Growing Shapes
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Iain Downer
Growing Shapes Aim My aim is to investigate and explain the growth of congruent shapes by using different methods. I am going to try and find out the formulae for the following in this order: • • • • • • •
Length Perimeter Area Number of shapes Number of Lines (inner, outer) Number of shapes added Number of vertices
At the end of my investigation I am going to look at my results for both squares and triangles and compare them to see if they have anything in common or are similar in any way. I also hope to extend my coursework and look at 3D shapes or other shapes such as pentagons, hexagons etc… I am going to work systematically and look to spot any patterns, and in the end, establish an algebraic formula for all things listed above.
The patterns for squares are as follows
1
2
3
4
Area = 1 Length = 1
5
Area = 5 Length = 3
Area = 13 Length = 5
Area = 41 Length = 9
The
patterns for triangles are as follows
1
Area = 25 Length = 7
Iain Downer
1
2
3
4
Area = 1 Width = 1 Area = 4 Width = 3 Area = 10 Width = 5
5
Area = 31 Width = 9
Area = 19 Width = 7
The pattern for squares grows by adding another square to each face every time:
The pattern for triangles grows by adding another triangle to each face every time:
Growing Shapes: Squares
2
Iain Downer Perimeter Pattern no. (n) 1 2 3 4 5
Perimeter 4 12 20 28 36
Pattern no. (n) 1 2 3 4 5
Perimeter 4 12 20 28 36
D1 8 8 8 8
As there are all 8’s in the D1 column, the formula contains 8n
Perimeter – 8n -4 -4 -4 -4 -4
Formula for perimeter of squares – Formula = 8n-4 Check When n = 5 Perimeter = 8n - 4 =8×5–4 = 36
Length and Width Pattern no. (n) 1 2 3 4 5
Length 1 3 5 7 9
D1 2 2 2 2
As there are all 2’s in the D1 column, the formula contains 2n Pattern no. (n) 1 2 3 4 5
Length 1 3 5 7 9
Length – 2n -1 -1 -1 -1 -1
Formula for length of squares – formula = 2n – 1 Check When n = 3 3
Iain Downer Length = 2n – 1 =2×3–1 =5
Number of Squares/Area Pattern no. (n) 1 2 3 4 5
No. of Squares 1 5 13 25 41
Pattern no. (n) 1-2 2-2 3-2 4-2 5
No. of Squares 1 5 13 25 41
D1
D2
4 8 12 16
4 4 4
As there are all 4’s in the D2 column, the formula contains 2n2.
No. of Squares – 2n2 -1 -3 -5 -7 -9
D1 As there are all -2’s in the D1 column, the formula contains 2n2 – 2n.
I need to substitute in the pattern number to find out what to do next. I am going to use 3 to start with. (2×32) – (2×3) = 12 To get from 12 to 13, I need to add 1. I am now going to substitute in 4 to see if it works as well. (2×42) – (2× 4) = 24 To get from 24 to 25, I need to add 1 so the formula = 2n2 – 2n + 1 and works for everything.
Check When n = 2 Area = 2n2 -2n + 1 = 2×22 – 4 +1 =5
I have to use another method to find the formula so I am going to use the odd numbers method.
4
Iain Downer
1
1 1
3 3
1 5
3
1 1
3
5
7
5
3
1
I notice that with pattern 4 in columns when n = 4: 2 lots of (1 + 3 + 5) = 2 x 9 1 lot of 7 = 1 x 7 2 x 9 = 2(n-1)2 7 = 2n – 1 Ts = 2(n-1)2 + 2n – 1 Number of Lines Pattern no. (n) 1 12 2 8 20 328 8 Pattern 8no. (n) 436 51 2 3 4 5
No. of lines 4 16 36 No. 64 of lines 4 100 16 36 64 100
D1 D2 As there are all 8’s in the D2 column, the formula contains 4n2. No. of lines - 4n2 0 0 0 0 0
The formula for number of lines = 4n2 Check Number of lines = 4n2 = 4 × 32 = 36
Number of Shapes Added I started with pattern number 2 because there is no previous shape to add the squares onto
5
Iain Downer Pattern no. (n) 42 43 44 45 6
No. shapes added 4 8 12 16 20
D1
As there are all 4’s in the D1 column, the formula contains 4n. Pattern no. (n) 2 3 4 5 6
No. shapes added 4 16 36 64 100
No. shapes added - 4n -4 -4 -4 -4 -4
The formula for number of shapes added = 4n – 4 Check Number of shapes added = 4n – 4 =4×3–4 =8
Number of outer vertices Pattern no. (n) 1 2 3 4 5
No. of outer vertices 4 8 12 16 20
Pattern no. (n) 1 2 3 4 5
No. of outer vertices 4 8 12 16 20
D1 4 4 4 4
As there are all 4’s in the D1 column, the formula contains 4n.
