23
Session 8 Proving Geometric Theorems Common Core High School Mathematics: Transforming Instructional Practice for a New Era 8.1
Session 8 Proving Geometric Theorems
Common Core High School Mathematics: Transforming Instructional Practice for a New Era
8.1
Learning Intentions & Success Criteria
Learning Intentions:
We are learning to deepen our understanding of the Common Core State Standards and the implications for teaching and learning mathematics.
Success Criteria:
We will be successful when we can describe how the content standards and math practice standards are evident in the implementation of a mathematical task.
8.2
Agenda
• Homework review and discussion• Proving triangle and circle theorems• Reading G-CO.9-13• Introducing similarity transformations• Break• “Prove that all circles are similar”• Reading G-SRT.1 and G-C.1• Homework and closing remarks
8.3
8.4
Homework Review and DiscussionActivity 1:
Table Discussion:Discuss your write up for the day 7 math task Patty Paper Geometry Open Investigations:
• Compare your strategies with others at your table.
• Reflect on how you might revise your own solution and/or presentation.
8.5
Proving Triangle and Circle Theorems
• Two geometric figures are defined to be congruent if there is a sequence of rigid motions (translations, rotations, reflections, and combinations of these) that carries one onto the other.
Activity 2:
(From the CCSSM Geometry overview)
Use this definition to prove that two line segments are congruent if and only if they have the same length.
8.6
Proving triangle and circle theorems
Theorem 1 (Isosceles triangle symmetry)Let ΔABC be isosceles, with . Then the angle bisector at A is the perpendicular bisector of .
ACAB
BC
Activity 2:
Find proofs of each of the following theorems, using either transformation or traditional methods.
8.7
Proving triangle and circle theoremsActivity 2:
Theorem 2 (SAS triangle congruence criterion)Let ΔABC and ΔDEF be two triangles, , , . Then ΔABC and ΔDEF are congruent.
DEAB DFAC
€
∠A ≅∠D
8.8
Proving triangle and circle theorems
Theorem 3 (Three points determine a circle)Let A, B and C be three non-collinear points. Then there is one and only one circle which contains A, B and C.
Theorem 4 (Circle intersection theorem)Two distinct circles can intersect in at most 2 points. If there is exactly one point of intersection, it lies on the line through the centres of the circles; if there are two intersection points, they are reflections of each other in this line.
Activity 2:
8.9
Proving triangle and circle theoremsActivity 2:
Theorem 5 (SSS triangle congruence criterion)Let ΔABC and ΔDEF be two triangles, with , , and . Then ΔABC and ΔDEF are congruent.
DEAB DFAC EFBC
8.10
Proving triangle and circle theorems
Theorem 6Any angle inscribed in a semicircle is a right angle
Theorem 7A tangent line to a circle is perpendicular to the radius through the point of tangency.
Activity 2:
8.11
Reading G-CO.9-13
• Read these standards from the high school Geometry conceptual category
• Turn and talk:• How do you see these standards in the
activity you have just completed?• How might you prove some of the theorems
mentioned in these standards that we have not covered today?
Activity 3:
8.12
Introducing Similarity Transformations
•With a partner, discuss your definition of a dilation.
Activity 4:
8.13
Introducing Similarity Transformations
• (From the CCSSM glossary) A dilation is a transformation that moves each point along the ray through the point emanating from a common center, and multiplies distances from the center by a common scale factor.
Figure source: http://www.regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm
Activity 4:
8.14
Introducing Similarity Transformations
• Two geometric figures are defined to be congruent if there is a sequence of rigid motions (translations, rotations, reflections, and combinations of these) that carries one onto the other.
• Two geometric figures are defined to be similar if there is a sequence of similarity transformations (rigid motions followed by dilations) that carries one onto the other.
Activity 4:
(From the CCSSM Geometry overview)
8.15
Introducing Similarity Transformations
• Read G-SRT.1
• Discuss how might you have students meet this standard in your classroom?
Activity 4:
Break
8.16
8.17
“Prove that all circles are similar”What Kind of a Standard is That?
•Read G-C.1
•What, precisely, is the student expectation in this standard?
Activity 5:
8.18
“Prove that all circles are similar”What Kind of a Standard is That?
Begin with congruence
• On patty paper, draw two circles that you believe to be congruent.
• Find a rigid motion (or a sequence of rigid motions) that carries one of your circles onto the other.
• How do you know your rigid motion works?
• Can you find a second rigid motion that carries one circle onto the other? If so, how many can you find?
Activity 5:
8.19
“Prove that all circles are similar”What Kind of a Standard is That?
Turning to similarity
•On a piece of paper, draw two circles that are not congruent.
•How can you show that your circles are similar?
Activity 5:
8.20
“Prove that all circles are similar”What Kind of a Standard is That?
• If two circles are congruent, this can be shown with a single translation.
• If two circles are not congruent, we have seen we can show they are similar with a sequence of translations and a dilation.
• Are the separate translations necessary, or can we always find a single dilation that will carry one circle onto the other?
• If so, how would we locate the centre of the dilation?
Activity 5:
8.21
Reading G-SRT.1 and G-C.1
• Read these standards from the high school Geometry conceptual category
• Turn and talk:• How do you see these standards in the
activity you have just completed?• How might you prove some of the theorems
mentioned in these standards that we have not covered today?
Activity 6:
Learning Intentions & Success Criteria
Learning Intentions:
We are learning to deepen our understanding of the Common Core State Standards and the implications for teaching and learning mathematics.
Success Criteria:
We will be successful when we can describe how the content standards and math practice standards are evident in the implementation of a mathematical task.
8.22
8.23
Homework and Closing RemarksActivity 7:
Homework (to be included in journal): • Day 8 Math Task: Tangent Line and Radius• Day 8 Class Reflection
Reading• Geometry, Big Idea 2: Geometry is about
working with variance and invariance, despite appearing to be about theorems
