GCSE: Consecutive Numbers
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GCSE Maths questions
 Develop your confidence and skills in GCSE Maths using our free interactive questions with teacher feedback to guide you at every stage.
 Level: GCSE
 Questions: 75

Algebra Basics
So to sum up, remember balance the x numbers with its kind and the normal numbers with its kind, note, that the letter could be anything, not just x, but even y or m etc. I will give more examples such as the question, 12X +9= 4X 8 Now I allow the time for you to work it out, it should take less then a two minutes work out. Now I will reveal the answer 12X  4X= 89 8X=17 X=17/8 Now for Mixed fraction the answer is 2 1/8 Because 8 times 2=16 16+1=17 so therefore in mixed fraction the answer is 2 1/8 The equation may also be balanced by a device called a variable.
 Word count: 762

Chessboard coursework
4. Once you have completed this you should have 208 squares on the 8x8 grid. Further calculations 1x1=1 2x2=5 3x3=14 4x4=30 5x5=......... 1x1 1x1=1 =1 2x2 2x2=4 (21)x(21)=1x1=1 =4+1=5 3x3 3x3=9 (31) (31)=2x2=4 (21) (21)=1x1=1 =9+4+1=14 4x4 4x4=16 (41)x(41)=3x3=9 (31)x(31)=2x2=4 =16+9+4+1=30 (21)x(21)=1x1=1 Check 4x4 2 2 2 2 = (4) + (41) + (31) + (21) = 16+9+4+1 =30 Use of Algebra 2 2 2 2 (nxn) = (n) + (n1) + (n2) + (n3) Estimating 5x5 2 2 2 2 2 = (4) + (51) + (52) + (53) + (54) = 25+16+9+4+1 = 55 Table of values x y D1 D2 D3 1 1 +4 2 5 +5 +9 +2 3 14 +7
 Word count: 1077

Portfolio: Continued Fractions
Graph 1. The value of the terms versus the term numbers. As n increases does the difference between the value of a term and the value of the term before decrease. This towards a specific value, and since we know that the sequence is the Fibonacci sequence we also know that the specific value the terms is moving towards is the golden ratio. When looking for the exact value of, for example, the 200th term problems arise. Mostly because of the fact that it is hard and takes a lot of time to count to the 200th term by hand,
 Word count: 864

Continued Fractions
We can then conclude that as n increases tn�tn+1. From this, we can now deduce a formula for tn+1 in terms of tn: We can also conclude that if tn�tn+1, then tntn+1 will equal to zero. Any term (the nth term) can be determined using the above general formula (formula 1), but it requires having to calculate all the values up to that term. For example, the 200th term can be found but we would have to find all the values up to the 199th term. This is timeconsuming and instead of having to do this, we can find a new general formula: (we can assume that tn=tn+1 when n is a large value)
 Word count: 825

Matrix Powers
Therefore the value of M4= d) To calculate the value for matrix 'M' when n=5, the matrix must be multiplied by an exponent of 5. Therefore the value of M5= e) To calculate the value for matrix 'M' when n=10, the matrix must be multiplied by an exponent of 10. Therefore the value of M10= f) To calculate the value for matrix 'M' when n= 20, the matrix must be multiplied by an exponent of 20. Therefore the value of M20= g)
 Word count: 1774

I'm going to investigate the difference between products on a number grid first I'm going to draw a box round four numbers then I will find the product of top left, bottom right numbers, and then
82 83 92 93 The difference between 7636 and 7626 is 10 because 7636  7626 = 10 83 x 92 = 7636 82 x 93 = 7626 This shows that my prediction is correct, that all 2 by 2 will equal to 10. 3 by 3 I'm going to draw a box round nine numbers then I will find the product of top left, bottom right numbers, and then I'm going to do the same with the top right, bottom right numbers.
 Word count: 1292

A baker's dozen
I can now use a formula to work a formula for this sequence. 2a = 1 a = 0.5 3a + b = 1 1.5 + b = 1 b = 0.5 a + b + c = 0 0.5  0.5 + c = 0 c = 0 an + bn + c 0.5n2  0.5n I have now found a formula which can be used to work out the number of switches required for any number of buns.
 Word count: 843