No. of outer vertices - 4n 0 0 0 0 0
Formula for number of outer vertices = 4n Check Number of outer vertices = 4n =4×3 6
Iain Downer = 12
Growing Shapes: Triangles Perimeter Pattern no. (n) 1 2 3 4 5 6 7
Perimeter 3 6 12 15 21 24 30
D1 3 6 3 6 3 6
D2
D3
3 -3 3 -3 3
-6 6 -6 6
I can tell that this method for working out the perimeter does not work because the numbers in the difference columns alternate between positive and negative. Instead of trying to find a formula for the all of the triangular patterns, I can try and find 2 separate formulas for the patterns where the added triangles are pointing up and the patterns where the triangles are pointing down. Pattern no. Pattern no. pointing up(n) 19 1 39 2 59 3 79 4 9Pattern no. pointing 5 up(n) Perimeter
Perimeter D1 3 As there are all 9’s in the 12 D1 column, the formula 21 contains 9n. 30 39 Perimeter – 9n
1 2 3 4 5
-6 -6 -6 -6 -6
3 12 21 30 39
Formula for the perimeter of a triangular pattern when the added squares are pointing up = 9n-6. Check Formula = 9n-6 = 9×3 – 6 = 21
7
Iain Downer
Pattern no. 29 49 96 89 10
Pattern no. pointing down(n) 1 2 3 4 5
D1
Perimeter 6 15 24 33 42
As there are all 9’s in the D1 column, the formula contains 9n. Pattern no. pointing down(n) 1 2 3 4 5
Perimeter 6 15 24 33 42
Perimeter – 9n -3 -3 -3 -3 -3
Formula for the perimeter of a triangular pattern when the added squares are pointing down = 9n-3. Check Formula = 9n-3 = 9×3 – 3 = 24
Number of Triangles/Area The yellow triangles represent the area
Pattern no. (n) 1 2 3 4 5
No. of Triangles 1 4 10 19 31
D1
D2
3 6 9 12
3 3 3
As there are all 3’s in the D2 column, the formula contains 1
8
1 2 n 2
Iain Downer
Pattern no.(n)
No. of Triangles
1 2 3 4 5
1 4 10 19 31
No. of Triangles - 1
1 2 n 2
-0.5 -2 -3.5 -5 -6.5
-1.5 -1.5 -1.5 -1.5
As there are all -1.5’s in the D1 column, the formula contains 1 Pattern no.(n)
No. of Triangles
1 2 3 4 5
1 4 10 19 31
D1
No. of Triangles - 1
1 2 1 n -1 n. 2 2
1 2 1 n -1 n 2 2
1 1 1 1 1
Formula for working out the Number of Triangles = 1
1 2- 1 n 1 n+1 2 2
Check Number of Triangles = 1
1 2 1 n -1 n + 1 2 2
= (32×1.5) - (1.5×3) +1 = 24
Number of lines The black lines represent the number of lines.
