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
ax  by  c
Family of Functions
What can we say about the family of linear
equations in standard form
ax  by  c
whose coefficients a, b, and c are in arithmetic
progression?
Arithmetic Progression
All lines appear
to intersect at
the point (-1, 2).
Can we prove
this?
Student Responses
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?
Arithmetic Progression
Sure enough, the
point (-1, 2)
must lie of all
lines in this
family.
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?”
Change function structure
• What would we see
if we graph these
families of
equations with a, b,
and c in arithmetic
progression.
ax  by  c
2
ax  by  c
2
ax   a  k  y  a  2k
2
ax 2   a  k  y  a  2k
ax   a  k  y  a  2k
3
ax 2   a  k  y 2  a  2k
a cos  x    a  k  y  a  2k
2
Change structure of coefficients
• What about a geometric progression?
ax  aky  ak
x  ky  k
2
2
What explains this graph?
Look for the Boundary Curve
Solve the System?
Quadratic in k
x  ky  k
2
k  yk  x  0
2
so
y  y  4x
k
2
2
Generalize and Expand
Or Generalize into Space
ax  by  cz  d
“Pythagorean” Coefficients
Linear Explorations
.
Dan Teague
NC School of Science and Mathematics
teague@ncssm.edu