Third Grade Unit 2: The Relationship Between Multiplication and Division5 weeksIn this unit students will: Begin to understand the concepts of multiplication and division Learn the basic facts of multiplication and their related division facts Apply properties of operations (commutative, associative, and distributive) as strategies to multiply and divide Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Fluently multiply and divide within 100, using strategies such as the patterns and relationships between multiplication and division Understand multiplication and division as inverse operations Solve problems and explain their processes of solving division problems that can also be represented as unknown factor multiplication problems. Unit Resources:

Unit 2 Overview Video Parent Letter (Spanish) Parent Guides Number Talks Vocabulary Cards Prerequisite Skills Assessment Sample Post Assessment Student-Friendly Standards Concept Map

Topic 1: Represent and Solve Problems Involving Multiplication and Division Big Ideas/Enduring Understandings: Multiplication and division are inverse operations. Multiplication and division can be modeled with arrays. There are two common situations where division may be used.

Partition (or fair-sharing) - given the total amount and the number of equal groups, determine how many/much in each group Measurement (or repeated subtraction) - given the total amount and the amount in a group, determine how many groups of the same size can be created.

Essential Questions: How are multiplication and division related? How can you write a mathematical sentence to represent a multiplication or division model we have made? How do estimation, multiplication, and division help us solve problems in everyday life?

Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.Represent and solve problems involving multiplication and division.

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MGSE3.OA.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.

MGSE3.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 partitioned equally into 8 shares (How many in each group?), or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each (How many groups can you make?). For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

MGSE3.OA.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.1

MGSE3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers using the inverse relationship of multiplication and division. For example, determine the unknown number that makes the equation true in each of the equations, 8 × ? = 48, 5 = □ ÷ 3, 6 × 6 =?.

Vertical Alignment Second-Grade Standards

Work with equal groups of objects to gain foundations for multiplication MGSE2.OA.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

MGSE2.OA.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.

Understand place value MGSE2.NBT.2 Skip-count by 5s, 10s, and

100s, 10 to 100 and 100 to 1000. Reason with shapes and their attributes MGSE2.G.2 Partition a rectangle into

rows and columns of same-size squares and count to find the total number of them.

Fourth-Grade StandardsUse the four operations with whole numbers to solve problems MGSE4.OA.1 Understand that a multiplicative comparison

is a situation in which one quantity is multiplied by a specified number to get another quantity. a. 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.

b. Represent verbal statements of multiplicative comparisons as multiplication equations

MGSE4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison. Use drawings and equations with a symbol or letter for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

MGSE4.OA.3 Solve multistep word problems 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 symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Fifth-Grade StandardsPerform operations with multi-digit whole numbers and with decimals to hundredths MGSE5.NBT.5 Fluently multiply multi-digit

whole numbers using the standard algorithm (or other strategies demonstrating understanding of multiplication) up to a 3 digit by 2 digit factor.

MGSE5.NBT.6 Fluently divide up to 4-digit dividends and 2-digit divisors by using at least one of the following methods: 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 or concrete models. (e.g., rectangular arrays, area models)

MGSE5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

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Multiplication and Division Instructional Strategies“Multiplication and division are commonly taught separately. However, it is very important to combine the two shortly after multiplication has been introduced. This will help the students to see the connection between the two.” (Van de Walle and Lovin, Teaching Student-Centered Mathematics 3-5, p. 60)

OA.1The standard interprets products of whole numbers. Students need to recognize multiplication as a means of determining the total number of objects when there are a specific number of groups with the same number of objects in each group. Multiplication requires students to think in terms of groups of things rather than individual things. At this level, Multiplication is seen as “groups of” and problems such as 5 x 7 refer to 5 groups of 7.

Sets of counters and tiles are great manipulatives to illustrate equal groups. This will aid students in solving both multiplication and division problems. They should represent the model first, then students should illustrate with drawings attaching the equations to all models.

This shows multiplication using grouping with 3 groups of 5 objects and can be written as 3 × 5.

Third graders begin division by sharing. Example: Three students need to share 12 trapezoids equally.