9
Iain Downer Pattern no. (n) 61 6 2 12 -3 3 3 15 3 6 4 21 5 24
No. of Lines 3 9 21 36 57
D1
D2
D3
This method doesn’t work because you never have all the same number in the difference column. This means that I have to work out the formula for the odd and even pattern numbers separately. This also means that I must extend the table using the pattern I have already found. Pattern no. (n) 1 2 3 4 5 6 7 8 9 10
No. of Lines 3 9 21 36 57 81 111 144 183 225
Even Pattern no. 2 4 6 8 10
(n) 1 2 3 4 5
D1 6 12 15 21 24 30 33 39 42
No. of lines 9 36 81 144 225
D2 6 3 6 3 6 3 6 3
D1 27 45 63 81
D3 -3 3 -3 3 -3 3 -3
D2 18 18 18
Formula for working out the number of lines in an even pattern number = 9n2 Check No. of lines = 9n2 = 32 × 9 = 81
Odd Pattern no. 118 18 336 18 554 18 772 9
(n) 1 2 3 4 5
No. of lines 3 21 57 111 183
D1
10
D2
Iain Downer
As there are all 18’s in the D2 column, the formula contains 9n2. Odd Pattern no. 1 3 5 7 9
(n) 1 2 3 4 5
No. of lines – 9n2 -6 -15 -24 -33 -42
No. of lines 3 21 57 111 183
D1 -9 -9 -9 -9
As there are all 9’s in the D1 column, the formula contains9n2 – 9n. Odd Pattern no. 1 3 5 7 9
(n) 1 2 3 4 5
No. of lines – 9n2- 9n +3 +3 +3 +3 +3
No. of lines 3 21 57 111 183
Formula for working out the number of lines in an even pattern number = 9n2 -9n+3. Check No. of lines = 9n2 -9n + 3 = (32 × 9) -9n +3 = 57
Number of Triangles Added The red squares represent the added triangles.
Pattern no. (n) 13 23 33 43 5
Triangles Added 0 3 6 9 12
D1
11
Iain Downer
As there are all 3’s in the D1 column, the formula contains 3n. Pattern no. (n) 1 2 3 4 5
Triangles Added 0 3 6 9 12
Triangles Added – 3n -3 -3 -3 -3 -3
Formula for working out the number of triangles added = 3n-3 Check Triangles added = 3n-3 =3×3–3 =6
Width
For width I am going to measure the widest part of the shape. Pattern no. (n) 1 2 3 4 5
Width 1 3 5 7 9
D1 2 2 2 2
As there are all 2’s in the D1 column, the formula contains 2n. Pattern no. (n) 1 2 3 4 5
Width 1 3 5 7 9
Width – 2n -1 -1 -1 -1 -1 12
Iain Downer
The formula for working out the width of the shape = 2n-1 Check Width = 2n-1 = 2×3 – 1 =5
Growing Shapes Extension: Pentagons
The pattern for pentagons is as follows:
You can’t draw them like this because regular pentagons do not tessellate.
You must draw them like this, using irregular pentagons.
13
Iain Downer
For pentagons, I am only going to work out the formula for the total pentagons and pentagons added.
Area/Number of Pentagons Pattern no. (n) 1 2 3 4 5
No. of Pentagons 1 6 16 31 51
D1
D2
5 10 15 20
5 5 5
As there are all 5’s in the D2 column, the formula contains 2.5n2. Pattern no. (n) 1 2 3 4 5
2.5n2 1 3 5 7 9
No. of Pentagons – 2.5n2 1.5 4 6.5 9 11.5
D1 -2.5 -2.5 -2.5 -2.5
As there are all -2.5’s in the D2 column, the formula contains 2.5n2 – 2.5n Pattern no. (n) 1 2 3 4 5
2.5n2 -2.5n 0 5 15 30 50
I can tell from the table that all I need to do is add 1 to get the same amount as the total pentagons so the formula = 2.5n2 – 2.5n + 1 Check No. of Pentagons = 2.5n2 – 2.5n + 1
14
Iain Downer = (2.5 x 32) – 2.5 x 3 + 1 = 16
Pentagons Added Pattern no. (n) 1 2 3 4 5
Pentagons Added 0 5 10 15 20
Pattern no. (n) 1 2 3 4 5
Pentagons added 1 3 5 7 9
D1 5 5 5 5
As there are all 5’s in the D1 column, the formula contains 5n
Pentagons added – 5n -5 -5 -5 -5 -5
The formula for working out the number of pentagons added = 5n-5. I have spotted a pattern for the number of shapes added for every shape. Squares added = 4n-4 Triangles added = 3n-3 Pentagons added = 5n-5 Therefore, I predict that the formula for working out the number of hexagons added = 6n-6. Check Pentagons added = 5n – 5 =5x3–5 = 10
15
Iain Downer
Growing Shapes Extension: Hexagons
The pattern for hexagons is as follows.
