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Grade 3 UNIT 1: Properties of Multiplication and Division and Problem Solving with Factors 2-5, & 10 Unit Instructional Days: 25

|Essential Question |Key Concepts |Cross Curricular Connections |

|How are models used to represent multiplication and |Multiplication and the meaning of the factors |Art: Represent a multiplication fact using household objects of their choice (ex: |

|division? |Division as an unknown factor problem |cotton balls, egg cartons, candy wrappers etc.). |

| |Analyze arrays to multiply using Units 2 and 3* | |

| |Division using units of 2 and 3 |Religion: As Catholics, we learn from Jesus how to love and be kind to one |

|Unit Vocabulary |Multiplication and division using units of 4 |another. Examine how kind acts grow – If one person does two good acts, then each |

| |Distributive property and problem solving using units of 2 – 5 and 10** |recipient does two good acts and so on, can you envision how good acts would |

|Array | |multiply?  Illustrate. Create word problems describing what was served at the Last|

|Column | |Supper. Ex: If each of the 12 apostles received two pieces of bread, how many |

|Commute |*Assessments |pieces were given out in total? (12 x 2 = 24) |

|Commutative Property |*Mid-Module Assessment: After Section C | |

|Equal groups |(2 days, included in Unit Instructional Days) |Science: Animal Leg Multiplication – calculate how many legs multiple sets of |

|Equations | |lions, spiders, centipedes, etc., have in total. Also, divide groups of animals |

|Distribute |**End-of-Module Assessment: after Section F (2days, included in Unit |into their correct habitat. Figure out how many are in each group. |

|Distributive Property |Instructional Days) | |

|Divide | | |

|Mathematical Practices |

|MP.1 Make sense of problems and persevere in solving them. Students model multiplication and division using the array model. They solve two-step mixed word problems and assess the reasonableness of their solutions. |

|MP.2 Reason abstractly and quantitatively. Students make sense of quantities and their relationships as they explore the properties of multiplication and division and the relationship between them. Students |

|decontextualize when representing equal group situations as multiplication, and when they represent division as partitioning objects into equal shares or as unknown factor problems. Students contextualize when they |

|consider the value of units and understand the meaning of the quantities as they compute. |

|MP.3 Construct viable arguments and critique the reasoning of others. Students represent and solve multiplication and division problems using arrays and equations. As they compare methods, they construct arguments and |

|critique the reasoning of others. This practice is particularly exemplified in daily application problems and problem-solving specific lessons in which students solve and reason with others about their work. |

|MP.4 Model with mathematics. Students represent equal groups using arrays and equations to multiply, divide, add, and subtract. |

|MP.7 Look for and make use of structure. Students notice structure when they represent quantities by using drawings and equations to represent the commutative and distributive properties. The relationship between |

|multiplication and division also highlights structure for students as they determine the unknown whole number in a multiplication or division statement. |

|Unit Outcome (Focus) |

|This 25-day unit begins the year by building on students’ fluency with addition and knowledge of arrays. In this unit the factors 2, 3, 4, 5, and 10 provide an entry point for moving into more difficult factors in |

|later units. |

UNIT 1 SECTION A: Understand Equal Groups of as Multiplication Instructional Days: 3

|Essential Question |Key Objectives |

| |Understand equal groups of as multiplication |

| |Relate multiplication to the array model |

| |Interpret the meaning of factors- the size of the group or the number of groups |

|How do we use models to represent multiplication and division? | |

|Comments |Standard No. |Standard |Priority |

|Lessons in Section A move students toward understanding familiar | |( Major Standard ( Supporting Standard ( Additional Standard | |

|repeated addition from Grade 2 in the form of array models, which| |( Standard ends at this grade ( Fluency Standard | |

|become a cornerstone of the module. Students use the language of | | | |

|multiplication as they understand what factors are and | | | |

|differentiate between the size of groups and the number of groups| | | |

|within a given context. | | | |

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| |2.OA.4 |Use repeated addition to find the total from a number of equal groups. |( |

