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"A series for teaching mathematics."--P. [1] of cover.
How do you refute the erroneous claim that all ratios are fractions? This book goes beyond a simple introduction to ratios, proportions, and proportional reasoning. It will help broaden and deepen your mathematical understanding of one of the most challenging topics for students-and teachers-to grasp. It will help you engage your students, anticipate their perplexities, help them avoid pitfalls, and dispel misconceptions. You will also learn to develop appropriate tasks, techniques, and tools for assessing your students' understanding of the topic.
Focuses on the specialized pedagogical content knowledge that you need to teach ratios and proportions 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 ratios and proportions.
Why do some equations have one solution, others two or even more solutions and some no solutions? Why do we sometimes need to ""switch"" the direction of an inequality symbol in solving an inequality? What could you say if a student described a function as an equation? How much do you know...and how much do you need to know? Helping your students develop a robust understanding of expressions, equations and functions requires that you understand this mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about expressions, equations and functions. It is organised around five big ideas, supported by multiple smaller, interconnected ideas - essential understandings. Taking you beyond a simple introduction to expressions, equations and functions, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students - and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls and dispel misconceptions. You will also learn to develop appropriate tasks, techniques and tools for assessing students' understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently.
How does working with data in statistics differ from working with numbers in mathematics? What is variability, and how can we describe and measure it? How can we display distributions of quantitative or categorical data? What is a data sample, and how can we choose one that will allow us to draw valid conclusions from data? How much do you know? and how much do you need to know? Helping your students develop a robust understanding of statistics requires that you understand fundamental statistical concepts deeply. But what does that mean? This book focuses on essential knowledge for mathematics teachers about statistics. It is organised around four big ideas, supported by multiple smaller, interconnected ideas. Taking you beyond a simple introduction to statistics, the book will broaden and deepen your understanding of one of the most challenging topics for students and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. You will also learn to develop appropriate tasks, techniques, and tools for assessing students’ understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently.
Although proportional reasoning is not formally introduced as a topic in the Common Core and other mathematics curricula until 6th grade, introducing its fundamental ideas in the early grades helps students develop essential skills in ratios, percentages, and other proportional representations when they reach the upper grades. The author takes this complex subject and crafts examples and questions that help teachers see the larger purpose in teaching concepts, such as unitizing, and how that understanding is essential for more complex ideas, such as ratios. Teachers and vertical teams can see how the concepts can build year after year. This new resource by well-known professional developer Marian Small suggests questions that are both interesting for students and useful for providing diagnostic information to teachers. Chapters are organized by grade level (K-8) around the Common Core State Standards for Mathematics to help teachers use the resource more easily.
What is the relationship between fractions and rational numbers? Can you explain why the product of two fractions between 0 and 1 is less than either factor? How are rational numbers related to irrational numbers, which your students will study in later grades? How much do you know… and how much do you need to know? Helping your upper elementary school students develop a robust understanding of rational numbers requires that you understand this mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about rational numbers. It is organised around four big ideas, supported by multiple smaller, interconnected ideas-essential understandings. Taking you beyond a simple introduction to rational numbers, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls and dispel misconceptions. You will also learn to develop appropriate tasks, techniques and tools for assessing students’ understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently.
Why are there so many formulas for area and volume, and why do some of them look alike? Why does one quadrilateral have no special name while another has several, like square, rectangle, rhombus, and parallelogram—and why are all these names useful? How much do you know … and how much do you need to know? Helping your students develop a robust understanding of geometry requires that you understand this mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about geometry. It is organized around four big ideas, supported by multiple smaller, interconnected ideas—essential understandings. Taking you beyond a simple introduction to geometry, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students—and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. You will also learn to develop appropriate tasks, techniques, and tools for assessing students’ understanding of the topic.
Like algebra at any level, early algebra is a way to explore, analyse, represent and generalise mathematical ideas and relationships. This book shows that children can and do engage in generalising about numbers and operations as their mathematical experiences expand. The authors identify and examine five big ideas and associated essential understandings for developing algebraic thinking in grades 3-5. The big ideas relate to the fundamental properties of number and operations, the use of the equals sign to represent equivalence, variables as efficient tools for representing mathematical ideas, quantitative reasoning as a way to understand mathematical relationships and functional thinking to generalise relationships between covarying quantities. The book examines challenges in teaching, learning and assessment and is interspersed with questions for teachers’ reflection.
This resource offers a groundbreaking effort to make mathematics education research on ratios and proportions readily accessible and understandable to preservice and in-service teachers of grades 6 to 8. Using extensive annotated samples of student work and based on research gathered in the Ongoing Assessment Project (OGAP), A Focus on Ratios and Proportions teaches readers how students develop understanding and fluency involving ratio and proportion concepts. Special features include: A close focus on student work, including 150+ annotated pieces of student work, to help teachers improve their ability to recognize, assess and monitor their students’ errors and misconceptions, as well as their developing conceptual understanding. A focus on the OGAP Ratios and Proportions Progression, based on research conducted with hundreds of teachers and thousands of pieces of student work. Sections on how Common Core State Standards for Math (CCSSM) are supported by math education research. Student work samples and vignettes to illuminate the research, as well as end of chapter Looking Back questions and Instructional Links, which allow teachers to analyze evidence of student thinking and strategies and consider instructional responses. An accompanying eResource, available online, offers an answer key as well as extensive explanation of the Looking Back questions. Like A Focus on Multiplication and Division and A Focus on Fractions, this book is designed to bridge the gap between what math education researchers know and what teachers need to know in order to better understand evidence in student work and make effective instructional decisions.