РефератыИностранный языкTsTshapes Essay Research Paper Looking at the

Tshapes Essay Research Paper Looking at the

T-shapes Essay, Research Paper


Looking at the 9-9 grid below and the T-shape drawn on it,


The total number of the numbers on the inside of the T-shape is called the T-total


123456789


101112131415161718


192021222324252627


282930313233343536


373839404142434445


464748495051525354


555657585960616263


646566676869707172


737475767778798081


828384858687888990


The t-total for this T-shape is:


1+2+3+11+20=37


So 37 = T-total


The number at the bottom is the T-number, So the T-number for this shape is 20


Aims:


1)Investigate the relationship between the T-total and the T-number


2)Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T-number and the grid size.


3)Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.


Aim 1- the solution


123456789


101112131415161718


192021222324252627


282930313233343536


373839404142434445


464748495051525354


555657585960616263


646566676869707172


737475767778798081


828384858687888990


T69= 50+51+52+60+69


=282


T22=3+4+5+13+22


=47


In the diagram below it shows the difference between the T-number and the other numbers. First is the T-shape in question:


123


11


20


This is the T-shape and here is the Difference T-shape:


N-19N-18N-17


N-9


N


This shows the difference N= T-number


In the T on the previous page I have noticed that the first difference from N is 9 which is also the Width of the square.


I?ll put that idea into another T. Note W= width number(9)


N-(2W-1)N-2WN-(2W+1)


N-W


N


This is the same thing as before but shown algebraically.


The formula for the Value of the T-total now is shown as:


5N-7W=T-total


Aim 2- different sizes and relationship


I know this works for the grid 9 by 9 but I?m not sure if it?ll work for any other grids.


Here is a test for a 10 by 10 grid


12345678910


11121314151617181920


21222324252627282930


31323334353637383940


41424344454647484950


51525354555657585960


61626364656667686970


71727374757677787980


81828384858687888990


919293949596979899100


T22=1+2+3+12+22


=42 I notice this is 5 more than 9 by 9


T69=48+49+50+59+69


=275Obviously no pattern

there.


Method test?


(695)-70=275 YES it worked


My method seems to have worked out as it is logical and fairly straight forward to explain.


12345678910


11121314151617181920


21222324252627282930


31323334353637383940


41424344454647484950


51525354555657585960


61626364656667686970


71727374757677787980


81828384858687888990


919293949596979899100


As there are 10 in each row it?s obvious that the row above will be 10 less than the row below. So 68 is 10 less than the T-number 78. If you calculate the whole T you realise that row 2 is 10 less than row 1 and row 3 is 20 less than row 1,but there are three relevant numbers in row 3 which are 19 less and 21 less than the T-number. These cancel out to form 20 each, So finally we get (110)+(610)=(710)


=70


Or =7W


12 by 12


123456789101112


131415161718192021222324


252627282930313233343536


373839404142434445464748


495051525354555657585960


616263646566676869707172


737475767778798081828384


858687888990919293949596


979899100101102103104105106107108


109110111112113114115116117118119120


121122123124125126127128129130131132


133134135136137138139140141142143144


T62=37+38+39+50+62 also =(625)-(712)


= 226 =226


T141=116+117+118+129+141 also =(1415)-(712)


=621 =621


Aim3- Transformations stretches and there effects on the formula


I?ll do this with a 12 by 12 first, as this will give me enough accuracy to start with.


123456789101112


131415161718192021222324


252627282930313233343536


373839404142434445464748


495051525354555657585960


616263646566676869707172


737475767778798081828384


858687888990919293949596


979899100101102103104105106107108


109110111112113114115116117118119120


121122123124125126127128129130131132


133134135136137138139140141142143144


Stretch A will be called ST64 as it starts at 64, it?s a stretch of 2 in both directions.


St64=26+27+28+29+30+40+52+64


=296


I think I can work out the formula using my previous method so:


12+24+36+(436)=216


21612=18


This means the formula is:


8N-18W=T-total


8N= number of integers in the T-shape


18W=difference number calculated


Conclusion:


The size of the T-shape calculates the number before N in the formula and the grid size calculates the value of W. the number before W is calculated by looking at the rows and finding how many rows away from the T-number they are. If the T is regular then the W number is negative but if the T is flipped upside down the W number is positive.

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