Danielle Dobitsch Honors Presentation April 2, 2015 EXPLORING A CONNECTION BETWEEN TRANSFORMATIONAL GEOMETRY AND MATRICES

 Finding connections between algebra and geometry  Common Core standards emphasize these connections  Big changes in standards by the Common Core in geometry PURPOSE

 Learning goals for what students should be able to do and know at each grade level (K-12)  In place for Mathematics and English Language Arts  Developed by building on standards that were in place in many states  Goal: to create high standards that are consistent across the states  Created to be realistic and practical for the classroom  Emphasize collaboration across states COMMON CORE STATE STANDARDS (CCSS)

 As of June 2014, 43 states have voluntarily adapted and are implementing the standards COMMON CORE STATE STANDARDS (CCSS)

 Greater focus on fewer topics  Connecting topics and thinking across grades  Pursuit of conceptual understanding, procedural fluency, and application KEY SHIFTS IN MATHEMATICS CCSS

 Deepening information on individual topics instead of rushing to cover as many topics as possible within the curriculum  Focusing more deeply on specific topics gives students a better chance to understand  Provides opportunities for a rich understanding of concepts, a well-developed procedural fluency, and time to apply mathematics to problems in and out of the classroom  For example:  In grade 7: Ratios and proportional relationships, and arithmetic of rational numbers  In grade 8: Linear algebra and linear functions KEY SHIFTS - GREATER FOCUS ON FEWER TOPICS

 Mathematics is made up of interconnected topics  Learning is now connected across grades  Students use prior knowledge from previous grades math classes to build on the mathematics they are learning  Each standard is not a new idea, but an extension of previous learning KEY SHIFTS - CONNECTING TOPICS AND THINKING ACROSS GRADES

 Pursuit of conceptual understanding, procedural fluency, and application with equal intensity  Conceptual understanding- learning mathematical ideas other than just facts and procedures, focusing on relationships and reasons  Procedural fluency- ability to use procedures flexibly, accurately, and efficiently  Application- use mathematics in situations that require mathematical reasoning while deepening conceptual understanding and procedural fluency KEY SHIFTS - PURSUIT OF CONCEPTUAL UNDERSTANDING, PROCEDURAL FLUENCY, AND APPLICATION

 Transformational approach to congruence and similarity  Standards begin in 8 th grade, focusing on geometric transformations with emphasis on properties, sequencing and their effect on 2-D figures in a coordinate plane GEOMETRY COMMON CORE SHIFTS

 8.G.A. Understand congruence and similarity using physical models, transparencies, or geometry software.  8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations  8.G.A.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.  8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.  8.G.A.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them 8 TH GRADE GEOMETRY STANDARD

 This content is likely to be new for many teachers  Traditional geometry in mathematics classes has been Euclidean based  Euclidean geometry concepts are not unifying throughout courses  CCSS took this into account when deciding to shift to congruency, symmetry, and similarity – more unifying concepts GEOMETRY COMMON CORE SHIFTS

 Euclidean geometry definition of congruence – two figures are congruent when their corresponding sides and angles are congruent  Transformational geometry definition of congruence – two figures are congruent if you can find an isometry that transforms one shape into the other  Different definitions means change in how to prove many geometric theorems  Transformational proofs are different than Euclidean proofs, but transformational proofs relate more to functions GEOMETRY SHIFT - CONGRUENCE

ISOMETRY (RIGID MOTION)

 M.C. Escher was a Dutch artist who incorporated mathematical depth in his work  Mathematical research for his art – plane-filling  Tiling shapes: “motifs”  Utilized the basic congruence-preserving transformations to produce his tilings- translations, rotations, reflections, and glide reflections.  Discovered tilings of many geometric figures STRIP PATTERNS(FRIEZES) – M.C. ESCHER

1.Use transparency strips to find and mark all symmetries of the five Inca patterns. 2.Which two of the five have the same symmetry group? 3.Try to draw one of these if you can:  A frieze pattern with exactly one reflection symmetry.  A frieze pattern with a glide reflection and at least one rotation.  A frieze pattern with rotation symmetry, and exactly one reflection. FRIEZE GROUP EXPLORATION

crystallographers’ notation SYMMETRY GROUPS OF STRIP TILINGS Symmetry Groups of Strip Tilings Four-Symbol NotationSymmetries Present p111 ● translation p1a1 ● translation ● glide reflection p1m1 ● translation ● horizontal reflection pm11 ● translation ● vertical reflection p112 ● translation ● rotation by 180 degrees pma2 ● translation ● rotation by 180 degrees ● vertical reflection ● glide reflection pmm2 ● translation ● rotation by 180 degrees ● vertical reflection ● horizontal reflection

 Translation  Singly generated  Z p111

 Translation  Glide reflection  Z p1a1

p1m1

pm11

p112

pma2

pmm2

GEOGEBRA STRIP TILINGS

 Lesson designed for a PreCalculus BC class of 32 students  Mostly 11 th grade, some 10 th, one 12 th grade  Common Core Standard addressed: “Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.”  Prerequisite knowledge: multiplying and adding matrices, determinant of 2x2 matrices  Relating the eight types of geometrical transformations and their compositions to their corresponding matrices and multiplications when applied to a figure on a plane  Connects linear algebra, matrices, and geometric transformations LESSON CONNECTING GEOMETRIC TRANSFORMATIONS AND MATRICES

TRANSFORMATION MATRICES

FINDING THESE TRANSFORMATION MATRICES

 Transforming the point [P] = [3 1], using the 8 transformation matrices.  Guided the students using r (0,0) and the equation [P] [T] = [P’]  We looked for the transformation matrix of r (0,0) by thinking about what we can multiply [3 1] by in order to get the P’ matrix of [-3 -1].  Transformation matrix:  similar to the identity matrix  matrix multiplication  only way we could get the newly transformed point matrix is by finding the transformation matrix that is made up of 0’s, 1’s and/or - 1’s based on the way matrix multiplication works. 1 ST ACTIVITY

1 ST ACTIVITY (CONTINUED)

WORKSHEET GIVEN TO STUDENTS

2 ND ACTIVITY

WORKSHEET GIVEN TO STUDENTS

GEOGEBRA

 Exploring transformations in depth gave me a lot more understanding of the meanings of the Common Core shifts  Technology can be useful in exploring a new topic and helping students understand composition of transformations  Seeing how the IC courses connect to the content I will teach made connections more explicit CONCLUSION

ANY QUESTIONS?