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| |3.OA.1 |Represent and solve problems involving multiplication and division. |( |

| |(DOK 2) |Interpret products of whole numbers, e.g., interpret 5 x 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 x 7. | |

| | | | |

| | |Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and | |

| |3.OA.3 |measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. |( |

| |(DOK 1) | | |

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UNIT 1 SECTION B: Division as an Unknown Factor Problem Instructional Days: 3

|Essential Question |Key Objectives |

| |Understand the meaning of the unknown as the size of the group in division |

| |Understand the meaning of the unknown as the number of groups in division |

| |Interpret the unknown in division using the array model |

|How do we use models to represent | |

|multiplication and division? | |

|Comments |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

| | |( Standard ends at this grade ( Fluency Standard | |

| Study of factors links Sections A and B; |3.OA.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 | ( |

|Section B extends to the study of |(DOK 2) |partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For | |

|division. Students understand division as | |example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. | |

|an unknown factor problem, and relate the | | | |

|meaning of unknown factors to either the | |Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. | |

|number or the size of groups (3.OA.2 & | | | |

|3.OA.6). By the end of section B students |3.OA.6 | |( |

|are aware of the fundamental connection |(DOK 1) |Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement | |

|between multiplication and division that | |quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Glossary, Table| |

|sets the foundation for the rest of the | |2.) | |

|Unit. |3.OA.3 | | |

| |(DOK 1) |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.4 | | |

| |(DOK 1) | |( |

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UNIT 1 *SECTION C: Analyze Arrays to Multiply Using Units of 2 and 3 Instructional Days: 4

|Essential Question | Key Objectives |

|How do we use models to represent |Demonstrate the commutativity of multiplication and practice related facts by skip-counting objects in array models. |

|multiplication and division? |Find related multiplication facts by adding and subtracting equal groups in array models. |

| |Model the distributive property with arrays to decompose units as a strategy to multiply |

|Comments |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

| | |( Standard ends at this grade ( Fluency Standard | |

|Students use the array model and familiar |3.OA.1 |Represent and solve problems involving multiplication and division. |( |

|skip-counting strategies to solidify their |(DOK2) |Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For | |

|understanding of multiplication and practice | |example, describe a context in which a total number of objects can be expressed as 5 x 7. | |

|related facts of 2 and 3. They become fluent | | | |

|enough with arithmetic patterns to “add” or | |Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) | |

|“subtract” groups from known products to | |Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found | |

|solve more complex multiplication problems |3.OA.5 |by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 |( |

|(3.OA.1, 3.OA.9). They apply their skills to |(DOK1) |= 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)3 | |

|word problems using drawings and equations | | | |

|with a symbol to find the unknown factor | |Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement | |

|(3.OA.3). This culminates in students using | |quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (See Glossary, | |

|arrays to model the distributive property as | |Table 2.) | |

|they decompose units to multiply (3.OA.5). |3.OA.3 | | |

| |(DOK1) |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.4 | | |

| |(DOK1) | |( |

UNIT 1 SECTION D: Division Using Units of 2 and 3 Instructional Days: 3

|Essential Question |Key Objectives |

| |Model division as the unknown factor in multiplication using arrays and tape diagrams |

| |Interpret the quotient as the number of groups or the number of objects in each group using units of 2 |

| |Interpret the quotient as the number of groups or the number of objects in each group using units of 3 |

|How do we use models to represent | |

|multiplication and division? | |

|Comment |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

| | | | |

| | |( Standard ends at this grade ( Fluency Standard | |

|In Section D students model, write and solve|3.OA.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 |( |

|division problems with 2 and 3 (3.OA.2). |(DOK2) |partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For | |

|Consistent skip-counting strategies and the | |example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. | |

|continued use of array models are pathways | | | |

|for students to naturally relate | |Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the | |

|multiplication and division. Modeling |3.OA.4 |unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ? |( |

|advances as students use tape diagrams to |(DOK1) | | |

|represent multiplication and division. A | |Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. | |

|final lesson in this topic solidifies a | | | |

|growing understanding of the relationship |3.OA.6 |Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing |( |

|between operations (3.OA.7). |(DOK1) |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.7 | | |

