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Math for America San Diego
Preparing for Transformational Geometry in High School
A Middle School IntroductionGenevieve Esmende & Kathleen Barry
Dr. Osvaldo Soto
Who we areMath for America San Diego – Noyce Program
Genevieve EsmendeNoyce Master Teaching [email protected]
Kathleen BarryNoyce Teaching [email protected]
Dr. Osvaldo SotoSenior Program [email protected]
How do we know these statements are true?
1. Vertical angles are congruent.
2. In an isosceles triangle, the two base angles are congruent.
Framing our Perspective
Harel, G. (2014). Common Core State Standards for Geometry: An Alternative Approach. Notices of the AMs, 61(1).
Questions
• What do these theorems have to do with transformational geometry?
• Why do we have to teach transformations?
• Why is transformational geometry in the standards?
• What’s the goal?
Aimlessness
We felt aimless with the given curriculum for transformations.• Why are we teaching them at all?• How do we transition from empirical to deductive
(from middle school to high school)?– Exploring transformation and writing rules on a coordinate
plane
Examples …
Middle School Textbook
High School TextbookNon Common Core
High School TextbookCommon Core
Reflection Lesson
• Exploration with the Mira.
• Played a game with their partner: Guess where I placed the line of reflection.
GSP Activity
Find my line of reflection.
Describe how you would accurately find the line of reflection.
You can draw at most two segments to help you.
Common Core Geometry Standards Middle School
Understand congruence and similarity using physical models, transparencies, or geometry software.
2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Middle School Transformation Activity
Find the lines of reflection to create the image from the given pre-image.
Compare with others. Did you draw your lines of reflection in the same place?
What do you notice about your lines in relation to your pre-image and image?
Student Work
Student Work
Student Work
Student Work
Properties of Rigid Transformations
Reflection, rotation, and translation …• Map lines to lines• Map parallel lines to parallel lines
• Preserve lengths of line segments• Preserve measures of angles
Instructional Principle: Experience before Label
Definition of Reflection
Definition of Translation
Definition of Rotation
Prove Geometry Theorems Using Transformations
1. Vertical angles are congruent.
Prove Geometry Theorems Using Transformations
1. Vertical angles are congruent. With a rotation
Prove Geometry Theorems Using Transformations
1. Vertical angles are congruent. With a reflection
Common Core Geometry Standards High School
Understand congruence in terms of rigid motions.
(Build on rigid motions as a familiar starting point for development of concept of geometry proof.)
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
High School Transformation Lesson
Use transformations to prove SAS.
Prove Geometry Theorems Using Transformations
2. In an isosceles triangle, the two base angles are congruent.
Take two and call us in the morning…
Thank you!
Genevieve EsmendeNoyce Master Teaching [email protected]
Kathleen BarryNoyce Teaching [email protected]
Dr. Osvaldo SotoSenior Program [email protected]
References
Cuoco, A. (2013). Congruence and Transformations. In Integrated CME project (p. 625). Boston, Mass.: Pearson.
Harel, G. (2014). Common Core State Standards for Geometry: An Alternative Approach. Notices of the AMs, 61(1).
Larson, R., Boswell, L., & Learning, L. (2012). Transformations. In Big ideas math: A common core curriculum. Erie, PA: Big Ideas Learning.
Yong, Darryl. Transformational Geometry: What’s New in the CCSS-M [Powerpoint slides].
Vu, Trang. Proving SAS Triangle Congruence using Transformations.