It is important for teachers to understand there are several ways in which think of multiplication:Multiplication is often thought of as repeated addition of equal groups. While this definition works for some sets of numbers, it is not particularly intuitive or meaningful when we think of multiplying 3 by 1/2, for example, or 5 by -2. In such cases, it may be helpful to widen the idea of grouping to include evaluation of

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part of a group. This concept is related to partitioning (which, in turn, is related to division).

Example: Three groups of five students can be read as 3 • 5, or 15 students, while half a group of 10 stars can be represented as 1/2 • 10, or 5 stars. These are examples of partitioning; each one of the three groups of five is part of the group of 15, and the group of 5 stars is part of the group of 10.

A second concept of multiplication is that of rate or price.

Example: If a car travels four hours at 50 miles per hour, then it travels a total of 4 • 50, or 200 miles; if CDs cost eight dollars each, then three CDs will cost 3 • $8, or $24.

A third concept of multiplication is that of multiplicative comparison.

Example: Sara has four CDs, Joanne has three times as many as Sara, and Sylvia has half as many as Sara. Thus, Joanne has 3 • 4, or 12 CDs, and Sylvia has 1/2 • 4, or 2 CDs.Example: Jim purchased 5 packages of muffins. Each package contained 3 muffins. How many muffins did Jim purchase? 5 groups of 3, 5 x 3 = 15.

Describe another situation where there would be 5 groups of 3 or 5 x 3.

Students recognize multiplication as a means to determine the total number of objects when there are a specific number of groups with the same number of objects in each group. Multiplication requires students to think in terms of groups of things rather than individual things. Students learn that the multiplication symbol ‘x’ means “groups of” and problems such as 5 x 7 refer to 5 groups of 7.

To further develop this understanding, students interpret a problem situation requiring multiplication using pictures, objects, words, numbers, and equations. Then, given a multiplication expression (e.g., 5 x 6) students interpret the expression using a multiplication context. They should begin to use the terms, factor and product, as they describe multiplication.

In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 x 5, and wrote equations to represent the sum. This strategy is a foundation for multiplication because students should make a connection between repeated addition and multiplication.

Students need to experience problem-solving involving equal groups (whole unknown or size of group is unknown) and multiplicative comparison (unknown product, group size unknown or number of groups unknown).

Student should be encouraged to solve these problems in different ways to show the same idea and be able to explain their thinking verbally and in written expression. Allowing students to present several different strategies provides the opportunity for them to compare strategies.

Sets of counters, number lines to skip count and relate to multiplication and arrays/area models will aid students in solving problems involving multiplication and division. Allow students to model problems using these tools.

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Students should represent the model used as a drawing or equation to find the solution.

Show a variety of models of multiplication. (i.e. 3 groups of 5 counters can be written as 3 × 5).

Provide a variety of contexts and tasks so that students will have ample opportunity to develop and use thinking strategies to support and reinforce learning of basic multiplication and division facts.

Ask students create multiplication problem situations in which they interpret the product of whole numbers as the total number of objects in a group. Ask them to write as an expression. Also, have students create division-problem situations in which they interpret the quotient of whole numbers as the number of shares.

Students can use known multiplication facts to determine the unknown fact in a multiplication or division problem. Have them write a multiplication or division equation and the related multiplication or division equation. For example, to determine the unknown whole number in 27 ÷ ? = 3, students should use knowledge of the related multiplication fact of 3 × 9 = 27. They should ask themselves questions such as, “How many 3s are in 27?” or “3 times what number is 27?” Have them justify their thinking with models or drawings.

OA.2This standard focuses on two distinct models of division: partition models and measurement (repeated subtraction) models. Partition models focus on the question, “How many in each group?” A context for partition models would be: There are 12 cookies on the counter. If you are

sharing the cookies equally among three bags, how many cookies will go in each bag? Measurement (repeated subtraction) models focus on the question, “How many groups can you make?” A context or measurement models would be: There

are 12 cookies on the counter. If you put 3 cookies in each bag, how many bags will you fill?

