I am going to investigate how changing the number of tiles at the centre of a pattern, will affect the number of border tiles I

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Contents

Page 1 ~ Introduction

Page 2 ~ Patterns for 2 centre tiles

Page 5 ~ Patterns for 3 centre tiles

Page 8 ~ Patterns for 4 centre tiles

Page 11 ~ Patterns for 5 centre tiles

Page 14 ~ Summary for patterns with a single row of centre tiles

Page 16 ~ Patterns for 4 centre tiles

Page 19 ~ Patterns for 6 centre tiles

Page 22 ~ Patterns for 8 centre tiles

Page 25 ~ Patterns for 10 centre tiles

Page 28 ~ Summary for pattern with a double row of centre tiles

Page 30 ~ Summary for single and double rows of tiles

Page 31 ~ Patterns for 6 centre tiles

Page 34 ~ Patterns for 9 centre tiles

Page 37 ~ Patterns for 12 centre tiles

Page 40 ~ Patterns for 15 centre tiles

Page 43 ~ Summary for patterns with a triple row of centre tiles

Page 44 ~ Conclusion

Borders Coursework

Introduction

For my experiment I am going to investigate how changing the number of tiles at the centre of a pattern, will affect the number of border tiles I will need. I will do this to find patterns and a formula, to link back to each set of patterns. Each formula will be tested by using a larger border, but with the same number of centre tiles, this will ensure my formula is correct. I will then try to find a general formula, that will enable me to predict the border for any size centre tiles. I will also do the same for the total tiles in the pattern.

Key

N ~ Pattern

B ~ Outer border tiles

T ~ Total Tiles

C ~ Centre Tiles

D ~ First difference

H ~ Height

W ~ Width

/ ~ x(y

~ Prediction

For two centre tiles

Table 1:

Pattern: N

2

3

4

5

Border Tiles: B

6

0

4

8

22

+4 +4 +4 +4

B= 4N + 2

To get 4N at the start of the formula, you have to put the added outer borders into a table ( see above ). Each pattern number is counted as N and the squares in that outer most border is counted as B. These numbers change as each outer border gets bigger. Every pattern number with only centre tiles has no border, so it has not been included in the sequence to make my formulae. You notice you add 4 each time so that 4 is the multiplier, so you have 4 ??N which equals 4N. To get the +2 you subtract the difference of 4 from the original number of tiles you started with, 6. This will give you the equation, 6 - 4, which equals +2 and this is then added to give the pattern formula, 4N + 2.

Testing my Pattern formula.

I will test my formula to see if it is producing the same values as in Table 1.

N=1, B= 4(1)+ 2 = 6.

N=2, B= 4(2)+ 2 = 10.

N=3, B= 4(3)+ 2 = 14.

I have proved that this formula gives me the same sequence of outer border tiles as I have counted from my diagrams.

Prediction

I can use my pattern formula to predict how many border tiles I would need for any pattern number, with 2 centre tiles. To test this hypothesis I will draw a larger diagram and count the outer border tiles to check my formula. If I was to let the pattern number be N= 7, (See below) then:

B= 4(7) + 2

= 28 +2

= 30

So I expect that pattern number 7 will have 30 outer border tiles.

I have drawn pattern number 7 and counted 30 border tiles, which confirms that my formula is correct.

Total Tiles

I am also going to investigate whether I can find a formula which give the total amount of tiles in my pattern, T, provided I have the pattern number.

Table 2

Pattern: N

2

3

4

5

Total Tiles: T

8

8

32

50

72

+10 +14 +18 +22

+4 +4 +4

Table two shows the pattern number and total amount of tiles in that particular pattern. There is no constant first difference until the second difference (which is constant). This tells you the formula is based on N². The second difference is 4 and you divide this by 2, to get 2. This is the multiplier of N², so you should now have 2N². To get the rest of the formula you first find the multiplier of the 2N( when you subtract it from the original total tiles number, 8 - 4 = 4, which you multiply by, N, to get 4N. You then take this away from the original number to give you the final number to add, 2. So the formulae for total tiles, T, for patterns with two centre tiles is:

T= 2N² + 4N + 2

Testing my formula.

I will test my formula to see if it is produces the same values as in Table 2.

N=1, T = 2(1)² + 4(1) + 2 = 8.

N=2, T = 2(2)² + 4(2) + 2 = 18.

N=3, T = 2(3)² + 4(3) + 3 = 32.

I have proved that this formula gives me the same sequence of total tiles as I have counted from my diagrams.

Prediction

I can use my Total Tiles formula to predict how many tiles I would need for any pattern number, with 2 centre tiles. To test this hypothesis I will draw a larger diagram and count the total amount of tiles to check my formula. If I was to let the pattern number be N= 7, (See below) then:

T= 2N² + 4N + 2

= 2(7)² + 4(7) + 2

= 2(49) + 30

= 128

So I expect that pattern number 7 will have 128 tiles.

I have drawn pattern 8 and counted 98 tiles, this confirms that my formula is correct.

For three centre tiles

Table 1

Pattern: N

2

3

4

5

Border Tiles: B

8

2

6

20

24

+4 +4 +4 +4

B= 4n + 4

To get 4N at the start of the formula, you have to put the added outer borders into a table ( see above ). Each pattern number is counted as N and the squares in that outer most border is counted as B. These numbers change as each outer border gets bigger. Every pattern number with only centre tiles has no border, so it has not been included in the sequence to make my formulae. You notice you add 4 each time so that 4 is the multiplier, so you have 4 ??N which equals 4N. To get the +4 you subtract the difference of 4 from the original number of tiles you started with, 8. This will give you the equation, 8 - 4, which equals +4 and this is then added to give the pattern formula, 4N + 4.

