by 16 to give me a product of 784. Finally, I will subtract 874 away from 784 to
give me a product difference of 90.
4 × 4 Grids:-
I have learnt from the above 4 × 4 grids that the difference between the
products has not changed throughout the following 3 cases I have completed.
I have found out that the difference between the products is always 90.
5 × 5 Grid Example:-
As can be seen above, a 5 × 5 square grid has been highlighted. I will use
this square grid below to help me reach a prediction for the product difference
of a 5 × 5 grid. I will firstly begin, by multiplying together the bottom left
number which is 45 by the top right number which is 9, The product comes to
405. I will then work out the next product number by multiplying together the
bottom right number by the top left number, which gives me a result of 245.
Conclusively, I will minus 245 away from 405 to give me a product difference of
160.
5 × 5 Grids:-
I have realised that from the 5 × 5 grids that I have drawn, there is always
a product difference of 160, the various different numbers in the box does
not affect the product difference of 160.
6 × 6 Grid Example:-
As can be seen in the following 10 × 10 grid, I have highlighted the 6 × 6
square grid, I will be predicting and working on. Below I have drawn a grid
and I will multiply 93 by 48 to give me a product of 4464. I will then multiply
98 by 43 to give me a product of 4214. Finally, I will subtract 4214 away from
4464 to give me a product difference of 250.
6 × 6 Grid Example:-
6 × 6 Grids:-
I have calculated the difference between the products for all of the 6 × 6
grids and now know the difference between the products is exactly 250. The
size of the grid does not affect the product difference of 250.
A Table to show the product differences of different grid sizes:-
Above, I have drawn a table to show the product differences of different grid
sizes. From these differences, I have worked out the rule is 10 n².
I worked this out by writing down the differences for each of the 5 different
grid sizes and then tried to find differences within the product differences,
from which I continued to do so until I reached 10n².
I will now use this rule to come up with a specific rule, to help me work
out if the product differences are correct.
Below I have drawn a Quadratic Sequence table to help me come up with a
rule; this rule will tell me how I will work out the algebra for each of the grid
sizes. This rule will also tell me if I have come up with the correct product
differences.
Quadratic Sequence!
I have now found out that the Nth term is 10n² - 20n +10. I will now use
this rule further to investigate and reach a prediction for the 7 × 7 grid. This
term will tell me if this is the correct rule to follow. I think that that the 7 × 7
grid will have a product difference of 360. I think this because:-
= 1 0 × 7 ² = 4 9 0
= 2 0 × 7 = 1 4 0
= 4 9 0 – 1 4 0 = 3 5 0
= 3 5 0 + 1 0
= 3 6 0
Above, I have attempted to prove in algebra that the 7 × 7 grid, product
difference always comes to 360.
Differences between products, with Algebra:-
2 × 2 Grid:-
I have once again proved, that the product difference in a 2 × 2 grid always
comes to 10. This algebra method above shows me that the change of
numbers in the same length and width grid has no impact on the product
difference, because the answer always comes to a certain number 10.
3 × 3 Grid: -
This algebra method, is once again correct because the product difference
always came to 40 in all the 3 × 3 grids and has now appeared above in
the algebra.
4 × 4 Grid:-
This 4 × 4 grid algebra shows me that 90 is the definite product difference
in all of the grids I have drawn without using algebra.
5 × 5 Grid:-
Above, I have drawn out a 5 × 5 square grid showing the algebra proving
to me that my all my 5 × 5 grids were correct as they all came out with a
product difference of 160, which this algebraic method shows.
6 × 6 Grid: -
250 is the accurate product difference for a 6 × 6 square grid. It is also
proved by this algebra method and my previous 4 grids drawn to prove
that this answer is correct.
7 × 7 Grid: -
This is the square grid which I predicted previously and the product
difference positively came to 360. I showed this by using algebra as
well as the rule to come up with this difference.
8 × 8 Grid:-
An 8 × 8 grid has been drawn to show the algebra, to prove that the product
difference is 490. I have correctly identified this answer on this algebra grid
because 1458 minus 968 is 490.
9 × 9 Grid:-
The algebra for this 9 × 9 grid is correct, I can see this because when
1089 is subtracted away from 1729, the product difference comes to 640.
10 × 10 Grid:-
This is the final algebra grid I have drawn. Although I have not drawn any
10 × 10 grids, I know definitely that the answer is 890. I know this because
the answer always comes from the previous numbers square number which
is multiplied by 10. In this case the previous number nines square number is
81, which is then multiplied by 10 to give me a product of 810.
Conclusion/ Evaluation!
I have completed my investigation on finding the difference between products
in square grids and have come to the conclusion that the correct rule was
established, 10n² - 20n +10 . In this investigation I had various different sized
square grids from which I had to multiply the bottom left number by the top
right number and multiply the bottom right number by the top left number.
