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CourseworkHelp
:Table Patterns
Q1. For the first part of this investigation into table patterns I am going to see what happens when you add the two numbers in opposite corners of squares which have a side 3 (which are drawn around nine numbers). Then I will investigate what happens when you subtract the smaller answer from the larger answer. After this we will take this investigation further. The squares are taken from the following ‘Multiplication Tables Grid’.
|
X |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
2 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
|
3 |
3 |
6 |
9 |
12 |
15 |
18 |
21 |
24 |
27 |
30 |
|
4 |
4 |
8 |
12 |
16 |
20 |
24 |
28 |
32 |
36 |
40 |
|
5 |
5 |
10 |
15 |
20 |
25 |
30 |
35 |
40 |
45 |
50 |
|
6 |
6 |
12 |
18 |
24 |
30 |
36 |
42 |
48 |
54 |
60 |
|
7 |
7 |
14 |
21 |
28 |
35 |
42 |
49 |
56 |
63 |
70 |
|
8 |
8 |
16 |
24 |
32 |
40 |
48 |
56 |
64 |
72 |
80 |
|
9 |
9 |
18 |
27 |
36 |
45 |
54 |
63 |
72 |
81 |
90 |
|
10 |
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
100 |
Example: Add the two numbers in the opposite corners.
|
8 |
10 |
12 |
|
12 |
15 |
18 |
|
16 |
20 |
24 |
8+24=32 and 12+16=28
Now, subtract the smaller answer from the larger answer.
32-28=4
Now, I will investigate this theory three times by picking out random number square and then I will make a hypothesis.
Square 1:
|
48 |
56 |
64 |
|
54 |
63 |
72 |
|
60 |
70 |
80 |
48+80=128 and 60+64=124
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 2
Therefore,
128-124=4
Well, we have the answer 4. I will try again and see whats answer will come out this time.
Square 2:
|
6 |
9 |
12 |
|
8 |
12 |
16 |
|
10 |
15 |
20 |
6+20=26 and 10+12=22
Therefore,
26-22=4
Okay, I am hoping that 4 is the next answer so I can make the hypothesis that I want to.
Square 3:
|
4 |
6 |
8 |
|
6 |
9 |
12 |
|
8 |
12 |
16 |
4+16=20 and 8+8=16
Therefore,
20-16=4
I am now in a position to make my hypothesis and then I will attempt to prove it using algebra.
Hypothesis
When you add the two numbers in opposite corners and subtract the smaller number from the larger number the answer is always 4. I have also found that the top left and bottom right numbers, when added up, is the larger number (this is the number is pink). I am now going to try to prove this statement by using algebraic terms.
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 3
This is my drawn hypothesis:
|
TL |
T |
TR |
|
ML |
M |
MR |
|
BL |
B |
BR |
(TL+BR)-(BL+TR)=4
'font-size:14.0pt; '>The Proving of My Hypothesis:
Starting at any point along one axis and assuming the value is x and starting at any point along the other axis and assuming the value is y then the diagram below shows the corresponding values inside the grid.
|
xy |
y(x+1) |
y(x+2) |
|
X(y+1) |
(y+1)(x+1) |
(y+1)(x+2) |
|
X(y+2) |
(y+2)(x+1) |
(y+2)(x+2) |
'font-size:14.0pt; '>Using these values I will test the hypothesis by creating an equation and solving it:
xy+(y+2)(x+2)=xy+xy+2y+2x+4
y(x+2)+x(y+2)=xy+2y+xy+2x
Therefore,
(xy+xy+2y+2x+4)- (xy+2y+xy+2x)=4
My hypothesis is now proven and this equation has proven that the answer is always 4.
Q2. For this second investigation into a square, which consists of 9 numbers, I am going to investigate what happens when the numbers in
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 4
the four corners is added together. Then I will add the middle numbers on each side together and compare them to the corner numbers.
Example: Add the numbers in the four corners.
|
8 |
10 |
12 |
|
12 |
15 |
18 |
|
16 |
20 |
24 |
8+12+16+24=60
Now, add the middle numbers on each side
|
8 |
10 |
12 |
|
12 |
15 |
18 |
|
16 |
20 |
24 |
10+12+18+20=60
I notice that both of the numbers equal 60. Also, on the corner numbers the top ones add up to 40 and the bottom to 20. Also, on the middle numbers the top middle and bottom add up to 30 as do the two side middles. I will now investigate this theory in three random attempts to see if any of these statements are right or could be right.
Square 1:
|
15 |
18 |
21 |
|
20 |
24 |
28 |
|
25 |
30 |
35 |
15+21+25+35=96
18+20+28+30=96
Both numbers are the same. I will try another two more times. However, I have already shown that on the corner numbers the top ones do not add up to 40 and the bottom ones to 20. Also, on the middle numbers the top middle and bottom do not add up to 30 as don’t the two side middles. So, I am left with the theory that the results are equal. Let’s carry on and see if that is true.
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 5
Square 2:
|
28 |
35 |
42 |
|
32 |
40 |
48 |
|
36 |
45 |
54 |
28+42+36+54=160
35+32+48+45=160
They are equal again. Let’s try once more.
Square 3:
|
56 |
63 |
70 |
|
64 |
72 |
80 |
|
72 |
81 |
90 |
56+70+72+90=288
63+64+80+81=288
They are both the same again. Now I am going to write a hypothesis and I am going to prove it using algebraic methods.
Hypothesis
My hypothesis is that when you add these two sets of numbers together the answers are both the same.
