Unit Summary
In Grade 5 Unit 5, students continue their study of fraction operations, deepening their understanding of fraction multiplication from Grade 4 and beginning to explore fraction division.
Students began learning about fractions very early, as described in the Unit 4 Unit Summary. Their exposure to fraction multiplication more specifically began in Grade 4, when they learned to multiply a fraction by a whole number, thinking of this as a whole number of groups of fractional-sized groups. For example, $${4\times {2\over3}}$$ is thought of as 4 groups of 2 thirds.
The unit begins with students developing a new understanding of fractions as division. In the past, they’ve thought of fractions as equal-sized partitions of wholes, but here they develop an understanding of a fraction as an operation itself and represent division problems as fractions (5.NF.3). Students now see that remainders can be interpreted in yet another way, namely divided by the divisor to result in a mixed-number quotient. For example, cutting a 7-foot string into 3 pieces of equal length results in pieces that are $$2\tfrac13$$ feet long. Then, students develop a new understanding of fraction multiplication as fractional parts of a group of a certain size (5.NF.4). For example, they interpret the expression $$\tfrac23\times4$$ as 2 thirds of 4, and thus can be seen as 2 parts of a partition of 4 into 3 equal parts. They also develop general methods to multiply fractions by whole numbers and fractions, including mixed numbers. Throughout this work, students develop an understanding of multiplication as scaling (5.NF.5), “an important opportunity for students to reason abstractly” (MP.2) as the Progressions notes (Progressions for the Common Core State Standards in Mathematics, Number and Operations - Fractions, 3-5, p. 14). Then, students explore division of a unit fraction by a whole number and a whole number by a unit fraction (5.NF.7), preparing students to divide with fractions in all cases in Grade 6 (6.NS.1). Next, students solve myriad word problems, seeing the strategies they used to solve word problems with whole numbers still apply but that special attention should be paid to the whole being discussed (5.NF.6, MP.4), as well as write and solve expressions involving fractions as a way to support the major work (5.OA.1, 5.OA.2). Finally, students make line plots to display a data set of measurements in fractions of a unit and solve problems involving information presented in line plots (5.MD.2), a supporting cluster standard that supports the major work of this and the past unit of using all four operations with fractions (5.NF).
In Unit 6, students will learn to multiply and divide decimals, relying on their understanding of these operations with fractions developed in this unit to do so. In Grade 6, students encounter the remaining cases of fraction division (6.NS.1). “Work with fractions and multiplication is a building block for work with ratios. In Grades 6 and 7, students use their understanding of wholes and parts to reason about ratios of two quantities, making and analyzing tables of equivalent ratios, and graphing pairs from these tables in the coordinate plane. These tables and graphs represent proportional relationships, which students see as functions in Grade 8” (NF Progression, p. 20). Students will further rely on this operational fluency throughout the remainder of their mathematical careers, from fractional coefficients in functions to the connection between irrational numbers and non-repeating decimals.
Pacing: 28 instructional days (25 lessons, 2 flex days, 1 assessment day)