Students explore multidigit division and its applications, which include interpreting a remainder in division word problems and using division to interpret a repeating pattern.
In this unit, students explore the concept of multidigit division and its applications, such as interpreting a remainder in division word problems and using division to determine the $$n^{\mathrm{th}}$$ term in a repeating shape pattern.
Students developed a foundational understanding of division in Grade 3, when they came to understand division in relation to equal groups, arrays, and area. They developed a variety of strategies to build towards fluency with division within 100, and they applied that knowledge to the context of one and twostep problems using the four operations. Students also came to understand the distributive property, which underpins the standard algorithm for division.
Just as at the beginning of the previous unit when students expanded their understanding of multiplication beyond Grade 3 understanding to include multiplicative comparison word problems, this unit starts off with the added complexity of division problems with remainders (4.OA.3). This is likely familiar to students from their own realworld experiences of trying to split quantities evenly, and thus the focus is on interpretation of those remainders in the context of various problems. Next, students focus on extending their procedural skill with division to include up to fourdigit dividends with onedigit divisors (4.NBT.6), representing these cases with base ten blocks, the area model, partial quotients, and finally the standard algorithm, making connections between all representations as they go. The use of the area model serves to help students conceptually understand division, and as a connection to their work with area and perimeter (4.MD.3), a supporting cluster standard. Lastly, armed with a deep understanding of all four operations spanned over the last three units, students solve multistep problems involving addition, subtraction, multiplication, and division, including their new problem situations such as multiplicative comparison and interpreting remainders (4.OA.3). They also explore number and shape patterns, using the four operations to draw conclusions about them (4.OA.5).
Throughout the unit, students are engaging with the mathematical practices in various ways. For example, students are seeing and making use of structure (MP.7) as they “decompos[e] the dividend into like baseten units and find the quotient unit by unit” (NBT Progressions, p. 16). Further, "by reasoning repeatedly (MP.8) about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities” (NBT Progression, p. 14). Lastly, as students solve multistep word problems involving addition, subtraction, and multiplication, they are modeling with mathematics (MP.4).
While students are encouraged throughout the unit to use models when appropriate to solve problems, their indepth experience with the place value system and multiple conceptual models and exposure to the division algorithms prepares them for extending these models to twodigit divisors in Grade 5 (5.NBT.6) and to fluency with the division algorithm in Grade 6 (6.NS.2). Every subsequent grade level depends on the understanding of multidigit division and its algorithms, making this unit an important one for students in Grade 4.
Pacing: 19 instructional days (16 lessons, 2 flex days, 1 assessment day)
This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.
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partial quotient  remainder 
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Models for up to 4digit by 1digit Division  Written Numerical Work 
Base ten block array (can be concrete or pictorial): 
Horizontally written partial quotients: $$65\div5=(50\div5)+(15\div5)$$ $$=10+3$$ $$=13$$ 
Area model with inefficient partial quotients: 
Vertically written inefficient partial quotients: 
Area model with most efficient partial quotients: 
Vertically written efficient partial quotients: 

Standard algorithm: 
4.OA.A.3
Solve division word problems with remainders.
4.OA.A.3
Solve division word problems that require the interpretation of the remainder.
4.NBT.B.6
Divide multiples of 10, 100, and 1,000 by onedigit numbers.
4.NBT.B.6
Divide two, three, and fourdigit numbers by onedigit numbers using a variety of simplifying strategies.
4.NBT.B.6
Solve twodigit dividend division problems with no remainder or a remainder in the ones place with smaller divisors and quotients.
4.NBT.B.6
Solve twodigit dividend division problems with a remainder in the tens and/or ones place with smaller divisors and quotients.
4.NBT.B.6
Solve twodigit dividend division problems with a remainder in any place with larger divisors and quotients.
4.NBT.B.6
Solve threedigit dividend division problems with a remainder in any place.
4.NBT.B.6
Solve fourdigit dividend division problems with a remainder in any place.
4.NBT.B.6
Solve two, three, and fourdigit dividend problems, including the special cases of having a 0 in the quotient or divisor, and assess the reasonableness of the quotient.
4.OA.A.3
4.MD.A.3
Apply the formulas for area and perimeter in realworld and mathematical problems involving all four operations.
4.OA.A.3
4.NBT.B.6
Solve twostep word problems, including those involving interpreting the remainder, and assess the reasonableness of answers.
4.OA.A.3
4.NBT.B.6
4.MD.A.3
Solve multistep word problems involving all four operations and assess the reasonableness of answers.
4.OA.C.5
Identify and extend growing number patterns.
4.OA.C.5
Identify and extend growing shape patterns.
4.OA.C.5
Identify and extend repeating shape patterns.
Key: Major Cluster Supporting Cluster Additional Cluster
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