# Voronoi - Lesson Plan

- Author:
- Eva Bauer-Öppinger, GeoGebra Team, Eva Ulbrich

## 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

## 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.

- ...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

*(--> 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:

*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:

## Switch from 2D to 3D

- teaching method: plenary
- duration:

*--> 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:

*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:

## 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.