• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See how many squares would be needed in order to construct any cross-built up in the way described in the investigation.

Extracts from this document...

Introduction

Khadija Patel                

Borders

Part 1-

Aim: -

My aim is to see how many squares would be needed in order to construct any cross-built up in the way described in the investigation.

Plan:-

  • To use simple techniques first and begin with a simple number.
  • Use examples, diagrams and label them.
  • To record all results found in a table
  • Extend investigation to 3-D

Method:-

To create a larger cross, surround the shaded squares with white squares. To begin my investigation I will start off with a simple number of squares.

Pattern 1- Number of shaded squares I have chosen is 1.

   Total number of squares = 4 + 1

                                        = 5

White squares = 4

By using the method stated in the investigation, I have constructed a larger cross. By using the same method I will again construct the next sequence of crosses.

Pattern 2-                                 Pattern 3- Total number of squares = 25

                                                        White squares = 12        

Total number of squares = 13

White squares = 8

Pattern 4-

                                                                 Total number = 41

                                                                White squares = 16

I have noticed that the squares going down are odd 1+3+1=5. 1+3+5+3+1=13. In each case 2 has been added to each pattern.1+3+5+7+5+3+1=25, like the previous pattern the addition has always been 2 to each digit.

...read more.

Middle

So in result the general formula for this would be n x 4 or 4n. I will prove this formula to be correct by choosing a formula at random. I have chosen the number 10. 4n= 4 x 10= 40 white squares.

I am now going to search for the formula the formula for the dark squares. I will the information from the result table and differences (from pg2) I have found that the number of shaded squares equals to the total from the previous squares. To see this look at page 2. I have decided to draw a new table similar to that of the previous page.

Pattern            Total Squares           Shaded Squares             White Squares               Difference in

Squares

1                  5                          1                           4                        

                                                                                         4

2                 13          5                           8                        

                                                                                         8

3                 2513                           12

                                                                                         12

4                 41        25                            16

Throughout the investigation I will use the following symbols for the formulae, these will not change. n = pattern number, d= dark squares, w = white squares.

Trying for a formula for shaded squares

I have found out the general formula to find the number of shaded squares which is 2nsquared – 2n + 1, because this is a quadratic form of ax +bx+c, I found a second difference between the pattern of shaded squares.

...read more.

Conclusion

Drawing simple diagrams at first can make it easier to find the answer for the investigation. Another good way is using different tables which hold a record of your findings take a part in finding an equation by recording the information on the table to justify your results. Using different techniques to tackle a problem is beneficial because of the reason that I drew a number of diagrams and a variety of assorted result tables to explain what I did. I also used different symbols and a plan to explain what I had done.

Equation 2n – 2n + 1 is very similar to equation 2n + 2n +1. I noticed this immediately after I had found the second equation. The difference between the two is the symbol before 2n. One is – (negative) and the other + (positive). This was because I had to add combine two equations together. The formula for white squares 4n and the formula for shaded squares 2n – 2n + 1, which in result gave 2n + 2n + 1.

The first equation I found was a linear equation, as 4n can be written as 4x. Equation 2 is a quadratic as it is in the form ax + bx+ c. the same with equation 3. I am planning to extend this investigation to 3D.

...read more.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE T-Total essays

  1. T-Shape investigation.

    1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 18 19 20 21 22 23 25 26 27 28 29 30 31 n 18 19 20 21 z 34 39 44 49 Right away I can see the t-total is increasing by 5.

  2. T-Total Investigation

    9 + 52 - 19 + 52 - 18 + 52 - 17 = 197 Thus proving this equation can be used to find the T-Total (t) by substituting x for the given T-Number. The equation can be simplified more: t = x + x - 9 + x -

  1. Investigation in to How many tiles and borders is needed for each pattern

    Anther formula I found was to work out the total tiles for each pattern For the nth term the formula is 2n +6n+5 Test The Formula: 1. Formula to calculate the number of borders is 4n+4 For example: when n=2 (4 2)+4 8+4=12 So then the 2nd pattern will have 12 borders.

  2. Math Coursework-T-Total Investigation

    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

  1. T-Total Investigation

    I will now try on an 8x8 grid. I will start off with the T-Number 30. 30 times 5 is 150. Now I will do the same as before to get the T-Total. 30, 30-8, 30-16, 30-17, 30-15. Now I have to minus 56.

  2. T-total Investigation

    We get 70 because it is the total of the differences between the T-no and the numbers in each of the other 4 squares. This is the next part of my formula. My formula is now 5T-70 I found out that my formula is correct.

  1. have been asked to find out how many squares would be needed to make ...

    This happens because you are simply adding on to this. This could be a useful fact in searching for a formula. I am now going to investigate any differences between the totals. First of all I will need to find some more totals, as the amount I have will not be conclusive.

  2. I am going to investigate how changing the number of tiles at the centre ...

    Table 2 Pattern: N 1 2 3 4 5 Total Tiles: T 11 23 39 59 83 +12 +16 +20 +24 +4 +4 +4 Table two shows the pattern number and total amount of tiles in that particular pattern. There is no constant first difference until the second difference (which is constant).

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work