Download Hyperbolic Geometry

7.1
WELCOME TO COMMON CORE HIGH SCHOOL
MATHEMATICS LEADERSHIP
SUMMER INSTITUTE 2015
SESSION 7 • 30 JUNE 2015
MODELING USING SIMILARITY; COPYING AND
BISECTING ANGLES; NON-EUCLIDEAN GEOMETRIES
7.2
TODAY’S AGENDA
 Homework Review and discussion
 G8 M3 L12: Modeling Using Similarity

Discussion
 Break
 G10 M1 L3: Copy and Bisect an Angle

Discussion
 Break
 Spherical geometry
 Lunch
 The parallel postulate and other axioms for geometry
 Daily journal
 Homework and closing remarks
7.3
ACTIVITY 1
HOMEWORK REVIEW AND DISCUSSION
7.4
ACTIVITY 1
HOMEWORK REVIEW AND DISCUSSION
Table discussion:
 Compare your answers to the “Extending the mathematics” prompt from our
last session.
 Identify common themes, as well as points of disagreement, in your
responses to the “Reflection on teaching” prompt.
7.5
ACTIVITY 2
MODELING USING SIMILARITY
ENGAGENY/COMMON CORE GRADE 8 MODULE 3, LESSON 12
7.6
ACTIVITY 2
MODELING USING SIMILARITY
Connecting the lesson goals with pedagogy
1.
Establish mathematics goals to focus learning
2.
Implement tasks that promote reasoning and
problem solving
Teaching Practices did you see modeled in
this lesson?
3.
Use and connect mathematical representations
4.
Facilitate meaningful mathematical discourse
5.
Pose purposeful questions
 How did the use of those practices support
6.
progress towards the lesson’s stated goals
(learning intentions and success criteria)?
Build procedural fluency from conceptual
understanding
7.
Support productive struggle in learning
mathematics
8.
Elicit and use evidence of student thinking
 Which of the Effective Mathematics
Break
7.8
ACTIVITY 3
COPYING AND BISECTING ANGLES
ENGAGENY/COMMON CORE GRADE 10 MODULE 1, LESSON 3
7.9
ACTIVITY 3
COPYING AND BISECTING ANGLES
Connecting the lesson goals with pedagogy
1.
Establish mathematics goals to focus learning
2.
Implement tasks that promote reasoning and
problem solving
Teaching Practices did you see modeled in
this lesson?
3.
Use and connect mathematical representations
4.
Facilitate meaningful mathematical discourse
5.
Pose purposeful questions
 How did the use of those practices support
6.
progress towards the lesson’s stated goals
(learning intentions and success criteria)?
Build procedural fluency from conceptual
understanding
7.
Support productive struggle in learning
mathematics
8.
Elicit and use evidence of student thinking
 Which of the Effective Mathematics
Break
7.11
ACTIVITY 4
SPHERICAL GEOMETRY
Session goals
 To explore an important non-Euclidean geometry
 To understand and prove the Triangle Angle Sum Theorem for spherical geometry
 To understand the importance of the Euclidean parallel postulate.
7.12
ACTIVITY 4
SPHERICAL GEOMETRY
Turn and talk:
 What is a “line”?
7.13
ACTIVITY 4
SPHERICAL GEOMETRY
Turn and talk:
 What is a “line” on a sphere?
 What is a “triangle” on a sphere?
Note: we are talking about a spherical surface, not a solid ball.
7.14
ACTIVITY 4
SPHERICAL GEOMETRY
Turn and talk:
On a sphere,
 Given two distinct points, is there a unique line that contains those two points?
(Do two points always determine a line?)
 Given a line L, and a point P not on L, is there a unique line passing through P
and parallel to L?
7.15
ACTIVITY 4
SPHERICAL GEOMETRY
Exploring angle sums of triangles on a sphere
 Using twine, or otherwise, construct a triangle on your beachball by
constructing three great circles.
 Measure the three angles of your triangle. Compare with your neighbouring
groups. What pattern(s) do you see?
7.16
ACTIVITY 4
SPHERICAL GEOMETRY
Exploring angle sums of triangles on a sphere
 Using twine, or otherwise, construct a triangle on your beachball by
constructing three great circles.
 Measure the three angles of your triangle. Compare with your neighbouring
groups. What pattern(s) do you see?
7.17
ACTIVITY 4
SPHERICAL GEOMETRY
The area of a lune
 A (spherical) lune is a region bounded by two
great circles on a sphere.
 If the angle between the great circles is
θradians, what is the area of the lune?
Hint: start with a “nice” value for θ, such as π/2,
π, or 2π, and recall the formula for the surface
area of a sphere.
7.18
ACTIVITY 4
SPHERICAL GEOMETRY
The area of a lune
 A (spherical) lune is a region bounded by
two great circles on a sphere.
 If the angle between the two great circles is
θradians, the area of the lune is 2R2θ
(where R is the radius of the sphere)
(From now on, we will take R = 1.)
7.19
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
 Any spherical triangle is the intersection of 3
lunes.
 The antipodal triangle is the intersection of the
3 antipodal lunes.
(Where are the lunes and the antipodal triangle
on your beachball?)
Picture credit:
http://www.uwosh.edu/faculty_staff/szydliks/elli
ptic/elliptic.htm
7.20
