Multiplication and Division, Part 2

Students deepen their understanding of multiplication and division, including their properties and extending their study of factors to include all units from 0 to 10, as well as multiples of 10 within 100.

Unit Summary

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 units, 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. Finally, the unit culminates with a focus on categorical data, where students draw and solve problems involving scaled picture graphs and scaled bar graphs, a nice application of the major work of multiplication and division.

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, upon which they will rely 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. Because of the increased difficulty of these facts, students will rely on both skip-counting (a Level 2 strategy) as well as converting 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 x 7, think of 5 x 7 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. Then, 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. Finally, students explore picture graphs in which each picture represents more than one object and bar graphs where the scale on the axis is more than 1, a key development from Grade 2 (3.MD.3). As the Progressions note, “these developments connect with the emphasis on multiplication in this grade” (MD Progression, p. 7). Students also solve one- and two-step word problems related to the data in these plots, relying on the extensive work students have done with word problems throughout the year. Thus, this supporting cluster standard nicely enhances the major work they’ve been working on throughout this and the previous unit.

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 equations 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’ understanding will be entirely dependent on their conceptual understanding and fluency with these operations—everything from multi-step multiplicative comparison words problems in Grade 4 to polynomial multiplication and division in Algebra 2, and lots in between. Thus, this unit culminates major work of Grade 3 as well as deeply important foundational work upon which students will rely for years to come.

Pacing: 31 instructional days (28 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 3rd Grade Scope and Sequence Recommended Adjustments.

Subscribe to Fishtank Plus to unlock access to additional resources for this unit, including:

Expanded Assessment Package
Problem Sets for Each Lesson
Student Handout Editor
Google Classroom Integration
Vocabulary Package

Assessment

This assessment accompanies Unit 3 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.
Read the following table that includes models used throughout the unit.

Equal groups	Example: 4 equal groups of 3 stars
Array	Example: 4 rows of 3
Tape diagram	Example: There are 4 bags with 3 plums in each bag. How many plums are there in all?
Concrete or pictorial base ten blocks	Example: Find $$2 \times 60$$ using base ten blocks.
Picture graph
Bar graph

Essential Understandings

Multiplication problems can be solved using a variety of strategies of increasing complexity, including making and counting all of the quantities involved in multiplication or division (Level 1 strategy), repeated counting on by a given number (Level 2), and using the properties of operations to compose and decompose unknown facts into known ones (Level 3).
Unknown numbers in a count sequence can be found by using mental strategies. “For example, in the count-by for 7, students might use the following mental decompositions of 7 to compose up to and then go over the next decade, e.g., 14 + 7 = 14 + 6 + 1 = 20 + 1 = 21” (OA Progression, p. 25).
Two, five, and ten are often helpful values to use when using properties to compose and decompose.
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.

Unit Materials, Representations and Tools

Arrays
Equal groups
Tape diagrams
Base ten blocks
Multiplication Chart
Picture graphs
Bar graphs

Vocabulary

distributive property

associative property

scale

key

parentheses

Related Teacher Tools:

3rd Grade Vocabulary Glossary

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.

View Preview

Lesson Map

Topic A: Introduction to The Properties of Operations

1

3.OA.B.5

3.OA.D.9

Study commutativity to find known facts of 6, 7, 8, and 9.

2

3.OA.A.4

3.OA.B.5

3.OA.C.7

3.OA.D.9

Understand the zero and identity properties of multiplication.

3

3.OA.B.5

3.OA.D.8

Understand the role of parentheses and apply to solving problems.

4

3.OA.B.5

3.OA.D.9

Introduce the distributive property of multiplication.

5

3.OA.B.5

3.OA.D.9

Introduce the associative property of multiplication.

Topic B: Multiplication and Division by 6 and 7

6

3.OA.A.4

3.OA.C.7

Skip-count to build fluency with multiplication facts using units of 6. Decompose an addend with respect to a ten when going over a decade in the skip-counting sequence.

