Focus on Congruence Mappings

Transcription

1 CAMT Conference for the Advancement of Mathematics Teaching July Susan May Course Program Specialist, Secondary Mathematics Anne Joyoprayitno Course Program Specialist, Secondary Mathematics Session facilitators Susan May Course Program Specialist Online Course Programs Charles A. Dana Center Anne Joyoprayitno Course Program Specialist Online Course Programs Charles A. Dana Center 2 1

2 CAMT The Charles A. Dana Center: What we do Collaborate with local and national organizations Strengthen mathematics and science education Promote college and career readiness Focus on equity and access for all students 3 The Charles A. Dana Center: How we do it Innovation Research & Practice Capacity Building High Academic Standards 4 2

3 CAMT A look at Texas Essential Knowledge and Skills (TEKS): Geometry (6) Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. What do these three approaches look like in your geometry class? Which process standards does deductive reasoning through these three approaches connect to? How? 5 What is a congruence mapping? Reflections, translations and rotations are called congruence mappings. In geometry, a congruence mapping is a movement of a set of points in such a way that the pre-image and the image are congruent. Under a congruence mapping, measurements such as distance, angle measure, and area do not change. Congruence mappings map a figure onto a congruent figure with corresponding sides and angles that are congruent. Congruence mappings also preserve collinearity and betweenness of points. 6 3

4 CAMT Defining reflections Point P is the image of a reflection across line m of point P if and only if the line m is the perpendicular bisector of segment PP. 7 Defining rotations Point P is the image of a rotation about center O by an angle measuring a of point P if and only if P O = PO and m<pop = a. P' P a O 8 4

5 CAMT Parallelograms: Geometry TEKS 6E (6E) The student is expected to prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems. 9 Parallelograms: Geometry TEKS 6E Create a parallelogram by rotating a triangle around a point P. Define your rotation (state the location of P and the angle of rotation). Prove your quadrilateral is a parallelogram. What other properties of parallelograms can you prove using your rotation? 10 5

6 CAMT Isosceles Triangle Theorem: Geometry TEKS 6D (6D) The student is expected to verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. 11 Isosceles Triangle Theorem: A transformational proof 12 6

7 CAMT An interesting problem Draw any figure on a sheet of Patty Paper. Copy the figure somewhere else on the Patty Paper. Consider changing the orientation of the image. Theorem: Any two congruent figures can be mapped onto each other by at most three reflections. 13 An interesting problem You know that figures that have the same size and shape are called congruent figures. Two figures are congruent if there exists a sequence of rigid transformations that maps one figure onto the other figure. In the diagram, ABC is congruent to DEF. 1. Find a sequence of rigid transformations that maps ABC onto DEF. Describe the transformations using correct mathematical language and notation. Show the transformations on the diagram. 2. Can you find a different sequence of rigid transformations that maps ABC onto DEF? 14 7

8 CAMT Want to learn more? Contact us by Susan May Anne Joyoprayitno 15 8