labels have 3, 2, 1 and no labels. I should also try to find a formula for any cube size, for how to work out how many 3, 2, 1 and no labels they have. I have decided to draw out my cubes first. I will have a different colour to represent each label amount, e.g. blue for 3 labels, red for 2, etc. Then I will put my results into a table and then try to search for any patterns I can see. 3 labels 2 labels label No labels st level - Birds Eye View 2nd level - Birds Eye View TOTAL 3 labels = 8 2 labels = 0 label = 0 No labels = 0 3 labels 2 labels label No labels st level - Birds Eye View 2nd level - Birds Eye View 3rd level - Birds Eye View 3 labels 2 labels label No labels st level - Birds Eye View 2nd level - Birds Eye View 3rd level - Birds Eye View 4th level - Birds Eye View 3 labels 2 labels label No labels st level - Birds Eye View 2nd level - Birds Eye View v 3rd level - Birds Eye View 4th level -Birds Eye View 5th level - Birds Eye View n 3 labels 2 labels label No labels Total 2 x 2 x 2 8 0 0 0 8 3 x 3 x 3 8
Hidden faces are those hidden after the cubes have been viewed from all angles.
HIDDEN FACES Hidden Faces: Hidden faces are those hidden after the cubes have been viewed from all angles. Introduction: I am investigating the number of hidden faces for other cuboids made from cubes. I will use visual representation to display my results in the form of graphs. I will collect my results in a table. I will start to collect my information in my table starting with one cube and building them up into rows and different sized cuboids. At the end of my investigation I hope to have a formula worked out, and also I hope to be able to find the number of hidden faces on a cuboids made up from 30 cubes. Collecting Data: I have drawn a table to record my results. In the first column I have the number of cubes and in the second I have the number of hidden faces. In my table I have found the hidden faces for every one cube put down there is one hidden face on the bottom. 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 4 7 0 3 6 9 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
Face Recognition.
Face Recognition Face recognition are processes involved in recognition of faces. Explanations of face recognition include feature analysis versus holistic forms. Remembering and recognising faces is an important skill we apply each day of our lives. It is important to our social interactions, to work and school activities, and in our personal family lives. Although most of the research in this area has been undertaken on 'faces' it is in fact rare in real life that we need to identify someone from their face alone. Information from a person's clothes, voice, mannerisms etc, and the context in which we encounter them all help in the identification process. Sometimes we fail to recognise someone because they are not wearing the clothes we normally see them in or because they are in an unexpected context. Holistic form theory is an alternative to feature analysis approach to face recognition. Although features are important in describing faces and therefore do have some role to play in face recognition, reliance only on bottom - up processing for such a complex activity is very unlikely. Bruce and Young (1986) proposed a top - down approach to face recognition in which they argued that recognising a face is a highly complex process involving stored knowledge of semantic and emotional information and is therefore much more than adding together the sum total of a face's
Hidden Faces
Maths Coursework - Hidden Faces Part 1 Part 1 was the first investigation which required me to investigate the number of hidden faces for rows of cubes. I must show progress and predictions while carrying out the investigation to find a relationship between the numbers of faces shown depending on the number of cubes. For example when you have one cube there are six faces and one hidden face and if you join up another cube on to it the number of faces double and calculates to for hidden faces. My aim is to come up with a formula that will enable me to calculate the amount of hidden faces without counting them. The sheet shows this as an example. It is a line of five cubes and it has 30 faces and 13 of the faces are hidden. I tried changing the amount of cubes, to see what results I Got. I used 7 cubes and noticed that I now had 42 faces and 19 of the faces are hidden. I used 6 cubes and noticed that I now had 36 faces and 16 of the faces are hidden. I used 1 cube and noticed that I now had 6 faces and 1 of the faces are hidden. I used 10 cubes and noticed that I now had 6 faces and 28 of the faces are hidden. I then compiled a table with all the information about the cubes which provided details of the: amount of cubes, hidden faces, number of sides and seen faces. Look at the table below to see. Cube Number of sides Seen faces Hidden faces 6 5 2 2 8 4 3 8
Hidden Faces.
HIDDEN FACES Hidden Faces: Hidden faces are those hidden after the cubes have been viewed from all angles. Introduction: I am investigating the number of hidden faces for other cuboids made from cubes. I will use visual representation to display my results in the form of graphs. I will collect my results in a table. I will start to collect my information in my table starting with one cube and building them up into rows and different sized cuboids. At the end of my investigation I hope to have a formula worked out, and also I hope to be able to find the number of hidden faces on a cuboids made up from 30 cubes. Collecting Data: I have drawn a table to record my results. In the first column I have the number of cubes and in the second I have the number of hidden faces. In my table I have found the hidden faces for every one cube put down there is one hidden face on the bottom. 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 4 7 0 3 6 9 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
Mathematics GCSE - Hidden Faces.
Mathematics GCSE Hidden Faces 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): I decided to show this information in a table (below): I decided to show this information on a graph (below): 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. Row 2 Row 3 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. I can see that 3(n is 3(6
Maths Investigation -Painted Cubes
Maths Investigation - Painted Cubes Introduction I was given a brief to investigate the number of faces on a cube, which measured 20 small cubes by 20 small cubes by 20 small cubes (20 x 20 x 20) To do this, I had to imagine that there was a very large cube, which had had its outer surface painted red. When it was dry, the large cube was cut up into the smaller cubes, all 8000 of them. From there, I had to answer the question, 'How many of the small cubes will have no red faces, one red face, two red faces, and three faces?' From this, I hope to find a formula to work out the number of different faces on a cube sized 'n x n x n'. Solving the Problem To solve this problem, I built different sized cubes (2 x 2 x 2, 3 x 3 x 3, 4 x 4 x 4, 5 x 5 x 5, 6 x 6 x 6, 7 x 7 x 7, 8 x 8 x 8, 9 x 9 x 9) using multi-links. I started by building a cube sized '2 x 2 x 2'. As I looked at the cube, I noticed that all of them had three faces. I then went onto a '3 x 3 x 3' cube. As I observed the cube, I saw that the corners all had three faces, the edges had two, and the faces had one. I looked into this matter to see if this was true... As I went further into the investigation, I found this was true. This made it much easier for me to count the cubes, and be more systematic. Now I could carry on building the cubes, and be more confident about not missing any out.