7

3.OA.A.4

3.OA.C.7

Skip-count to build fluency with multiplication facts using units of 7. Decompose an addend with respect to a ten when going over a decade in the skip-counting sequence.

8

3.OA.B.5

3.OA.C.7

3.OA.D.9

Use the associative property as a strategy to multiply by units of 6.

9

3.OA.B.5

3.OA.C.7

3.OA.D.9

Use the distributive property as a strategy to multiply by units of 6.

10

3.OA.B.5

3.OA.C.7

3.OA.D.9

Use the distributive property as a strategy to multiply by units of 7.

11

3.OA.A.3

3.OA.D.8

Solve one- and two-step word problems involving units up to 7.

Topic C: Multiplication and Division by 8 and 9

12

3.OA.A.4

3.OA.C.7

Skip-count to build fluency with multiplication facts using units of 8 and 9. Decompose an addend with respect to a ten when going over a decade in the skip-counting sequence.

13

3.OA.B.5

3.OA.C.7

3.OA.D.9

Use the associative property as a strategy to multiply by units of 8 and 9.

14

3.OA.B.5

3.OA.C.7

3.OA.D.9

Use the distributive property as a strategy to multiply by units of 8 and 9.

15

3.OA.B.5

3.OA.C.7

3.OA.D.9

Use the distributive property as a strategy to multiply by units of 8 and 9, including the subtractive use of the distributive property.

16

3.OA.A.3

3.OA.D.8

Solve one- and two-step word problems involving units up to 9.

Topic D: Multiplication and Division by Values Greater than 10

17

3.OA.A.4

3.OA.B.5

3.NBT.A.3

Multiply one-digit whole numbers by multiples of 10.

18

3.OA.A.4

3.OA.B.5

3.NBT.A.3

Multiply one-digit whole numbers by two-digit whole numbers using the associative and distributive properties.

Topic E: Two-Step Word Problems and Patterns in Arithmetic

19

3.OA.D.8

Solve two-step word problems involving all four operations and assess the reasonableness of answers.

20

3.OA.D.8

3.NBT.A.3

Solve two-step word problems involving all four operations and assess the reasonableness of answers.

21

3.OA.D.9

Identify patterns in single or multiple rows/columns of the multiplication table.

22

3.OA.D.9

Identify other patterns in the multiplication table.

23

3.OA.D.9

Identify arithmetic patterns and explain them using properties of operations.

Topic F: Scaled Picture and Bar Graphs

24

3.MD.B.3

Create scaled picture graphs where the scale is provided.

25

3.MD.B.3

Create scaled picture graphs where the scale must be determined.

26

3.MD.B.3

Create scaled bar graphs where the scale is provided.

27

3.MD.B.3

Create scaled bar graphs where the scale must be determined.

28

3.OA.D.8

3.MD.B.3

Solve one- and two-step word problems using information presented in scaled picture and bar graphs.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Measurement and Data

3.MD.B.3 — Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Measurement and Data

3.MD.B.3 — Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

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.

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.

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.

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 — 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 = ?.

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 — 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.)

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.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.

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 — 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).

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).
3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Operations and Algebraic Thinking

3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Foundational Standards

Measurement and Data

2.MD.D.10

Measurement and Data

2.MD.D.10 — Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Number and Operations in Base Ten

2.NBT.A.2

Number and Operations in Base Ten

2.NBT.A.2 — Count within 1000; skip-count by 5s, 10s, and 100s.
3.NBT.A.2

Number and Operations in Base Ten

3.NBT.A.2 — Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Operations and Algebraic Thinking

2.OA.A.1

Operations and Algebraic Thinking

2.OA.A.1 — Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
2.OA.C.3

Operations and Algebraic Thinking

2.OA.C.3 — Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
2.OA.C.4

Operations and Algebraic Thinking

2.OA.C.4 — Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
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.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.

Future Standards

Number and Operations in Base Ten

4.NBT.B.5

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.
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.

Number and Operations—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.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.
5.NF.B.7

Number and Operations—Fractions

5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade

Operations and Algebraic Thinking

4.OA.A.1

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

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

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

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.

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.