Modeling Relationships in Space & Time

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] To complement this GeoGebra book please look at the THEME entitled "Images for the Eye & the Mind's Eye" in the THEMES sections of the mathMINDhabits website.

Table of Contents

  1. Motion problems
    1. The Princess and the Key -
    2. Meeting Up - on the road with Red & Blue
    3. Recovering from a long lunch
    4. Jack & Jill
    5. two lap average
    6. two lap average - revisited
    7. A Run for Two
    8. Ship rendezvous
    9. Keeping Pace - A Balanced Assessment Task
    10. Projectile w/ air resistance
    11. Horse on road and field
    12. Carpe diem - a day clock
    13. v(t) = dx(t) / dt - beyond linear functions
    14. Two Lap Average [for teachers]
    15. TWO MOTIONS - one path in space
  2. Planets
    1. Two Planets & A Sun - A Digital Triptych
    2. planetary orbits in two frames
  3. Modeling spatial relationships
    1. Sum of polygonal areas
    2. Sum of Circle Areas
    3. Fuel Consumption Calculator - length/volume ratio
    4. Rain gutter
    5. Shortest ladder
    6. The Sliding Ladder Problem
    7. More about ladders
    8. Erehwon Coinage
    9. A Different Kind of Analog Clock
    10. Bicycle Chain Math - A Balanced Assessment Task
    11. Optimally Locating Public Services
  4. Other sundry modeling issues
    1. A Leaky Bathtub - competing rates & equilibrium
    2. Retirement & leaky bathtubs; 2 metaphors - 1model
    3. good fits - bad models ?
    4. CHALLENGE - How can f(x) = f'(x) ?

1.

Motion problems

  • 1. The Princess and the Key -
  • 2. Meeting Up - on the road with Red & Blue
  • 3. Recovering from a long lunch
  • 4. Jack & Jill
  • 5. two lap average
  • 6. two lap average - revisited
  • 7. A Run for Two
  • 8. Ship rendezvous
  • 9. Keeping Pace - A Balanced Assessment Task
  • 10. Projectile w/ air resistance
  • 11. Horse on road and field
  • 12. Carpe diem - a day clock
  • 13. v(t) = dx(t) / dt - beyond linear functions
  • 14. Two Lap Average [for teachers]
  • 15. TWO MOTIONS - one path in space

1.1.

The Princess and the Key -

A beautiful princess is locked in a tower. [br][br]The lock to the tower can only be opened from the inside. [br][br]The princess’ lover has the key but he is outside the tower.[br][br]Talking to one another through a window they decide that he should toss the key up to her.[br][br]There are windows in the tower. They are trying to decide which window she should lean out and how fast he should throw the key up.[br][br]• Use this applet to decide on a combinations of window height and initial speed that will enable the princess to grab the tossed up key. Indicate when and for how long the key is just outside the window.[br][br]• What strategy would you advise to maximize the possibility of the key being caught?[br][br]• Knowing that the time in seconds for which the height of the key in meters is a maximum [br]is initialspeed / 9.8 write an expression for the height of the key as a function of time.
– Judah L Schwartz

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1.7.

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1.8.

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1.9.

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1.10.

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1.11.

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1.12.

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1.13.

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1.14.

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1.15.

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2.

Planets

  • 1. Two Planets & A Sun - A Digital Triptych
  • 2. planetary orbits in two frames

2.1.

Two Planets & A Sun - A Digital Triptych

The motions of a Sun and two planets as seen in each of their coordinate systems.[br][br]You can animate the motion by clicking on the lower left hand corner of the screen.[br][br]You can show the trajectories by checking the trajectories checkbox.[br][br]You can show the distances between the bodies by checking the distance checkbox.[br][br]Does this help you understand the retrograde motion of Mars?[br][br]Why does the triangle of distances look the same in all three coordinate systems?
– Judah L Schwartz

2.2.

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3.

Modeling spatial relationships

  • 1. Sum of polygonal areas
  • 2. Sum of Circle Areas
  • 3. Fuel Consumption Calculator - length/volume ratio
  • 4. Rain gutter
  • 5. Shortest ladder
  • 6. The Sliding Ladder Problem
  • 7. More about ladders
  • 8. Erehwon Coinage
  • 9. A Different Kind of Analog Clock
  • 10. Bicycle Chain Math - A Balanced Assessment Task
  • 11. Optimally Locating Public Services

3.1.

Sum of polygonal areas

The GOLD dot is connected to all four vertices of a rectangle. [br]You can set the size & shape of the rectangles by dragging the GRAY dot.[br]You can place the GOLD dot anywhere in the rectangle.[br][br]The length of the segment from each vertex of the rectangle to the GOLD dot is the side of a regular polygon.[br][br]The sum of the areas of the BLUE polygons is equal to the sum of the areas of the GREEN polygons no matter where you place the GOLD dot. [br][br]Can you prove this? What happens when you drag the GOLD dot outside the rectangle? 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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3.8.

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3.9.

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3.10.

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3.11.

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4.

Other sundry modeling issues

  • 1. A Leaky Bathtub - competing rates & equilibrium
  • 2. Retirement & leaky bathtubs; 2 metaphors - 1model
  • 3. good fits - bad models ?
  • 4. CHALLENGE - How can f(x) = f'(x) ?

4.1.

A Leaky Bathtub - competing rates & equilibrium

If water comes into a leaky bathtub at a constant rate and leaks out a rate proportional to the height of the water in the tub, then an equilibrium depth of water in the tub can be reached. You can use this applet to explore how the equilibrium depth depends on the rate of flow into and the rate of flow out of the tub. Clearly, the leaky bathtub is a metaphor for many equilibrium situations that arise because of competing rates. Can you think of others? You can watch an animation by clicking on the "play" icon in the lower left hand corner of either window.
– Judah L Schwartz

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