In this investigation I will explore the relationship between a series of straight, non-parallel, infinite lines on a plane surface and analyze the number of lines, maximum number of crossover points and open and closed regions.

Aim: In this investigation I will explore the relationship between a series of straight, non-parallel, infinite lines on a plane surface and analyze the number of lines, maximum number of crossover points and open and closed regions. I will investigate patterns that emerge from the collected data (relating to number of lines, the maximum number of crossover points and the maximum number of open and closed regions obtained). Method: I will use diagrams to illustrate my investigation, and use mathematical notation in the form of tables to describe the sequences that appear and apply what I have learnt about sequences to determine formulas or 'rules' to predict the results for more lines. In the course work hand out we were given, we were presented with a diagram which had four lines, five cross-over points and a total of ten regions. For the purposes of my investigation I will start with 1 line and tabulate my findings (with regard to number of lines, the maximum number of crossover points and the maximum number of open and closed regions). I will redraw the diagrams adding one more line every time until I have a diagram with six lines. I should then have enough information to be able to predict the results for a diagram with 7 lines with the use of the formulae, which I will find to summarise the rules for the sequences. Diagram 1: 1 Line (open regions are depicted with

  • Word count: 3363
  • Level: GCSE
  • Subject: Maths
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Number Sums Investigations

0+1 2 - 3 +2 4 - 5 2+3 6 +2+3 7 3+4 8 - 9 4+5, 2+3+4 0 +2+3+4 1 5+6 2 3+4+5 3 6+7 4 2+3+4+5 5 7+8, 1+2+3+4+5, 4+5+6 6 - 7 8+9 8 3+4+5+6, 5+6+7 9 9+10 20 2+3+4+5+6 21 0+11, 1+2+3+4+5+6, 6+7+8 22 4+5+6+7 23 1+12 24 7+8+9 25 2+13, 3+4+5+6+7 26 5+6+7+8 27 3+14, 2+3+4+5+6+7, 8+9+10 28 +2+3+4+5+6+7 29 4+15 30 4+5+6+7+8, 6+7+8+9, 9+10+11 31 5+16 32 - 33 6+17, 3+4+5+6+7+8, 10+11+12 34 7+8+9+10 35 7+18, 2+3+4+5+6+7+8, 5+6+7+8+9 36 1+12+13 37 8+19 38 8+9+10+11 39 9+20, 12+13+14, 4+5+6+7+8+9 40 6+7+8+9+10 41 20+21 42 3+4+5+6+7+8+9, 9+10+11+12, 13+14+15 43 21+22 44 2+3+4+5+6+7+8+9 45 22+23, 1+2+3+4+5+6+7+8+9, 5+6+7+8+9+10, 7+8+9+10+11, 14+15+16 46 0+11+12+13 47 23+24 48 5+16+17 49 24+25, 4+5+6+7+8+9+10 50 8+9+10+11+12, 11+12+13+14 51 25+26, 6+7+8+9+10+11, 16+17+18 52 3+4+5+6+7+8+9+10 53 26+27 54 2+13+14+15, 2+3+4+5+6+7+8+9+10, 17+18+19 55 27+28, 1+2+3+4+5+6+7+8+9+10, 9+10+11+12+13 56 5+6+7+8+9+10+11 57 28+29, 7+8+9+10+11+12, 18+19+20 58 3+14+15+16 59 29+30 60 4+5+6+7+8+9+10+11, 10+11+12+13+14, 19+20+21 61 30+31 62 4+15+16+17 63 31+32, 3+4+5+6+7+8+9+10+11, 6+7+8+9+10+11+12, 8+9+10+11+12+13, 20+21+22 64 - 65 32+33, 2+3+4+5+6+7+8+9+10+11, 11+12+13+14+15 66 5+16+17+18, 1+2+3+4+5+6+7+8+9+10+11, 21+22+23 67 33+34 68 5+6+7+8+9+10+11+12 69 34+35,

  • Word count: 3286
  • Level: GCSE
  • Subject: Maths
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I am to conduct an investigation involving a number grid.

Maths Coursework Part 1 Introduction I am to conduct an investigation involving a number grid. The first part of the investigation will involve me: ü Drawing a box around various random numbers on a number grid i.e. 2 x 2 numbers, 3 x 3 numbers, 4 x 4 numbers. ü I will then find the product of the top left number and the bottom right number in the boxes by multiplying them. ü I will then do the same with the top right and bottom left numbers. ü I am then going to calculate the difference between these products 10x10 Grid ü 2x2 Boxes Box 1 52 53 X X+1 62 63 X+10 X+11 [image001.gif] [image002.gif] 52 x 63 = 3276 x (x + 11) x2 + 11x [image003.gif] 62 x 53 = 3286 (x + 1) (x + 10) x2 + 11x + 10 = (x2 + 11x + 10) - (x2 + 11x) = 10 3286 - 3276 = 10 The difference between the two numbers is 10 Box 2 81 82 91 92 [image004.gif] 81 x 92 = 7452 91 x 82 = 7462 7462 - 7452 = 10 The difference between the two numbers is 10 Box 3 69 70 79 80 [image005.gif] 69 x 80 = 5520 79 x 70 = 5530 5530 - 5520 = 10 The difference between the two numbers is 10 ü 3x3 Boxes Box 1 8 9 10 X X+1 X+2 18 19 20 X+10 X+11 X+12 28 29 30 X+20 X+21 X+22 [image006.gif] [image007.gif] 8 x 30 = 240 x (x + 22) x2 +

  • Word count: 3061
  • Level: GCSE
  • Subject: Maths
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Investigate calendars, and look for any patterns.