For hexagons I am only going to work out the area. Area/Number of Hexagons I have looked at the formulae for working out the area of squares, triangles and pentagons and I have spotted a pattern. Area of triangles = 1.5n2-1.5n+1 Area of Squares = 2n2-2n+1 Area of pentagons = 2.52-2.5n+1 After looking at these formulae, I predict that the formula for working out the total number of hexagons will be 3n2 – 3n +1 Pattern no. (n) 1 2 3 4 5
No. of Hexagons 1 7 19 37 61
D1
D2
6 12 18 24
6 6 6
As there are all 6’s in the D2 column, the formula contains 3n2. Pattern no. (n) 1 3 2 3 3 3 43 5
3n2 1 3 5 7 9
No. of Hexagons – 3n2 -2 -5 -8 -11 -14 16
D1
Iain Downer
As there are all 3’s in the D1 column, the formula contains 3n2 – 3n Pattern no. (n) 1 2 3 4 5
3n2-3n 0 6 18 36 60
I can see from the table that I need to add 1 to each to get the same amount as the pattern number so the formula = 3n2 – 3n + 1 Check No. of hexagons = 3n2 – 3n + 1 = (3 x 32) – (3 x 3) +1 = 19
Growing Shapes Extension: 3D Shapes For 3D shapes, I am only going to work out the number of cubes. The pattern for 3D shapes is as follows.
17
Iain Downer I drew the shapes using individual cubes which I found on AutoShapes in Word. Area/Number of Cubes Pattern no. (n) 1 2 3 4 5
No. of Cubes 1 7 25 63 129
D1 6 18 38 66
D2 12 20 28
D3 8 8
As there are all 8‘s in the D3 column, the formula contains 1 Pattern no. (n) No. of cubes 1
1
2
7
3
25
4
63
5
129
No. of cubes – 1 1 3 2 -3 3
-
1 3 n 3
1 3 n 3
D1
1 3 1 -7 3 1 -11 3 1 -15 3
-3
-11 1 3 2 -37 3
-22
As there are all -4’s in the D2 column, the formula contains 1 Pattern no. (n) No. of cubes 1
1
2
7
3
25
4
63
5
129
As there are all -2
D2
No. of cubes – 1 2 3 1 4 3
1
-4 -4 -4
1 3 n – 2n2 3
1 3 n – 2n2 3
D1
2 3 2 2 3 2 2 3 2 2 3
2
7 2 3 1 12 3
9
2 1 2 in the D1 column, the formula contains 1 n3 – 2n2 + 2 n 3 3 3
18
Iain Downer Pattern no. (n) No. of cubes 1 2 3 4 5
1 7 25 63 129
No. of cubes – 1
1 3 2 n – 2n2 + 2 n 3 3
2 8 16 64 130
I can see from the table that I need to -1 so the formula = 1
1 3 2 n – 2n2 + 2 n - 1 3 3
Conclusion Here is a table showing all the formulas I have managed to work out during my investigation. Squares – Perimeter Squares – Area Squares – No. of lines Squares – Length Squares – Shapes added Squares – No. of outer vertices Triangles – Perimeter Triangles – Area
8n-4 2n2-2n+1 4n2 2n-1 4n-4 4n Pointing up 9n-6, Pointing down 9n-3
Triangles – No. of lines Triangles – Width Triangles – Shapes added Pentagons – Area Pentagons – Shapes added Hexagons – Area 3D shapes – Area
9n2 2n-1 3n-3 2.5n2-2.5n+1 5n-5 3n2-3n+1
1
1
19
1 2 1 n -1 n + 1 2 2
1 3 2 n -2n2+2 n-1 3 3
Growing Shapes Aim My aim is to investigate and explain the growth of congruent shapes by using different methods. I am going to try and find out the formulae for the following in this order: • • • • • • •
Length Perimeter Area Number of shapes Number of Lines (inner, outer) Number of shapes added Number of vertices
At the end of my investigation I am going to look at my results for both squares and triangles and compare them to see if they have anything in common or are similar in any way. I also hope to extend my coursework and look at 3D shapes or other shapes such as pentagons, hexagons etc… I am going to work systematically and look to spot any patterns, and in the end, establish an algebraic formula for all things listed above.