Students need to recognize the operation of division in two different types of situations. One situation requires determining how many groups and the other situation requires sharing (determining how many in each group). Students should be exposed to appropriate terminology (quotient, dividend, divisor, and factor).

To develop this understanding, students interpret a problem situation requiring division using pictures, objects, words, numbers, and equations. Given a division expression (e.g., 24 ÷ 6) students interpret the expression in contexts that require both interpretations of division.

Common Multiplication and Division Situations:

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OA.3This standard references various strategies that can be used to solve word problems involving multiplication & division. Students should apply their skills to solve word problems. Students should use a variety of representations for creating and solving one-step word problems, such as: If you divide 4 packs of 9 brownies among 6 people, how many brownies does each person receive? (4 x 9 = 36, 36 ÷ 6 = 6).

See the table above for examples of a variety of problem solving contexts, in which students need to find the product, the group size, or the number of groups. Students should be given ample experiences to explore and make sense of ALL the different problem structures.

Examples of Multiplication:6

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There are 24 desks in the classroom. If the teacher puts 6 desks in each row, how many rows are there? This task can be solved by drawing an array by putting 6 desks in each row. This is an array model.

This task can also be solved by drawing pictures of equal groups. 4 groups of 6 equals 24 objects

A student could also reason through the problem mentally or verbally, “I know 6 and 6 are 12. 12 and 12 are 24. Therefore, there are 4 groups of 6 giving a total of 24 desks in the classroom.”

A number line could also be used to show jumps of equal distance.

Students in third grade students should use a variety of pictures, such as stars, boxes, flowers to represent unknown numbers (variables). Letters are also introduced to represent unknowns in third grade.

Examples of Division:There are some students at recess. The teacher divides the class into 4 lines with 6 students in each line. Write a division equation for this story and determine how many students are in the class (n ÷ 4 = 6. There are 24 students in the class).

Determining the number of objects in each share (partitive division, where the size of the groups is unknown):The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many hair clips will each person receive?Determining the number of shares (measurement division, where the number of groups is unknown).

Students use a variety of representations for creating and solving one-step word problems, i.e., numbers, words, pictures, physical objects, or equations. They use multiplication and division of whole numbers up to 10 x10. Students explain their thinking, show their work by using at least one representation, and verify that their answer is reasonable.

Word problems may be represented in multiple ways:

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Examples of Division Problems:Determining the number of objects in each share (partitive division, where the size of the groups is unknown):Example: The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many hair clips will each person receive?

Determining the number of shares (measurement division, where the number of groups is unknown):Example: Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, how many days will the bananas last?

Solution: The bananas will last for 6 days.

OA.4This standard refers to the equations for the different types of multiplication and division problem structures (see the table above). The easiest problem structure includes Unknown Product (3 x 6 = ? or 18 ÷ 3 = 6). The more difficult problem structures include Group Size Unknown (3 x ? = 18 or 18 ÷ 3 = 6) or Number of Groups Unknown (? x 6 = 18, 18 ÷ 6 = 3).

The focus of 3.OA.4 goes beyond the traditional notion of fact families, by having students explore the inverse relationship of multiplication and division. Students apply their understanding of the meaning of the equal sign as ”the same value as” to interpret an equation with an unknown. When given 4 x ? = 40, they might think:

4 groups of some number is the same as 40

Students apply their understanding of the meaning of the equal sign as ”the same value as” to interpret an equation with an unknown. When given 4 x ? = 40, they might think:

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4 groups of some number is the same as 40 4 times some number is the same as 40 I know that 4 groups of 10 is 40 so the unknown number is 10 The missing factor is 10 because 4 times 10 equals 40.

Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.

Example:Solve the equations below:24 = ? x 672 ÷ = 9

Melisa has 3 bags. There are 4 marbles in each bag. How many marbles does Melisa have altogether? 3 x 4 = m

This standard is strongly connected to 3.OA.3 when students solve problems and determine unknowns in equations.