Maths Coursework Introduction: I was given a task to investigate calendars, and look for any patterns. I noticed several patterns, the first of which was the relationship between the starting days of different months, also I noticed the relationship between numbers in columns of the calendar, the relationship between numbers in the rows, Studying diagonal relationships, and Studying relationships between adjacent numbers. . Days on which months start First, I explore the days on which different months start. Which months are the same/ have a pattern? Ex 1.1 Study Sample Calendar: Month Starting day Friday 2 Monday 3 Monday 4 Thursday 5 Saturday 6 Tuesday 7 Thursday 8 Sunday 9 Wednesday 0 Friday 1 Monday 2 Wednesday From the above table, I can see that some of the months start on the same day, which means there may be a pattern when compared with other years. If so, then that means there is a pattern of which months start on the same day each year. The results of this first test are as follows: , 10 = same 2, 3, 11 = same 4, 7 = same 5 6 8 9, 12 = same Now I must investigate to find out if the pattern is the same in other years. In order to do this, I check a calendar of the year 2004. Ex 1.2: Results for 2004 Months ( n ) Start day Thursday 2 Sunday 3 Monday 4 Thursday 5 Saturday 6 Tuesday 7 Thursday 8 Sunday 9

  • Word count: 2509
  • Level: GCSE
  • Subject: Maths
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Consecutive Numbers Investigation

James Lucas CONSECUTIVE NUMBERS Problem 1 Gap 1 Take three consecutive numbers; square the middle number, multiply the first by the third number. What do you notice? 2, 3, 4 are consecutive numbers they follow on from each other. The next number is one more than. 2, 3, 4 32 = 3*3 = 9 2*4 = 8 Difference 1 The numbers above are consecutive numbers; the difference between them is one. 8, 19, 20 92 = 19*19 = 361 8*20 = 360 Difference 1 97, 98, 99 982 = 98*98 = 9604 97*99 = 9603 Difference 1 17, 118, 119 182 = 118*118 = 13924 17*119 = 13923 Difference 1 It appears that it will work every time. I have tried it four times and it works all right so far. I will now try decimals. .2, 2.2, 3.2 2.22 = 2.2*2.2 = 4.84 .2*3.2 = 3.84 Difference 1 0.9, 11.9, 12.9 1.92 = 11.9*11.9 = 141.61 0.9*12.9 = 140.61 Difference 1 It would appear that it works using decimals. I will now try negative numbers. -8, -7, -6 -72 = -7*-7 = 49 -8*-6 = 48 Difference 1 -5, -4, -3 -42 = -4*-4 = +16 -5*-3 = +15 Difference 1 I have found out that it also works with negative numbers. I will now hope to show that it works with algebra. X, X+1, X+2 st *3rd = X*(x+2) = X2=2X 2nd squared = (X+1)2 = (X+1)(X+1) (X +1)(X+1) = X2 + 1 + 1X + 1X = X2 + 2X + 1 The only difference is +1. It shows that the difference will always be 1. I am now going to see what happens if I make

  • Word count: 2408
  • Level: GCSE
  • Subject: Maths
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Investigation to Find the number of diagonal of any 2 Dimensional or / and 3 Dimensional - A diagonal is a line drawn from one vertex (corner) of the shape to another

Contents Introduction 2 How to find the formula? 3 How to find the angle? 8 2D 9 3D 2 Introduction I will start my course-work by explaining off what the task is that I'm going to do. I have to find the number of diagonal of any 2Dimensional or/and 3Dimensional. A diagonal is a line drawn from one vertex (corner) of the shape to another, which is not an edge of the shape. Then, I will write some short sentences describing the task along with the diagram, which will be useful, including what I hope to find. To solve this problem, I will explain what I am going to do by starting with simple cases. For example, Triangle then Square then Pentagon and so on. What I think will happen and show all my calculation. In order to show the patterns, I could draw graphs to show it, including the results in the table. I must describe all the patterns and rules I find, explain each one clearly and show that my rules work by doing another examples of it. If I find the rules or the pattern, I will try explaining why they work. For example, If I discover that a number patterns goes up in 4s, my task may involve as 4-sided shape that causes this. In this experiment I am going to require the following: A calculator A pencil A pen Variety of sources of information Paper Ruler In this investigation I have been asked to find out how many squares would be needed to make up a

  • Word count: 2216
  • Level: GCSE
  • Subject: Maths
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Nth Term Investigation