The patterns for squares are as follows
1
2
3
4
Area = 1 Length = 1
5
Area = 5 Length = 3
Area = 13 Length = 5
Area = 41 Length = 9
The
patterns for triangles are as follows
1
Area = 25 Length = 7
Iain Downer
1
2
3
4
Area = 1 Width = 1 Area = 4 Width = 3 Area = 10 Width = 5
5
Area = 31 Width = 9
Area = 19 Width = 7
The pattern for squares grows by adding another square to each face every time:
The pattern for triangles grows by adding another triangle to each face every time:
Growing Shapes: Squares
2
Iain Downer Perimeter Pattern no. (n) 1 2 3 4 5
Perimeter 4 12 20 28 36
Pattern no. (n) 1 2 3 4 5
Perimeter 4 12 20 28 36
D1 8 8 8 8
As there are all 8’s in the D1 column, the formula contains 8n
Perimeter – 8n -4 -4 -4 -4 -4
Formula for perimeter of squares – Formula = 8n-4 Check When n = 5 Perimeter = 8n - 4 =8×5–4 = 36
Length and Width Pattern no. (n) 1 2 3 4 5
Length 1 3 5 7 9
D1 2 2 2 2
As there are all 2’s in the D1 column, the formula contains 2n Pattern no. (n) 1 2 3 4 5
Length 1 3 5 7 9
Length – 2n -1 -1 -1 -1 -1
Formula for length of squares – formula = 2n – 1 Check When n = 3 3
Iain Downer Length = 2n – 1 =2×3–1 =5
Number of Squares/Area Pattern no. (n) 1 2 3 4 5
No. of Squares 1 5 13 25 41
Pattern no. (n) 1-2 2-2 3-2 4-2 5
No. of Squares 1 5 13 25 41
D1
D2
4 8 12 16
4 4 4
As there are all 4’s in the D2 column, the formula contains 2n2.
No. of Squares – 2n2 -1 -3 -5 -7 -9
D1 As there are all -2’s in the D1 column, the formula contains 2n2 – 2n.
I need to substitute in the pattern number to find out what to do next. I am going to use 3 to start with. (2×32) – (2×3) = 12 To get from 12 to 13, I need to add 1. I am now going to substitute in 4 to see if it works as well. (2×42) – (2× 4) = 24 To get from 24 to 25, I need to add 1 so the formula = 2n2 – 2n + 1 and works for everything.
Check When n = 2 Area = 2n2 -2n + 1 = 2×22 – 4 +1 =5
I have to use another method to find the formula so I am going to use the odd numbers method.
4
Iain Downer
1
1 1
3 3
1 5
3
1 1
3
5
7
5
3
1
I notice that with pattern 4 in columns when n = 4: 2 lots of (1 + 3 + 5) = 2 x 9 1 lot of 7 = 1 x 7 2 x 9 = 2(n-1)2 7 = 2n – 1 Ts = 2(n-1)2 + 2n – 1 Number of Lines Pattern no. (n) 1 12 2 8 20 328 8 Pattern 8no. (n) 436 51 2 3 4 5
No. of lines 4 16 36 No. 64 of lines 4 100 16 36 64 100
D1 D2 As there are all 8’s in the D2 column, the formula contains 4n2. No. of lines - 4n2 0 0 0 0 0
The formula for number of lines = 4n2 Check Number of lines = 4n2 = 4 × 32 = 36
Number of Shapes Added I started with pattern number 2 because there is no previous shape to add the squares onto
5
Iain Downer Pattern no. (n) 42 43 44 45 6
No. shapes added 4 8 12 16 20
D1
As there are all 4’s in the D1 column, the formula contains 4n. Pattern no. (n) 2 3 4 5 6
No. shapes added 4 16 36 64 100
No. shapes added - 4n -4 -4 -4 -4 -4
The formula for number of shapes added = 4n – 4 Check Number of shapes added = 4n – 4 =4×3–4 =8
Number of outer vertices Pattern no. (n) 1 2 3 4 5
No. of outer vertices 4 8 12 16 20
Pattern no. (n) 1 2 3 4 5
No. of outer vertices 4 8 12 16 20
D1 4 4 4 4
As there are all 4’s in the D1 column, the formula contains 4n.