Students should also experience creating story problems for given equations. When crafting story problems, they should carefully consider the question(s) to be asked and answered to write an appropriate equation.

Students may approach the same story problem differently and write either a multiplication equation or division equation .

Engage NY Lessons are included in the topic file to support instruction:Topic Overview (lesson 1-3)Lesson 1: Understand equal groups of as multiplicationLesson 2: Relate multiplication to the array modelLesson 3: Interpret the meaning of factors – the size of the group or the number of groups

Topic Overview (lesson 4-6)Lesson 4: Understand the meaning of the unknown as the size of the group in divisionLesson 5: Understand the meaning of the unknown as the number of groups in divisionLesson 6: Interpret the unknown in division using the array model

Topic Overview (Lesson 11-13)Lesson 11: Model division as the unknown factor in multiplication using arrays and tape diagramsLesson 12: Interpret the quotient as the number of groups or the number of objects in each group using units of 2Lesson 13: Interpret the quotient as the number of groups or the number of objects in each group using units of 3

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Topic Overview (Lesson 14-17)Lesson 14: Skip-count objects in models to build fluency with multiplication facts using units of 4Lesson 15: Relate arrays to tape diagrams to model the commutative property of multiplicationLesson 16: Use the distributive property as a strategy to find related multiplication factsLesson 17: Model the relationship between multiplication and divisionCommon MisconceptionsSome common misconceptions that students may have are thinking a symbol (? or ) is always the place for the answer. This is especially true when the problem is written as 15 ÷ 3 =? or 15 = x 3. Students also think that 3 ÷ 15 = 5 and 15 ÷ 3 = 5 are the same equations. The use of models is essential in helping students eliminate this understanding.

Another key misconception is that the use of a symbol to represent a number once cannot be used to represent another number in a different problem/situation. Presenting students with multiple situations in which they select the symbol and explain what it represents will counter this misconception.DifferentiationIncrease the RigorOA.1 Teacher shows an array. Have students list all equations that it represents. (EX. 4 rows, 5 columns: 4x5, 5x4, 20/4, 20/5, 4+4+4+4+4, 5+5+5+5) Create 3 different multiplications arrays that represent 30. Which model for multiplication do you like best? The product is 48. What could the factors be? Explain the relationship between a factor and product. What happens to the product when one of the factors increases? (for example, when 3x7 becomes 3x8)

OA.2 I have 24 total marshmallows divided into more than one bag. Each bag has the same amount. How many bags could I have? How many would be in each bag? How many ways can I share 36 equally? Explain the relationship between multiplication and division? Use models, drawings, and/or examples to support your answer. Think of an example in life when 42 ÷ 6 would be used with the sharing method of division (42 into 6 equal groups) and when it would be used with the

measurement method (42 divided into groups of 6). Note: students do not need to know the vocabulary of sharing and measurement methods, but should be exposed to both.

If the quotient is 6, what could your possible one-digit dividend and one-digit divisor be? List 3 different possibilities. How many people could share 32 M&M’s equally? Explain all possibilities.OA.3 Write a word problem that the number sentence 72 ÷ 9 could be used to solve. Write a word problem with the product of 35. We need 52 juice boxes for our class party. Juice boxes come in packs of 6 or 8. How many packs of each do you need to have enough for each student? Jim purchased 5 packages of muffins. Each package contained 3 muffins. Describe another situation where there would be 5 groups of 3 or 5 x 3. The monkey keeper at the zoo needs 7 apples a day to help feed the monkeys. She has 50 apples at the start of the week. Will she have enough apples for the

entire 7 day week ? Explain your reasoning.

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Lisa has 30 shoes and says that she owns 12 pairs. Explain why she is correct or incorrect.OA.4 Write four equations with n, where n would solve all of them. (ie. 4 x ___= 8, 10 = 5 x ___, ___x 6 = 12, 16 =___x 8) Alesia has 32 cupcakes to sell at the bake sale. What are possible package arrangements that she could make? Write an equation for each arrangement. The classroom has 30 desks. How many different ways could you arrange the desks? Explain to a friend how you found n in the equation 4 x n = 40. There are 50 ants on the ant hill. If there are 5 groups of 4 red ants, how many possible equal groups of black ants are there?

