Unit 3 extends the study of factors from 2, 3, 4, 5, and 10, which students explored in Unit 2, to include all units from 0 to 10, as well as multiples of 10 within 100. To work with these more challenging factors, students will rely on skip-counting (a Level 2 strategy) and converting to an easier problem (a Level 3 strategy dependent on the properties of operations). They then will apply their understanding of all four operations to two-step word problems as well as arithmetic patterns at the end of the unit.
Topic A begins by reminding students of the commutative property they learned in Unit 2, as well as introducing them to the distributive and associative properties, which they will rely upon for many of the strategies they learn for the larger factors. In order to be able to use these properties, they need to understand how to compute with a factor of 1, which they explore along with 0, as well as understand how to use parentheses. They’ll then explore the factors of 6, 7, 8, and 9 in Topics B and C. In Unit 2, when students were first introduced to multiplication and division with the easier factors of 2-5 and 10, students could count all of the objects to find the product, a Level 1 strategy, and used skip-counting as their fluency developed. But, because of the increased difficulty of 6–9 facts, students will not only rely on skip-counting (a Level 2 strategy), but also convert to an easier problem (a Level 3 strategy). Converting to an easier problem is dependent on the properties of operations (e.g., to find $$6\times7$$, think of $$5\times7$$ and add a group of 7 is dependent on the distributive property). Thus, students will work with the properties extensively throughout the unit, with their understanding of them and notation related to them growing more complex and abstract throughout the unit. In Topic D, students will multiply one-digit numbers by multiples of 10 and by two-digit numbers using the associative property. Finally, students solve two-step word problems involving all four operations, assessing the reasonableness of their answer, and identify arithmetic patterns and explain them using the properties of operations.
In Unit 3, students deepen their understanding of multiplication and division, including their properties. "Mathematically proficient students at the elementary grades use structures such as…the properties of operations…to solve problems" (MP.7) (Standards for Mathematical Practice: Commentary and Elaborations for K–5, p. 9). Students use the properties of operations to convert computations to an easier problem (a Level 3 strategy), as well as construct and critique the reasoning of others regarding the properties of operations (MP.3). Lastly, students model with mathematics with these new operations, solving one- and two-step word problems using them (MP.4).
Students’ understanding of multiplication and division will further develop in Unit 4, when students study area. Students will also use their understanding of these operations in Unit 7 when they apply them in the context of measurement word problems. In subsequent years, students will depend on their conceptual understanding and fluency with these operations to apply and extend them in a variety of ways—everything from multi-step multiplicative comparison words problems in 4th grade to polynomial multiplication and division in Algebra 2, and lots in between. Thus, this unit sets the seal on major work of 3rd grade as well as deeply important foundational work upon which students will rely for years to come.
Pacing: 26 instructional days (23 lessons, 2 flex days, 1 assessment day)