Unit Summary
In Unit 6, students extend and deepen Grade 1 work with understanding halves and fourths/quarters (1.G.3) as well as Grade 2 practice with equal shares of halves, thirds, and fourths (2.G.3) to understanding fractions as equal partitions of a whole. Their knowledge becomes more formal as they work with area models and the number line. Throughout the unit, students have multiple experiences working with the Grade 3 specified fractional units of halves, thirds, fourths, sixths, and eighths. To build flexible thinking about fractions, students are exposed to additional fractional units such as fifths, ninths, and tenths.
Students begin the unit by partitioning different models of wholes into equal parts (e.g., concrete fraction strips and pictorial area models) (3.G.2), allowing this supporting cluster content to enhance the major work of Grade 3 with fractions. They identify and count equal parts as 1 half, 1 fourth, 1 third, 1 sixth, and 1 eighth in unit form before introduction to the unit fraction $$\frac{1}{b}$$ (3.NF.1). Then, they make copies of unit fractions to build non-unit fractions, understanding unit fractions as the basic building blocks that compose other fractions (3.NF.1). Next, students transfer their work to the number line. They begin by using the interval from 0 to 1 as the whole and then extend to mark fractions beyond a whole. Noticing that some fractions with different units are placed at the exact same point on the number line, they come to understand equivalent fractions (3.NF.3a). Students recognize that whole numbers can be written as fractions, including writing 1 as $$\frac{1}{1}, \frac{2}{2}, \frac{3}{3}$$, etc., as well as writing whole numbers as fractions with a denominator of 1, e.g., 2 as $$ \frac{2}{1}$$, 3 as $$\frac{3}{1}$$, etc. Lastly, students use their understanding of the number of units and the size of each unit to compare fractions in simple cases, such as when dealing with common numerators or common denominators by reasoning about their size (3.NF.3d). Lastly, students measure lengths with fractional units and use data generated by measuring multiple objects to create line plots (3.MD.4). Students “use their developing knowledge of fractions and number lines to extend their work from the previous grade by working with measurement data involving fractional measurement values” (MD Progression, p. 10), thus using this supporting cluster work to enhance the major work of fractions.
This unit affords ample opportunity for students to engage with the Standards for Mathematical Practice. Students will develop an extensive toolbox of ways to model fractions, including area models, tape diagrams, and number lines (MP.5), choosing one model over another to represent a problem based on its inherent advantages and disadvantages. Students construct viable arguments and critique the reasoning of others as they explain why fractions are equivalent and justify their conclusions of a comparison with a visual fraction model (MP.3). They attend to precision as they come to more deeply understand what is meant by equal parts, and being sure to specify the whole when discussing equivalence and comparison (MP.6). Lastly, in the context of line plots, “measuring and recording data require attention to precision (MP.6)” (MD Progression, p. 3).
Unfortunately, “the topic of fractions is where students often give up trying to understand mathematics and instead resort to rules” (Van de Walle, p. 203). Thus, this unit places a strong emphasis on developing conceptual understanding of fractions, using the number line to represent fractions and to aid in students' understanding of fractions as numbers. With this strong foundation, students will operate on fractions in Grades 4 and 5 (4.NF.3—4, 5.NF.1—7) and apply this understanding in a variety of contexts, such as proportional reasoning in middle school and interpreting functions in high school, among many others.
Pacing: 31 instructional days (28 lessons, 2 flex days, 1 assessment day)