1. Taking Math Beyond the Classroom: Some high school math fair ideas
Katrina Palmer
Appalachian State University
October 13, 2005

2. Outline Matrices (Algebra II) Hyperbola (Integrated Math IV)
Tomography
Area of a Triangle
Hyperbola (Integrated Math IV)
Geophone

3. Algebra II State Objective 1.04
“Operate with matrices to model and solve problems”
Assumptions:
- Students know what a matrix is (a table of numbers)
- Students know how to add, subtract, mult. matrices
- Students know how to find the determinant and the
inverse of a 2 by 2 matrix

4. What is tomography? yarn demo
“Recover interior structure of a body using external measurements”

5. Why is tomography important?
can’t see inside things
can’t always “cut” your way in
where might it be used?

6. How could you “look inside”
What kind of things (xrays, light, robot, etc) can be sent through?

7. X-ray, CT (CAT) scan, PET scan

8. CT (computed tomography) details
What is sent through the body?
What kind of data is collected?
A detector detects
how much energy is
left in the ray
X-ray contains
a certain amount
of energy
So we know how much was absorbed

9. From this data, how do we create an image?
Mathematically, what is an image?
the values of the matrix are typically between 0 and 255 (where 0 represents black and 255 is white)

10. Mathematical solution: Solve: w + x = 3 y + z = 1 x + z = 2 1 2
sources
Possible Solutions: w x y z 3 2 1 1
detectors 2
Mathematical solution:
Solve: w + x = 3
y + z = 1
x + z = 2 1 2

11. Some Questions Review of solving systems of equations
Given n equations and n unknowns, how many solutions could there be?
How many unknowns do we have?
How many data points should we collect?

12. Small Example w x y z 1. How many solutions are there if a) v = 1
b) v = 2
For what values of v are there no solutions?
For what values of v are there exactly one solution?
For what values of v are there more than one solution?
sources w x y z 3 1
detectors v 2

13. How can mathematicians tell if their program works?
Don’t know what brain looks like.
Create ‘fake’ brain slice (phantom image)
Calculate (‘measure’) the line densities in many directions of this new (known) object
Use program to ‘reconstruct’ from ‘measured’ line sums
Compare reconstruction to original

14. Given the image, find the line sums
Assumptions:
Each square is one unit of length
Density for white squares is 1 (it absorbs1 unit of energy)
Density for black squares is 0 (it doesn’t absorb any energy)

15. Harder Line Sums
density=1.2
Review:
What is the equation of a circle
with radius 8?
What is the distance between two
points, (x1,y1) and (x2, y2)? 5 L A B 3 C D
Line Sum for line L= 1.2*Dist(AB)+0.8*Dist(BC)+1.2*Dist(CD)
density=0.8

16. And Even Harder Line Sums
density=1.2
- No line sums are the same
Skills needed
a) intersection line/ellipse
b) intersection line/circle 5 2 4
density=0.8

17. Projects for Tomography
Create formula(s) for line sums given a pixelated image
Create formula(s) for line sums given a continuous image
Find an automated way to calculate a certain number of line sums by rotating around the image

18. Area of a triangle
Problem: Given 3 points, find the area of the triangle.

19. A2 A1
Area of Triangle= A1+A2-A3 A3

20. Area of Trapezoid:
For this trapezoid:
Area of this trapezoid:

22. Area of Triangle= A1+A2-A3

23. Extensions
What happens when one or more points have a negative component? Does this change the formula?
Does the formula work if you are given 3 colinear points?

24. More Extensions Use Determinant to test is 3 points are collinear
Use Determinant to find the volume of a tetrahedron
Use Determinant to test if 4 points are coplanar

25. Integrated Mathematics IV State Objective 2.01
“Use the quadratic relations (parabola, circle, ellipse, hyperbola) to model and solve problems”

26. Locating Rocks
How can we use math to locate rock under the earth’s surface?
When you locate a spot, how do you know how deep to dig?

27. Assumptions: - homogeneous subsurface
- geophone receives signal after t seconds

28. Some Questions
Which variables do we have in real life and which variables are we looking for?
Write an expression for s in terms of d and X.
Write an expression for the velocity in terms of s and T.
Find the equation of a hyperbola where the two variables are the two we can not realistically measure.

29. QUESTION: A reflected signal is received 91 meters
from the source after 1.1 seconds. The reflected
signal is received at a second geophone 200 meters
away from the source after 2.1 seconds. Find the
depth and the velocity.

30. Solution: The intersection of two hyperbolas
(30.5, 90.6)

31. Extensions
Generalize the solution. Given that the first and second geophones are a distance of X1 and X2 from the source, and the signal is received t1 seconds at the first geophone and after t2 seconds at the second geophone.
Will a solution always exist?

32. Some other interesting topics
Cubic filters (used to increase depth of field for iris recognition)
Edge detection (convolution - matrix/vector multiply)
Cryptography (multiply by a matrix and it’s inverse)

33. Cubic Filters for Iris Recognition

34. Edge Detection/ De-blurring

35. Edge Detection Activities
explore relationship between convolution and matrix-vector multiplication
experiment with edge-detection
Why do some kernels detect edges that go from black to white from left to right while others detect edges that go from white to black from left to right?

36. Cryptography Use a coding matrix A to code a message
Use the inverse of A to decode the message

37. References Tomography Image De-blurring
Area of Triangle & Cryptography
Elementary Linear Algebra, Larson/Edwards/Falvo 5th Edition
Geophone
Inverse Problems, Charles W. Groetsch