The Nine Point Circle: Implications for the Secondary Mathematics Classroom |
The Curriculum and Evaluations Standards (1989), proposed by NCTM, espouse fourteen curriculum standards for grades 9-12 , which include
Mathematics as Problem Solving
Mathematics as Communication
Mathematics as Reasoning
Mathematical Connections
Algebra
Functions
Geometry from a Synthetic Perspective
Geometry from an Algebraic Perspective
Trigonometry
Statistics
Probability
Discrete Mathematics
Conceptual Underpinnings of Calculus
Mathematical Structure (p. 123).
Students should be provided opportunities that are abundant in meeting these standards. Many of these standards could be met by learning about the nine point circle and doing activities that pertain to the nine point circle.
Simply having students construct the nine point circle (by hand or GSP) allows students opportunities to learn, use, review, and integrate a wide variety of geometrical concepts such as triangle, altitude, midpoint, and orthocenter. Constructing the nine point circle provides a way for students to see mathematical connections between some basic geometrical concepts and how they tie together.
Integrating the history of the nine point circle gives students opportunities to see the evolution of a mathematical concept, how mathematics can be used as communication , and how mathematics can be viewed through a different lens.
When students make connections between pedal triangles and the nine point circle, they are unraveling a new way of seeing mathematics. This activity (Pedal Triangles and the Nine Point Circle) connects many geometric concepts, such as centroid, incenter, incircle, orthocenter, orthic triangle, circumcenter, circumcircle, and the nine-point circle. This would be a great geometry activity for an overall review and summation of the main geometrical concepts previously mentioned. It is here that students can be shown the beauty of mathematics and how it is all connected.
The activity Altitudes, Orthocenters, and the Nine-Point Circle is a nice extension that ties together the concepts of altitude, orthocenter, centroid, Nine-Point Circle, and the Euler Line. The construction stresses the interrelatedness of all these geometrical concepts.
Giving students activities to prove the nine point circle helps students develop their problem solving abilities, allows students to see mathematics as problem solving, and provides a basis for developing mathematics as communication and reasoning.
Nine point circle activities are also a great way to integrate geometry from both perspectives: synthetic and algebraic.
Incorporating the nine point circle into the mathematics curriculum gives students ample opportunities to work with a mathematical and geometrical concept that is new and exciting. Nine point circle activities are numerous and varied and can be used at both the middle and high school level.
In summary, integrating the nine point circle into a geometry classroom provides students and teachers new and exciting ways to develop geometrical concepts, meet the Standards, and sheds a new light on the geometry curriculum.
References
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
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