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As a middle grades math teacher, instructional coach, and tutor who works with students across all grades, I witness the struggle with fractions at every level. There are two primary sources of difficulty with fraction operations: The first is recognizing factors hidden in numerators and denominators; the second is choosing the correct rule and applying it properly. "Fractions with Prime Factor Tiles" is a revolutionary tool for teaching fraction operations that corrects both of these deficiencies through hands-on activities. Numerators and denominators are expressed as products of prime factors with color-coded tiles that make the common and non-common factors of numbers visually obvious. Rules for operations are taught and reinforced through physical manipulation of the tiles in a manner that is logical and intuitive.

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When I was teaching in the high school, we taught a unit on rational expressions and equations. In simple terms a rational expression is a fraction that has numbers and variables in the numerator, denominator, or both. Because rational expressions behave a lot like fractions, I usually started this unit with a day or two of review of fractions to help students build confidence with this foundational concept. Every year, I was surprised how many students struggled with fraction concepts, and it was clear to me that we needed to do more to build their conceptual understanding in the early grades. Generally, we are doing better with this, using more and different models to help students really understand the relationships between the part and the whole and also between fractions. Number lines help build conceptual understanding of fraction relationships and area models are useful tools for both relationships and operations. Another tool that helps students build an understanding of both fraction relationships and operations are interlocking fraction circles. The short video below explains how these circles support students’ understanding of fractions.

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As I travel around the country working with teachers, I frequently hear, “My students just don’t like fractions.” Teachers are right, fractions are confusing. Let’s explore the foundational understandings and possible misconceptions that students may go through as they are learning about fractions. As articulated in the progression on “Number and Operations-Fractions, 3-5,” fractional understanding begins in Grades 1 and 2 as students partition shapes. This is certainly a logical beginning as students have had experiences identifying when their share is smaller than someone else’s!  In Grade 3, students begin considering breaking a whole into equal parts.  Students work with wholes that are varying shapes such as rectangles or circles  and the focus is placed on equal parts.  The emphasis in Grade 3 is on unit fractions (i.e., fractions having 1 as the numerator). Just as our whole numbers are composed by combining 1s, fractions can be similarly constructed by combining unit fractions. For example, ¾ = ¼ + ¼ + ¼.

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"Would you rather have half of one chocolate bar or a quarter of a different chocolate bar? Most popular answer: It depends on the size of the chocolate bars! As teachers know, the relative size of fractions depends on how the whole is defined. Authors Jim Callahan and Marilynn Varricchio address these common problems with fractions in their new book Fractions Made Easy (Didax, 2016). Drawing on material from the book, we will focus on how visual models can be used to support a solid conceptual understanding of fractions in third grade.

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