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• Level: GCSE
• Subject: Maths
• Word count: 1567

# T shapes. Once I have all of the results, I shall work out the nth term and investigate the relationship between the T number and the T total.

Extracts from this document...

Introduction

Kate Stewart

I am going to draw several T shapes on a 9 by 9 number grid and get the total of the numbers inside the T shape, which is called the T total, get the T number, which is the bottom number in the T shape and note them down in a table of results. I will repeat this a few times and note down the other results I get in the same table.

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Table of results.

 T number T total 20 37 +5 21 42 +5 22 47 +5 23 52 +5 24 57 n 5n-63

5n-63 is the general rule.

Predict and test.

T25 = 57 + 5 = 62.

You get 57 from the table and you +5 because the pattern goes up in 5. It is just an easier way of working out a prediction.

62 is my prediction for the answer.

I chose T25 because it is the next consecutive number in the sequence.

I am now going to test it to see if my prediction was correct.

 6 7 8 15 16 17 24 25 26

T25 = 6+7+8+16+25 = 62

My prediction was correct. This means that the general rule is correct. However, I am going to try again with another T shape just to make sure the general rule actually works.

T68 = (68x5) – 63 = 277.

277 is my prediction and 68 is my T number.

I am now going to test it out to see if my prediction is correct. If it is correct, then the general rule is correct.

 49 50 51 58 59 60 67 68 69

T68 = 49+50+51+59+68= 277

Therefore, as my prediction was correct, the general rule is also correct.

In general.

 n-19 n-18 n-17 n-9 n

T number = n

T total = n+n-9+n-18+n-17+n-19

=5n-63.

You have 5n because there are 5 squares and it is -63 because all of the – numbers added together = -63.

I am going to draw several T shapes on several random number grids of different sizes to translate the T shapes to different positions and then when I have the results, I will investigate the relationships between the T total, the T number and the grid size.

8 by 8 number grid.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Conclusion

 33 34 35 41 42 43 49 50 51

T50=33+34+35+42+50=194

As my second prediction was also correct, this means that my general rule is also correct.

In general.

 n-17 n-16 n-15 n-8 n

T number = n

T total = n+n-8+n-17+n-16+n-15

= 5n-56.

It is 5n because there are 5 squares and it is -56 because all the minus numbers added together make -56.

6 by 6 number grid.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
 T number T total 14 28 +5 15 33 +5 16 38 +5 17 43 n 5n-42

5n-42 is the general rule.

Predict and test.

T20=100-42=58

58 is my prediction.

 7 8 9 13 14 15 19 20 21

T20=7+8+9+14+20=58.

My prediction was correct so that means that my general rule may also be correct but I will try it out again just to be sure.

T35= (35x5)-42=133

 22 23 24 28 29 30 34 35 36

T35= 35+29+23+22+24=133

My prediction was correct again and so therefore is my general rule.

In general.

 n-13 n-12 n-11 n-6 n
 Grid size = g T total 9 5n-63 +7 8 5n-56 +14 6 5n-42 g 5n-7g

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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