No. of outer vertices - 4n 0 0 0 0 0
Formula for number of outer vertices = 4n Check Number of outer vertices = 4n =4×3 6
Iain Downer = 12
Growing Shapes: Triangles Perimeter Pattern no. (n) 1 2 3 4 5 6 7
Perimeter 3 6 12 15 21 24 30
D1 3 6 3 6 3 6
D2
D3
3 -3 3 -3 3
-6 6 -6 6
I can tell that this method for working out the perimeter does not work because the numbers in the difference columns alternate between positive and negative. Instead of trying to find a formula for the all of the triangular patterns, I can try and find 2 separate formulas for the patterns where the added triangles are pointing up and the patterns where the triangles are pointing down. Pattern no. Pattern no. pointing up(n) 19 1 39 2 59 3 79 4 9Pattern no. pointing 5 up(n) Perimeter
Perimeter D1 3 As there are all 9’s in the 12 D1 column, the formula 21 contains 9n. 30 39 Perimeter – 9n
1 2 3 4 5
-6 -6 -6 -6 -6
3 12 21 30 39
Formula for the perimeter of a triangular pattern when the added squares are pointing up = 9n-6. Check Formula = 9n-6 = 9×3 – 6 = 21
7
Iain Downer
Pattern no. 29 49 96 89 10
Pattern no. pointing down(n) 1 2 3 4 5
D1
Perimeter 6 15 24 33 42
As there are all 9’s in the D1 column, the formula contains 9n. Pattern no. pointing down(n) 1 2 3 4 5
Perimeter 6 15 24 33 42
Perimeter – 9n -3 -3 -3 -3 -3
Formula for the perimeter of a triangular pattern when the added squares are pointing down = 9n-3. Check Formula = 9n-3 = 9×3 – 3 = 24
Number of Triangles/Area The yellow triangles represent the area
Pattern no. (n) 1 2 3 4 5
No. of Triangles 1 4 10 19 31
D1
D2
3 6 9 12
3 3 3
As there are all 3’s in the D2 column, the formula contains 1
8
1 2 n 2
Iain Downer
Pattern no.(n)
No. of Triangles
1 2 3 4 5
1 4 10 19 31
No. of Triangles - 1
1 2 n 2
-0.5 -2 -3.5 -5 -6.5
-1.5 -1.5 -1.5 -1.5
As there are all -1.5’s in the D1 column, the formula contains 1 Pattern no.(n)
No. of Triangles
1 2 3 4 5
1 4 10 19 31
D1
No. of Triangles - 1
1 2 1 n -1 n. 2 2
1 2 1 n -1 n 2 2
1 1 1 1 1
Formula for working out the Number of Triangles = 1
1 2- 1 n 1 n+1 2 2
Check Number of Triangles = 1
1 2 1 n -1 n + 1 2 2
= (32×1.5) - (1.5×3) +1 = 24
Number of lines The black lines represent the number of lines.