| |(DOK1) |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.3 |Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown | |

| |(DOK1) |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, i.e., Order of Operations.) | |

| |3.OA.8 | | |

| |(DOK2) | |( |

UNIT 1 SECTION E: Multiplication and Division Using Units of 4 Instructional Days: 4

|Essential Question |Key Objectives |

|How do we use models to represent multiplication |Skip –count objects in models to build fluency in multiplication facts using units of 4 |

|and division? |Relate arrays to tape diagrams to model the commutative property of multiplication |

| |Use the distributive property as a strategy to find related multiplication facts |

| |Model the relationship between multiplication and division |

|Comment |Standard No. | Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

| | |( Standard ends at this grade ( Fluency Standard | |

|Section E shifts students from simple |3.OA.5 |Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) |( |

|understanding to analyzing the relationship |(DOK1) |Examples: 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| |

|between multiplication and division. Practice of | |× 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and | |

|both operations is combined—this time using units| |8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)3 | |

|of 4—and a lesson is explicitly dedicated to | | | |

|modeling the connection between them (3.OA.7). | |Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing | |

|Skip-counting, the distributive property, arrays,|3.OA.7 |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 |( |

|number bonds and tape diagrams are tools for both|(DOK1) |one-digit numbers. | |

|operations (3.OA.1, 3.OA.2, 3.OA.9). A final | | | |

|lesson invites students to explore their work | |Interpret products of whole numbers. |( |

|with arrays and related facts through the lens of|3.OA.1 | | |

|the commutative property as it relates to |(DOK2) | | |

|multiplication (3.OA.5). | |Interpret whole-number quotients of whole numbers. |( |

| |3.OA.2 | | |

| |(DOK2) | | |

| | |Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement |( |

| |3.OA.3 |quantities. | |

| |(DOK1) | | |

| | |Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the |( |

| |3.OA.4 |unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ? | |

| |(DOK1) | | |

| | |Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. |( |

| | |(DOK1) | |

| | | | |

| |3.OA.6 | | |

UNIT 1 *SECTION F: Distribute Property and Problem Solving Using Units of 2 – 5 and 10 Instructional Days: 4

|Essential Question |Key Concept |

|How do we use models to represent multiplication and |Apply the distributive property to decompose units |

|division? |Solve two-step word problems involving multiplication and division and assess the reasonableness of answers |

| |Solve two-step word problem involving all four operations and assess the reasonableness of answers |

|Comment |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

| | | | |

| | |( Standard ends at this grade ( Fluency Standard | |

|Section F introduces the factors 5 and 10, familiar |3.OA.3 |Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement | |

|from skip-counting in Grade 2. Students apply the |(DOK1) |quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. |( |

|multiplication and division strategies they have used | |Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) | |

|to mixed practice with all of the factors included in | |Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by | |

|Module 1 (3.OA.1, 3.OA.2, 3.OA.3). Students model |3.OA.5 |3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 |( |

|relationships between factors, analyzing the |(DOK1) |and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) | |

|arithmetic patterns that emerge to compose and | |Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing| |

|decompose numbers as they further explore the | |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 | |

|relationship between multiplication and division | |one-digit numbers. | |

|(3.OA.3, 3.OA.5, 3.OA.7, 3.OA.9). Students apply the | |Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the | |

|tools and concepts they have learned to |3.OA.7 |unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This |( |

|problem-solving with multi-step word problems using |(DOK1) |standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform | |

|all four operations (3.OA.3, 3.OA.8). They demonstrate| |operations in the conventional order when there are no parentheses to specify a particular order, i.e., Order of Operations.) | |

|the flexibility of their thinking as they assess the | |Interpret products of whole numbers. (DOK2) | |

|reasonableness of their answers for a variety of |3.OA.8 |Interpret whole-number quotients of whole numbers.(DOK2) |( |

|problem types. |(DOK2) |Determine the unknown whole number in a multiplication or division equation relating three whole numbers. (DOK1) | |

| | | | |

| | |Understand division as an unknown-factor problem. (DOK1) | |

| | | | |

| | | |( |

| |3.OA.1 | |( |

| |3.OA.2 | |( |

| |3.OA.4 | | |

| | | |( |

| | | | |

| |3.OA.6 | | |

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Possible Activities (Applicable for ALL Sections)

FACT FAMILY ACTIVITY: Provide students with four numbers, three of which are part of a fact family and one does not belong. The students must correctly determine which number does not belong, then write a multiplication fact and draw an array using the three numbers in the fact family. Great worksheets and activities found at math- click on Fact Family link on the left.

