A Gooooooal in Geometry!!

A Gooooooal in Geometry!!Arturo BenitezArturo Benitez

Roosevelt High School Math DepartmentRoosevelt High School Math Department

North East Independent School DistrictNorth East Independent School District

San Antonio, TexasSan Antonio, Texas

Sy-Bor WenSy-Bor Wen

Department of Mechanical Engineering,Department of Mechanical Engineering, Texas A&M University, College StationTexas A&M University, College Station

• The research question is …The research question is …

……how do we control short burst of energy how do we control short burst of energy at close proximity lasers to create nano-at close proximity lasers to create nano-patterning and geometric patterns?patterning and geometric patterns?

• Analysis of nano-patterning through near Analysis of nano-patterning through near field effects with femtosecond and field effects with femtosecond and nanosecond lasers on semiconducting and nanosecond lasers on semiconducting and metallic targetsmetallic targets

1.1. What is the societal need that the research What is the societal need that the research is trying to address?is trying to address?– Make things smaller; optics, electronics, Make things smaller; optics, electronics,

medical, etc.medical, etc.

2.2.What is the bottleneck that lead to the What is the bottleneck that lead to the research?research?– Rayleigh diffraction theorem < Rayleigh diffraction theorem < λλ/2/2– The science at the nano-scale works differently.The science at the nano-scale works differently.

3.3.What is the Research Question?What is the Research Question?– How do we manufacture, design and engineer How do we manufacture, design and engineer

materials in the nano-scale?materials in the nano-scale?

Laser Set-Up

Activity SheetReaching 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).

Coordinates

A ( , )

B ( , )

C ( , )

Length of Sides

AB = ______

BC = ______

AC = ______

Measurement of Angles

M A =

M B =

M C =

1 unit = 10 meters

( 0, 0)

Coordinates

A ( 20,100 )

B ( 45,120 )

C ( 20,120 )

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°

A

BC

32m

25m

20m

20

25

1 unit = 10 meters

( 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


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

A

BC

49.2m

45m20

m

20

45

1 unit = 10 meters

( 0, 0)

67.4m

30m

73.8mθ

θ

E F

30

x

hyp

30

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

A

B C

54.1m

45m

30m

1 unit = 10 meters

( 0, 0)

90m60m

54.1m

θ

θ

EF

108.1m

30m

45m MN

AE = 60 m a² + b² = c²

EF = 90 m 60² + 90² = c²

AF = 108.1 m c = 108.1 m

NF = 30 m a² + b² = c²

NM = 45 m 30² + 45² = c²

MF = 54.1 m c = 54.1 m


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°

30

45Y

90

30

XTan 56.3° =

X =73.8 m

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

A

BC

31.6

m

10m30m

1 unit = 10 meters

( 0, 0)

5m

10m67

.1m

θ θ

EF

11.2m

30m60

m

M

N

AE = 10 m a² + b² = c²

EF = 5 m 10² + 5² = c²

AF = 11.2 m c = 11.2 m

NF = 30 m a² + b² = c²

NM = 60 m 30² + 60² = c²

MF = 67.1 m c = 67.1 m


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 °

20

1010

X

Y

30Tan 26.6° =

Y = 59.9 m

Conclusion : The player’s passes do not conclude in a goal.

Core ElementsIn 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 Plan Day Topic Instructional Activities TAKS TEK

SResources

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


G11.C Holt Geometry TI 84 Calculators

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


G11.C TI 84 Calculators

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


G11.C TI 84 Calculators World Cup Field


Instructional Plan Day Topic Instructional Activities TAKS TEKS Resources

5 Solve Complex Trigonometric Ratios

Multiple Reflection Goal (reflections using mirrors)


Students work in groups of 3 or 4 to solve more complex trigonometric ratios.




6 Solve Complex Trigonometric Ratios

Multiple Reflection Goal (reflections using mirrors)


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


7 Trigonometric Ratios

Post-Test Objectives 6,7 and 8



Pre-Test / Post-TestSample Question

1. 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-TestSample Question

Exit Level Spring 2009 TAKS Test

• The Dwight LookThe Dwight LookCollege of EngineeringCollege of EngineeringTexas A&M UniversityTexas A&M University

• Dr. Robin AutenriethDr. Robin Autenrieth• Dr. Cheryl PageDr. Cheryl Page• Mr. Matthew Pariyothorn Mr. Matthew Pariyothorn

• The NationalThe NationalScience Foundation Science Foundation

• ChevronChevron

• Texas Workforce CommissionTexas Workforce Commission

• Nuclear Power InstituteNuclear Power Institute

A Gooooooal in Geometry!!

Documents

Transcript of A Gooooooal in Geometry!!