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• Level: GCSE
• Subject: Maths
• Word count: 1433

# Investigation in to How many tiles and borders is needed for each pattern

Extracts from this document...

Introduction

GCSE Maths Coursework                                                                Tanbir Hussain 10R

Borders

Introduction: In this I am going to find out how many tiles and borders is needed for each pattern. I am also going to make predictions. I am gong to work out a formula for the tiles and borders needed for the nth pattern. I will also work out a formula for the total tiles for each pattern.

## Diagrams:

1.

4 Borders                                               5.

5 Tiles

9 Altogether

2.

12 Borders

13 Tiles

25 Altogether

24 Borders

61 Tiles

85 Altogether

3.

6.

16 Borders

25 Tiles

41 Altogether

4.

20 Borders

41 Tiles

61 Altogether

28 Borders

85 Tiles

113 Altogether

7.

Middle

## Test Prediction:

I have tested the prediction by showing anther diagram of the 8th pattern. This pattern has 145 tiles, 36 borders and a total of 191 tiles together.

36 Borders

145 Ties

191 Altogether

Patterns:

I have noticed a pattern where the total tile of the 1st pattern is the number of tiles for the 2nd pattern. This applies for each of the patterns.

 Pattern No No Of Tiles No Of Borders Total Tiles 1 5 8 13 2 13 12 25

I also have noticed that if you add pattern No.1s border with the 1st difference you can get the second patterns number of borders.

For example:

n   Tiles  Borders   1st Difference

1     5       8 +4    8+4=12

2    13     12

3    25     16              +4    16+4=20

4    41     20

Formula:

I found out the formulas was a quadratic because it has a 2nd difference. For example in No of tiles:

5    13    25    41

8    12    16

4      4

I worked out the formula for the tiles, borders and the total tiles altogether. I used the general equation that is an +bn+c to work out the tiles and borders for the nth

1. From the table I have found a formula to calculate the border.

For the nth term the formula is 4n+4

1. Anther formula I found was to work out the number of tiles for the nth term.

For the nth term the formula is 2n+2n+1

1. Anther formula I found was to work out the total tiles for each pattern

For the nth term the formula is 2n +6n+5

Test The Formula:

1. Formula to calculate the number of borders is 4n+4

For example: when n=2

(4 2)+4

8+4=12

So then the 2nd pattern will have 12 borders.

2. Formula to calculate the number of tiles is 2n+2n+1

For example: When n=3

(3 ) 2=18

2  3=6

+1

=18+6+1

=25

So then for the 3rd pattern you will get 25 tiles.

3. Formula to calculate the total tiles for each pattern is 2n +6n+5

For example: When n=4

(4 ) 2=32

6  4=24

+5

=32+24+5

=61

Conclusion:

I have finished this part of my investigation and here is my result:

Here is the list of patterns I found:

n   Tiles  Borders   1st Difference

1     5       8 +4             8+4=12

2    13     12

3    25     16              +4           16+4=20

4    41     20

Looking at my patterns I found the following formulas:

1. To work out the number of tiles for the nth term the formula is

2n +2n+1

2. To work out the number of borders of the nth term the formula is

4n+4.

3. To work out the total borders for the nth term the formula is

2n +6n+5

In this in part of my investigation I have learnt that how many number of tiles and the number of borders is needed for any pattern that I have investigated.

I found this investigation quite easy but I had a little bit of trouble on finding the formula but eventually I solved by using the formula

an +bn+c.

a= ½ the 2second difference          b= the total of                        c is equal to when

a+c=1st difference                  n=0

Extension On Borders

Introduction:

In this part of my investigation I am going to extend the patterns. I am going to repeat the procedure with the extended patterns.

Diagrams:

1.                                                                 5.

10 Borders

8 Tiles

18 Altogether

2.

14 Borders

18 Tiles

32 Altogether

26 Borders

3.                                                                        72 Tiles

Conclusion

+4n+2

For example: When n=5

(5 ) 2=50

4  5=20

+2

=50+20+2

=72

So then for the 5th pattern you will get 72 tiles.

3. Formula to calculate the total tiles for each pattern is 2n +8n+8

For example: When n=6

(6 ) 2=72

8  6=48

+8

=72+48+8

=128

Conclusion:

I have finished this part of my investigation and here is my result:

Here is the list of patterns I found:

n   Tiles  Borders   1st Difference

1     8       10 +4    10+4=14

2    18      14

3    32      18              +4    18+4=22

4    50      22

Looking at my patterns I found the following formulas:

1. To work out the number of borders of the nth term the formula is

4n+6.

2. To work out the number of tiles for the nth term the formula is

2n +4n+12

3. To work out the total borders for the nth term the formula is

2n +8n+8

In this in part of my investigation I have learnt that how many number of tiles and the number of borders is needed for any pattern that I have investigated.

I found this investigation quite easy but I had a little bit of trouble on finding the formula but eventually I solved by using the formula

an +bn+c.

a= ½ the 2second difference          b= the total of                        c is equal to when

a+c=1st difference                 n=0

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