To investigate the areas of different shapes when they are joined together on square dotted paper

Shapes with 2 dots inside: -

Shape

No. of dots inside

No. of dots outside

Area (cm²)

6

7

8

9

0

2

2

2

2

2

6

0

9

8

8

4

6

5.5

5

5

From the table we can see that there is no direct pattern between the number of dots inside, outside or the area. I have noticed that in all the shapes tested the area is 1 more than half the number of dots on the outside. From that we can draw up the formula: -

Testing the Formula: -

As you can see the formula works as it could predict the area of that shape before I counted it. I have noticed that when you ad another dot to the outside the area increases by 0.5 cm². This is summarised in the table below: -

No. of dots on the outside

Area (cm²)

8

5

9

5.5

0

6

This is because when you add another dot to the shape it brings the shape out by two lines, which join at the new dot. Those two lines have an area of 0.5 cm². This is shown in the diagrams below: -

Shapes with 3 dots inside: -

Shape

No. of dots inside

No. of dots outside

Area (cm²)

1

2

3

4

5

3

3

3

3

3

2

8

0

5

9

8

6

7

9.5

6.5

From the table we can see that there is no direct link between the number of dots inside, outside or the area. I have noticed that in all cases the area is 2 more than half the number of dots on the outside. From that I can draw up the formula: -

Testing the Formula: -

As you can see the formula works as it could predict the area of that shape before I counted it.

Explaining Patterns and Formulas: -

In the first test (1 dot on the inside) the area worked out to be half that of the number of dots on the outside. The effect of adding another dot on the inside meant that the formula must be +1 at the end. Also the effect of adding 2 more dots meant that the formula would also be the same as the first one but +2 at the end. This is summarised in the table below: -

No. of dots inside

No. added to end of formula

0

2

3

2

I have also noticed that the number you have to add to the end of the formula is always 1 less than the number of dots in the middle. That would bring the number of dots in the middle into the equation as well but I can't see how it fits into it yet. I predict that when you have 4 dots in the middle the formula will be the same as the others but with +3 at the end. I am now going to test this prediction on shapes with 4 dots on the inside: -

Testing shapes with 4 dots in the middle: -

Shape

No. of dots inside

No. of dots outside

Area (cm²)

6

7

8

9

20

4

4

4

4

4

4

2

4

3

4

0

9

0

9.5

0

As you can see from the table the area is 3 more than half the number of dots on the outside. From that we can draw up the formula: -

Testing the formula: -

As you can see the formula works as I was able to predict the area of the shape before counting it by hand.

From testing different numbers of dots on the inside I have been able to draw up formulas that work and spot patterns between different numbers of dots on the inside and their formulas. There is definitely a link between the individual formulas but I am still unsuccessful in spotting that link. I have found formulas that work for individual numbers of dots on the inside but not yet one that works for any number of dots on the inside or in fact, any shape. I am now going to attack the problem by testing shapes with different areas. Maybe the formulas from that will point me more in the direction of finding the overall formula.

Shapes with areas of 2 cm²: -

Shape

No. of dots inside

No. of dots outside

Area (cm²)

21

22

23

24

25

0

0

0

0

0

6

6

6

6

6

2

2

2

2

2

From the table we can see that there is a relationship between the number of dots on the outside and the area. The area is 1 less than half the number of dots on the outside. Therefore we can draw up the formula: -

Testing the formula: -

As you can see the formula works as when it was tested it was able to give me the area without counting it. I am now going to test shapes of area 3 cm²: -

Shapes with areas of 3 cm²: -

Shape

No. of dots inside

No. of dots outside

Area (cm²)

26

27

28

29

30

0

0

0

0

0

8

8

8

8

8

3

3

3

3

3

From the table we can see that again the area is 1 less than half the number of dots on the outside. That would make the formula the same as the last one. Just to make sure that is the right formula I will test it on another shape: -

This proves that this is the right formula for shapes with areas of 3 cm². So far, when I have been testing the areas of shapes, both formulas have been: -

I predict that the formula for areas of 4 cm² will be the same as the two previous ones. I will now try and prove this by testing shapes with areas of 4 cm²: -

Shapes with areas of 4 cm²: -

Shapes

No. of dots inside

No. of dots outside

Area (cm²)

31

32

33

34

35

2

0

0

6

8

0

8

0

4

4

4

4

4

As you can see in the table my prediction was wrong. It only works for certain shapes (33 and 35). To find the formula for these shapes I must compare the tables of when I was changing the number of dots in the middle with the tables of when I was changing the area. When I was looking at the table for 1, 2 and 3 dots inside I noticed that the number that you had to add on the end of the formula was 1 less than the number of dots on the inside.

No. of dots inside

No. added to end of formula

0

2

3

2

The formulas for the tables of changing the number of dots on the inside are all '+' at the end. The formulas for the tables of changing the area were all '-' at the end. In the changing the area tables the number of dots on the inside for 2 and 3 cm² were all '0'. I believe this is why it is -1 at the of the formula as -1 is 1 less than 0.

Shape

No. of dots inside

Number added to end of formula

22

0

-1

28

0

-1

The same rules apply to the formulas of the changing the area tables; the number you add on to the end of the formula is 1 less than the number of dots on the inside. I am going to test this on shapes 32 and 33, as they were the ones that were odd results in the table compared to the others.

Shape 32: -

As you can see the number I had to add on to the formula was 1 less than the number of dots in the middle. I will now test this on shape 33. I predict that the number you have to add on the end will be -1 as it is 1 less than '0'.

Shape 33: -

As you can see my prediction was correct the number I had to ad on the end was 1 less than the number of dots on the inside. When the number of dots 0n the inside is '0' then the number you have to add on the end of the formula is -1. This is the same as saying add 1 to the end of the formula as a '+' minus a '-' equals a '+'.

Taking into consideration that the number you add on to the end of the formula is 1 less than the number of dots on the inside I am able to draw up what I believe is the final formula: -

In algebraic terms it can be written: -

I will now test this formula on a series of shapes to see if it is able to find the area of any shape on squared paper.

In this shape there are 12 dots on the outside and 4 dots on the inside. I predict that the area will be: -

After counting the squares by hand I can say that the area was the same as what I had stated in my prediction. My formula works for this shape. I will now test it on a more complicated shape to see if it works.

In this shape there are 24 dots on the outside and 5 dots on the inside. I predict that the area of this shape will be: -

After counting the number of squares by hand the area was the same as what I predicted. I will now test the formula on a shape with irregular angles to make sure it can work for all shapes.

In this shape there are 12 dots on the outside and 17 dots on the inside. I predict that the area of this shape will be: -

After counting the shape by hand I can conclude that my prediction was right as it could work out the area of the shape. I believe that I have proved enough that this formula works for every single shape. The reason why the formula is dots outside divided by 2 is because, as I previously stated, as you increase the number of dots on the outside by one you increase the area by 0.5. Timesing by 0.5 is the same as dividing by 2. The reason that is +(dots in -1) is because 1 dot can touch the corners of up to 4 squares. So for every internal dot there is there can be 4 times as many squares. The reason why you take away 1 from the number of internal dots is shown below: -

As I stated in my introduction I was planning on testing shapes with irregular angles such as 30º and 75º. I do not feel that it is necessary to test those as I have found and tested the formula that works for all shapes. I have also already proven that it works on shapes with irregular angles.