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

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Linear ExplorationsTeaching High School
Mathematics Beautiful Lessons Found on the
Scenic Route

• Dan Teague
• NC School of Science and Mathematics
• teague_at_ncssm.edu

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Linear Equation in Standard Form
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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?

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Arithmetic Progression

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

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Student Responses
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Read the Equation

• How many ks are there on the right side of the
  equation?
• How many as are there on the right side of the
  equation?

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Arithmetic Progression

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

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Think like a mathematician

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

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Change function structure

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

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Change structure of coefficients

• What about a geometric progression?

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What explains this graph?
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Look for the Boundary Curve
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Solve the System?
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Quadratic in k
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Generalize and Expand
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Or Generalize into Space
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Pythagorean Coefficients
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Linear Explorations.

• Dan Teague
• NC School of Science and Mathematics
• teague_at_ncssm.edu