# Lattice Land - Lattice Triangles

**Subject:** Mathematics

**Time:** 3-4 Class Periods (45 minutes each)

**Level:** Geometry

## Lesson Overview

Lattice Land is intended for users at any level of fluency with mathematics to explore a “Mathland” (other Mathlands include Flatland, or Donald Duck in Math-magic Land). In this Mathland, the "lattice" is an ordered array of dots on a plane, such that there is one dot at each coordinate (x,y), where x and y are integers. In set notation, {(x,y)|x,y∈Z}. In Lattice Land, users may only move from dot to dot, which restricts the number and the type of geometric objects. Thus, in the Lattice Land environment, there are countably many solutions to complex questions; whereas in continuous geometry there is usually one, none, or infinite solutions. This discrete way of looking at geometry allows users to more easily explore counting theories, area and perimeter, and pattern recognition. These lessons focus on all kinds of triangles. You may opt to use all of these lessons, or pick and choose the ones best suited to the needs of your class. The first focuses on types of triangles: equilateral, isosceles, scalene, right, acute, obtuse. Students should all be challenged to draw at least one example of each. Encourage them to notice that some triangles can satisfy two categories, for example an isosceles right triangle, while one triangle is impossible to draw in Lattice Land. The teacher and class may choose how much time to devote to defining categories, or proving why there does not exist an equilateral lattice triangle. The second lesson focuses on areas of triangles: different triangles with the same base and height, area formulas, and the relationship between area and perimeter. Students should be encouraged to explore and come up with their own theories and definitions to share with their groups and class. The third lesson focuses on dissection. This is a useful and fairly creative skill that may not necessarily be a required aspect of the curriculum, but comes up frequently on exams like the ACT and SAT. This lesson asks students for creative approaches to finding areas on their own.**Prerequisites:**

- Students must know basic geometric terms: point, line, segment, polygon, angles.
- It helps if students have seen the Pythagorean Theorem before, but it is not necessary.

**Pre-Class Preparation:**

- Make sure that all computers are working, and have the latest version of NetLogo installed.
- Make sure that the Lattice Land files are easily accessible. If they are on a cloud account, ensure that all computers have internet access.
- Familiarize yourself with Lattice Land. It helps to look at all the modules in order to plan ahead.

## Compatible With

mac

windows

laptops

chrome books

phones

tablets

## Standards

**Computational Thinking in STEM**

- Computational Problem Solving Practices
- Assessing Different Approaches/Solutions to a Problem
- Developing Modular Computational Solutions
- Systems Thinking Practices
- Defining Systems and Managing Complexity
- Understanding the Relationships within a System

## Comments, Feedback, and Questions