1. A Gooooooal in Geometry!!
Arturo Benitez
Roosevelt High School Math Department
North East Independent School District
San Antonio, Texas
Sy-Bor Wen
Department of Mechanical Engineering,
Texas A&M University, College Station

2. The research question is …
…how do we control short burst of energy at close proximity lasers to create nano-patterning and geometric patterns?
Analysis of nano-patterning through near field effects with femtosecond and nanosecond lasers on semiconducting and metallic targets

3. 2. What is the bottleneck that lead to the research?
1. What is the societal need that the research is trying to address?
Make things smaller; optics, electronics, medical, etc.
2. What is the bottleneck that lead to the research?
Rayleigh diffraction theorem < λ/2
The science at the nano-scale works differently.
3. What is the Research Question?
How do we manufacture, design and engineer materials in the nano-scale?

4. Laser Set-Up

5. Activity Sheet Reaching your Goal: Trigonometric Ratios
Objective: To use trigonometric ratios to find measurement of angles and length of sides; to take a shot at the goal.
Materials:
Pencil
Color pencil
Ruler
Recording Sheets
Calculator
Instructions:
1. Place player on a random coordinate (Cartesian Plane).
2. Identify player by labeling him/her as point A.
3. Draw a line to the center of the goal and label that point B.
4. From point A, draw a perpendicular line segment to the end line closes to the goal and label that point C.
5. Connect point C and point B with a line segment.
6. Identify coordinates A, B, and C on the recording sheet.
7. Find the lengths of the sides AB (Hint: Use Pythagorean Theorem).
8. Find the lengths of the BC (Hint: Use units on graph).
9. Find the lengths of the AC (Hint: Use unit on graph).
10. Find the measurement of angle A. (Hint: Use trigonometric ratios).
11. Find the measurement of angle B. (Hint: Use supplementary angle with A).
12. Find the measurement of angle C. (Hint: Right angle).

6. 1 unit = 10 meters
Coordinates
A ( , )
B ( , )
C ( , )
Length of Sides
AB = ______
BC = ______
AC = ______
Measurement of Angles
M A =
M B =
M C =
( 0, 0)

7. 1 unit = 10 meters 25m C B 20m 32m A ( 0, 0) Coordinates A ( 20,100 )
Length of Sides
AB = 32m a² + b² = c²
BC = 25m ² + 20² = c²
AC = 20m c = 32m
Measurements of Angles
M A = 51.3°
M B = 38.7°
M C = 90°
Tan θ =
θ = 51.3°
90° ° = 38.7°
25m C B
20m
32m A
( 0, 0)

8. 1 unit = 10 meters C 45m B 20m 49.2m θ A θ 73.8m 30m E F 67.4m ( 0, 0)
Coordinates
A ( 0,100 ) E ( 0,70 )
B ( 45,120 ) F ( 67.4,70 )
C ( 0,120 )
Length of Sides
AB = 49.2 m a² + b² = c²
BC = 45 m ² + 20² = c²
AC = 20 m c = 49.2 m
Measurements of Angles
m BAC = 66° m FAE = 66°
m B = 24° m F = 24°
m C = 90° m E = 90°
Tan θ = Tan 66° =
θ = 66° x = 67.4m
90° - 66° = 24° Cos 66° =
Hyp =73.8 m C
45m B
20m
49.2m θ A θ
73.8m
30m E F
67.4m
( 0, 0)

9. 1 unit = 10 meters B 45m C 30m 54.1m θ A θ 108.1m 60m
Coordinates
A ( 90,90 ) E ( 90,30 ) M (45,0)
B ( 45,120 ) F ( 0,30 ) N (0,0)
C ( 90,120 )
Length of Sides
BC = 45 m a² + b² = c²
AC = 30 m 45² + 30² = c²
AB = 54.1 m c = 54.1 m B
45m C
30m
54.1m θ A θ
108.1m
60m
AE = 60 m a² + b² = c²
EF = 90 m ² + 90² = c²
AF = m c = m F E
90m
NF = 30 m a² + b² = c²
NM = 45 m ² + 45² = c²
MF = 54.1 m c = 54.1 m
54.1m
30m N M
( 0, 0)
45m

10. Measurements of Angles
m BAC = 56.3° m FAE = 56.3°
m B = 33.7° m F = 33.7°
m C = 90° m E = 90°
Tan θ = Tan 56.3° =
θ = 56.3° Y = 60m
90° ° = 33.7°
Tan 56.3° =
X =73.8 m

