Beyond Invert and Multiply: Making Sense of Fraction Computation, Grades 3–6


AAP Pre K–12 Learning 2016 Revere Awards Winner

Beyond Invert & Multiply builds on the foundational understandings that are described in Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, Second Edition, and applies them to situations involving fraction computation. You’ll find:

Classroom Scenarios that address common challenges students (and teachers!) encounter as they learn to operate with fractions;

What’s the Math? sections intended to clarify the mathematics;

What’s the Research? insights that show why “that’s the way we’ve always done things” might not be the most successful approach;

Classroom Activities with friendly, step-by-step instructions and video clips demonstrating how to teach the concepts;

Reproducibles that are also available in a downloadable, printable format;

Study Questions at the end of each chapter that encourage readers to set concrete goals for implementing what they’ve read; and

Connections to the Common Core for those states implementing the Common Core State Standards for Mathematics.

icon pdf How Beyond Pizzas & Pies and Beyond Invert & Multiply work together

Julie McNamara
240 Pages

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Julie McNamara is an Assistant Professor of Mathematics Education at California State University, East Bay in Hayward, California. She has also served as Mathematics Specialist with TeachingWorks at the University of Michigan. Julie spent two years supporting elementary and middle school teachers as a fulltime designer and provider of mathematics professional development for Math Solutions, and continues to consult for Math Solutions part-time. She has also taught mathematics methods in the elementary credential programs at the University of California, Berkeley and at Saint Mary’s College in Moraga, California. She is a former classroom teacher in the San Francisco Bay Area. Her experience in the classroom led her to pursue graduate studies in the Development in Mathematics and Science program at the University of California, Berkeley, and to earn her master’s degree and Ph.D. in mathematics education. Her research focuses on the teaching and learning of mathematics concepts that are foundational but considered "hard to teach and hard to learn."

Julie is co-author of Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, 2010, as well as Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fractions Sense, Second Edition, 2015. In addition, Julie has written a follow up fraction book, Beyond Invert & Multiply: Making Sense of Fraction Computation, Grades 3-6, 2015, all published by Math Solutions.

"Now we can ask why! Beyond Invert & Multiply is a centerpiece for our district’s fraction instruction and professional development. It is the perfect blend of content and pedagogy complemented by meaningful video clips. It is hard to imagine a classroom, school, or district that won’t benefit from such an exceptional resource."

John SanGiovanni, elementary mathematics instructional facilitator
Howard County Public School System, Howard County, Maryland

"Operations with fractions—a thorny topic in elementary mathematics to teach and to learn—is thoroughly demystified in Beyond Invert & Multiply. I've used the videos in professional development sessions; they are indispensable to show how you can present a problem to students and have them respond, knowing that misconceptions will arise."

Josh Rosen, math specialist
Dobbs Ferry, New York

"The bite-sized chapters in Beyond Invert & Multiply provide just the right amount of detail to allow busy practitioners access to big ideas. The activities are engaging and include clips of real classrooms using the tasks so that readers can see them in action!"

Nicole Garcia, Mathematics Research and Design Specialist
TeachingWorks, University of Michigan

"Beyond Invert & Multiply is a playbook for exemplary mathematics instruction! McNamara addresses the gap between teacher content knowledge and pedagogy while supporting educators endeavoring to meet the academic rigor of the Common Core State Standards. Emphasis on developing students' understanding of fractions is the game-changing approach essential for future success in mathematics."

Noncy Fields, fourth-grade teacher
Ypsilanti Community Schools, Ypsilanti, Michigan

"Many teachers are wondering, even after studying the Common Core State Standards for Mathematics carefully, "How am I supposed to teach something I don't yet fully understand?" McNamara has anticipated our situation and designed Beyond Invert & Multiply to lead us, step-by-important step, into greater understanding of the standards and ways we might teach them effectively. I want every teacher and coach to have a copy of this resource and to wear it out: mark it, bend its spine, and mine it for all its riches and classroom videos."

Lori MacDonald, Coach, K–5 Mathematics
Berkeley Unified School District, Berkeley, California

"Using a combination of articulate descriptions, classroom discussion transcripts, diagrams, and videos, Beyond Invert & Multiply delivers clear guidance on difficult mathematics topics. Whatever your learning style, this resource will hit home."

Arjan Khalsa, CEO Conceptua Math

"Another gem from the author of Beyond Pizzas & Pies! Beyond Invert & Multiply emphasizes the importance of constructive struggling—the "golden zone" for teachers and their students. The resource offers a refreshingly concise, straightforward, and accessible approach to building the foundation that all children need in order to be successful with deeper applications of mathematics in middle school and beyond."

Ginny Gillespie, fourth-grade teacher
Malcolm X Elementary, Berkeley, California

"Beyond Invert & Multiply is a 'can’t teach without' resource for fraction instruction. The classroom videos, text, and ready-to-use instructional activities clearly illustrate and explain how foundations of fraction sense and fraction computation are developed and supported through 'best practice.'"

Angela Waltrup, elementary mathematics teacher specialist
Frederick County Public Schools, Frederick County, Maryland

"Beyond Invert & Multiply is exactly what educators need for guidance in implementing the Common Core Standards. The resource helps us clearly focus on making meaning, applying learning to real-life applications, and supporting students in really articulating their thinking—whether it be in small groups, in writing, or to the whole class. Consider Beyond Invert and Multiply your road map for daily best teaching practices."

Jonathan Mayer, Principal
Claremont Middle School, Oakland, California

"Beyond Invert & Multiply offers a perfect balance of relevant background information, research, and activities to help teachers create the opportunities our young mathematicians require in order to develop an understanding of fractions as numbers. The layout of the text along with the succinct videos make this resource well-suited for coming together to learn with colleagues."

Justin Gregory Johns, K–5 mathematics instructional coach
American Embassy School, New Delhi, India

"One reason for the persistent performance gap in mathematics is an over-emphasis on teaching for procedural fluency when it comes to fractions. Beyond Invert & Multiply offers teachers of all students specific research-based strategies for teaching what rational numbers are and how to perform operations using rational numbers in ways that develop conceptual understanding. This understanding will serve students well as they progress through the K–12 mathematics pipeline."

Kyndall Brown, Executive Director
California Mathematics Project

"McNamara not only has a command of mathematics content, pedagogy, and children's thinking; she masterfully weaves these areas together in this rich and accessible resource. Beyond Invert & Multiply will serve as a powerful resource for in-service and pre-service teachers both in developing their own content knowledge and also in supporting their students' deep conceptual understandings of fractions."

Darrell Earnest, Assistant Professor
University of Massachusetts, Amherst

"Looking for a resource to help as you implement the Number and Operations—Fractions content domain of the Common Core State Standards for Mathematics? Here it is . . . Beyond Invert & Multiply not only helps teachers identify what’s important about operations involving fractions, but also provides contexts, classroom video clips, and activities—all backed up by research."

Francis (Skip) Fennell, L. Stanley Bowlsbey Professor of Education and
Graduate and Professional Studies McDaniel College, Westminster, Maryland

"Beyond Invert & Multiply has made it possible for me to teach operations with fractions without 'tricks.' Instead, my students have a conceptual understanding of huge ideas like why dividing fractions actually results in larger numbers. The activities are perfect for all my students: ELLs, gifted, RSP, and general education. Thank you, Julie, for writing this!"

Morri Spang, fifth-grade teacher
San Pedro St. Elementary School, Los Angeles, California
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