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Linear Explorations in High School Mathematics: Seeing Beauty on the Scenic Route

In this engaging book, Dan Teague from NC School of Science and Mathematics delves into the realm of linear equations in standard form, focusing on arithmetic progressions and the intriguing intersections they create. From challenging proofs to changing parameters, students are encouraged to think like mathematicians and explore the possibilities within the linear family of functions.

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Linear Explorations in High School Mathematics: Seeing Beauty 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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