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
There are useful pictures and animations of the situation on these websites:
http://www.uwosh.edu/faculty_staff/szydliks/elliptic/elliptic.htm
http://math.rice.edu/~pcmi/sphere/gos4.html#1
7.21
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
 Denote the original spherical triangle by T, and the antipodal triangle by T’.
 There are 6 lunes on your beachball (3 lunes whose intersection is T, and 3 antipodal
lunes). For a given point P on the sphere,
 If P is in the triangle T, how many lunes contain P?
 If P is in the triangle T’, how many lunescontain P?
 If P is in neither T nor T’, how many lunes contain P?
7.22
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
Are you convinced?
 The 6 lunes cover the entire sphere without overlapping, except that the
triangles T and T’ are each covered 3 times.
7.23
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
Now add up the area of the 6 lunes in 2 different ways. (Notation: L1, L2 and L3
are the 3 original lunes; primes denote their antipodal lunes.)
But the area of T’ is equal to the area of T, and the area of a lune is equal to the
area of the antipodal lune, so
7.24
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
Since we are taking R = 1, the area of each lune is twice the radian measure of
the angle between the great circles forming the lune; i.e. twice the measure of
the corresponding angle of T. If the three angles of T measure α, β, and
γradians, then,
This is Girard’s theorem: the angle sum of a spherical triangle (on a sphere of
radius 1) is equal to π + the area of the triangle.
7.25
ACTIVITY 4
SPHERICAL GEOMETRY
The Spherical Triangle Angle Sum Theorem (Girard’s Theorem)
Exercise:
Show that on a sphere of radius R, the angle sum of a triangle T is
7.26
ACTIVITY 4
SPHERICAL GEOMETRY
Hyperbolic Geometry
Hyperbolic geometry is the geometry of a “saddle surface,” a portion of which is
shown in the center figure below.
Picture credit: http://www.thephysicsmill.com/2013/03/17/for-there-we-are-captured-the-geometry-of-spacetime/
7.27
ACTIVITY 4
SPHERICAL GEOMETRY
Hyperbolic Geometry
Hyperbolic geometry is in many ways the “opposite” of spherical geometry. For
example:
 Given any line L and any point P, there are infinitely many lines through P that
are parallel to L.
 The angle sum of a hyperbolic triangle is always less than π radians, and the
difference from π is proportional to the area of the triangle.
7.28
ACTIVITY 4
SPHERICAL GEOMETRY
Hyperbolic Geometry
Hyperbolic geometry has many important applications, but most people first
encounter it through the prints of M.C. Escher, for example the famous Angels
and Devils:
https://www.youtube.com/watch?v=QnSIWe_o15g
7.29
ACTIVITY 4
SPHERICAL GEOMETRY
What aspects of teaching and learning geometry did our work with spherical and
hyperbolic geometry illuminate for you?
What aspects of this work might be useful in your classroom teaching?
Lunch
7.31
ACTIVITY 5
THE PARALLEL POSTULATE AND OTHER AXIOMS FOR GEOMETRY
Turn and talk:
 What is an “axiom”?
 How do axioms differ from definitions?
7.32
ACTIVITY 5
THE PARALLEL POSTULATE AND OTHER AXIOMS FOR GEOMETRY
Euclid’s 5 postulates (axioms) - modified
 Postulate 1 Given any two points, there is a unique line which contains both of them.
 Postulate 2 A line segment can be extended indefinitely in either direction.
 Postulate 3 A circle can be drawn with any given center and radius.
 Postulate 4 All right angles are congruent.
 Postulate 5 Given any line, and any point not on the line, there is a unique line through the
given point and parallel to the given line.
(For the original versions, and the rest of Euclid’s Elements, see
http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html)
7.33
ACTIVITY 5
THE PARALLEL POSTULATE AND OTHER AXIOMS FOR GEOMETRY
Turn and talk:
 What are Euclid’s axioms saying?
 Why are they axioms, and not definitions?
 Which of them are true in spherical geometry? In hyperbolic geometry?
7.34
ACTIVITY 5
DAILY JOURNAL
7.35
ACTIVITY 5
DAILY JOURNAL
Take a few moments to reflect and write on today’s activities.
7.36
ACTIVITY 6
HOMEWORK AND CLOSING REMARKS
 G8 M3 L12 handout: Page 9, Question (3); EngageNY G10 M1 L3: Exit ticket.
 Extending the mathematics:
Any proof of the Pythagorean theorem that we have seen relies at some point
on the (Euclidean) triangle sum theorem. If that theorem is not true in
Spherical geometry, what happens to the Pythagorean Theorem? (Do
squares—or rectangles—exist in spherical geometry?)
 Reflecting on teaching:
Spherical geometry is not a standard topic in high school geometry, and it is
not mentioned in the Common Core standards. Do you think your students
could benefit from exposure to this topic? If so, how? (If not, why not?)