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 ENGAGE NY /COMMON CORE GRADE 8 MODULE 3, LESSON 12

7.6 ACTIVITY 2 MODELING USING SIMILARITY Connecting the lesson goals with pedagogy  Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?  How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)? 1. Establish mathematics goals to focus learning 2. Implement tasks that promote reasoning and problem solving 3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse 5. Pose purposeful questions 6. Build procedural fluency from conceptual understanding 7. Support productive struggle in learning mathematics 8. Elicit and use evidence of student thinking

Break

7.8 ACTIVITY 3 COPYING AND BISECTING ANGLES ENGAGE NY /COMMON CORE GRADE 10 MODULE 1, LESSON 3

7.9 ACTIVITY 3 COPYING AND BISECTING ANGLES Connecting the lesson goals with pedagogy  Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?  How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)? 1. Establish mathematics goals to focus learning 2. Implement tasks that promote reasoning and problem solving 3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse 5. Pose purposeful questions 6. Build procedural fluency from conceptual understanding 7. Support productive struggle in learning mathematics 8. Elicit and use evidence of student thinking

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 2R 2 θ (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: 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:

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: L 1, L 2 and L 3 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:

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:

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  What is an “axiom”?  How do axioms differ from definitions? ACTIVITY 5 THE PARALLEL POSTULATE AND OTHER AXIOMS FOR GEOMETRY Turn and talk:

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

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 Take a few moments to reflect and write on today’s activities. ACTIVITY 5 DAILY JOURNAL

7.36  G8 M3 L12 handout: Page 9, Question (3); Engage NY 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?) ACTIVITY 6 HOMEWORK AND CLOSING REMARKS