Testing my Pattern formula.

I will test my formula to see if it is producing the same values as in Table 1.

N=1, B= 4(1) + 4 = 8.

N=2, B= 4(2) + 4 = 12.

N=3, B= 4(3) + 4 = 16.

I have proved that this formula gives me the same sequence of outer border tiles as I have counted from my diagrams.

Prediction

I can use my pattern formula to predict how many border tiles I would need for any pattern number, with 3 centre tiles. To test this hypothesis I will draw a larger diagram and count the outer border tiles to check my formula. If I was to let the pattern number be N= 7, (See below) then:

B= 4(7) + 4

= 32

So I expect that pattern number 7 will have 32 outer border tiles.

I have drawn pattern number 7 and counted 32 border tiles, which confirms that my formula is correct.

Total Tiles

I am also going to investigate whether I can find a formula which give the total amount of tiles in my pattern, T, provided I have the pattern number.

Table 2

Pattern: N

2

3

4

5

Total Tiles: T

1

23

39

59

83

+12 +16 +20 +24

+4 +4 +4

Table two shows the pattern number and total amount of tiles in that particular pattern. There is no constant first difference until the second difference (which is constant). This tells you the formula is based on N². The second difference is 4 and you divide this by 2, to get 2. This is the multiplier of N², so you should now have 2N². You will then need to find the second part of the equation, this will be the difference between 2N( subtracted from original number and 2(2)( subtracted from the original number. The final part of the equation is to add on the number of centre tiles. So the formula for total tiles, T, for patterns with 3 centre tiles is:

T= 2N² + 6N + 3

Testing my formula.

I will test my formula to see if it is produces the same values as in Table 2.

N=1, T = 2(1)² + 6(1) + 3 = 11.

N=2, T = 2(2)² + 6(2) + 3 = 23.

N=3, T = 2(3)² + 6(1) + 3 = 39.

This formula gives me the same sequence of total tiles as I counted from my diagrams.

Prediction

I can use my Total Tiles formula to predict how many tiles I would need for any pattern number, with 3 centre tiles. To test this hypothesis I will draw a larger diagram and count the total amount of tiles to check my formula. If I was to let the pattern number be N= 7, (See below) then:

T= 2N² + 6N + 3

= 2(7)² + 6(7) + 3

= 2(49) + 45

= 143

So I expect that pattern number 7 will have 143 tiles.

I have drawn pattern 7 and counted 143 tiles, this confirms that my formula is correct.

For four centre tiles

Table 1

Pattern: N

2

3

4

5

Border Tiles: B

0

4

8

22

26

+4 +4 +4 +4

B= 4N + 6

To get 4N at the start of the formula, you have to put the added outer borders into a table ( see above ). Each pattern number is counted as N and the squares in that outer most border is counted as B. These numbers change as each outer border gets bigger. Every pattern number with only centre tiles has no border, so it has not been included in the sequence to make my formulae. You notice you add 4 each time so that 4 is the multiplier, so you have 4 ??N which equals 4N. To get the +6 you subtract the difference of 4 from the original number of tiles you started with, 10. This will give you the equation, 10 - 4, which equals +6 and this is then added to give the pattern formula, 4N + 6.

Testing my Pattern formula.

I will test my formula to see if it is producing the same values as in Table 1.

N=1, B= 4(1) + 6 = 10.

N=2, B= 4(2) + 6 = 14.

N=3, B= 4(3) + 6 = 18.

I have proved that this formula gives me the same sequence of outer border tiles as I have counted from my diagrams.

Prediction

I can use my pattern formula to predict how many border tiles I would need for any pattern number, with 4 centre tiles. To test this hypothesis I will draw a larger diagram and count the outer border tiles to check my formula. If I was to let the pattern number be N= 7, (See below) then:

B= 4(7) + 6

= 28 +6

= 34

So I expect that pattern number 7 will have 34 outer border tiles.

I have drawn pattern number 7 and counted 30 border tiles, which confirms that my formula is correct.

Total Tiles

I am also going to investigate whether I can find a formula which give the total amount of tiles in my pattern, T, provided I have the pattern number.

Table 2

Pattern: N

2

3

4

5

Total Tiles: T

4

28

46

68

94

+14 +18 +22 +26

+4 +4 +4

Table two shows the pattern number and total amount of tiles in that particular pattern. There is no constant first difference until the second difference (which is constant). This tells you the formula is based on N². The second difference is 4 and you divide this by 2, to get 2. This is the multiplier of N², so you should now have 2N². You will then need to find the second part of the equation, this will be the difference between 2N( subtracted from original number and 2(2)( subtracted from the original number. The final part of the equation is to add on the number of centre tiles. So the formulae for total tiles, T, for patterns with 4 centre tiles is:

T= 2N² + 8N + 4

Testing my formula.

I will test my formula to see if it is produces the same values as in Table 2.

N=1, T = 2(1)² + 4(1) + 6 = 14.

N=2, T = 2(2)² + 4(2) + 6 = 28.

N=3, T = 2(3)² + 4(3) + 6 = 46.
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I have proved that this formula gives me the same sequence of total tiles as I have counted from my diagrams.

Prediction

I can use my Total Tiles formula to predict how many tiles I would need for any pattern number, with 4 centre tiles. To test this hypothesis I will draw a larger diagram and count the total amount of tiles to check my formula. If I was to let the pattern number be N= 7, (See below) then:

T= 2N²

= 2(7)²

= 2(49)

= 98

So I expect ...

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