9
Iain Downer Pattern no. (n) 61 6 2 12 -3 3 3 15 3 6 4 21 5 24
No. of Lines 3 9 21 36 57
D1
D2
D3
This method doesn’t work because you never have all the same number in the difference column. This means that I have to work out the formula for the odd and even pattern numbers separately. This also means that I must extend the table using the pattern I have already found. Pattern no. (n) 1 2 3 4 5 6 7 8 9 10
No. of Lines 3 9 21 36 57 81 111 144 183 225
Even Pattern no. 2 4 6 8 10
(n) 1 2 3 4 5
D1 6 12 15 21 24 30 33 39 42
No. of lines 9 36 81 144 225
D2 6 3 6 3 6 3 6 3
D1 27 45 63 81
D3 -3 3 -3 3 -3 3 -3
D2 18 18 18
Formula for working out the number of lines in an even pattern number = 9n2 Check No. of lines = 9n2 = 32 × 9 = 81
Odd Pattern no. 118 18 336 18 554 18 772 9
(n) 1 2 3 4 5
No. of lines 3 21 57 111 183
D1
10
D2
Iain Downer
As there are all 18’s in the D2 column, the formula contains 9n2. Odd Pattern no. 1 3 5 7 9
(n) 1 2 3 4 5
No. of lines – 9n2 -6 -15 -24 -33 -42
No. of lines 3 21 57 111 183
D1 -9 -9 -9 -9
As there are all 9’s in the D1 column, the formula contains9n2 – 9n. Odd Pattern no. 1 3 5 7 9
(n) 1 2 3 4 5
No. of lines – 9n2- 9n +3 +3 +3 +3 +3
No. of lines 3 21 57 111 183
Formula for working out the number of lines in an even pattern number = 9n2 -9n+3. Check No. of lines = 9n2 -9n + 3 = (32 × 9) -9n +3 = 57
Number of Triangles Added The red squares represent the added triangles.
Pattern no. (n) 13 23 33 43 5
Triangles Added 0 3 6 9 12
D1
11
Iain Downer
As there are all 3’s in the D1 column, the formula contains 3n. Pattern no. (n) 1 2 3 4 5
Triangles Added 0 3 6 9 12
Triangles Added – 3n -3 -3 -3 -3 -3
Formula for working out the number of triangles added = 3n-3 Check Triangles added = 3n-3 =3×3–3 =6
Width
For width I am going to measure the widest part of the shape. Pattern no. (n) 1 2 3 4 5
Width 1 3 5 7 9
D1 2 2 2 2
As there are all 2’s in the D1 column, the formula contains 2n. Pattern no. (n) 1 2 3 4 5
Width 1 3 5 7 9
Width – 2n -1 -1 -1 -1 -1 12
Iain Downer
The formula for working out the width of the shape = 2n-1 Check Width = 2n-1 = 2×3 – 1 =5
Growing Shapes Extension: Pentagons
The pattern for pentagons is as follows:
You can’t draw them like this because regular pentagons do not tessellate.
You must draw them like this, using irregular pentagons.
13
Iain Downer
For pentagons, I am only going to work out the formula for the total pentagons and pentagons added.
Area/Number of Pentagons Pattern no. (n) 1 2 3 4 5
No. of Pentagons 1 6 16 31 51
D1
D2
5 10 15 20
5 5 5
As there are all 5’s in the D2 column, the formula contains 2.5n2. Pattern no. (n) 1 2 3 4 5
2.5n2 1 3 5 7 9
No. of Pentagons – 2.5n2 1.5 4 6.5 9 11.5
D1 -2.5 -2.5 -2.5 -2.5
As there are all -2.5’s in the D2 column, the formula contains 2.5n2 – 2.5n Pattern no. (n) 1 2 3 4 5
2.5n2 -2.5n 0 5 15 30 50
I can tell from the table that all I need to do is add 1 to get the same amount as the total pentagons so the formula = 2.5n2 – 2.5n + 1 Check No. of Pentagons = 2.5n2 – 2.5n + 1
14
Iain Downer = (2.5 x 32) – 2.5 x 3 + 1 = 16
Pentagons Added Pattern no. (n) 1 2 3 4 5
Pentagons Added 0 5 10 15 20
Pattern no. (n) 1 2 3 4 5
Pentagons added 1 3 5 7 9
D1 5 5 5 5
As there are all 5’s in the D1 column, the formula contains 5n
Pentagons added – 5n -5 -5 -5 -5 -5
The formula for working out the number of pentagons added = 5n-5. I have spotted a pattern for the number of shapes added for every shape. Squares added = 4n-4 Triangles added = 3n-3 Pentagons added = 5n-5 Therefore, I predict that the formula for working out the number of hexagons added = 6n-6. Check Pentagons added = 5n – 5 =5x3–5 = 10
15
Iain Downer
Growing Shapes Extension: Hexagons
The pattern for hexagons is as follows.
