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Teaching Arithmetic: Lessons for Extending Division, Grades 4–5


Through these lessons, students begin to understand the rules of divisibility and the connection between multiplication and division. Additionally, students build their understanding of the relationships among dividends, divisors, quotients, and remainders.

This book is part of the Teaching Arithmetic, Complete Series.

Maryann Wickett and Marilyn Burns
296 Pages

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A Remainder of One
by Marilyn Burns

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Review by Lisa Pallett, Math Support Teacher at Longfellow Elementary in Columbia, Maryland.  From the Fall 2004 issue of Intersection, a newsletter of the Exxon Mobil Corporation and the National Council of Teachers of Mathematics (NCTM). The review has been reprinted with permission from Intersection, © 2004, by Exxon Mobil Corporation.

Lessons for Extending Division, by Maryann Wickett and Marilyn Burns, is an excellent resource that should be on the bookshelf of every fourth and fifth grade teacher. It includes outstanding lessons that integrate problem solving into teaching arithmetic and demonstrate the value of student communication for building understanding.

Each chapter has an engaging lesson, including lessons that introduce division, lessons on patterns with divisors, dividends and quotients, lessons on divisibility rules, lessons that teach long division in context, and lessons for computational practice. Additional activities and assessment ideas are also included in the back of the book. Motivating games and lessons that integrate measurement and literature are included. The lessons and activities extend division and encourage students to analyze division in depth.

This teacher-friendly book is organized into sections, which make this resource easy to use and understand. An overview allows teachers to quickly identify the goals of the lesson to determine if it is appropriate for their students. The lessons have clear, step-by-step instructions. Teachers will want to be sure to read the “Teaching Notes” and “The Lesson” to gain as much insight as possible into the lesson before teaching. The “Teaching Notes” provide the background information necessary to have an in-depth conversation with the students. A vignette of the lesson actually being taught in the classroom is provided in “The Lesson.” This provides clear examples of how the author implemented the lesson, as well as student reactions. Engaging questioning is modeled for teachers and lively discussions are provided that demonstrate quality learning taking place.

Another significant attribute of this book is the intentional effort the authors make to encourage students to use mathematical vocabulary. This ensures that they use the correct terminology for dividend, divisor, quotient, and remainder, and that they have a true understanding of number meaning.

The book not only includes lessons for students on division, but lessons for teachers. Infused in the lessons are strategies for classroom management that teachers may find valuable in many curricular areas.

A non-traditional algorithm for long division is presented to students. The author demonstrates a delicate balance between introducing the algorithm while still allowing students the flexibility to solve problems in a meaningful way. The algorithm relies on students’ number sense more than memorized procedures, which leads to greater understanding.

If teachers choose to use this book they will need to spend a significant amount of time reading this resource. However, they will find it time well spent if their students demonstrate the understanding of division that the students in the vignettes have. Lessons for Extending Division should be required reading for all fourth and fifth grade teachers.

Review by Margaret Godfrey, K–12 supervisor, Hunter College, New York, New York. From the April 2006 issue of Teaching Children Mathematics, copyright 2006 by the National Council of Teachers of Mathematics (NCTM). All rights reserved. For use associated with the Math Solutions Web site at only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

This volume offers many specific lessons and techniques for teaching division. The authors use their extensive experience in working with children to select specific lessons and dialogue when presenting concepts and practical problems.

Four introductory lessons are considered crucial. Two lessons use patterns and encourage students to see relationships. The third chapter connects division and multiplication and includes a creative approach to a story as an alternative to the standard algorithm. Then two chapters emphasize patterns, asking students to explore examples where (1) the dividend and (2) the quotient “stay the same.” Real-life manipulatives are used to show that some things can be broken into fractional pieces (brownies) but some cannot (a marker).

Four chapters on divisibility rules are followed by three chapters using manipulatives, yarn, and beans. A grape licorice (yarn) activity starts with divisors that have no remainder and extends divisions with remainders. The classroom dialogue explores several children’s thinking processes. Three chapters describe games children can play to stretch their knowledge of division. A final chapter presents division riddles and develops logic. The book also includes additional activities and suggested assessments along with blackline masters.

The book sets forth a structure to help teachers plan and teach lessons in division. This is accomplished through step-by-step directions, specification of materials, teaching notes, and extended lesson dialogue followed by questions and thoughtful authors’ answers. This book is also valuable to fourth- and fifth grade teachers because division is a continuing source of difficulty for students. It presents ideas that can strengthen any elementary or middle school teacher’s repertoire of techniques for teaching division.

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This book builds on the lessons included in Teaching Arithmetic: Lessons for Introducing Division, Grades 3–4 with activities and strategies for solving division problems with large numbers.

A Remainder of One
by Marilyn Burns

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