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

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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.

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.

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.

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