Unit 2 opens students’ eyes to some of the most important content students will learn in 3rd grade—multiplication and division. In this unit, "students begin developing these concepts by working with numbers with which they are more familiar, such as 2s, 5s, and 10s, in addition to numbers that are easily skip-counted, such as 3s and 4s," allowing the cognitive demand to be on the meaning of multiplication and division themselves rather than computing (CCSS Toolbox, Sequenced Units for the Common Core State Standards in Mathematics Grade 3). Then in Unit 3, students will work on the more challenging units of 0, 1, 6–9, and multiples of 10.
In 2nd grade, students learned to count objects in arrays using repeated addition (2.OA.4) to gain a foundation for multiplication. They’ve also done extensive work on one- and two-step word problems involving addition and subtraction, having encountered all of the problem types that involve those operations (2.OA.1). Students will rely on this foundational equal-groups understanding and orientation to solving contextual problems extensively in this unit.
At the start of this unit, students gain an understanding of multiplication and division in the context of equal group and array problems in Topic A. To keep the focus on the conceptual understanding of multiplication and division (3.OA.1, 3.OA.2), Topic A does not discuss specific strategies to solve, and thus students may count all objects (a Level 1 strategy) or remember their skip-counting and repeated addition (Level 2 strategies) from 2nd grade to find an unknown product. In Topics B and C, however, the focus turns to developing more efficient strategies for solving multiplication and division, including skip-counting and repeated addition (Level 2 strategies) as well as "just knowing" the facts, which works toward the goal that "by the end of 3rd grade, [students] know from memory all products of two single-digit numbers and related division facts" (3.OA.7). As the Operations and Algebraic Thinking Progression states, "reaching fluency in single-digit multiplications and related division may be quite time consuming because there are no general strategies for multiplying or dividing all single-digit numbers as there are for addition or subtraction" (OA Progression, p. 22). Because "there are many patterns and strategies dependent upon specific numbers," they first work with factors of 2, 5, and 10 in Topic B, since they learned these skip-counting sequences in 2nd grade. Then in Topic C, they work with the new factors of 3 and 4. Only then, when students have developed more familiarity with these factors, will students solve more complex and/or abstract problems with them, including determining the unknown whole number in a multiplication or division equation relating three whole numbers (3.OA.4) and solving two-step word problems using all four operations (3.OA.3, 3.OA.8), assessing the reasonableness of their answers for a variety of problem types in Topic D. Finally, the unit culminates with a focus on categorical data, where students draw and solve problems involving scaled picture graphs and scaled bar graphs, a nice application of the major work of multiplication and division. As the Progressions note, "these developments connect with the emphasis on multiplication in this grade" (MD Progression, p. 7). Students also solve one- and two-step word problems related to the data in these plots, relying on the extensive work students have done with word problems throughout the year. This supporting cluster standard nicely enhances the major work they’ve been working on throughout the unit.
Throughout the unit, students engage in a variety of mathematical practices. The unit pays particular attention to reasoning abstractly and quantitatively, as students come to understand the meaning of multiplication and division and the abstract symbols used to represent them (MP.2). Further, students model with mathematics with these new operations, solving one- and two-step word problems involving them (MP.4).
This introduction to multiplication and division is further deepened in Unit 3, when students will explore the more difficult factors of 0, 1, 6–9, and multiples of 10. Then, in Unit 4, students will explore area and its connections to these operations. In 4th grade, their understanding of multiplication and division will be even more rich, when they will come to understand multiplicative comparison and solve word problems involving it (4.OA.1, 4.OA.2). Further, they’ll solve multi-step word problems involving all four operations, at times needing to interpret a remainder in the context of the problem (4.OA.3). Lastly, students will extend their computation prowess to multi-digit numbers, multiplying a whole number of up to four digits by a one-digit whole number and two two-digit numbers, as well as dividing up to four-digit dividends by a one-digit divisor (4.NBT.5, 4.NBT.6). Because multiplication and division provide a foundation for myriad algebraic and geometric topics from linear functions to trigonometry, this content is critical for all future mathematical learning.
Pacing: 24 instructional days (21 lessons, 2 flex days, 1 assessment day)