fractals – lesson 4

Fractals – Lesson 4

The perimeter and area of the Von Koch Snowflake

CEDS – Study Plus in Cornwall

Learning objectives

We are learning to:

• Understand the difference between term-to-term rules and position-to-term rules.

• Write an expression to describe the nth term of a sequence.

CEDS – Study Plus in Cornwall

Learning outcome

By the end of the lesson you will:

• Understand the difference between term-to-term and position-to-term rules.

• Have derived the position-to-term rule for the number of edges and the length of each edge of the Von Koch Snowflake.

• Have made a discovery about the curious properties of the Von Koch snowflake.

CEDS – Study Plus in Cornwall

Lesson 4 - Overview

1. Patterns in sequences

2. Position-to-term rules of the Von Koch Snowflake

3. A curious property!

4. Thinking about our learning!• Displays and narration

CEDS – Study Plus in Cornwall

1. Patterns in sequences

1, 2, 4, 8, … … …

Term to term rule

CEDS – Study Plus in Cornwall

1. Patterns in sequences

10, 13, 16, 19 … … …

Term to term rule

CEDS – Study Plus in Cornwall

1. Patterns in sequences

… … … … … …

Term to term rule

multiply by ½

CEDS – Study Plus in Cornwall

2. The Von Koch Snowflake

Pattern Number

Number of edges

Length of each edge

Total Perimeter

1 3 9 27

2 12 3 36

3 48 1 48

4 192 1/3 64

5 768 1/9 85 1/3

CEDS – Study Plus in Cornwall

2. Von Koch Snowflake

Pattern Number

Number of edges Length of each edge

1 3 9

2 12 3

3 48 1

4 192 1/3

5 768 1/9

10

CEDS – Study Plus in Cornwall

2. Von Koch Snowflake

CEDS – Study Plus in Cornwall

Nth terms

Number of edges sequence

3, 12, 48, 192, … … 3 x 4n-1

Length of each edge sequence

9, 3, 1, 1/3, … … 9 ÷ 3n-1

CEDS – Study Plus in Cornwall

Perimeter of the Von Koch Snowflake

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

1 2 3 4

Pattern Number

Len

gth

in

cm

Sequence showing the length of the perimeter

27, 36, 48, 64, …

× 3/4

× 4/3

CEDS – Study Plus in Cornwall

What is the perimeter of

pattern number 6?

CEDS – Study Plus in Cornwall

CEDS – Study Plus in Cornwall

Learning outcome

By the end of the lesson you will:

• Understand the difference between term-to-term and position-to-term rules.

• Have derived the position-to-term rule for the number of edges and the length of each edge of the Von Koch Snowflake.

• Have made a discovery about the curious properties of the Von Koch snowflake.