[list][*]Topic: Voronoi[br][/*][*]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 [/*][/list]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.

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

Which competencies could students learn by the activity or lesson?[br][br]The students...[br][list][*]...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. [br][br][/*][*]...gain another perspective on mathematic concepts by using 3D printed materials. [br][br][/*][*]...train to present/defend their ideas in front of a group. [/*][*]...train to find an agreement with a few other students. [/*][/list]

The steps in more detail follow below. [br][list][*]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[/*][/list]

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

Start a dialog between you and the students and find out if the students still have some knowledge about perpendicular bisectors. [br][br][list][*]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.[br][br][/*][/list]

Where do I get my mail from?[br][br]Students get a picture of a map with 3 dots and should answer questions:[br][br][list][*]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.[/*][/list]

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

[list][*]teaching method: direct instruction[/*][*]duration: [/*][/list][br]Students should note some important information concerning voronoi into their exercise books:[br][br][i]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.[br][br][/i]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. [i](--> special cases included!)[br][br][/i]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.

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

[list][*]teaching method: plenary[/*][*]duration:[/*][/list][br]Consider the following situation: [br]Colour is poured evenly onto a plate at the points. Where the different colours meet, straight lines are created. ([i]--> show gif below)[/i]

For representing the switch to 3D, the students should play with the GeoGebra Applet in capter 5.1.

[list][*]method: think-pair-share[/*][*]duration:[/*][/list][br]THINK:[br]The students should experiment with the applet ([url=https://www.geogebra.org/m/HNxm3pxf]Voronoi-Cones[/url]) and they should come up with a connection to the gif above. [br][br]PAIR:[br]Talk to a partner and share your thoughts.[br][br]SHARE:[br]Some groups intoduce their ideas to the class.[br][br]The teacher leads this class discussion and summerizes it at the end. [br]([i]talk about voronoi-CONES and its connection to a 2D voronoi-diagram if you view it from straight above.)[br][br][/i]Teacher also show the students some 3D printed voronoi models (picture below).

[list][*]teaching method: group work[/*][*]duration:[/*][/list][br]Students solve the 2D voronoi puzzle: each puzzle piece is a single voronoi cell. [br]Students solve the 3D voronoi puzzle: each puzzle piece is part of a cone.

In following lessons the students could form groups of 4 people and try out some of the [url=https://docs.google.com/document/d/1VRDI7jihgccvHTJZ1JC4hR96xAogsz5MwBIebtlj50M/edit]game variations[/url] with voronoi-cones.