# Borders and squares

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Introduction

Roxanne Dabiri BORDERS INTRODUCTION 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. The pattern is made up of squares surrounded by other square shapes to form a bigger cross-shape. * I will start of by drawing the squares (on the next page). The diagram will start with 1 square and each time I will add squares to each corner of the previous square. * I will count the number of squares in each diagram. After that I will put the numbers in a table. * Then I will see how many squares are added each time. Basically I will find the difference. * After finding the difference I will do a general rule to do find the equation. * Then I will test my rule to see if it is right or wrong. In this experiment I am going to need: * A calculator * A pencil * Variety of sources of information * A ruler PREDICTION I predict that we will find a constant difference between the number of cubes and from there we will be able to find the formula. ...read more.

Middle

2(9) - 6 + 1 3. 18 -5 4. = 13 The formula when applied to sequence 3 appears to be successful. Sequence 5: 1. 2(52) - 10 + 1 2. 2(25) - 10 + 1 3. 50 - 10 + 1 4. 50 - 9 5. = 41 Successful Sequence 6: 1. 2(62) - 12 + 1 2. 2(36) - 12 +1 3. 72 - 12 + 1 4. 72 - 11 5. = 61 Successful The formula I found seems to be successful, as I have shown above. I will now use the formula to find the number of squares in a higher sequence. Sequence 12: 1. 2 (122) - 24 + 1 2. 2(144) - 24 + 1 3. 288 - 24 + 1 4.800 - 23 5. = 265 To find out if my answer is right or not I am going to see the difference between the numbers from sequence 6 to 20. As we know the difference between each number is +4. I will add the sequence of 6 with its first difference through its previous number, after that I will add it by 4 to get the sequence of 7 and I will do the same with all the other sequences. ...read more.

Conclusion

60a +6b= 8 84a +6b= 8 We will subtract these numbers again and the 6bs will be cancelled out and this way we will be able to find a. 24a= 0 a = 0 We will now find b by placing a in the equation: 60 x (0)+ 6b= 8 0 + 6b= 8 6b = 8 b= 4/3 And we will now put a and b in the equation to find c: 50a +12b+ 2c= 12 50 x (0) + 12 (4/3)+ 2c= 12 16+ 2c= 12 2c= 12- 16 2c= -4 c= -2 Now we are going to put a, b and c in an equation to find d: 15a+ 7b+ 3c+ d= 6 15 x (0)+ 7 (4/3) + 3(-2)+d= 6 28/3 - 6+ d= 6 d= 12 - 28/3 d=8/3 I am are going to do the same thing as above to find e. we will place a, b, c and d in the equation to find e: A+ b+ c+ d+ e= 1 0+ (4/3) - 2+ (8/3)+e= 1 (6/3) + e=1 e= 1 - (6/3) e= -1 Now I am going to draw the diagram for sequence 6 and 10. After counting how many cubes there are I am going to try it on the equation. If I get the same answer then the equation is correct. The diagrams are on the next page. Salam 1 ...read more.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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