READ THE BOOK ONE HUNDRED HUNGRY ANTS to the class. 100 hungry ants smell a picnic, and line up single file. The smallest ant suggests that it would be faster if they go by 2s, then 4s, then 5s, and then 10s. Can they get to the picnic in time? This book can be used to introduce division and the factors of 100. Ant cutouts could also be arranged in arrays to show rectangular and square arrangements, as well as the commutative property. Follow up activities can be found online. Great online follow up activities for One Hundred Hungry Ants: math. Click on Mathematics and Literature link and then select Multiplication and Division to find the resources for Hungry Ants.

MODELING MULTIPLICATION ACTIVITIES: Ask students to arrange a set of 24, 48, or 64 unifix cubes into as many equal groups as possible. They can also draw the arrays on paper if unifix cubes are not available. An interactive website is available. Interactive Student Activity: Space Arrays: Google search “space arrays” to find a great interactive student website on understanding arrays.

MODELING DIVISION ACTIVITIES: Ask students to arrange 24 (48, 64) unifix cubes into as many equal groups as possible. They can also draw groups on paper if unifix cubes are not available. Check out nlvm.usu.edu to support multiplying and dividing numbers by an area representation. Click on Number & Operations Grades Pre-K–2 for the Rectangle Multiplication and Rectangle Division models.

PROBLEM-SOLVING ACTIVITIES: Combine students in groups to go through the problem-solving process using standards of practice as a reference. Ex: Ruth, Roz, and Gladys bought two boxes of golf balls. Each box contains five sleeves of three balls in each. How many golf balls will each person get? Write to help explain your best thinking using words, numbers, or pictures. (Answer: 10 golf balls)

Ex: There are 10 children in the art club. The teacher put them into groups, so that each group has the same number of children. How many groups could there be? How many children will be in each group? Write to help explain your best thinking using words, numbers, or pictures. (Answer: 2 groups of 5) Check out drycreek for great problems solving problems. Click on Family Resources and then Math WASL Prompts.

MATCHING WORD PROBLEMS: Ask students to create word problems that match given the multiplication problems: i.e., 6 x ? = 42; 4 x 3 = ?; ? x 7 = 49. Ask students to create word problems that match given division problems: i.e., 64 ÷ 8 = ? ; ? ÷ 4 = 8; 24 ÷ ? = 8. Ex: Tommy bought 56 light bulbs. If each pack of light bulbs contained 8 bulbs, how many packs did he buy? (Answer: 7 packs) For additional word problems with unknowns visit . Click on State Standards/Common Core/Third Grade/3.OA.4 or any other third grade standard.

MULTIPLICATION MADNESS: A great game to practice multiplication facts and develop reasoning and problem solving skills. Materials: Two sets of colored markers (chips), two paper clips. Player A places a paper clip on two numbers at the bottom of the game board (or on the same two numbers). The students will say the product and cover the matched product in a square on the game board with a game chip. Player B may move only ONE of the paper clips to a different number. She/he multiplies the two numbers together and places a different colored chip on the board. Play continues in this manner until one player has four chips in a row vertically, horizontally, or diagonally. Multiplication Madness game boards can be downloaded for free at . Click on the link Math Games/Activities on the top right.

1. MULTIPLICATION: Some students may be challenged to explore arrays using two-digit numbers.

2. DIVISION: Some students may be challenged to explore equal groups using two digit numbers.

3. MULTIPLICATION FACTS: Challenge students to see how many multiplication facts they can solve in one minute (goal is 28).

4. DIVISION FACTS: Challenge students to see how many division facts they can solve in one minute (goal is 28). Flashcards are available online (see right).

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