# Finding multiple paths to solve a task

The task: Show different ways to construct an equilateral triangle based on a given vertex C and one its side lying on a given line. A crucial stage to make sense of what the task involves is to problematize its statement. That is, to think of the task it in terms of questions in order to identify relevant information and comprehend what is asked. For instance, what data or information is provided? What does it mean that one side of the triangles lies on a given line? Where is the third vertex located? What properties does an equilateral triangle hold? The ned figure shows a point C as the given vertex and a line l. Then, the goal is where to locate segment AB on line l, so that triangle ABC is equilateral.
Discussion and reflections In posing the initial task we made explicit that the goal was not only to solve the task, but, to look for different ways to construct the triangle. That is, it is important to recognize that looking for several ways to represent and explore a task is an essential strategy to engage students in mathematical thinking. What distinguish, in terms of mathematical concepts and reasoning, the approaches to the task and why are they important? A key idea to identify and analyse multiple forms to represent a task is to think of involved mathematical objects in terms of their mathematical properties or relationships. For instance, in some of the approaches it was important to know that in any equilateral triangle, the circumcenter, incenter, and orthocentre coincide. The first and second approaches initially focus on constructing a triangle holding partial condition as a means to find a path to solve the task. The use of a Dynamic Geometry Environment enhances the implementation of this problem solving strategy since students have the opportunity to visualize some objects’ behaviours. In many of the approaches, moving points and finding loci within the representation became a tool to construct the equilateral triangle.