For hexagons I am only going to work out the area. Area/Number of Hexagons I have looked at the formulae for working out the area of squares, triangles and pentagons and I have spotted a pattern. Area of triangles = 1.5n2-1.5n+1 Area of Squares = 2n2-2n+1 Area of pentagons = 2.52-2.5n+1 After looking at these formulae, I predict that the formula for working out the total number of hexagons will be 3n2 – 3n +1 Pattern no. (n) 1 2 3 4 5
No. of Hexagons 1 7 19 37 61
D1
D2
6 12 18 24
6 6 6
As there are all 6’s in the D2 column, the formula contains 3n2. Pattern no. (n) 1 3 2 3 3 3 43 5
3n2 1 3 5 7 9
No. of Hexagons – 3n2 -2 -5 -8 -11 -14 16
D1
Iain Downer
As there are all 3’s in the D1 column, the formula contains 3n2 – 3n Pattern no. (n) 1 2 3 4 5
3n2-3n 0 6 18 36 60
I can see from the table that I need to add 1 to each to get the same amount as the pattern number so the formula = 3n2 – 3n + 1 Check No. of hexagons = 3n2 – 3n + 1 = (3 x 32) – (3 x 3) +1 = 19
Growing Shapes Extension: 3D Shapes For 3D shapes, I am only going to work out the number of cubes. The pattern for 3D shapes is as follows.
17
Iain Downer I drew the shapes using individual cubes which I found on AutoShapes in Word. Area/Number of Cubes Pattern no. (n) 1 2 3 4 5
No. of Cubes 1 7 25 63 129
D1 6 18 38 66
D2 12 20 28
D3 8 8
As there are all 8‘s in the D3 column, the formula contains 1 Pattern no. (n) No. of cubes 1
1
2
7
3
25
4
63
5
129
No. of cubes – 1 1 3 2 -3 3
-
1 3 n 3
1 3 n 3
D1
1 3 1 -7 3 1 -11 3 1 -15 3
-3
-11 1 3 2 -37 3
-22
As there are all -4’s in the D2 column, the formula contains 1 Pattern no. (n) No. of cubes 1
1
2
7
3
25
4
63
5
129
As there are all -2
D2
No. of cubes – 1 2 3 1 4 3
1
-4 -4 -4
1 3 n – 2n2 3
1 3 n – 2n2 3
D1
2 3 2 2 3 2 2 3 2 2 3
2
7 2 3 1 12 3
9
2 1 2 in the D1 column, the formula contains 1 n3 – 2n2 + 2 n 3 3 3
18
Iain Downer Pattern no. (n) No. of cubes 1 2 3 4 5
1 7 25 63 129
No. of cubes – 1
1 3 2 n – 2n2 + 2 n 3 3
2 8 16 64 130
I can see from the table that I need to -1 so the formula = 1
1 3 2 n – 2n2 + 2 n - 1 3 3
Conclusion Here is a table showing all the formulas I have managed to work out during my investigation. Squares – Perimeter Squares – Area Squares – No. of lines Squares – Length Squares – Shapes added Squares – No. of outer vertices Triangles – Perimeter Triangles – Area
8n-4 2n2-2n+1 4n2 2n-1 4n-4 4n Pointing up 9n-6, Pointing down 9n-3
Triangles – No. of lines Triangles – Width Triangles – Shapes added Pentagons – Area Pentagons – Shapes added Hexagons – Area 3D shapes – Area
9n2 2n-1 3n-3 2.5n2-2.5n+1 5n-5 3n2-3n+1
1
1
19
1 2 1 n -1 n + 1 2 2
1 3 2 n -2n2+2 n-1 3 3
