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# Maths Number Grids/Sequences

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Introduction

Number Grid Investigation

Introduction:

The coursework task is to investigate the patterns generated from using rules in a square grid. The grid provides a structured approach to learning number relationships.

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In accordance with the task there are four steps to follow:

1. A box is drawn round four numbers [see grid above].
2. Find the product of the top left number and the bottom right number.
3. Do the same with the top right number and the bottom left number.
4. Calculate the difference between these numbers.

From the example in the above grid: [45 x 56 = 2520  –  46 x 55 = 2530] I find that the product difference for the diagonal square [2x2] is 10.

I am going to start by investigating as to whether or not the location of the 2x2 square on the grid is significant.

1. 81 x 92 = 7452  - 82 x 91 = 7462.             Product difference is 10.
2. 9 x 20 = 180 – 10 x 19 = 190.                    Product difference is 10.
3. 12 x 23 = 276 – 13 x 22 = 286.                  Product difference is 10.

From the worked examples A, B & C I find that the product difference is 10. Taking these results into account, I predict for any 2x2 square the result will always be 10.

Below is a table of results for 2x2 squares that were randomly chosen from the 10x10 grid.

Table 1. 2x2 results.

 1st No. multiplication 2nd  No. multiplication Difference 5 x 16 = 80 6 x 15 = 90 10 17 x 28 = 476 18 x 27 = 486 10 32 x 43 = 1376 33 x 42 = 1386 10 68 x 79 = 5372 69 x 78 = 5382 10

The results from T.1 show that the difference is a constant 10, so my prediction was correct.

In Mathematics, Algebra is designed to help solve certain types of problems; letters can be used to represent values, which are usually unknown. I will now attempt to prove my results algebraically. Let `N` represent the number in the top left of the square.

## N+10

N+11

This can be expressed into the equation:

( N + 1)

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Recap method: The top left x the bottom right  - the top right x the bottom left.

2 x2 results

1. ( 2x 9) – ( 1x 10) = 8
2. ( 29x 36) – ( 28x 37) = 8
3. ( 56x 63) – ( 55x 64) = 8

From the results A, B & C first observations enable me to predict that for any 2x2 square chosen randomly from the 8x8 grid will give a product difference of 8. I will now produce a second set of results to help my investigation.

Table.3. 2x2 results

 1st No. multiplication 2nd No. multiplication Difference 34x43 =1462 35x42=1470 8 5x14 = 70 6x13 = 78 8 51x60 = 3060 52x59 = 3068 8 7x16 = 112 8x15 =120 8

Once again the results in T.3 prove my prediction was correct – the constant product difference is 8. Therefore, the equation can be shown algebraically, let `N` represent the number in the top left of the square.

## N

N+1

N+8

N+9

This can be expressed into the equation:

( N+ 1) ( N+ 8) – N ( N + 9)

Multiply out the brackets:

N² + 8N + N +8 – N²– 9N

Simplified to:

N² + 9N + 8 –N² -9N  =   8

To reiterate, when subtracted the ‘N` cancels out leaving the number 8. Once again this is the size of the grid I am working with, and also it is the constant difference from a 2x2 square.

3x3 Results

1. ( 7x 21 ) – ( 5x 23 ) = 32
2. ( 20x 34) – ( 18x 36) = 32
3. ( 40x 54 ) – ( 38x 56) = 32

Algebraic table for 3x3:

## N

N+2

N+16

N+18

Equation:

(N + 2) ( N + 16) – N ( N + 18)

N² +2N +16N +32 – N² + 18N

+ 18N + 32 – 18N  = 32

4x4 Results

1. ( 4x 25 ) – ( 1x 28 ) = 72
2. ( 24x 45) – ( 21x 48 ) = 72
3. ( 36x 57 ) – ( 33x 60) = 72

Algebraic table for 4x4:

 N N+3 N+24 N+ 27

Equation:

( N + 3 ) ( N + 24) – N ( N + 27)

N² + 3N + 24N + 72 – N² + 27N

+27N + 72 – 27N = 72

Table 4. Results for 8x8 grid.

