# Multi-Digit Multiplication

Students deepen their understanding of multiplication by exploring factors and multiples, multiplicative comparison, as well as multi-digit multiplication.

## Unit Summary

In Grade 4 Unit 2, students multiply up to four-digit numbers by one-digit numbers, relying on their understanding of place value and properties of operations, as well as visual models like an area model, to solve.

As a foundation for their multi-year work with multiplication and division, students in Grade 2 learned to partition a rectangle into rows and columns and write a repeated addition sentence to determine the total. They also skip-counted by 5s, 10s, and 100s. Then, in Grade 3, students developed a conceptual understanding of multiplication and division in relation to equal groups, arrays, and area. They developed a variety of strategies to build toward fluency with multiplication and division within 100 and applied that knowledge to the context of one- and two-step problems using the four operations.

To begin the unit, students extend their understanding of multiplication situations that they learned in Grade 3 to include multiplicative comparison using the words “times as many.” Next, to continue to refresh students’ work in Grades 2 and 3 on skip-counting and basic multiplication facts and extend it further to values they have not yet worked with, students investigate factors and multiples within 100, as well as prime and composite numbers (4.OA.4). Thus, this supporting cluster content serves as a foundation for the major work with multiplication and division with larger quantities. Tangentially, it will also support the major work in Unit 5 to recognize and generate equivalent fractions. Then, students move into two-digit by one-digit, three-digit by one-digit, four-digit by one-digit, and two-digit by two-digit multiplication, using the area model, partial products, 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 multiplication and as a connection to their work with area and perimeter (4.MD.3), a supporting cluster standard. Finally, with a full understanding of all multiplication cases, they then apply their new multiplication skills to solve multi-step word problems using multiplication, addition, and subtraction, including cases involving multiplicative comparison (4.NBT.5, 4.OA.3, 4.MD.3), allowing for many opportunities to connect content across multiple domains.

This unit affords lots of opportunities to deepen students’ mathematical practices. For example, “when students decompose numbers into sums of multiples of base-ten units to multiply them, they are seeing and making use of structure (MP.7). Students “reason 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 multi-step word problems involving addition, subtraction, and multiplication, they are modeling with mathematics (MP.4).

Students’ work in this unit will prepare them for fluency with the multiplication algorithm in Grade 5 (5.NBT.5). Students also learn about new applications of multiplication in future grades, including scaling quantities up and down in Grade 5 (5.NF.5), all the way up to rates and slopes in the middle grades (6.RP, 7.RP). Every subsequent grade level depends on the understanding of multiplication and its algorithm, making this unit an important one for students in Grade 4.

Pacing: 26 instructional days (23 lessons, 2 flex days, 1 assessment day)

## Assessment

This assessment accompanies Unit 2 and should be given on the suggested assessment day or after completing the unit.

### Fishtank Plus

#### Expanded Assessment Package

Learn how to use these tools with our Guide to Assessments

## Unit Prep

### Essential Understandings

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• Every counting number is divisible by 1 and itself, and some counting numbers are also divisible by other numbers. Some counting numbers have exactly two factors (prime numbers); others have more than two (composite numbers). The numbers 0 and 1 are special cases in that they are neither prime nor composite. The product of any nonzero number and any other nonzero number is divisible by each number and is called a multiple of each number.
• In an additive comparison, the underlying question is what amount would be added to one quantity in order to result in the other. In a multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in the other.
• One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1,000. Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role played by the distributive property.
• Rounding numbers can help one to determine whether an answer is reasonable based on whether the estimate is close to the computed answer or not.
• Making sense of problems and persevering to solve them is an important practice when solving word problems. Key words do not always indicate the correct operation.

### Vocabulary

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 composite number partial product factor pair prime number multiple square number

### Unit Materials, Representations and Tools

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• Grid paper (optional)
• Base ten blocks (optional)
• Dice (optional: used for game in Problem Set but can be adapted)
• Square tiles or other counters (optional: suggested to use for Anchor Task)
• Tape diagrams
• Area models, partial products algorithm, standard algorithm

### Intellectual Prep

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#### Intellectual Prep for All Units

• Read and annotate “Unit Summary” and “Essential Understandings” portion of the unit plan.
• Do all the Target Tasks and annotate them with the “Unit Summary” and “Essential Understandings” in mind.
• Take the unit assessment.

#### Unit-Specific Intellectual Prep

• Read pp. 14–15 in Progressions for the Common Core State Standards in Mathematics Number and Operations in Base Ten, K-5 (starting at the section titled “Use place value understanding and properties of operations to perform multi-digit arithmetic”).
• Read the document “Situation Types for Operations in Word Problems” for multiplication and division. Identify the word problem types of any applicable assessment questions.
• Read the following table that includes models used throughout the unit.
 Models for up to 4-digit by 1-digit Multiplication Written Numerical Work Base ten block array (remedial - can be concrete or pictorial) Horizontally written partial products Graph paper array (remedial) Vertically written partial products Area model Vertically written partial products Standard algorithm Models for up to 2-digit by 2-digit Multiplication Written Numerical Work Base ten block array (remedial - can be concrete or pictorial) Horizontally written partial products Graph paper array (remedial) Vertically written partial products Area model Vertically written partial products Standard algorithm ## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### Measurement and Data
• 4.MD.A.3 — Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

##### Number and Operations in Base Ten
• 4.NBT.B.5 — Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

##### Operations and Algebraic Thinking
• 4.OA.A.1 — Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

• 4.OA.A.2 — Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

• 4.OA.A.3 — Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

• 4.OA.B.4 — Find all factor pairs for a whole number in the range 1—100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1—100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1—100 is prime or composite.

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• 3.MD.C.7.B

• 3.MD.D.8

• 3.NBT.A.3

• 4.NBT.A.1

• 4.NBT.A.2

• 4.NBT.A.3

• 4.NBT.B.4

• 3.OA.A.1

• 3.OA.A.2

• 3.OA.A.3

• 3.OA.A.4

• 3.OA.B.5

• 3.OA.B.6

• 3.OA.C.7

• 3.OA.D.8

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• 4.NBT.B.6

• 5.NBT.B.5

• 4.NF.A.1

• 4.NF.B.4

• 5.NF.B.3

• 5.NF.B.4

• 5.NF.B.5

• 5.NF.B.5.A

• 5.NF.B.5.B

• 5.NF.B.6

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.