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Linear Explorations Teaching High School Mathematics: Beautiful Lessons Found on the Scenic Route

Linear Explorations Teaching High School Mathematics: Beautiful Lessons Found on the Scenic Route. Dan Teague NC School of Science and Mathematics teague@ncssm.edu. Linear Equation in Standard Form. Family of Functions. What can we say about the family of linear equations in standard form

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Linear Explorations Teaching High School Mathematics: Beautiful Lessons Found on the Scenic Route

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  1. Linear ExplorationsTeaching High School Mathematics: Beautiful Lessons Found on the Scenic Route Dan Teague NC School of Science and Mathematics teague@ncssm.edu

  2. Linear Equation in Standard Form

  3. Family of Functions What can we say about the family of linear equations in standard form whose coefficients a, b, and c are in arithmetic progression?

  4. Arithmetic Progression All lines appear to intersect at the point (-1, 2). Can we prove this?

  5. Student Responses

  6. Read the Equation • How many k’s are there on the right side of the equation? • How many a’s are there on the right side of the equation?

  7. Arithmetic Progression Sure enough, the point (-1, 2) must lie of all lines in this family.

  8. Think like a mathematician We have seen an interesting result and we have a proof that convinces us our observations were correct. Now, modify the problem. Change the conditions and ask “what other interesting results can be found?”

  9. Change function structure • What would we see if we graph these families of equations with a, b, and c in arithmetic progression.

  10. Change structure of coefficients • What about a geometric progression?

  11. What explains this graph?

  12. Look for the Boundary Curve

  13. Solve the System?

  14. Quadratic in k

  15. Generalize and Expand

  16. Or Generalize into Space

  17. “Pythagorean” Coefficients

  18. Linear Explorations. Dan Teague NC School of Science and Mathematics teague@ncssm.edu

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