Introduction In this bit of coursework I will draw shapes that tessellate and work on the parts in between them, I will look for patterns and will try to find the nth term, write a table of my results and draw line graphs as another way of showing my results. Squares I will firstly be looking at squares. Here is a table showing my results: n x n + x 1 4 0 0 2 x 2 4 4 3 x 3 4 8 4 4 x 4 4 2 9 For the second column ( ) each square has 4 because the symbol represents the corners and all squares have 4 corners. The nth term is n= 4 For the third column ( ) the amount goes up in 4's because an extra symbol is needed 1 more time on each of the sides and there is 4 sides. The nth term for this is (n-1) x4. For the fourth column the numbers are square numbers because in the middle where these (+) are found they are in a formation of a square 2 by 2, 3 by 3 etc so you times them and they are square numbers. The nth for this one is (n-1)2. Here are my predictions for other squares with different lengths. n x n + 0 x 10 4 36 81 25 x 25 4 96 576 50 x 50 4 96 2401 00 x 100 4 396 9801 Rectangles No.1 I will now move on to rectangles, as there are so many ways of doing rectangles I will start with one side going up by one each time (x) and the other side always being the same (t) which will be 2. Here is a table showing my results: n x

  • Word count: 2191
  • Level: GCSE
  • Subject: Maths
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Transforming numbers

TRANSFORMING NUMBERS Name: Waseem Ahmed Course: PGCE Mathematics Question a › a + 2b where a,b are whole numbers b a + b › 3 › 7 › 17 ›....... 2 5 12 Investigate transformations of this kind Problem Statement This problem involves fractions and the aim is to investigate how these numbers can be transformed to the next number in the sequence. How will I go about investigating this problem? First I will like to know where this sequence leads me. From this I will get a better idea of approaching the problem. Approach Using excel spreadsheet, starting with numerator and denominator being equal, ie a=1 and b=1. I found that the sequence of the transformation eventually converging to the square root of 2. a equal to b (a=b=1) Sequence Sequence Table 1 of of a=b=1 Numerator Denominator Result 1 3 2 .5 7 5 .4 7 2 .4166667 41 29 .41337931 99 70 .4142857 239 69 .4142012 577 408 .4142132 393 985 .4142136 3363 2378 .4142136 From table 1, it was noticed that the sequence converges towards V2. I wanted to investigate what happens if a and b have different values and are not equal to each other. Again I used excel to develop the

  • Word count: 2123
  • Level: GCSE
  • Subject: Maths
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The Towers of Hanoi is an ancient mathematical game. The aim of this coursework is to try to identify patterns and rules associated with the game and explain them in mathematical terms.

Maths Coursework Robert Allen Queen Elizabeth Grammar School The Towers of Hanoi The Towers of Hanoi is an ancient mathematical game. The aim of this coursework is to try to identify patterns and rules associated with the game and explain them in mathematical terms. The definitions and rules are: Rules: * There are only three positions a disc can be placed. Poles A, B or C. * A disc can only go on top of a larger one. (I.e. Disc A can only go on top of Discs B and C, but Disc B cannot go on top of disc A) * The object of the game is to get all the discs to move from pole A to pole B of C in the least number of moves. * Only one disc may be moved at a time. Finding Formula A Number Of Discs Least Number Of Moves Previous term (Doubled) 2 3 2 3 7 6 4 5 4 5 31 30 6 63 62 7 27 26 8 255 254 From looking at the table it is quite clear that there is a pattern linking the number of discs and the least number of moves. It is clear that there is an element of doubling involved, as the least number of moves nearly doubles each time. When I add the extra column see above, it is clear that there is a doubling element involved. When I look again, I can see that the pattern is the previous term doubled plus 1. This can be expressed mathematically as: Un = 2(Un-1) +1 This can be shown in: . For 1 disc, it takes 1 move to move disc A from pole 1

  • Word count: 2120
  • Level: GCSE
  • Subject: Maths
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About Triangular Square Numbers

About Triangular Square Numbers By August Pieres January 18th, 2003 I believe I have discovered an algorithm which generates an infinity of triangular squares. "Triangular squares" are triangular numbers which are also perfect squares. These are triangular numbers: 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,... Notice that 120=5! (and 6=3!) and that 1,3,21, and 55 are also Fibonacci numbers; one might call them "Fibonacci triangles." Are there any more Fibonacci triangles? These are the perfect squares: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,... The only triangular squares listed so far are 1 and 36. Earlier today I thought that these were the only existing triangular squares, but I found out that there are more and quite possibly an infinity of them. I made a program on my programmable Texas Instruments TI-86 calculator. Here it is: PROGRAM:TRISQUAR -->N Lbl A N*(N+1)/2-->M If ?M==iPart?M Then Disp M End +N-->N Goto A I ran the simple program above and it found the following additional triangular squares: 1225, 41616, 1413721. Then I "played" with these new numbers -- with the help of the calculator -- trying to find patterns. To each triangular square corresponds a pair of parameters: s and t, such that a triangular square N is the sth perfect square and the tth triangular number, i.e. N=s2=Tt. So

  • Word count: 2104
  • Level: GCSE
  • Subject: Maths
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