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• Level: GCSE
• Subject: Maths
• Word count: 1381

Investingating L-totals

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Introduction

Practice coursework

INVESTIGATING “L” TOTALS

 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

An 8x8 grid is a grid that is 8 rows across by 8 rows down. The numbers go up in sequences of 1.

The “L” total is the total of all of the numbers in the L shape. The “L” shape is the shape below. Its name comes from the shape that it is.

 1 9 17 18

=45

 2 10 18 19

=49

 3 11 19 20

=53

 4 12 20 21

=57

I have found that my numbers go up in sequences of 4 if add them together

To identify where I placed my “L” on the grid I am going to use the bottom left number

E.g.

 34 42 50 51

=177

I am going to come up with a table of results for different numbers

THESE ARE MY RESULTS FOR THE L NUMBER BEING 17 GOING UP TO 24.

 L NUMBER L TOTAL 17 45 18 49 19 53 20 57 21 61 22 65 23 69 24 73

I have found that my totals increase in sequences of 4

I am now going to miss a few out and carry on at number 33

First off I am going to predict that number 33’s L total is 109

 17 25 33 34

33+34+17+25 = 109

I have found that however much further up the table that you go then they still go up in 4’s. It is also always going to be an odd number.

 L-16 L-8 L L+1

L+1+L+L-8+L-16

4L-24+1

4L-23

Middle

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I am going to add 1+2+11+20

=This total equals 34

I am now going to add 2+3+12+21

This total equals 38

I am now going to add 3+4+13+22

This total equals 42

The formula for this is 4l+3g-1

I am now going to flip the “L” shape so that it is the other way around – this “L” shape will be coloured blue. The “L” number is the one in the top right. It is highlighted blue

I am now going to add 2+11+20+19

The total of this is 52

I am now going to add 3 +12+21+20

This total is 56

I am concluding that this is increases in sequences of four

The formula for this box is 4L+3g-1. This is the same as the sequence above. This means that their totals will be the same

I am now going to see what happens to the sequences if I increase the size of the table so that it is 9X9. This means that it will be nine squares across and 9 squares down

 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

I am now going to add up the numbers in the “L” shape. The “L” number is the one highlighted in yellow

1+10+19+20 = 50

I am now going to add the next lot of numbers

2+11+20+21 = 54

I am now going to, again add the next sequence of numbers

3+12+21+22 = 58

I am going to make a table of results to show you what I mean

 “L” number Total 19 50 20 54 21 58

I am concluding that this sequence also goes up in fours. This surprised me however because I would expect it to go up in fives because I have added one more column of numbers onto the table. The total is always an even number.

The algebra of this sequence is:

 L-18 L-9 L L+1

Conclusion

 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
 L-25 L-24 L-16 L-17 L-8 L-7 L-6 L-5 L L+1 L+2 L+3

I have worked out that this formula is:

12L-107+6

12L-101

12L-6G+6

I am going to try to work out the total of the numbers in the box

This total equals 202

I am now going to add 2+3+10+11+18+19+20+21+26+27+28+29

This total equals 214

I am now going to add 3+4+11+12+19+20+21+22+27+28+29+30

This total equals 226

I have concluded that it goes up in sequences of 12

I am now going to try to adjust the size of my “L” so that it is 5 down and 5 across

 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

I am going to try to work out the total of the numbers in the box

I am going to add 1+2+9+10+17+18+25+26+27+28+29+33+34+35+36+37

The total of this is 367

I am now going to move the “L” shape along one space so now I am going to add 2+3+10+11+18+19+26+27+28+29+30+34+35+36+37+38

The total of this is 383

I am now going to move the “L” shape along one space so now I am going to add 3+4+11+12+19+20+27+28+29+30+31+35+36+37+38+39

The total of this is 399

I have concluded that the totals go up in sequences of 16

If I have an L, shape like this:

 L-32 L-31 L-24 L-23 L-16 L-15 L-8 L-7 L-6 L-5 L-4 L L+1 L+2 L+3 L+4

I have worked out a formula for this:

16L-168+4

16L-164

Or:

16L-8g+6

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