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

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

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

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

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

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

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

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

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)

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

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?

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

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

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)

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

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

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

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

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

Area of Trapezoid: For this trapezoid: Area of this trapezoid:

Area of Triangle= A1+A2-A3

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?

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

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

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?

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

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.

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.

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

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?

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)

Cubic Filters for Iris Recognition

Edge Detection/ De-blurring

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?

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

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