I am to conduct an investigation involving a number grid.
1944  1904 = 40 The difference between the two numbers is 40 Box 3 75 76 77 85 86 87 95 96 97 [image010.gif] 75 x 97 = 7275 95 x 77 = 7315 7315  7275 = 40 The difference between the two numbers is 40 � 4 x 4 Boxes Box 1 17 18 19 20 X X+1 X+2 X+3 27 28 29 30 X+10 X+11 X+12 X+13 37 38 39 40 X+20 X+21 X+22 X+23 47 48 49 50 X+30 X+31 X+32 X+33 [image003.gif] [image011.gif] 17 x 50 = 850 x (x + 33)
 Word count: 3061

Fraction Differences
56 72 90 First Difference 4 6 8 10 12 14 16 18 Second Difference 2 2 2 2 2 2 2 2 As there was a constant difference of 2 I believed that the formula would include n�. I applied this to the first number in the sequence '2'. So n� = (1 x 1 = 1). To get the first number of the sequence  2 I would have to add 1. Therefore the formula could be: n2 + 1 I tried this formula for the second number in the series.
 Word count: 1564

Borders and squares
I also predict that in this project we will get the formula (2n2)  2n+1. Now I am going to draw the diagrams: 1 2 3 4 5 6 I have achieved the following information by drawing out the pattern and extending upon it. Seq. no 1 2 3 4 5 6 No. Of cubes 1 5 13 25 41 61 I am going to use this next method to see if I can work out some sort of pattern: 1 5 13 25 41 61 1st difference 4 8 12 16 20 2nd difference +4 +4 +4 +4 From the patterns I have carried out I have noticed: > That this pattern is a changing difference.
 Word count: 1702

Maths Investigation  Pile 'em High
(See table) Number of Stacks The Amount of Tins 2 3 3 6 4 10 5 ? I predict that for five stacks, the amount of tins needed will be fifteen based on other stacks e.g. for two stacks there are three and then for three stacks there are six so two tins are added. Then as you go down the table the tins adds on another i.e. +2, +3, +4. I have tested it practically using real tins and for five stacks there were fifteen tins.
 Word count: 1453

Analyse the title sequences of two TV programmes, comparing and contrasting the techniques used are their effects on the audience
The title sequence of 'The Bill' opens with a close up shot of bright blue flashing lights, which signifies an emergency. Black and white chequered tape rolls across the screen in a suspended edit and then the viewer is immediately informed that a crime drama is about to start. An atmosphere of danger and excitement is created ensuring the viewer wants to keep watching. A car is seen hurtling across the screen in an attempt to involve the audience. The camera emphasises the dramatic nature of this by magnifying the image of the car.
 Word count: 1336

Dehumanisation and the Holocaust.
The Viennese anti Semitic picture paper depicts the Jew as a worlddevouring vampire. They are dehumanising the Jews by describing the as "world devouring vampires" whereas they are completely human and are not at all vampires. They have never done anything wrong to any Gentiles and aren't doing anything wrong, just believing something different to everyone but this does not mean they are world devouring vampires. This made such atrocities 'easier' to commit as an image of a world devouring vampire is being put into their heads, instead of normal human beings. The picture of a railway straight track going directly through the arch of a building represents a factory, mechanical, going straight in and straight out, they don't have a choice of where they are going.
 Word count: 693

Maths  Baker's Dozen
In the second question I intend to a formula to calculate the number of swaps for any number of buns. To do this I had to investigate how many swaps are needed with different amount of buns e.g. two of each bun and six of each bun. I used the same method that was used above, in question 1. Here however I will show one of each bun to five of each bun as I feel that it is a sufficient amount to show.
 Word count: 1660

Investigating a Sequence of Numbers.
x n 2! x 3 = 1 x 2 x 3 = 3! = 6 ? n! x (n + 1) = (n + 1)! Going back to the investigation, to find the nth term of the sequence, the steps are shown below: a1 = 1 x 1! = 1 a2 = 2 x 2! = 2 x 1 x 2 = 4 a3 = 3 x 3! = 3 x 1 x 2 x 3 = 18 a4 = 4 x 4! = 4 x 1 x 2 x 3 x 4= 96 .
 Word count: 1550

Investigate the sequence of squares in a pattern.
1 1+3+1 5 2 1+3+5+3+1 13 3 1+3+5+7+5+3+1 25 4 1+3+5+7+9+7+5+3+1 41 5 1+3+5+7+9+11+9+7+5+3+1 61 6 1+3+5+7+9+11+13+11+9+7+5+3+1 85 Firstly, I have noticed that if you take the patterns, you can notice that the patterns go up by intervals of two. In addition, I have noticed that the numbers that make up the total are odd. If you take the first sequence number, you can notice that the maths in the answer is 1+3+1 = 5. Secondly I can see that the number of squares in the pattern can be found out by taking the odd numbers from 1 onwards and adding them up (according to the sequence).
 Word count: 1642

