William Alston. G.C.S.E Maths Coursework. 1st October 2001
Introduction: -
We are to investigate the areas of different shapes when they are joined together on square dotted paper. To start off with, in this investigation, I will be looking at regular shapes (those with 45° and 90° angles). To break it up into simple parts I will test 1 thing at a time using 5 different shapes. I will start by making regular shapes with 1 dot inside and then 2 dots and then 3 dots on the inside. Then I will make regular shapes and change the area each time e.g. 4, 5 and 6cm's². Also I will try different numbers of dots on the outside if I feel it is necessary to my investigation. That is something that I will be deciding as I go along. Then I will look at irregular shapes (those with angles of say 30º or 75º). After I have drawn the 5 shapes I will put them in a table, so they are easy, to Asses. I will try to draw up a formula for those shapes then I will test the formula by drawing another shape and working out the formula before physically counting the area. Throughout the investigation I will be carrying out an on-going evaluation, as this will help me in finding formulas and noticing patterns. I will also be predicting shape areas and explaining patterns and formulas and trying to justify why the formulas work. I will also be explaining the decisions that I have made. I ultimately aim to find a formula that is able to give me the area of any shape drawn on dotted paper, then testing it to see if it works on any shape (the nth term).
Equipment I will need: -
Calculator, pens, pencils, rubber and square dotted paper.
Algebra notations: -
a=Area
do=dots outside
di=dots inside
Shapes with 1 dot inside: -
Shape
No. of dots inside
No of dots outside
Area (cm²)
2
3
4
5
4
7
6
2
8
2
3.5
3
6
4
From this first table I can see that there is a relationship between the dots on the outside and the area. It shows us that the area is half the number of dots on the outside. From that we can draw up a formula that is: -
Testing the formula: -
As you can see the formula works, as it was able to predict the area of that shape before I counted it. I am now going to test shapes with 2 dots on the inside.
Introduction: -
We are to investigate the areas of different shapes when they are joined together on square dotted paper. To start off with, in this investigation, I will be looking at regular shapes (those with 45° and 90° angles). To break it up into simple parts I will test 1 thing at a time using 5 different shapes. I will start by making regular shapes with 1 dot inside and then 2 dots and then 3 dots on the inside. Then I will make regular shapes and change the area each time e.g. 4, 5 and 6cm's². Also I will try different numbers of dots on the outside if I feel it is necessary to my investigation. That is something that I will be deciding as I go along. Then I will look at irregular shapes (those with angles of say 30º or 75º). After I have drawn the 5 shapes I will put them in a table, so they are easy, to Asses. I will try to draw up a formula for those shapes then I will test the formula by drawing another shape and working out the formula before physically counting the area. Throughout the investigation I will be carrying out an on-going evaluation, as this will help me in finding formulas and noticing patterns. I will also be predicting shape areas and explaining patterns and formulas and trying to justify why the formulas work. I will also be explaining the decisions that I have made. I ultimately aim to find a formula that is able to give me the area of any shape drawn on dotted paper, then testing it to see if it works on any shape (the nth term).
Equipment I will need: -
Calculator, pens, pencils, rubber and square dotted paper.
Algebra notations: -
a=Area
do=dots outside
di=dots inside
Shapes with 1 dot inside: -
Shape
No. of dots inside
No of dots outside
Area (cm²)
2
3
4
5
4
7
6
2
8
2
3.5
3
6
4
From this first table I can see that there is a relationship between the dots on the outside and the area. It shows us that the area is half the number of dots on the outside. From that we can draw up a formula that is: -
Testing the formula: -
As you can see the formula works, as it was able to predict the area of that shape before I counted it. I am now going to test shapes with 2 dots on the inside.