Functions on Shapes in the Plane
A geometric construction or a series of them may be thought of a function. Here is an example: Construct the three medians of a triangle. Reflect the point of concurrence of the medians over each of the sides of the triangle. These operations define a new triangle. Taken together these operations are a function from the domain of point triples in the plane to the range of point triples in the plane. This collection of applets allows for the exploration of such functions on shapes. [i][b] This collection of tasks is part of a series entitled "Making and Exploring Conjectures - Expanding Teachers' and Students' Mathematical 'Habits of Mind'". They are designed primarily to help teachers think more deeply about aspects of mathematics that they know and are expected to teach. Along with his colleagues and students, Judah L. Schwartz has been writing prize-winning educational software for more than thirty years. For several decades he taught physics at the Massachusetts Institute of Technology and taught and did research on mathematics and science education at the Harvard Graduate School of Education. After retiring from MIT and Harvard, he taught pre-service and in-service teachers of mathematics for nine years at Tufts University. Professor Schwartz and colleagues are available to conduct small scale workshops to help people think more creatively about the mathematics they know. Interested schools, school systems and organizations can contact him at math.habits.of.mind@gmail.com[/b][/i]
Зміст
- Polygons
- Seven Functions on a Triangle
- An interesting Function on the space of quadrilaterals
- '...in a square'
- Taking Sides
- Beyond the Median
- Build a Triangle out of medians
- Build parent triangle out of its medians
- Quad in a Quad
- Napoleon's Theorem
- Point in an Equilateral Triangle
- Circles & ellipses
- Three Tangent Circles
- Five circles in a square - adapted from Polya
- Yin Yang Yin
- Elliptical Billiard Table
- Rescue at sea from an elliptical island
- Burning tent on an elliptical island
- Orthogonal Circles
- Three Orthogonal Circles on an Isosceles Triangle
- Sundry other
Seven Functions on a Triangle
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Draw the three medians in a triangle. The three points at which the medians intercept the sides form a new triangle. Similarly, for the angle bisectors and for altitudes (in non-obtuse triangles). This means that you can think of each of these constructions as a function from a domain of triples of points in the plane to a range of triples of points in the plane. You can use this environment to explore the relationship between the original triangle and the one formed by the intersections of the medians, angle bisectors and altitudes with the sides of the triangle. Here is another set of functions on a triangles - It is well known that the angle bisectors of the angles of a triangle all meet at a point, as do the altitudes and the medians. In each case reflect the meeting point over the three sides of the triangle - the resulting three points form a new triangle. You can use this environment to explore the relationship between the original triangle and the one formed by reflecting the three, in general different, points of concurrence over the sides of the triangle. Finally, you can reflect each of the vertices of the triangles over the side opposite it. Once again a new triangle is formed. Drag the points A, B, and C to make the triangle you want to examine. What conjectures can you make about these seven functions? Do these functions have inverses? What happen when you compose these functions with themselves? with one another? What conjectures would you like to explore? Which conjectures do you think might be easy to prove? |
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Three Tangent Circles
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from Polya - Mathematical Discovery Three circles, each of which is tangent to the other two have their centers on a line. If the radius of the surrounding circle is R and the length of the vertical (black) segment is T what is the size of the green area? You can explore this problem by moving the WHITE dot, but it remains for you to prove your method will work for any set of three mutually tangent circles like this. |
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