Functions on the Domain of Space - Polygons

Judah L Schwartz, Apr 29, 2016

[b]This is a series of applets drawn from the website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/b]

Table of Contents

  1. Medians, Altitudes & Angle Bisectors
    1. Students define a median
    2. Build a triangle from medians?
    3. Build a Triangle from Medians - II
    4. Beyond the Median
    5. The Marion Walter Theorem
    6. A generalization of the Marion Walter Theorem
    7. Build a triangle from Angle Bisectors?
    8. Build a triangle from Altitudes?
  2. Reasoning about rates
    1. Reasoning about rates - overlapping squares
    2. Reasoning about rates - yellow bowties I
    3. Reasoning about rates - yellow bowties II
  3. Polygons as closed chains of linked segments
    1. Point in an Equilateral Triangle
    2. Point in a Square
    3. Point in a Regular Pentagon
    4. Point in an Equilateral Triangle - revisited
    5. Quadrilateral factory ?
    6. Polygon Explorer
    7. Polygon Synthesizer
  4. Curios
    1. Quadrature of the Rectangle
    2. Quadrature of the Triangle
    3. Quadrature of the Lune

1.

Medians, Altitudes & Angle Bisectors

  • 1. Students define a median
  • 2. Build a triangle from medians?
  • 3. Build a Triangle from Medians - II
  • 4. Beyond the Median
  • 5. The Marion Walter Theorem
  • 6. A generalization of the Marion Walter Theorem
  • 7. Build a triangle from Angle Bisectors?
  • 8. Build a triangle from Altitudes?

1.1.

Students define a median

Ninth grade students using a dynamic geometry program are asked to explore possible definitions of median in the triangle formed by the white dots.[br]When the students choose a vertex and ask the program to produce a median, the program responds by drawing the white line and the gold dot (see below).[br][br]Students explore the values of areas, angles and side lengths and suggest the following “definitions” – [br][br][i][b]A – a median divides the side opposite the vertex in two equal pieces[br][br]B – a median divides the area of the triangle into two equal areas[/b][/i][br][br]Can you show that these definitions are equivalent – if either one is true then the other is as well?[br][br]Why might definition A be preferable to definition B ? Why might definition B be preferable to definition A ?[br][br]Can the definition of median be generalized to polygons with more than three sides?
Students define a median
– Judah L Schwartz

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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2.

Reasoning about rates

  • 1. Reasoning about rates - overlapping squares
  • 2. Reasoning about rates - yellow bowties I
  • 3. Reasoning about rates - yellow bowties II

2.1.

Reasoning about rates - overlapping squares

Two identical squares can be made to overlap in many ways. Here are two such methods.[br][br]Varying the overlap slider varies the degree of overlap. In the right hand panel the[br]area in common for each method is plotted as a function of the degree of overlap.[br][br]Which function corresponds to which method? How do you know?[br][br]Why is one function linear and the other not?[br][br]What is the non-linear function? How do you know?[br][br]What conjectures do you have about the nature of the common area function for [br]convex polygons other than squares?[br][br]What conjectures do you have about the common area function if the two squares[br]are not related by a simple translation along a line of symmetry?[br][color=#ff0000][i][b][br]What questions could/would you ask your students based on this applet?[/b][/i][/color]
– Judah L Schwartz

2.2.

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2.3.

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3.

Polygons as closed chains of linked segments

  • 1. Point in an Equilateral Triangle
  • 2. Point in a Square
  • 3. Point in a Regular Pentagon
  • 4. Point in an Equilateral Triangle - revisited
  • 5. Quadrilateral factory ?
  • 6. Polygon Explorer
  • 7. Polygon Synthesizer

3.1.

Point in an Equilateral Triangle

The GOLD point is chosen at random inside the equilateral triangle. [[i]You can change the size of the equilateral triangle by dragging the BLACK dots.[/i]] The red, blue and green segments are lines drawn from the GOLD point and perpendicular to each of the sides. Their lengths vary in size as you move the GOLD point from place to place inside the triangle. However, the sum of their lengths is constant. Why? How is the sum of the lengths related to the size of the triangle? Why? Can you prove it? Would a similar thing be true in a square? Why or why not? What about other regular polygons with an odd number of sides? with an even number of sides? If the GOLD point is dragged outside the triangle the relationship still holds if the distance from the GOLD point to the triangle is counted as negative! Why?
– Judah L Schwartz

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3.5.

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3.6.

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3.7.

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4.

Curios

  • 1. Quadrature of the Rectangle
  • 2. Quadrature of the Triangle
  • 3. Quadrature of the Lune

4.1.

Quadrature of the Rectangle

The quadrature of a rectangle is the construction of a square having the same area as the rectangle using only a ruler and compass - sometimes referred to as "squaring the rectangle".[br][br]Drag the YELLOW dot.[br][br]Why are the GREEN & BLUE areas equal?
– Judah L Schwartz

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