This is my drawn hypothesis:
|
TL |
T |
TR |
|
ML |
M |
MR |
|
BL |
B |
BR |
(T+ML+MR+B)=(TL+TR+BR+BL
Also
(T+ML+MR+B)-(TL+TR+BR+BL=0
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 6
The Proving of My Hypothesis
Starting at any point along one axis and assuming the value is x and starting at any point along the other axis and assuming the value is y then the diagram below shows the corresponding values inside the grid.
|
xy |
y(x+1) |
Y(x+2) |
|
X(y+1) |
(y+1)(x+1) |
(y+1)(x+2) |
|
x(y+2) |
(y+2)(x+1) |
(y+2)(x+2) |
'font-size:14.0pt; '>Using these values I will test the hypothesis by creating an equation and solving it:
(xy+ y(x+2)+ x(y+2)+ (y+2)(x+2))=(y(x+1)+x(y+1)+(y+1)(x+2)+(y+2)(x+1))
(xy+(xy+2y)+(xy+2x)+(xy+2x+2y+4)=(xy+y)+(xy+x)+(xy+x+2y+2)+(xy+2x+y+2))
 |
 |
 |
 |
 |
|||||
4xy+4y+4x+4=4xy+4y+4x+4
4'font-size:16.0pt; '>=4
The equations are equal so I have now proven my hypothesis that the numbers always are equal to each other.
Q3. I will now try to find two patterns of my own then I will investigate and prove that these patterns are always true.
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 7
Pattern 1:
Example:
|
8 |
10 |
12 |
|
12 |
15 |
18 |
|
16 |
20 |
24 |
Add the two purple numbers and then add together the two blue numbers.
12+18=30 10+20=30
The numbers both equal 30. This could mean that they are the same all of time. However, this could just be a coincidence so I will check it three times.
Square 1:
|
24 |
32 |
40 |
|
27 |
36 |
45 |
|
30 |
40 |
50 |
27+45=72 40+32=72
The numbers are equal.
Square 2:
|
28 |
32 |
36 |
|
35 |
40 |
45 |
|
42 |
48 |
54 |
35+45=80 32+48=80
Again the numbers are equal.
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 8
Square 3:
|
56 |
63 |
70 |
|
64 |
72 |
80 |
|
72 |
81 |
90 |
64+80=144 63+81=144
The numbers are again equal now I am going to make my hypothesis.
Hypothesis:
My hypothesis is that when you add the middle right and the middle left numbers together they equal the centre top added to the centre bottom.
This is my drawn hypothesis:
|
TL |
T |
TR |
|
ML |
M |
MR |
|
BL |
B |
BR |
(ML+MR) =(T+B)
Also
(ML+MR)-(T+B)=0
The Proving of My Hypothesis
Starting at any point along one axis and assuming the value is x and starting at any point along the other axis and assuming the value is y then the diagram below shows the corresponding values inside the grid.
|
xy |
y(x+1) |
Y(x+2) |
|
X(y+1) |
(y+1)(x+1) |
(y+1)(x+2) |
|
x(y+2) |
(y+2)(x+1) |
(y+2)(x+2) |
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 9
'font-size:14.0pt; '>Using these values I will test the hypothesis by creating an equation and solving it:
x(y+1)+(y+1)(x+2)=y(x+1)+(y+2)(x+1)
(xy+x+xy+x+2y+2)=(xy+y+xy+y+2x+2)
 |
|||||||||||
 |
 |
 |
 |
 |
|||||||
2xy+2x+2y+2=2xy+2x+2y+2
2=2
The equations are equal so I have now proven my hypothesis that the numbers are always equal to each other.
Pattern 2:
Example:
|
8 |
10 |
12 |
|
12 |
15 |
18 |
|
16 |
20 |
24 |
Multiply the top left number by the bottom right number and compare it to the bottom left number multiplied by the top right number.
8x24=192
16x12=192
I notice that the numbers are both the same now I am going to repeat this, with different random squares, another three times.
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 10
Square 1:
|
4 |
8 |
12 |
|
5 |
10 |
15 |
|
6 |
12 |
18 |
4x18=72
6x12=72
The answers are both the same.
Square 2:
|
14 |
16 |
18 |
|
21 |
24 |
27 |
|
28 |
32 |
36 |
14x36=504
28x18=504
The answers are both the same again.
Square 3:
|
24 |
30 |
36 |
|
28 |
35 |
42 |
|
32 |
40 |
48 |
24x48=1152
32x36=1152
The products are both the same again meaning I am going to now make my hypothesis.
2000 Maths GCSE Coursework Ben Blackmore 10 Shirley Mr. Wellings
Page 11
Hypothesis:
My hypothesis is that when you multiply the top left and the bottom right numbers together there product equals the product of the top right and the bottom left.
This is my drawn hypothesis:
|
TL |
T |
TR |
|
ML |
M |
MR |
|
BL |
B |
BR |
(TLxBR)=(BLxTR)
Also
(TLxBR)-( BLxTR)=0
The Proving of My Hypothesis
Starting at any point along one axis and assuming the value is x and starting at any point along the other axis and assuming the value is y then the diagram below shows the corresponding values inside the grid.
|
xy |
y(x+1) |
Y(x+2) |
|
X(y+1) |
(y+1)(x+1) |
(y+1)(x+2) |
|
x(y+2) |
(y+2)(x+1) |
(y+2)(x+2) |
'font-size:14.0pt; '>Using these values I will test the hypothesis by creating an equation and solving it:
(xy) x ((y+2)(x+2))=(x(y+2)) x (y(x+2))
(xy) x (xy+2x+2y+4)=(xy+2x) x (xy+2y)
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 |
 |
 |
 |
 |
 |
||||||||||
x2y2+2x2+2xy2+4xy= x2y2+2x2+2xy2+4xy
Therefore both sides are equal to each other so I have proved my hypothesis that both sides are the same.
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