This result is made because we are just adding onto it.
Differences
From the quadratic sequence, we see that the main difference is 4. The first formula I will try to find is the formula for the surrounding white squares.
Formula for white squares
Pattern
'font-size:12.0pt; '>1 x 4 = 4 white squares
'font-size:12.0pt; '>2 x 4 = 8 white squares
'font-size:12.0pt; '>3 x 4 = 12 white squares
N = pattern number
D = dark squares
W = white squares
I believe that the formula is 4 x the pattern number or 4N. The following is an example to test if I am right:
4N = number of white squares
N = 10
4 x 10 = 40
This gives me the correct amount of white squares; this is true because of the table of new orders gives me this answer.
Formula for black squares
Total number of squares – 4N = number of dark squares
Adding to my list of algebraic letters I will add the letters TNS.
N = pattern number
D = dark squares
W = white squares
T = total number of squares
I am now going to complete a table to show the development of the pattern number (N) against the total number of squares (T).
3D Structure of pattern 1
1=1
+
1+3+1=5
+
1=1
7 cubes
3D Structure of pattern 2
1=1
+
1+3+1=5
+
1+3+5+3+1=13
+
1+3+1=5
+
1=1
25 cubes
3D Structure of pattern 3
1=1
+
1+3+1=5
+
1+3+5+3+1=13
+
1+3+5+7+5+3+1=25
+
1+3+5+3+1=13
+
1+3+1=5
63 cubes
3D Structure of pattern 4
1=1
+
1+3+1=5
+
1+3+5+3+1=13
+
1+3=5+7+5+3+1=25
+
1+3+5+7+9+7+5+3+1=41
+
1+3+5+7+5+3+1=25
+
1+3+5+3+1=13
+
1+3+1=5
129 cubes
3D Structure of pattern 5
1=1
1+3+1=5
+
1+3+5+3+1=13
+
1+3+5+7+5+3 +1=25
+
1+3+5+7+9+7+5+3+1=41
+
1+3+5+7+9+11+9+7+5+3+1=61
+
1+3+5+7+9+7+5+3+1=41
+
1+3+5+7+5+3+1=25
+
1+3+5+3+1=13
+
1+3+1=5
231 cubes
This is a predicted table of the next 5 nth term numbers. The table shows the number of cubes that would occur if the nth terms continued. This could be used to construct the rest of the 3D cubes, as a result, the table shows continuous sequences which refer to my prediction in my hypotheses.
D. Overall summary and interpretation
Taking part in this investigation allowed me to understand mathematical methods that are used in both 2D and 3D sequences. Doing this states the acknowledgment of geometric formulas in the sequences. I have displayed how the 2D and 3D sequences interpret each other in different ways such as geometric and numerical. In doing this investigation I have found that there is a link between some of the squares, such as black squares and the total of squares.
My prediction was supported with the evidence of my mathematical equations and methods; this is because they show my nth term integers which allow me to make out the next sequences for the structures. As a result, this supports my evidence; my sequences interpreted each other in the use if quadratic and cubical equation. Again this supports my predictions.
If my coursework was to be read by another candidate, they would benefit from it because will enable them to understand the context of the investigation. This is due to the way I have displayed the diagrams, shapes, formulas and the 2 and 3D structures. On the other hand, the investigation could have been extended by adding other sequences, and explaining them in-depth formulas such as geometrical. Doing this would give me a greater understanding of the context of the study. Also, other readers would find it easy to understand because it explains the main aims and objectives of the investigation. Also, I have displayed the main features of the investigation such as the diagrams and structure of the shapes because it is stated in the intro.
The coursework doesn’t consist of irrelevant diagrams, the diagrams state the structure of the 2D and 3D symmetrical shapes. None of the tables are irrelevant because the tables are used to specify the nth term integers. This coursework has many mathematical equations and consists of statistical methods involved in geometrical equations.
To conclude, I believe that I have understood the main factors of the investigation, the use of geometrical, numerical, and mathematical methods allow me to recognize and summarise the use of my methods throughout the investigation.
Borders Coursework
- Introduction
Throughout this coursework, I will give and explain the task taken out based upon two different squares, white and black. I have been asked to find out the differences in the patterns of the colored squares. My investigation is to discover how many squares would be needed to make a cross-shape built, not only will my coursework consist of 2D diagrams, but 3D also. This will imply ma methods in a geometrical equation for the shapes. My coursework will consist of statistical equations and formulas to solve the 2D and 3D sequences; this will specify my understanding about the context of the investigation. An example of the diagrams being produced is below:
From this shape, we see that there are 4 white squares and 1 black square. Making 5 squares, this type of method will be shown throughout the investigation as the squares will grow in size.
- Hypotheses
I will be predicting the methods I wan to use to investigate the statistical formulas and equations equivalent to the sequences of the squares. Also, the nth term will be used for the colored squares; nonetheless it is quite clear that there is a prospect for the nth term used in each of the squares.
1. There will be an nth term for the number of white squares
2. There will be an nth term for the number of black squares
3. There will also be an nth term for the amount of colored squares
These are my theories which will be shown throughout the investigation as evidence.
C. Plan of action
My plan of action will be to give indication to the methods and formulas that I will use. Algebraic expressions such as nth term will be used because it will be reliable towards my investigation.
Here are some of the expressions that will be used:
Quadratic formula- an2 + bn + c
Cubic formula- an3+ bn2+ cn + d
Using 3D formulas to find invisible patterns.
I will predict the variable sequences and compare results to analyze and explain them. Formulas for the patterns will be involved such as cubic and quadratic. Having this will help me to understand the concepts of my geometric investigation.