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)

For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 4th Grade Scope and Sequence Recommended Adjustments.

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Assessment

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

Unit Prep

Intellectual Prep

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

Essential Understandings

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

prime number

composite number

partial product

square number

multiple

factor pair

Related Teacher Tools:

4th Grade Vocabulary Glossary

Unit Materials, Representations and Tools

Hundreds Chart
Square inch tiles or other counters
Dice
Base ten blocks
Tape diagrams
Area models
Partial products algorithm
Standard algorithm for multiplication

Additional Unit Practice

Unit Practice

With Fishtank Plus you can access our Daily Word Problem Practice and our content-aligned Fluency Activities created to help students strengthen their application and fluency skills.

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Lesson Map

Topic A: Multiplicative Comparison

1

4.OA.A.1

4.OA.A.2

Solve multiplicative comparison problems. Distinguish multiplicative comparison from additive comparison.

2

4.OA.A.1

4.OA.A.2

Solve multiplicative comparison problems with a smaller unknown, including cases that involve difficult language.

3

4.OA.A.1

4.OA.A.2

Solve multiplicative comparison problems with an unknown multiplier. Interpret a multiplication equation as a comparison.

Topic B: Factors and Multiples

4

4.OA.B.4

Identify multiples and determine if a whole number is a multiple of another number.

5

4.OA.A.3

4.OA.B.4

Explore patterns in multiples of various whole numbers.

6

4.OA.A.3

4.OA.B.4

Find factor pairs for numbers to 100 and recognize that a whole number is a multiple of each of its factors.

7

4.OA.B.4

Determine whether a given number is prime or composite.

8

4.OA.B.4

Solve contextual problems that involve factors, multiples, prime, composite, and/or square numbers.

Topic C: Multiplication of up to Four-Digit Whole Numbers by One-Digit Whole Numbers

9

4.NBT.B.5

Multiply 10, 100, and 1,000 by one- and two-digit numbers.

10

4.NBT.B.5

Multiply multiples of 10, 100, and 1,000 by one-digit numbers. Estimate multi-digit products by rounding numbers to their largest place value.

11

4.NBT.B.5

Multiply two-, three-, and four-digit numbers by one-digit numbers using a variety of mental strategies.

12

4.NBT.B.5

Multiply two-digit numbers by one-digit numbers.

13

4.NBT.B.5

Multiply three-digit numbers by one-digit numbers.

14

4.NBT.B.5

Multiply four-digit numbers by one-digit numbers.

15

4.NBT.B.5

Multiply four-digit, three-digit, and two-digit numbers by one-digit numbers and assess the reasonableness of the product.

Topic D: Multiplication of Two-Digit Whole Numbers by Two-Digit Whole Numbers

16

4.NBT.B.5

Multiply two-digit multiples of 10 by two-digit multiples of 10. Estimate multi-digit products by rounding numbers to their largest place value.

17

4.NBT.B.5

Multiply two-digit multiples of 10 by two-digit numbers.

18

4.NBT.B.5

Multiply two-digit numbers by two-digit numbers using a variety of mental strategies.

19

4.NBT.B.5

Multiply two-digit by two-digit numbers using four partial products.

20

4.NBT.B.5

Multiply two-digit by two-digit numbers using two partial products and assess the reasonableness of the product.

Topic E: Multi-Step Word Problems

21

4.OA.A.3

4.MD.A.3

Abstract the formulas for the area and perimeter of a rectangle and apply those formulas in real-world and mathematical problems involving multiplication, addition, and subtraction.

22

4.OA.A.2

4.OA.A.3

Solve two-step word problems involving multiplication, addition, and subtraction and assess the reasonableness of answers.

23

4.OA.A.3

Solve multi-step word problems involving multiplication, addition, and subtraction and assess the reasonableness of answers.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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.

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.

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.

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.

Operations and Algebraic Thinking

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.

Operations and Algebraic Thinking

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.

Operations and Algebraic Thinking

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.

Foundational Standards

Measurement and Data

3.MD.C.7.B

Measurement and Data

3.MD.C.7.B — Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.D.8

Measurement and Data

3.MD.D.8 — Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

Number and Operations in Base Ten

3.NBT.A.3

Number and Operations in Base Ten

3.NBT.A.3 — Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
4.NBT.A.1

Number and Operations in Base Ten

4.NBT.A.1 — Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
4.NBT.A.2

Number and Operations in Base Ten

4.NBT.A.2 — Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.A.3

Number and Operations in Base Ten

4.NBT.A.3 — Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.B.4

Number and Operations in Base Ten

4.NBT.B.4 — Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Operations and Algebraic Thinking

3.OA.A.1

Operations and Algebraic Thinking

3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.A.2

Operations and Algebraic Thinking

3.OA.A.2 — Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.A.3

Operations and Algebraic Thinking

3.OA.A.3 — Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
3.OA.A.4

Operations and Algebraic Thinking

3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?.
3.OA.B.5

Operations and Algebraic Thinking

3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
3.OA.B.6

Operations and Algebraic Thinking

3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
3.OA.C.7

Operations and Algebraic Thinking

3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.OA.D.8

Operations and Algebraic Thinking

3.OA.D.8 — Solve two-step word problems using the four operations. 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. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Future Standards

Number and Operations in Base Ten

4.NBT.B.6

Number and Operations in Base Ten

4.NBT.B.6 — Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.B.5

Number and Operations in Base Ten

5.NBT.B.5 — Fluently multiply multi-digit whole numbers using the standard algorithm.

Number and Operations—Fractions

4.NF.A.1

Number and Operations—Fractions

4.NF.A.1 — Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
4.NF.B.4

Number and Operations—Fractions

4.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
5.NF.B.3

Number and Operations—Fractions

5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4

Number and Operations—Fractions

5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.5

Number and Operations—Fractions

5.NF.B.5 — Interpret multiplication as scaling (resizing), by:
5.NF.B.5.A

Number and Operations—Fractions

5.NF.B.5.A — Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5.NF.B.5.B

Number and Operations—Fractions

5.NF.B.5.B — Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
5.NF.B.6

Number and Operations—Fractions

5.NF.B.6 — Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

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.