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Figuring Out Fluency – Operations With Rational Numbers and Algebraic Equations

Figuring Out Fluency – Operations With Rational Numbers and Algebraic Equations
A Classroom Companion

August 2022 | 232 pages | Corwin

“This book is awesome! What stood out to me was the deep understanding I was able to have about what fluency actually means. Too often the message has been fluency and accuracy, especially at the middle school level. By providing teachers with tools for building fluency with integers, expressions, and algebra, this book shifts that message to also focus on flexibility and strategy selection.”

Lindsey Henderson
Secondary Mathematics Specialist, Utah State Board of Education 
Salt Lake City, UT

Because fluency practice is not a worksheet.

Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. It is not about speed or recall. Real fluency is about choosing strategies that are efficient, flexible, lead to accurate solutions, and are appropriate for the given situation. Developing fluency is also a matter of equity and access for all learners.

The landmark book Figuring Out Fluency in Mathematics Teaching and Learning offered educators the inspiration to develop a deeper understanding of procedural fluency, along with a plethora of pragmatic tools for shifting classrooms toward a fluency approach. Now, teachers have the chance to apply that inspiration through explicit instruction and practice every day with the classroom companion Figuring Out Fluency—Operations With Rational Numbers and Algebraic Equations. With this book, teachers can

  • Dive deeper into the Significant Strategies for fluency explained in the anchor book as they apply to rational number operations
  • Explore how these strategies can be applied for proportional reasoning, solving equations for unknowns, and solving systems of linear equations
  • Access over 100 classroom-ready activities, including worked examples, routines, and games.
  • Find activities to explicitly teach students how to use and choose strategies to operate on rational numbers and solve algebraic equations
  • Download all of the needed support tools, game boards, and other resources from the companion website for immediate implementation

Give each and every student the knowledge and power to become skilled and confident mathematical thinkers and doers.

Part 1: Figuring Out Fluency: Key Ideas
Part 2: Strategy Modules
Module 1: reasoning Strategies for Addition
Module 2: Reasoning Strategies for Subtraction
Module 3: Reasoning Strategies for Multiplication
Module 4: Reasoning Strategies for Division
Module 5: Reasoning Strategies for Ratios and Proportions
Module 6: Strategies for Solving Equations for an Unknown
Module 7: Strategies for Solving Systems of Linear Equations
Part 3: Putting it All Together: Developing Fluency

Free resources

Activity 7.7: "Would You Rather..." Routine

Activity 7.7: "Would You Rather..." Routine

There are two versions of this routine with systems of linear equations—choosing different strategies (e.g., Use Tables or Use Graphs) and making choices within a strategy (e.g., For which variable will I substitute? Or Which variables will I eliminate?).

Read Now

Activity 6.10: The Transformer Game

Activity 6.10: The Transformer Game

This game is a fun way to give students opportunities to choose and use different basic transformations to start solving equations.

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Activity 5.7: Worked Examples For Ratios and Proportions

Activity 5.7: Worked Examples For Ratios and Proportions

Use this activity for worked examples for ratios and proportions. Correctly worked examples and partially solved worked examples benefit students so that they can understand strategy.


ACTIVITY 4.11: Quotient Connect Game

ACTIVITY 4.11: Quotient Connect Game

This game is an engaging way to practice division using Partial Quotients. It is played similarly to the classic board game Boggle as students find quotients through connected digits.

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This book places a premium on mathematical thinking. The authors make the compelling case using myriad examples that fluency requires reasoning and relies on conceptual understanding. If you want your students to confidently choose and flexibly use a range of strategies when working with rational numbers and algebraic equations, dig into the treasure trove of engaging activities in this book!

Grace Kelemanik
Natick, MA

Sixth grade is a challenging time for fluency development—some students are still working on basic fact fluency, while others are starting to engage in a more sophisticated way of algebraic thinking. This book is an excellent resource for scaffolding middle school students on a personal level wherever they may be on the fluency continuum!

Kiersten Campbell
Clinton, UT

Figuring Out Fluency is a necessary resource for every middle school mathematics educator seeking to grow their knowledge base, learn effective teaching strategies, and gain access to teaching activities they can immediately implement.

Kristopher J. Childs
Equity, and Social Justice Officer—Open Up Resources
Winter Garden, FL

This book is exactly what the mathematics world needs. Even as a veteran teacher, I need to be reminded that students need explicit strategy instruction. Students need to see that there are efficient ways to solve problems and the standard algorithm isn’t always the best. This companion book is overflowing with ideas for the classroom teacher to improve mathematical fluency!

Elizabeth Uden
Olathe, KS

Despite striving for students to develop fluency with number and operations, inconsistent definitions of what it means to be fluent and a lack of understanding to implement effective fluency-building practices can leave you feeling defeated. Enter the action-oriented Figuring Out Fluency book that clarifies what it means to be fluent with number and operations while providing content specific examples that can be implemented in your classroom to begin seeing the fruit of your professional learning efforts.

Kyle Pearce
Belle River, Ontario, Canada

This is a book you keep on your teacher-desk! It’s full of classroom activities for your classroom while at the same time, it teaches you how to build your own activities so you can build mathematical fluency for your students.

Jon Orr
Lambton Kent District School Board
Chatham, Ontario, Canada

This book is like a full-course meal. Each course complements the other, from the amuse-bouche to the main course where each module, from operations of whole numbers to solving systems, leaves the reader satisfied with figuring out fluency. Lastly, as sweet as the dessert, developing fluency puts it all together. This is the most practical math book for secondary instruction of this time.

Valentin Sotomayor
Seminole County Public Schools
Seminole County, FL

When most educators hear “fluency,” they immediately think of elementary school. The authors do a masterful job of pushing us to think of fluency at the middle grades as well and provide concrete examples of what it looks like. The activities are ready to implement immediately to bring to life the key ideas around fluency for rational numbers and algebraic equations.

Kevin J. Dykema
Mattawan, MI

This book is the instruction tool I have been looking for with essential insights to effective mathematical teaching without unnecessary details. Filled with explicit ideas on how to build fluency, the book merges conceptual and procedural learning goals perfectly. A must-have tool for both educators and students looking for a deeper understanding of mathematics.

Kirk Redford
Clearfield, UT

This brilliantly crafted book stresses the importance of teaching mathematical fluency as equipping learners with multiple strategies that are not only
efficient and flexible but also lead to accurate solutions and are appropriate for the given situation. The fun and engaging activities, examples, and games all stress that fluency is also a matter of equity and access for all learners.

Kanita K DuCloux
Western Kentucky University
Bowling Green, KY Bay_

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ISBN: 9781071825181
CAD$ 47.95

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