PROVING SIMILARITY VIA TRANSFORMATIONS

Dilation is a Non-Rigid Transformation that preserves angle, but involves a scaling factor that affects the distance, which results in images that are similar to the original shape.

G-SRT Cluster Headings dealing with Similarity:

• Understand Similarity in terms of similarity transformations

• Prove theorems involving similarity

PROVING SIMILARITY VIA TRANSFORMATIONS

From a transformational perspective…

Two shapes are defined to be similar to each other if there is a sequence of rigid motions followed by a non-rigid dilation that carries one onto the other.

A dilation formalizes the idea of scale factor studied in Middle School.

PROVE SIMILARITY BY TRANSFORMATIONSWhat non-rigid transformation

proves that these triangles

are similar?

What is the center of dilation?

What is the scale factor of the

Dilation?

FIND SCALE FACTORS GIVEN A TRANSFORMATION

www.ck12.org Similarity Transformations Created by: Jacelyn O'Roark

CIRCLES IN ANALYTIC GEOMETRY

G-GPE (Expressing Geometric Properties with Equations) Derive the equation of a circle given center (3,-2) and radius 6

using the Pythagorean Theorem

Complete the square to find the center and radius of a circle with equation x2 + y2 – 6x – 2y = 26

Think of the time spent in Algebra I on factoringVersus completing the square to solve quadraticEquations. What % of quadratics can be solvedby factoring? What % of quadratics can be Solved by completing the square?Is completing the square using the area modelmore intuitive for students?

CONIC SECTIONS – CIRCLES AND PARABOLAS

• Translate between the geometric description and the equation for a conic section • Derive the equation of a parabola given a focus and directrix• Parabola – Note: completing the square to find the vertex of a parabola is

in the Functions Standards

(+) Ellipses and Hyperbolas in Honors or Year 4

Sketch and derive the equation for the parabola withFocus at (0,2) and directrix at y = -2

Find the vertex of the parabola with equationY = x2 + 5x + 7

VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS

• Identify the shapes of 2-dimensional cross sections of 3-dimensional objects

VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS

• Identify 3-dimensional shapes generated by rotations of 2-dimensional objects

http://www.math.wpi.edu/Course_Materials/MA1022C11/volrev/node1.html

NORTH COUNTRY INSERVICE OUTLINE

• Review with Agreed Upon Expectations from 2-15-13 Inservice – Share Experiences

• Review of CCSSM Practice Standards – Share Experiences

• Presentation of How Geometry Unfolds over K – 12 in CCSSM

• Focus on Volume Standard in HS Geometry

• Develop one unit focusing on HS Volume Standard and Practice Standards

HS.GMD.A.1

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.