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

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

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?

Laser Set-Up

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).

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)

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 25² + 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° - 51.3° = 38.7° 25m C B 20m 32m A ( 0, 0)

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 5² + 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)

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 60² + 90² = c² AF = 108.1 m c = 108.1 m F E 90m NF = 30 m a² + b² = c² NM = 45 m 30² + 45² = c² MF = 54.1 m c = 54.1 m 54.1m 30m N M ( 0, 0) 45m

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° - 56.3° = 33.7° Tan 56.3° = X =73.8 m

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 10² + 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 = 5 m 10² + 5² = c² AF = 11.2 m c = 11.2 m M NF = 30 m a² + b² = c² NM = 60 m 30² + 60² = c² MF = 67.1 m c = 67.1 m ( 0, 0)

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° - 26.6 ° = 63.4 ° Tan 26.6° = Y = 59.9 m Conclusion : The player’s passes do not conclude in a goal.

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.

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 (45-45-90 and 30-60-90) 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

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.

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

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

Pre-Test / Post-Test Sample Question Find the length of CB. Round to the nearest tenth. A. 37.0 miles B. 16.2 miles C. 6.1 miles D. 68.0 miles miles

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

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