Geometry introduces geometric reasoning as a method for problem solving. In this course, you will explore the properties of geometric figures such as triangles, quadrilaterals, and other polygons. You will make constructions using pencil and paper, and also dynamic software, and you will practice using mathematical language to express ideas and justify your reasoning. Some important geometric ideas such as symmetry, similarity, and trigonometry will also be examined. Lastly, you will begin to explore the basis of formal mathematical proofs and solid geometry. The course material progresses from more visual, intuitive ways of solving problems to more formal explorations of geometric ideas, properties, and, finally, proofs.
The course consists of 10 sessions, each with a half hour of video programming, problem-solving activities provided online and in a print guide, and interactive activities and demonstrations on the Web. Although each session includes suggested times for how long it may take to complete all of the required activities, these times are approximate. Some activities may take longer. You should allow at least two and a half hours for each session. The 10th session explores ways to apply the concepts of geometry you've learned in K-8 classrooms. You should complete the sessions sequentially.
Session 1: What Is Geometry?
Explore the basics of geometric thinking using rich visualization problems and mathematical language. Use your intuition and visual tools for geometric construction. Reflect on the basic objects of geometry and their representation.
Session 2: Triangles and Quadrilaterals
Learn about the classifications of triangles, their different properties, and relationships between them. Examine concepts such as triangle inequality, triangle rigidity, and side-side-side congruence, and look at the conditions that cause them. Compare how these concepts apply to quadrilaterals. Explore properties of triangles and quadrilaterals through practical applications such as building structures.
Session 3: Polygons
Explore the properties of polygons through puzzles and games; then proceed into a more formal classification of polygons. Look at mathematical definitions more formally, and explore how terms can have different but equivalent definitions.
Session 4: Parallel Lines and Circles
Use dynamic geometry software to construct figures with given characteristics, such as segments that are perpendicular, parallel, or of equal length, and to examine the properties of parallel lines and circles. Look past formal definitions and discover the properties and relationships among geometric figures for yourself.
Session 5: Dissections and Proof
Review and explore transformations such as translation, reflection, and rotation. Apply these ideas to solve more complex geometric problems. Use your knowledge of properties of figures to reason through, solve, and justify your solutions to problems. Analyze and prove the midline theorem.
Session 6: The Pythagorean Theorem
Continue to examine the idea of mathematical proof. Look at several geometric or algebraic proofs of one of the most famous theorems in mathematics: the Pythagorean theorem. Explore different applications of the Pythagorean theorem, such as the distance formula.
Session 7: Symmetry
Investigate symmetry, one of the most important ideas in mathematics. Explore geometric notions of symmetry by creating designs and examining their properties. Investigate line symmetry and rotation symmetry; then learn about frieze patterns.
Session 8: Similarity
Examine your intuitive notions of what makes a "good copy," and then progress toward a more formal definition of similarity. Explore similar triangles, and look into some applications of similar triangles, including trigonometry.
Session 9: Solids
Explore various aspects of solid geometry. Examine Platonic solids and why there are a finite number of them. Investigate nets and cross sections for solids as a way of establishing the relationships between two-dimensional and three-dimensional geometry.
Session 10: Classroom Case Studies
Explore how the concepts developed in this course can be applied at different grade levels through case studies of K-2, 3-5, and 6-8 teachers (former course participants), all of whom have adapted their new knowledge to their classrooms. Select video 10 for K-5 teachers and videos 11 and 12 for 6-8 teachers.