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Lattice Land - Triangles Dissection

If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web

WHAT IS IT?

Lattice Land - Triangles Dissection is one of several models in the Lattice Land software suite. Lattice Land is an interactive MathLand, a microworld in which students can uncover advanced mathematical thinking through play, conjecture, and experimentation. It provides another entryway into geometry, investigating the geometry of a discrete lattice of points. In Lattice Land, there is no one right answer and no pre-determined pathway you must travel. However, even seemingly trivial exercises can quickly become rich explorations.

A lattice is an array of dots on a plane such that there is one dot at each coordinate (x,y), where x and y are integers. Thus each dot on the lattice is one unit away from each of its four closest neighbors (one above, one below, one to the left, and one to the right). A lattice triangle is a triangle whose vertices fall on dots of the lattice.

The setup of this model resembles a traditional GeoBoard with 25 pegs and 16 square units. In this triangles dissection model, you can click and drag the vertices of a lattice triangle to explore all possible triangles within this space. You may also draw segments to dissect the space (and the triangles in it) in various ways, to help develop ways in which to think about area. The Triangles Dissection model may be used following, or in lieu of, the Triangles Explore model.

HOW IT WORKS

We've implemented a lattice in NetLogo by using agents called DOTS sprouted at the center of each patch. The segments between the dots are simply edges or links. The environment then responds to click-and-drag of the mouse.

In this model, the lattice is restricted to a 4 unit by 4 unit lattice. Additionally, the user can only work with the triangle, which is randomly generated at setup. This model mimics the appearance and functionality of the popular geometry manipulative Geoboard. Students can make triangles with areas ranging from 0.5 square units to 8 square units, and every 0.5 square-unit increment in between.

You can also draw segments that highlight the base and height of the triangle, or draw a rectangle that contains the triangle as an exercise in dissection.

This model uses Heron's Formula to calculate area of a triangle:

• Area = sqrt(s(s-a)(s-b)(s-c))
• where s is the semiperimeter (half the perimeter) of the triangle

HOW TO USE IT

The SETUP button creates a world with the given dimensions and size set by the sliders.

Select an ACTION.

• MODIFY TRIANGLE allows you to click and hold any vertex of the triangle to any other dot on the lattice.
• DRAW SEGMENT allows you to join any two dots selected consecutively by a line segment.
• DELETE SEGMENT allows you to delete any line segment you have drawn by selecting either endpoint of the segment. Note that this will delete ALL segments that share the selected endpoint.

Press the GO button to run the selected ACTION.

Press CHECK-AREA to verify area calculations for the triangle.

Look at the three LENGTH OF SIDE monitors to track the lengths of sides S1, S2, and S3, labeled on the triangle.

THINGS TO NOTICE

Notice that some triangles have no sides that are parallel to either the x- or y-axes. What implications does this mean for application of the familiar 1/2 * base * height formula?

THINGS TO TRY

Notice that you can produce a triangle with every possible area in between (at 0.5 square unit increments). How many different triangles can you create on the GeoBoard with the same area? What generalizations can be made about triangles that share the same area?

Can you find the area of every triangle without using the CHECK-AREA button? Using the Pythagorean Theorem? Using dissection? How can we use the DRAW-SEGMENT action to outline a box (of known area around any triangle, and how can that help us reason about area?

EXTENDING THE MODEL

Add other polygons to the model. Explore triangulation by dissecting each polygon into triangles and calculating their areas. Is triangulation possible with every polygon (including concave and convex polygons)?

NETLOGO FEATURES

The DRAW-SEGMENT procedure makes use of two anonymous procedures to actually draw on the screen.

This model uses continuous updates, rather than tick-based updates. This means that the model does not update at regular time intervals (ticks) dictated by the code. Instead, this model updates when the user performs an action. Thus, the depth of inquiry into the mathematics of Lattice Land is dictated by the user: nothing (other than the lattice) is generated until the user draws something.

RELATED MODELS

• Lattice Land - Triangles Explore
• Lattice Land - Explore

HOW TO CITE

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

• Pei, C. and Wilensky, U. (2017). NetLogo Lattice Land - Triangles Dissection model. http://ccl.northwestern.edu/netlogo/models/LatticeLand-TrianglesDissection. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

Please cite the NetLogo software as:

• Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

COPYRIGHT AND LICENSE

Copyright 2017 Uri Wilensky.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

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