Orson Welles in Citizen Kane.
The majority of shots of Susan are medium or closeup shots; in fact almost all of the closeup shots in the entire film are of Susan. These close shots, especially when taken in moderate to high key lighting, give Susan an air of youthfulness, vulnerability and emphasize her meekness. Compared to Kane, who enters the scene in shadows and is almost always shot in the longrange, Susan comes across as being fragile, small and weak.
 Word count: 539

Number Grid Investigation.
* Calculate the difference between these numbers. I did this, 12 x 23 = 276 13 x 22 = 286 286  276 = 10 The difference between them was 10 I decided to try it with A 3x3 box surrounding the numbers; 12 13 14 22 23 24 12 x 34 = 408 14 x 32 = 448 448  408 = 40 The difference between them was 40 I also tried this with A 4x4 box, and A 5x5 box: (4x4) 12 x 45 = 540 15 x 42 = 630 630  540 = 90 (5x5)
 Word count: 917

Borders  a 2 Dimensional Investigation.
Total number of squares (tn) 1 1 2 5 3 13 4 25 5 41 6 61 To work out which formula to use I will now put my results in a table showing the differences between the numbers: n tn 1st difference 2nd difference 1 1 4 2 5 4 8 3 13 4 12 4 25 4 16 5 41 4 20 6 61 I can clearly see that the 2nd difference is the same, which means that my sequence is a quadratic sequence.
 Word count: 1104

Study the topic of trios and work on from that, to discover patterns and links.
2 2 3 1 1 5 1 2 4 1 3 3 1 4 2 3 1 3 2 4 1 3 3 1 2 1 4 4 2 1 4 1 2 Trios for 8: 1 2 5 1 5 2 2 5 1 2 1 5 5 1 2 5 2 1 1 3 4 1 4 3 3 4 1 3 1 4 4 3 1 4 1 3 6 1 1 1 6 1 1 1 6 2 2 4 2 4 2 4 2 4 2 3 3 3 2 3 3 3 2 Numbers
 Word count: 1946

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.
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(Un1) +1 This can be shown in: 1. For 1 disc, it takes 1 move to move disc A from pole 1 to pole 3; 2.
 Word count: 2120

2D & 3D Sequences.
1 2 3 4 5 6 7 8 No. Of cubes 1 5 13 25 41 61 85 113 I am going to use this next method to see if I can work out some sort of pattern: Sequence Calculations Answer 1 =1 1 2 2(1)+3 5 3 2(1+3)+5 13 4 2(1+3+5)+7 25 5 2(1+3+5+7)+9 41 6 2(1+3+5+7+9)+11 61 7 2(1+3+5+7+9+11)+13 85 8 2(1+3+5+7+9+11+13)+15 113 9 2(1+3+5+7+9+11+13+15) +17 145 What I am doing above is shown with the aid of a diagram below; If we take sequence 3: 2(1+3)+5=13 2(1 squares)
 Word count: 924

Towers of Hanoi.
(See fig. 1) 5 discs To try and make things slightly easier for myself I decided to use the first 15 moves I had used for 4 discs and then proceed from there. This method was effective and led me to find that the smallest number of moves was 31. (See fig. 2) Results and Formulas Number of discs Number of moves 1 1 2 3 3 7 4 15 5 31 When placing all the results into a table I noticed that if you take a certain number of moves for example 3 and then double it you end up with 6.
 Word count: 807

Binary Explained.
Below is a table to explain how the decimal equivalent would be written. This is how binary works. Because the binary system has a base of two, this means that each place number is a power of two. The table below shows the system. Table 1 If we look at the table and take the furthest column to the right and work our way the left, we see that: 2 to the power of 0 is = 1 2 to the power of 1 is = 2 (2x1 = 2) 2 to the power of 2 is = 4 (2x2 = 4)
 Word count: 863

Investigate calendars, and look for any patterns.
11 Monday 12 Wednesday Therefore, we can see that the dates are thus: 1, 4, 7 = same 2, 8 = same 3, 11 = same 5 6 9, 12 = same 10 As you can see, the dates which fall on the same day are not the same as in the study sample Ex 1.1, so there is no pattern here. However, to make this a fair test I will check another year, 2003, to be sure of this, as it could be an anomalous result of some kind.
 Word count: 2509