NetLogo Models Library:
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.
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:
The SETUP button creates a world with the given dimensions and size set by the sliders.
Select an ACTION.
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.
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?
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?
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)?
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.
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Copyright 2017 Uri Wilensky.
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