# Voronoi - Lesson Plan

## Voronoi - Lesson Plan

• Topic: Voronoi
• Grade, subject: 6th grade or older, mathematics
• Duration: 1 - 2 units (à 50 min)
• Material for students: exercise books, sheet of paper with sets of points, pencil, ruler, ...
• Additional materials: 3D-printed materials
The students get to know what voronoi diagrams are, how they are used in strategical planning and where they are found in nature. 3D-printed materials are included, to show students how 2D voronoi-diagrams can be extended to a 3D puzzle.

## Prior knowledge of the students

The students already know/learned...
• ...how to use a GeoGebra Application
• ...what a perpendicular bisector is and how it is constructed.
The teacher should know...
• ...how use the command "Voronoi(<List of Points>)" in GeoGebra

## Gained competencies and skills by the activity or lesson

Which competencies could students learn by the activity or lesson? The students...
• ...train constructing perpendicular bisectors maually and by using GeoGebra.
• ...know characteristics of perpendicular bisectors and additionally of voronoi-diagrams.
• ...learn about the applications of voronoi-diagrams in the real world.
• ...gain another perspective on mathematic concepts by using 3D printed materials.
• ...train to present/defend their ideas in front of a group.
• ...train to find an agreement with a few other students.

## Lesson Plan - Overview

The steps in more detail follow below.
• Introduction
• activating pre-knowledge
• Example 1: post office
• Example 2: post office
• Theoretical input
• Finding examples
• Switch from 2D to 3D
• PUZZLES: 2D and 3D
• Follow up information

## Introduction

• teaching method: direct instruction
• duration: ~3 min
The teacher explains the topic: What do have the line-up at a football match and the wing of insects in common? Where is there a mathematical background? What is this background? (--> perpendicular bisectors!)

## activating pre-knowledge

Start a dialog between you and the students and find out if the students still have some knowledge about perpendicular bisectors.
• if YES: Emphasis that every dot on the perpendicular bisector has the same distance to the two dots it belongs to.
• if NO: Explain how it is constructed and do exersices.

## Example 1: post office

Where do I get my mail from? Students get a picture of a map with 3 dots and should answer questions:
• Consider that you are living in _____. Which post office is the nearest to you?
• Consider that you are working in the post office ____. Find at least two addresses which you are delivering to.
• Are there addresses which have the same seperation to more offices? Can you find a specific address?
• Try to find areas around every post office, where each office is responsible to deliever the mail to.

## Example 2: post office (OR as differentation to example 1)

Similar task as in example 1, just with more dots (e.g. 5). (You can decide appropriately to you students if it is necessary to do one more similar example.)

## Theoretical input

• teaching method: direct instruction
• duration:
Students should note some important information concerning voronoi into their exercise books: We are given a set of different locations. A voronoi-region of one of the locations - a so-called center - is the set of all points in a plane, which are closer to this one location than to any other location. The voronoi-regions for all the locations is called a voronoi-diagram. Every student get pictures of sets ot points (similar to the picture below) and should think how the voronoi-diagrams will look like and sketch them. (--> special cases included!) To compare the solutions the teacher opens the examples in GeoGebra and shows them the right voronoi-diagrams solved with the command "voronoi-diagram" in GeoGebra.

## Finding examples

• teaching method: individual work & plenary
• duration:
Observe your environment: Can you find examples in the nature? Teacher explains where it can be used: strategic positioning of locations (e.g. supermarkets, post office, ...)

## Switch from 2D to 3D

• teaching method: plenary
• duration:
Consider the following situation: Colour is poured evenly onto a plate at the points. Where the different colours meet, straight lines are created. (--> show gif below)
For representing the switch to 3D, the students should play with the GeoGebra Applet in capter 5.1.
• method: think-pair-share
• duration:
THINK: The students should experiment with the applet (Voronoi-Cones﻿) and they should come up with a connection to the gif above. PAIR: Talk to a partner and share your thoughts. SHARE: Some groups intoduce their ideas to the class. The teacher leads this class discussion and summerizes it at the end. (talk about voronoi-CONES and its connection to a 2D voronoi-diagram if you view it from straight above.) Teacher also show the students some 3D printed voronoi models (picture below).

## Puzzles: 2D & 3D

• teaching method: group work
• duration:
Students solve the 2D voronoi puzzle: each puzzle piece is a single voronoi cell. Students solve the 3D voronoi puzzle: each puzzle piece is part of a cone.

## Follow up information

In following lessons the students could form groups of 4 people and try out some of the game variations﻿ with voronoi-cones.