Students deepen their understanding of fraction multiplication and begin to explore to fraction division (and fractions as division), as well as apply this new understanding to the context of line plots.
In Grade 5 Unit 5, students continue their exploration with fraction operations, deepening their understanding of fraction multiplication from Grade 4 and introducing them to fraction division.
Students began learning about fractions very early, as described in the Unit 4 Unit Summary. However, students’ exposure to fraction multiplication only began in Grade 4, when they learned to multiply a fraction by a whole number, interpreting this as repeated addition. For example, $${4\times {2\over3}}$$ is thought of as 4 copies of 2 thirds. This understanding is reliant on an understanding of multiplication as equal groups (3.OA.1). In Grade 4, however, students also developed an understanding of multiplicative comparison (4.OA.1), which will be of particular importance to the new ways in which students will interpret fraction multiplication in this unit.
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. Then, students develop a new understanding of fraction multiplication as fractional parts of a set of a certain size (5.NF.4), which is a new interpretation of multiplicative comparison. Students use this understanding to 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). Then, students also 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: 27 instructional days (24 lessons, 2 flex days, 1 assessment day)
This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.
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5.NF.B.3
Model fractions as division using area models and solve word problems involving division of whole numbers with answers in the form of fractions or mixed numbers.
5.NF.B.3
Model fractions as division using tape diagrams and solve word problems involving division of whole numbers with answers in the form of fractions or mixed numbers.
5.NF.B.4.A
5.NF.B.6
Multiply a fraction by a whole number using set models.
5.NF.B.4.A
5.NF.B.6
Multiply a fraction by a whole number using tape diagrams and number lines.
5.NF.B.4.A
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
Relate multiplication of a fraction by a whole number to multiplication of a whole number by a fraction and use this to develop a general method to multiply any fraction by any whole number (or vice versa).
5.OA.A.2
5.NF.B.4
5.NF.B.6
Solve real-world problems involving multiplication of fractions and whole numbers and create real-world contexts for expressions involving multiplication of fractions and whole numbers.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Multiply a fraction by a fraction without subdivisions using tape diagrams and number lines.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Multiply a fraction by a fraction with subdivisions using tape diagrams and number lines.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Multiply a fraction by a fraction with more complicated subdivisions using an area model.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Develop a general method to multiply a fraction by a fraction.
5.OA.A.2
5.NF.B.4
5.NF.B.6
Solve real-world problems involving multiplication of fractions with fractions and create real-world contexts for expressions involving multiplication of fractions with fractions.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Multiply mixed numbers by whole numbers.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Multiply mixed numbers by fractions.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Multiply mixed numbers by mixed numbers.
5.NF.B.4
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
5.NF.B.6
Develop a general method to multiply with mixed numbers.
5.OA.A.2
5.NF.B.4
5.NF.B.6
Solve real-world problems involving multiplication with mixed numbers and create real-world contexts for expressions involving multiplication with mixed numbers.
5.NF.B.5
5.NF.B.5.A
5.NF.B.5.B
Interpret multiplication as scaling.
5.NF.B.7.A
5.NF.B.7.C
Divide a unit fraction by a whole number.
5.NF.B.7.B
5.NF.B.7.C
Divide a whole number by a unit fraction.
5.NF.B.7.C
Solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions.
5.NF.B.3
5.NF.B.6
5.NF.B.7
Solve real-world problems involving multiplication and division with fractions.
5.OA.A.1
5.OA.A.2
5.NF.B.3
5.NF.B.4
5.NF.B.5
5.NF.B.6
5.NF.B.7
Write and evaluate numerical expressions involving operations with fractions.
Key: Major Cluster Supporting Cluster Additional Cluster
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