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by Monica Neagoy
Table of Contents
This ASCD Study Guide is meant to enhance your understanding of the concepts and practical ideas presented in Unpacking Fractions: Classroom-Tested Strategies to Build Students' Mathematical Understanding, an ASCD book written by Monica Neagoy. You can use this study guide either with teachers if you are a professor, teacher leader, or coach, or with younger students if you are a classroom teacher. You may already have posed to yourself or a colleague many of the questions included here, and others will become evident as you read through the book. You may choose to answer the questions before tackling each chapter of the book or answer them as you finish the chapter. Each chapter is rich in information—content, pedagogy, and research—so take the time to digest one at a time. The study questions provided address some of the most salient ideas in the book. They do not address all of the notions covered in the book.
Claudia, a 4th grade student, once said to me, "I understand how to take three parts out of five parts to get the fraction ^{3}⁄_{5}, but I don't understand the fraction ^{7}⁄_{5}. How can I take seven parts out of five parts?" What would you say and do to help Claudia find meaning in the fraction ^{7}⁄_{5}?
When showing the fraction ¼, we almost always (and, therefore, so do our students), begin with a whole (either continuous or discrete), partition it into four equal parts, and then designate one of those parts in some way. What students don't see enough of is teachers iterating to show ¼. Reflect on how you might model fractions ⅓ or ⅔ using two gray marbles and any number of white marbles. Since you cannot partition the marbles, think about how you might iterate to model the desired fractions.
Can you explain how the fraction represented by this figure could be ¾ or ¼? Could the figure represent ½ or ^{3}⁄_{2}? Explain. And what about ⅓?
Consider these two and a half discs:
Suppose you used Cuisenaire rods to model the comparison between two juice concentrations. Juice 1 was made with seven parts pure orange juice and nine parts water, whereas Juice 2 was made with eight parts pure orange juice and ten parts water
Knowing that most students will say, "The two juice mixtures have the same orange concentration—or are just as "orangey"—how would you guide them to see otherwise?
For many students, knowing an operation is usually synonymous with knowing how to carry out the algorithm. Reflect for a moment on your own repertoire of contexts and real-world situations that can help students find meaning in the four fraction operations well before the teaching and learning of the algorithms for fraction computation. Do you have one or more for each operation?
A common and unfortunate way we explain how to multiply a decimal by a power of 10 is to say, "Move the decimal point to the right n places," where n is the number of zeros in the power of 10. Reflect on what really happens, and devise a more mathematical explanation for multiplying a decimal by a power of 10. Describe representations you might use to help students visualize your explanations.
Reflect on the journey students will travel from 2nd or 3rd grade to 8th or 9th grade—from fractions to linear functions. Can you see in your own mind how these fundamental notions build on one another and at each level become more sophisticated? Once you do, you will appreciate the importance to laying a solid foundation for fraction understanding early on.
Unpacking Fractions: Classroom-Tested Strategies to Build Students' Mathematical Understanding was written by Monica Neagoy. This 285-page, 8" x 10" book (Stock #115071; ISBN-13: 978-1-4166-2123-2) is available from ASCD for $31.16 (ASCD member) or $38.95 (nonmember).Copyright © 2017 by ASCD. To order a copy, call ASCD at 1-800-933-2723 (in Virginia 1-703-578-9600) and press 2 for the Service Center. Or buy the book from ASCD's Online Store.
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