Generalizing

Judah L Schwartz, Apr 20, 2016

Table of Contents

  1. polygons
    1. Taking Sides
    2. Polygons in polygons
    3. Sum of polygonal areas
  2. circles
    1. Disc rolling on a Disc
    2. Disc rolling inside a Disc
  3. functions
    1. Build & Fit Linear Functions
    2. Build & Fit - Quadratic Functions
    3. Powers of Complex Numbers

1.

polygons

  • 1. Taking Sides
  • 2. Polygons in polygons
  • 3. Sum of polygonal areas

1.1.

Taking Sides

Here is a regular polygon of n sides inscribed in a unit circle. In the limit of a very large number of sides the area and perimeter of the polygon approach those of the circle.[br][br]Write an expression for A(n), the area of an n sided regular polygon inscribed in a unit circle.[br]Write an expression for P(n), the perimeter of an n sided regular polygon inscribed in a unit circle.[br][br]Contrast the rates at which A(n) and P(n) approach their limits. [br][br][br]Challenges:[br][br]The number of sides, n, grows while the length of each side, S gets smaller and smaller. How does the product of n and S behave? How do you know? Can you prove it?[br][br]The area of a [i]UNIT[/i] circle is [math]π[/math] and its perimeter is [math]2π[/math]. How do you convince a student that[br]the area of a circle is [b][i]NOT[/i][/b] half its perimeter?[br][br][color=#ff0000][i][b]What other questions [could,would] you ask you students based on this applet ?[/b][/i][/color]
– Judah L Schwartz

1.2.

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1.3.

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2.

circles

  • 1. Disc rolling on a Disc
  • 2. Disc rolling inside a Disc

2.1.

Disc rolling on a Disc

A GOLD disc rolls without slipping around the edge of a SILVER disc.[br][br]If the discs are the same size, how many revolutions does the GOLD disc make as it rolls around the SILVER disc exactly once?[br][br]You may find the result surprising? Can you explain it?[br][br]What happens if the discs are not the same size – if the radius of the GOLD is half that of the SILVER? twice? three times?[br][color=#ff0000][i][b][br]What questions could/would you ask of your students based on this applet?[/b][/i][/color]
– Judah L Schwartz

2.2.

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3.

functions

  • 1. Build & Fit Linear Functions
  • 2. Build & Fit - Quadratic Functions
  • 3. Powers of Complex Numbers

3.1.

Build & Fit Linear Functions

You can make a linear function by dragging the two RED points almost anywhere on the screen (why almost?). The function will have the form - linear(x) = Ux + C, where U is a real number and the value of C is a real number. Now try to build the same linear function by combining the appropriate amounts of linear term and constant term using the BLUE sliders. Try to do this without displaying the functional form of the RED linear function you have made. You can check your work by showing the expressions for the functions on the screen. CHALLENGE - A student asks "How do you know that you can always make a linear function given two points?" What do you respond? Can two points define any other kind of function? Would rise/slant or run/slant be as good a measure of slope as rise/run? Why or why not?
– Judah L Schwartz

3.2.

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3.3.

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