Students could then find an efficient route for map B. They should eventually conclude that no routes in map B satisfy the conditions of the problem. To extend this investigation, students could look for efficient paths in other situations or they might change the conditions of the map B problem to find the pathway with the least backtracking. Such an investigation in the middle grades is a precursor of later work with Hamiltonian circuits, a foundation for work with sophisticated networks. Exploring the Pythagorean Relationship Visual demonstrations can help students analyze and explain mathematical relationships. Students could replicate some of the other visual demonstrations of the relationship using dynamic geometry software or paper-cutting procedures, and then discuss the associated reasoning.
Geometric models are also useful in representing other algebraic relationships, such as identities.
Students could then partition the square as shown in fig. 6.22a, calculate the area of each section, and finally represent the total area. Nature, art, and the sciences provide opportunities for the observation and the subsequent exploration of geometry concepts and patterns as well as for appreciating and understanding the beauty and utility of geometry.