# Fractal curves - points and edges

## Fractal game workshop plan

Materials: puzzle-parts (each pupil gets a package à 55 pieces) possibly a beamer for presenting the photos of the puzzles Learning outcomes: Recognising repetitions and symmetries, sharpening perception of the environment, recognising that sequences and series are also a way of repetition, learning mathematical descriptions of repetitions through fractals, patterns, enlargements and reductions, ... Curricula:
• primary school (VS, 6-10 years): symmetries, perception, surfaces, shapes, ...
• lower secondary school (Sek I, 10-14 years): (Self-)similarity, uniform scaling
• upper secondary school (Sek II, 14-18 years): fractals, sequences and series
Duration: 2 school lessons à 50 minutes (on different days) Teaching Methods:
• teacher-centered instruction (frontal)
• individual work
• plenary
• group work

## Introduction

• teaching method: frontal
• duration: 10 min + 5 min time buffer
• materials: photos, pictures, videos, vegetables, pine cones, ...
The teacher explains that in nature sometimes things repeat themselves and that this can basically be described mathematically. Pictures/videos/... are being viewed. further ideas for an extended instruction: primary school:
• consider if you include the video!?
• students should bring objects such as pine cones, ...
• pass objects through, let them discover regularities and repetitions
lower secondary school
• show video, show pictures of objects that contain repetitions
• show scaling using GeoGebra
﻿ ﻿ ﻿﻿ ﻿ ﻿- ﻿square & rectangle, composed/decomposed surface ﻿ ﻿ ﻿ ﻿ ﻿- ﻿cutting surfaces into smaller self-similar parts upper secondary school
• show video, show pictures of objects that contain repetition
• explaining mathematical concepts of sequences and series
• explaining the meaning of endless repetition, Mandelbrot set, ...

## Puzzling alone - find your personal solution

• teaching method: individual work
• duration: 20 min + 10 min
• material: puzzle
Handing out the puzzle. Explaining the task: "Use this puzzle to make a repetition or pattern that you consider beautifully". (Help may only be given "technically" and not "contentwise".) After half the time, announce the time. After 20 minutes, give another 10 minutes extra time. If there are specially fast students, the teacher goes there, takes a photo and asks students to find a new solution.

• teaching methods: plenary discussion
• duration: 15 Minuten
• material: finished puzzle-results, beamer (for presenting photos of further puzzle-results)
The students present their results to the rest of the class and explain how they solved the puzzle and what they like about their results (and why).

## Connecting puzzles

• teaching method: group work
• duration: 30 min
• material: puzzle
Students meet in teams or are divided into teams by the teacher. In this unit the students either try to combine their two results or to work out a new solution for the task together.

## Follow up after the workshop

primary school "Geometric forms arise from different (other) geometric forms", no longer just "the round must go into the round". lower secondary school students should work with applets of e.g. Diego Lieban: cutting up triangles, ... so that the result is self-similar. upper secondary school continue with the topics "sequences and series"