32 Borders
113 Tiles
145 Altogether
Table Of Results:
Here are the table of results in my diagrams.
Prediction:
I predict that the next pattern will have 145 tiles and 36 borders.
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.
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
- From the table I have found a formula to calculate the border.
For the nth term the formula is 4n+4
- 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
- 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 2 second 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
98 Altogether
18 Borders 6.
32 Tiles
50 Altogether
4. 22 Borders
50 Tiles
72 Altogether
30 Borders
98 Tiles
128 Altogether
34 Tiles
128 Tiles
162 Altogether
Table Of Results:
Here are the table of results in my diagrams.
Prediction:
I predict that the next pattern will have 162 tiles
and 38 borders.
Test Prediction:
I have tested the prediction by showing anther diagram
of the 8th pattern.
This pattern has 162 tiles, 38 borders and a total of
200 tiles together.
162 Tiles
38 Borders
200 Altogether
Patterns:
Again the patterns I noticed are the same where the total tile of the 1st pattern is the number of tiles for the 2nd pattern. This applies for each of the patterns.
For example:
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
Formula:
I found out the formulas was a quadratic because it has a 2nd difference. For example in No of tiles:
8 18 32 50
10 14 18
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+6
2. Anther formula I found was to work out the number of tiles for the nth term.
For the nth term the formula is 2n +4n+2
3.Anther formula I found was to work out the total tiles for each pattern
For the nth term the formula is 2n +8n+8
Test The Formula:
1. Formula to calculate the number of borders is 4n+6
For example: when n=4
(4 4)+6
16+6=22
So then the 4th pattern will have 22 borders.
2.Formula to calculate the number of tiles is 2n +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 2 second difference b= the total of c is equal to when
a+c=1st difference n=0