In this introduction to polygons, a triangle convinces a shapeshifter to make him a quadrilateral and later a pentagon, but discovers that where angles and sides are concerned, more isn't always better.
Learn about the specialized pedagogical content knowledge you need to teach geometry effectively in grades 6-8. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with geometry-not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts.
Geometry, of all the branches of mathematics, is the one that is most easily visualized by making something. However, it is all too easy to reduce it to reams of formulas to memorize and proofs to replicate. This book aims to take geometry back to its practical roots with 3D printed models and puzzles as well as demonstrations with household objects like flashlights and paper towel tubes. This is not a traditional geometry textbook, but rather builds up understanding of geometry concepts encountered primarily in middle school while also bringing in elements of concepts normally learned much later. Some of the models are counterintuitive, and figuring out how and why they work will both entertain and give insights. Two final chapters suggesting open-ended projects in astronomy and physics, and art and architecture, allow for deeper understanding and integration of the learning in the rest of the book.
Do your students suppose that 1/3 is greater than 1/2, since 3 is greater than 2? Do they believe that having “halves” means having two, and only two, congruent “pieces” of a whole? What tasks can you offer—what questions can you ask—to determine what your students know or don’t know—and move them forward in their thinking? This book focuses on the specialised pedagogical content knowledge that you need to teach fractions effectively in grades 3–5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with fractions—not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of fractions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning.
The author presents his advice and practice routines for composing and capturing photographs of people, illustrated with examples from his own practice sessions and from his extensive work photographing weddings around the world.
A variety of stimulating, curriculum-correlated activities help learners succeed in the 2nd grade math classroom, and teacher support makes it easy to implement mathematics standards. Geometry offers narrow focus on the concepts and skills that help develop a strong foundation in mathematics. Valuable pre- and post-assessments aid teachers in individualizing instruction, diagnosing the areas where students are struggling, and measuring achievement.
Architectural practices worldwide have to deal with increasingly complex design requirements. How do practices acquire the ability to do so? The Changing Shape of Practice provides a handbook of examples for practices that wish to integrate more research into their work and a reference book for students that seek to prepare themselves for the changing shape of practice in architecture. It addresses the increasing integration of research undertaken in architectural practices of different sizes ranging from small to very large practices from the UK, USA, Europe and Asia. The book is organized according to the size of the practices which is significant in that it addresses the different structures and resourcing requirements that are enabled by specific practice sizes, as this determines and constrains the type, scope and modes of research available to a given practice. The practices covered include: Woods Bagot Perkins + Will White AECOM UN Studio Shop Architects PLP Architecture Kieran Timberlake 3XN ONL AZPML Thomas Herzog + Partners Herreros Arquitectos Spacescape OCEAN Design Research Association By taking stock of the current shape of practice, the book provides essential information for professional architects who are integrating research into their practice.
This unique volume surveys recent research on spatial visualization in mathematics in the fields of cognitive psychology and mathematics education. The general topic of spatial skill and mathematics has a long research tradition, but has been gaining attention in recent years, although much of this research happens in disconnected subfields. This volume aims to promote interaction between researchers, not only to provide a more comprehensive view of spatial visualization and mathematics, but also to stimulate innovative new directions in research based on a more coordinated effort. It features ten chapters authored by leading researchers in cognitive psychology and mathematics education, as well as includes dynamic commentaries by mathematics education researchers on cognitive psychology chapters, and by cognitive psychologists on mathematics education chapters. Among the topics included: From intuitive spatial measurement to understanding of units. Spatial reasoning: a critical problem-solving tool in children’s mathematics strategy tool-kit. What processes underlie the relation between spatial skill and mathematics? Learning with and from drawing in early years geometry. Communication of visual information and complexity of reasoning by mathematically talented students. Visualizing Mathematics makes substantial progress in understanding the role of spatial reasoning in mathematical thought and in connecting various subfields of research. It promises to make an impact among psychologists, education scholars, and mathematics educators in the convergence of psychology and education.