11. 1 unit = 10 meters B C N 30m F E A 60m 67.1m AE = 10 m a² + b² = c²
Coordinates
A ( 40,100 ) E( 40,110 ) N (10,110)
B ( 50,120 ) F( 30,110 ) M (10,40)
C ( 40,120 )
Length of Sides
BC = 10 m a² + b² = c²
AC = 20 m ² + 20² = c²
AB = 31.6 m c = 31.6 m B 5m C
10m
30m N
30m F E
31.6m θ θ
10m A
11.2m
60m
67.1m
AE = 10 m a² + b² = c²
EF = m 10² + 5² = c²
AF = 11.2 m c = 11.2 m M
NF = 30 m a² + b² = c²
NM = 60 m ² + 60² = c²
MF = 67.1 m c = 67.1 m
( 0, 0)

12. Measurements of Angles
m BAC = 26.6° m FAE = 26.6°
m B = 63.4° m F = 63.4°
m C = 90° m E = 90°
Tan θ = Tan 26.6° =
θ = 26.6° X = 5 m
90° ° = 63.4 °
Tan 26.6° =
Y = 59.9 m
Conclusion : The player’s passes do not conclude in a goal.

13. Core Elements
In applying these concepts to high school math courses, we will target Geometry and Advanced Mathematical Decision-Making.
The mathematical core elements translated in this lesson:
graphing on a Cartesian plane
slope of the line
angles of incidence
angles of reflection
trigonometric ratios
sine, cosine, tangent, parallel lines, alternate interior angles, inverse sine and cosine, and inverse tangent.

14. The pertinent TEKS standards which can be associated with this core element are:
(G2) Geometric structure. The student is expected to:
(B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.
(G4) Geometric structure. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.
(G5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to:
(D)identify and apply patterns from right triangles to solve meaningful problems, including special right triangles ( and ) and triangles whose sides are Pythagorean triples.
(G7) Dimensionality and the geometry of location. The student is expected to:
(A) use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures;
(B) use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and
(C) derive and use formulas involving length, slope, and midpoint.
(G8) Congruence and the geometry of size. The student is expected to:
(C) derive, extend, and use the Pythagorean Theorem;
(G11) Similarity and the geometry of shape. The student is expected to:
(C) develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods

15. TAKS Objectives
Objective 6: The student will demonstrate an understanding of geometric relationships and spatial reasoning.
Objective 7: The student will demonstrate an understanding of two- and three-dimensional representations of geometric relationships and shapes.
Objective 8: The student will demonstrate an understanding of the concepts and uses of measurement and similarity.

16. Instructional Activities
Instructional Plan
Day
Topic
Instructional Activities
TAKS
TEKS
Resources 1
8-2 Trigonometric Ratios
Power Point Presentation on Trigonometric Ratios
Students work independently to solve trigonometric ratios
Objectives 6,7 and 8
G11.C
Holt Geometry
TI 84 Calculators 2
8-3 Solving Right Triangles
Power Point Presentation on Solving Right Triangles
Students work independently to solve right triangles
Pre-Test 3
Demonstration of Fundamental Trigonometric Ratios
World Cup Goals Clip
“Reaching Your Goal” Power Point
Demonstration of “Reaching Your Goal”
Begin “Reaching Your Goal” Activity Sheet
Students work independently
World Cup Video Clip
World Cup Field Demonstration
Protractor
Laser 4
Solve Intermediate Trigonometric Ratios
Finish “Reaching Your Goal” Activity Sheet
Teacher will monitor student success on solving trig ratios and then allow student to score their goal on demonstration field.
Students work with partners

17. Instructional Activities
Instructional Plan
Day
Topic
Instructional Activities
TAKS
TEKS
Resources 5
Solve Complex Trigonometr ic Ratios
Multiple Reflection Goal (reflections using mirrors)
Teacher will monitor student success on solving trig ratios and then allow student to score their goal on demonstration field.
Students work in groups of 3 or 4 to solve more complex trigonometric ratios.
Objectives 6,7 and 8
G11.C
TI 84 Calculators
World Cup Field Demonstration
Protractor
Laser 6
Objectives 6,7 and 8 7
Trigonometr ic Ratios
Post-Test

18. Pre-Test / Post-Test Sample Question
Find the length of CB. Round to the nearest tenth.
A miles
B miles
C miles
D miles
miles

19. Pre-Test / Post-Test Sample Question
Exit Level Spring 2009 TAKS Test

20. The Dwight Look College of Engineering Texas A&M University
Dr. Robin Autenrieth
Dr. Cheryl Page
Mr. Matthew Pariyothorn
    The National Science Foundation
  Chevron
  Texas Workforce Commission
  Nuclear Power Institute