Transcript M - Texas A&M University

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 nanopatterning and geometric patterns?
• Analysis of nano-patterning through near
field effects with femtosecond and
nanosecond lasers on semiconducting and
metallic targets
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=
( 0, 0)
MC=
1 unit = 10 meters
20m
C
A
25m
B
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°
25
Tan θ =
20
θ = 51.3°
( 0, 0)
90° - 51.3° = 38.7°
1 unit = 10 meters
20m
C
B
A ( 0,100 )
E ( 0,70 )
B ( 45,120 )
F ( 67.4,70 )
θ
C ( 0,120 )
θ
Length of Sides
AB = 49.2 m
30m
A
45m
Coordinates
E
67.4m
F
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 =
θ = 66°
45
20
90° - 66° = 24°
( 0, 0)
Tan 66° =
x
30
x = 67.4m
30
hyp
Hyp =73.8 m
Cos 66° =
1 unit = 10 meters
B
Coordinates
45m
C
30m
θ
A ( 90,90 ) E ( 90,30 ) M (45,0)
B ( 45,120 ) F ( 0,30 )
C ( 90,120 )
Length of Sides
A
θ
BC = 45 m
a² + b² = c²
AC = 30 m
45² + 30² = c²
AB = 54.1 m
60m
30m
90m
N
( 0, 0)
45m
M
c = 54.1 m
AE = 60 m
a² + b² = c²
EF = 90 m
60² + 90² = c²
AF = 108.1 m
F
N (0,0)
c = 108.1 m
E
NF = 30 m
a² + b² = c²
NM = 45 m
30² + 45² = c²
MF = 54.1 m
c = 54.1 m
Measurements of Angles
m  BAC = 56.3°
m  FAE = 56.3°
Tan 56.3° =
m  B = 33.7°
m  F = 33.7°
X =73.8 m
m C = 90°
m  E = 90°
Tan θ = 45
Tan 56.3° =
θ = 56.3°
Y = 60m
30
90° - 56.3° = 33.7°
90
Y
X
30
1 unit = 10 meters
5m
C
Coordinates
10m
30m
N 30m
θ θ
C ( 40,120 )
10m
60m
11.2m
A ( 40,100 ) E( 40,110 ) N (10,110)
B ( 50,120 ) F( 30,110 ) M (10,40)
F E
A
B
Length of Sides
BC = 10 m
a² + b² = c²
AC = 20 m
10² + 20² = c²
AB = 31.6 m
AE = 10 m
EF =
M
5m
AF = 11.2 m
NF = 30 m
( 0, 0)
c = 31.6 m
a² + b² = c²
10² + 5² = c²
c = 11.2 m
a² + b² = c²
NM = 60 m
30² + 60² = c²
MF = 67.1 m
c = 67.1 m
Measurements of Angles
m  BAC = 26.6°
m  FAE = 26.6°
Tan 26.6° =
m  B = 63.4°
m  F = 63.4°
Y = 59.9 m
m  C = 90°
m  E = 90°
Tan θ =
10
20
θ = 26.6°
Tan 26.6° =
30
Y
X
10
X=5m
90° - 26.6 ° = 63.4 °
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 Plan
Day
1
2
Topic


Instructional Activities
8-2
Trigonometric
Ratios


Students work independently to solve
trigonometric ratios
8-3 Solving
Right Triangles

Power Point Presentation on Solving Right
Triangles


3
4


Demonstration
of Fundamental
Trigonometric
Ratios
Solve
Intermediate
Trigonometric
Ratios
Power Point Presentation on
Trigonometric Ratios
TAKS
TEK
S
Objectives
6,7 and 8
G11.C
Objectives
6,7 and 8
G11.C
Objectives
6,7 and 8
G11.C
Resources




Holt Geometry
TI 84 Calculators
Holt Geometry
TI 84 Calculators
Students work independently to solve
right triangles
Pre-Test

World Cup Goals Clip

TI 84 Calculators

“Reaching Your Goal” Power Point

Demonstration of “Reaching Your Goal”

World Cup Video Clip


Begin “Reaching Your Goal” Activity
Sheet


World Cup Field
Demonstration
Protractor
Laser

Students work independently

Finish “Reaching Your Goal” Activity
Sheet

TI 84 Calculators


World Cup Field
Demonstration
Protractor
Laser

Teacher will monitor student success on
solving trig ratios and then allow student
to score their goal on demonstration field.

Students work with partners
Objectives
6,7 and 8
G11.C

Instructional Plan
Day
5
6
7
Topic



Solve
Complex
Trigonometr
ic Ratios
Solve
Complex
Trigonometr
ic Ratios
Trigonometr
ic Ratios

Instructional Activities
TAKS
TEKS
Multiple Reflection Goal (reflections
using mirrors)
Objective
s 6,7 and
8
G11.C

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.

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.

Post-Test
Objectives
6,7 and 8
Objective
s 6,7 and
8
G11.C
G11.C
Resources

TI 84 Calculators


World Cup Field
Demonstration
Protractor
Laser

TI 84 Calculators


World Cup Field
Demonstration
Protractor
Laser

TI 84 Calculators


World Cup Field
Demonstration
Protractor
Laser



Pre-Test / Post-Test
Sample Question
1. Find the length of CB. Round to the nearest tenth.
miles
A. 37.0 miles
B. 16.2 miles
C. 6.1 miles
D. 68.0 miles
Pre-Test / Post-Test
Sample Question
Exit Level Spring 2009
TAKS Test
•
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