 Square selection size Product difference Difference between each difference p.d. Increase 2 x 2 8 3 x 3 32 24 16 4 x 4 72 40 16 5 x 5 128 56 16 6 x 6 200 72 16 7 x 7 288 88 16

By investigating further and looking at my results for T.4. I have established there is a formula for an nth term for any square selection size within an 8x8 grid.

 1st 2nd 3rd 4th (2x2) (3x3) (4x4) (5x5) 8 32 72 128 24 40 56 16 16

Nth term

Product difference

p.d.

Increase

The formula for finding any term in any size from the 8x8 grid is 8n² E.g.

• 2nd term  -  8 x 2² = 32  (correct, see T.4. 3x3)
• 4th term  -  8 x 4² =  128 (correct, see T.4. 5x5)
• 7th term  -  8 x 7²=  392 (correct, 7th term is an 8x8 square( 8x5 - 1x64 = 392)

I can conclude that the general formula for finding out product difference in an 8x8 grid is 8n²

Again from the results, I can see there is a pattern for each square size and grid size. This means I shall try my overall formula for a `s`  sized square on a `G` sized grid.

Formula is : G ( s – 1)²

8 (grid size) ( 4 (square size) – 1) ²

8 (4 – 1) ² = 72

The overall formula proves to be correct, see T4, a 4 x 4 square has a product difference of 72.

.

9x9 Grid

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Recap method: The top left x the bottom right – the top right x the bottom left.

2x2 Results

1. ( 2x 10) – ( 1x 11) = 9
2. ( 25 x 33 ) – ( 24x 34 ) = 9
3. ( 48x 56) – ( 47x 57 ) = 9

Results A, B & C show a product difference of 9. After further calculations of randomly chosen 2x2 squares prove that there is a constant product difference of 9.

I now present the equation algebraically, let `N` represent the number in the top left square.

## N

N+1

N+9

N+10

Equation:

( N + 1) ( N + 8) – N ( N + 9)

N² + N +9N + 9 – N² + 10N

+ 10N + 9 –10N  =  9

3x3 Results

1. ( 9x 27) – ( 7x 27) = 36
2. ( 57x 73) – ( 55x 75) = 36
3. ( 63x 79) – ( 61x 81) = 36

Algebraic table for 3x3:

#### N

N+2

N+18

N+20

Conclusion

3 x D = 20n + 20

I now have a formula for a 3 x D rectangle.

E.g. 3 x 7 – 5th term in the sequence – 20 x 5 + 20 = 120.

7 x 21 – 1 x 27 = 120.

My formula of  20n + 20 is correct.

From this I can begin to see the product differences are increasing by 10`s, and also the formulas are going up by 10`s. I can predict that the formula for finding a 4 x? rectangle will be 30 n + 30, but I will have to test this to see if this is true.

1. 4x 3 = 60
2. 4 x 4 = 90
3. 4 x 5 = 120

My prediction was correct. You need to multiply the nth term by 30 this time, and then add 30 which is the difference between each difference. I can now construct a formula for a 4 x D rectangle: 4 x D = 30 n + 30.

E.g. 4 x 6 – 4th term in the sequence – 30 x 4 + 30 = 150

6 x 31 – 1 x 36 = 150

M y formula of 30n + 30 is correct.

These workings out and formulas give me a rule for any sized rectangle, providing I know the width. Possibly there could be a rule for find any width size rectangle with an unknown sized length?

## Conclusion

In this project I have found that number grids are an extremely powerful tool for a wide range of maths concepts such as, number patterns, problem solving and investigation. I have successfully predicted what can come next in square selection sized differences, nth terms and algebraic equations that simplify expressions.

There are many different rules formulas within grid sizes and square or rectangle selections. If I were to extend this project I would possibly change the original rules with regards to diagonal multiplication and subtraction, to see if there are any different patterns, or I would change the grid sizes and rectangle sizes or perhaps try different shapes.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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