Module 1, Topic A begins with a conceptual exploration of the multiplicative patterns of the base ten systems. This exploration extends the place value work done with multi-digit whole numbers in Grade 4 to larger multi-digit whole numbers and decimals. Students use place value disks and a place value chart to build the place value chart from millions to thousandths. They compose and decompose units crossing the decimal with a view toward extending their knowledge of the 10 times as large and 1/10 as large relationships among whole number places to that of adjacent decimal places. This concrete experience is linked to the effects on the product when multiplying any number by a power of ten. For example, students notice that multiplying 0.4 by 1,000 shifts the position of the digits to the left three places, changing the digits’ relationships to the decimal point and producing a product with a value that is 10 10 10 as large (400.0) (5.NBT.2). Students explain these changes in value and shifts in position in terms of place value. Additionally, students learn a new and more efficient way to represent place value units using exponents (e.g., 1 thousand = 1,000 = 103) (5.NBT.2). Conversions among metric units such as kilometers, meters, and centimeters give students an opportunity to apply these extended place value relationships and exponents in a meaningful context by exploring word problems in the last lesson of Topic A (5.MD.1).
In Module 1, Topics D through F mark a shift from the opening topics of Module 1. From this point to the conclusion of the module, students begin to use base ten understanding of adjacent units and whole-number algorithms to reason about and perform decimal fraction operations—addition and subtraction in Topic D, multiplication in Topic E, and division in Topic F (5.NBT.7).
In Topic D, unit form provides the connection that allows students to use what they know about general methods for addition and subtraction with whole numbers to reason about decimal addition and subtraction (e.g., 7 tens + 8 tens = 15 tens = 150 is analogous to 7 tenths + 8 tenths = 15 tenths = 1.5). Place value charts and disks (both concrete and pictorial representations) and the relationship between addition and subtraction are used to provide a bridge for relating such understandings to a written method. Real-world contexts provide opportunities for students to apply their knowledge of decimal addition and subtraction as well in Topic D.
A focus on reasoning about the multiplication of a decimal fraction by a one-digit whole number in Module 1, Topic E provides the link that connects Grade 4 multiplication work and Grade 5 fluency with multi-digit multiplication. Place value understanding of whole-number multiplication coupled with an area model of the distributive property is used to help students build direct parallels between whole-number products and the products of one-digit multipliers and decimals (5.NBT.7). Once the decimal has been placed, students use an estimation-based strategy to confirm the reasonableness of the product through place value reasoning. Word problems provide a context within which students can reason about products.
Module 2, Topic A begins a sequential study of multiplication that culminates in Topic D. In order to link prior learning from Grade 4 Module 1 and Grade 5 Module 1 and to set the stage for solidifying the standard multiplication algorithm, students begin at the concrete–pictorial level. They use place value disks to model multi-digit multiplication of place value units, for example, 42 × 10, 42 × 100, 42 × 1,000, leading quickly to problems such as 42 × 30, 42 × 300, and 42 × 3,000 (5.NBT.1, 5.NBT.2). Students then round factors in Lesson 2 and discuss the reasonableness of their products. Throughout Topic A, students evaluate and write simple expressions to record their calculations using the associative property and parentheses to record the relevant order of calculations (5.OA.1).
Module 2, Topics E through H provide a parallel sequence for division to that offered in Topics A to D for multiplication. Topic E begins concretely with place value disks as an introduction to division with multi-digit whole numbers (5.NBT.6). In Lesson 17, 420 ÷ 60 is interpreted as 420 ÷ 10 ÷ 6. Next, students round dividends and 2-digit divisors to nearby multiples of ten in order to estimate single-digit quotients (e.g., 431 ÷ 58 ≈ 420 ÷ 60 = 7) and then multi-digit quotients. This work is done horizontally, outside the context of the written vertical method.
Module 2, Topic G uses the knowledge students have accumulated about whole number division with double-digit divisors and extends it to the division of decimals by double-digit divisors (5.NBT.7). Parallels between sharing or grouping whole number units and sharing or grouping decimal units are the emphasis of Topic G. Students quickly surmise that the concepts of division remain the same regardless of the size of the units being shared or grouped. Placement of the decimal point in quotients is based on students’ reasoning about when wholes are being shared or grouped and when the part being shared or grouped transitions into fractional parts. Students reason about remainders in a deeper way than in previous grades. Students consider cases in which remainders expressed as whole numbers appear to be equivalent; however, equivalence is disproven when such remainders are decomposed as decimal units and shared or grouped.
