Formulating Measures

Judah L Schwartz, Apr 19, 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] For background on this collection of applets, please read the essay entitled "[i]Formulating Measures: toward modeling in the K-12 science and mathematics curriculum[/i]" on the mathMINDhabits website.

Table of Contents

  1. Formulating measures from given attributes
    1. Isosceles triangle space
    2. Isosceles Triangles - revisited
    3. Polya's Triangle Space
    4. Rectangles in the {Perimeter, Area} plane
    5. parallelogram space
    6. kite space
    7. Circles & ellipses in the {radius, eccentricity} plane
  2. Formulating Measures from user-defined attributes
    1. 'SQUARE-ness' of Rectangles
    2. SQUARE-ness revisited
    3. Rectangularity of Parallelograms
    4. Rhomboidicity of Kites
    5. Squareness of Parallelograms
    6. 'EQUILATERAL-ness' of Triangles
    7. 'SMOOTH-ness' of spheres
    8. Disc-ness
    9. Crowds
    10. Crowded-ness
    11. a road up a mountain
    12. another road up a mountain
    13. The "cyclicity" of quadrilaterals

1.

Formulating measures from given attributes

The applets in this chapter all have two panels - one in which an image of a geometric shape is displayed and a second panel in which is displayed a coordinate plane with axes {attribute1, attribute2} of the geometric shape in question. Geometric shapes with specific values of attribute1 and attribute2 are represented as points in the coordinate plane. A measure can be thought of as a function on {attribute1, attribute2} space.

  • 1. Isosceles triangle space
  • 2. Isosceles Triangles - revisited
  • 3. Polya's Triangle Space
  • 4. Rectangles in the {Perimeter, Area} plane
  • 5. parallelogram space
  • 6. kite space
  • 7. Circles & ellipses in the {radius, eccentricity} plane

1.1.

Isosceles triangle space

In the right hand panel choose a property of a set of isosceles triangles[br]- constant area[br]- constant perimeter[br]- constant altitude/base[br][br]GREEN line shows all the isosceles triangles with that area[br] area corresponding to the large GREEN dot is shown in the left panel[br]BLUE line shows all the isosceles triangles with that perimeter[br] perimeter corresponding to the large BLUE dot is shown in the left panel[br]YELLOW line shows all the isosceles triangles with that altitude/base ratio[br] altitude/base ratio corresponding to the large YELLOW dot is shown in the left panel[br][br]Drag the WHITE dot to set the value of the property you have chosen.[br][br]Drag the colored dot (color depends on your choice of property)[br]to generate of set of isosceles triangles with a constant value of that property.[br][br]Can you explain the shapes of the curves in the {base, altitude} plane [right panel] ?[br][br]Can you write an expression for Area as a function of base length for all three cases?[br]Can you write an expression for Perimeter as a function of base length for all three cases?[br]Can you write an expression for altitude/base ratio as a function of base length for all three cases?
– Judah L Schwartz

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1.7.

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2.

Formulating Measures from user-defined attributes

The applets in this chapter while posing problem situations similar to the ones in Chapter I, all require the user to decide on the attributes to be used in defining measures.

  • 1. 'SQUARE-ness' of Rectangles
  • 2. SQUARE-ness revisited
  • 3. Rectangularity of Parallelograms
  • 4. Rhomboidicity of Kites
  • 5. Squareness of Parallelograms
  • 6. 'EQUILATERAL-ness' of Triangles
  • 7. 'SMOOTH-ness' of spheres
  • 8. Disc-ness
  • 9. Crowds
  • 10. Crowded-ness
  • 11. a road up a mountain
  • 12. another road up a mountain
  • 13. The "cyclicity" of quadrilaterals

2.1.

'SQUARE-ness' of Rectangles

If you drag the ORANGE dot you can produce different rectangles. Devise a [br]measure of 'Squareness' that allows you to arrange any collection of [br]rectangles in order of 'Squareness'.[br][br]How many such measures can you devise?[br]Which of these measures do you prefer? Under what circumstances?
– Judah L Schwartz

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2.11.

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2.13.

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