Modeling - non-standard & standard problems

Judah L Schwartz, Feb 3, 2014

This collection is a potpourri of problems that require the formulation of measures, combining those measures into functions and relating those functions to one another to form equations and/or inequalities. Many of these problems are standard algebra word problems that are approached in non-standard graphical ways. [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]


  • 1. 'SQUARE-ness' of Rectangles
  • 2. Squareness of Parallelograms
  • 3. 'EQUILATERAL-ness' of Triangles
  • 4. Sum of Circle Areas
  • 5. Erehwon Coinage
  • 6. Adult tickets & Child tickets
  • 7. Coffee Mixture Problem
  • 8. Coin problem workbench
  • 9. Meeting Up - on the road with Red & Blue
  • 10. Recovering from a long lunch
  • 11. combining rates
  • 12. A puzzling problem
  • 13. The Sliding Ladder Problem
  • 14. The Princess and the Key
  • 15. Horse on road and field
  • 16. Two Planets & A Sun - A Digital Triptych
  • 17. planetary orbits in two frames
  • 18. Keeping Pace - A Balanced Assessment Task
  • 19. Crowded-ness

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'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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