After finding the products, I subtracted away the first product found from the
second product found, from this I worked out the product difference for each
of the square grids. I have also realised that the different numbers put in the
same sized grid do not affect the product difference found. The coursework
I have produced, does meet the requirements of my aim. My aim was to find
the product difference in various different square grids. During this coursework
I did not come across any anomalous results.
Maths Coursework – Algebra Extension!
Objective: - I will carry out an investigation to reveal the relationship
between the differences of products found in different sized
rectangle grids.
I have completed the first part of the coursework, now I will continue this
investigation by extending this coursework and examining the product
difference between different sized rectangle grids. I will begin this investigation
by starting off with a 2 × 4 rectangle grid.
Method:-
1. Firstly, I will find a rectangle grid anywhere on the 10 × 10 square grid.
2. I will then multiply the bottom left number by the top right number to
give me the product number.
3. I will then find out the second product number in the rectangle by
multiplying the bottom right number by the top left number.
4. I will then subtract the first product away from the second product, to give
me the specific product difference.
5. Finally, I will use algebra, to prove and support my answer is correct.
I will carry on this investigation by starting off with a 2 × 3 grid and doing
3 examples of each grid. I will then carry on this study until I have completed
3 examples of 4 × 8 grids. After this I will make a prediction for the 5 × 10 grid
and test it to find out the nth term for the rectangle grids. I will then test the rule
further to see if it works with all the other previous rectangle grids.
2 × 4 Grid Example:-
2 × 4 Grids:-
From the following 2 × 4 rectangle grids, I have
found out that the difference between the
products is always 20. I have also uncovered that
the different numbers present in the boxes does not
affect the difference of 20.
3 × 6 Grid Example:-
3 × 6 Grids:-
From the following 3 cases, I have
found out that all the rectangular
3 × 6 grids, have a product difference
of 100. I can now also see that , in
these rectangle grids, the numbers
present in the boxes does not affect
the product difference of 100.
4 × 8 Grid Example:-
4 × 8 Grids:-
I have learnt from my 4 × 8 rectangle grids
that the difference between the products
has not changed throughout the following 3
cases I have completed. I have found out
that the difference between the products is
always 210.
A table to show product differences in Rectangles:-
Above, I have drawn a table to show the product differences of various sized
Rectangles. I have also found out by inspection that the length and the width is
always minus 1 multiplied by 10. The length and the width of the rectangles does not affect the rule found, as they it is not taken into consideration. The rule found is always 10n. I will now use this rule to come up with a rule.
I have now found out that the Nth term is ( L – 1 ) ( W – 1 ) × 10. I will now
use this rule further to investigate and reach a prediction for the 5 × 10 grid.
This Nth term will tell me if this is the correct rule to follow. I predict that the
product difference found will be 360.
5 × 10 Grid Prediction:-
This 5 × 10 grid algebra shows me that the product difference in all of the same length and width of rectangular grids is always 360.
Algebra Method:-
1. Find the length of the rectangular grid and minus 1.
2. Find the width of the rectangular grid and once again minus 1.
3. Multiply the answer found from the length and the width together and then
multiply this answer by 10, to give us the product difference.
Differences between products, with Algebra:-
2 × 4 Grid:-
( L – 1 ) ( W – 1) × 10
( 2 – 1 ) ( 4 - 1 ) × 10
= 1 × 3 × 1 0 = 3 0
I have proved that the product difference in a 2 × 4 grid is always 30. This
algebra method above shows me that the different numbers in the same
rectangular grid has no impact on the product difference, because the result
is always 30.
3 × 6 Grid:-
( L – 1 ) ( W – 1) × 1 0
( 3 – 1) ( 6 – 1 ) × 1 0
= 2 × 5 × 10 = 1 0 0
3 × 6 Grid Algebra:-
This 3 × 6 grid above shows the algebra for it, the product difference for this
grid always has a product difference of 100.
4 × 8 Grid:-
( L – 1 ) ( W – 1) × 1 0
( 4 – 1) ( 8 – 1 ) × 1 0
= 3 × 7 × 1 0 = 2 1 0
Above, I have drawn a 4 × 8 rectangular grid, which shows me the total
product difference of 210.
Conclusion/Evaluation!
I have completed my investigation on finding the difference between
products in rectangles and have come to the conclusion that the correct
rule was established, ( L – 1 ) ( W – 1) × 10. In this investigation I had various
different sized rectangles and I had to multiply the bottom left number by the
top right number and multiply the bottom right number by the top left number.
After I did this, I subtracted away the first product found from the second
product found, from this I worked out the product difference for each rectangle
gird. I have also realised that the different numbers put in the same sized
grid do not affect the product difference. The coursework I have produced,
does meet the requirements of my aim. My aim was to find the product
difference in different sized rectangle grids. During this coursework I did not
come across any anomalous results.