Accelerated InterventionThe Intervention Table provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand’s Numeracy Project.

Cluster of Standards Name of Intervention Snapshot of summary or

Student I can statement. . . Materials Master

Operations and

Algebraic Thinking

Represent and solve problems involving

multiplication and division.

MGSE3.OA.1MGSE3.OA.2MGSE3.OA.3MGSE3.OA.4

Five Sweets Per Packet Solve multiplication problems by skip counting in twos, fives, and tens

Number Strips Solve multiplication problems by skip counting in twos, fives, and tens MM 6-1

Animal Arrays Solve multiplication problems by using repeated additionMM 5-2MM 6-2

Multiplication or Out Solve multiplication problems by using repeated addition MM 5-2MM 6-2

Twos, Fives, and Tens Solve multiplication problems by using repeated addition

A Little Bit More/A Little Bit Less Derive multiplication facts from 2, 5, and 10 times tables

Fun With Fives Derive multiplication facts from 2, 5, and 10 times tables MM 4-5

Biscuit Boxes Solve Division Problems by Sharing

Pirate Crews Solve Division Problems by Sharing MM 4-6

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Long Jumps Solve division by using multiplication facts MM 4-8

Third Grade Math Triumphs for Intervention (came with the My Math adoption):2.1: Equal Groups2.2: Repeated Addition and Skip-Counting2.3: ArraysEvidence of LearningBy the completion of this topic, students should be able to: multiply and divide within 100, using strategies such as the patterns and relationships between multiplication and division use estimation to determine reasonableness of products and quotients computedAdditional Assessments Shared Assessments: See assessment folder.Purchased ResourcesMy Math:Chapter 4: Understanding Multiplication4.1 Hands on modeling4.2 Repeated addition4.3 Arrays4.4 Arrays and multiplication

Chapter 5: Understanding Division5.1 Hands on modeling5.2 Equal shares5.3 Relating to subtraction5.4 Relating to multiplication5.5 Inverse operations5.6 Problem solving

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.do

Teacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarhttp://www.exemplarslibrary.com/User: Cobb EmailPassword: First Name Camping (OA.2 & OA.7) Equal Snacks (OA.2 & OA.7) Great Pizza Dilemma (OA.2 & OA.7) Mrs. Hasson’s Decorating Dilemma (OA.2 & OA.7) Hot Dogs for a Picnic (OA.3)

Think Math (previous adoption)Chapter 2: Multiplication2.1 Rectangular Arrays2.2 Arrays of Square Tiles2.6 Pairing Objects2.7 Listing Combinations2.8 Using Multiplication2.11 Separating Arrays

Web Resources K-5 Math Teaching Resources http://www.k-5mathteachingresources.com/3rd-grade-number-activities.htmlOA.1Relate Addition and MultiplicationArray Picture Cards

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OA.2Identify the UnknownOA.3Word Problems (Arrays) Set 1OA.4Missing Numbers: DivisionIllustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources. https://www.illustrativemathematics.org/OA.2Fish TanksMarkers in BoxesOA.3Gifts from Grandma (variation 1)Two Interpretations of DivisionAnalyzing Word Problems Involving MultiplicationOA.4Finding the Unknown in a Division EquationNational Council of Teachers of Mathematics, Illuminations: http://illuminations.nctm.org/Default.aspxExploring Equal SetsMultiplication: It’s In The CardsAll About MultiplicationLearn Zillion: https://learnzillion.com/resources/73932OA.1Interpret Products by Drawing PicturesInterpret Products Using Repeated AdditionInterpret Products Using ArraysInterpret Products Using a Number LineOA.2Solve Division Problems by Drawing PicturesDivide Using a Sharing ModelUse Repeated Subtraction for DivisionVisualizing a Division Word ProblemCreate a Story Problem Based on a Division Expression Using a Sharing ModelCreate a Story Problem Based on a Division Expression Using Repeated SubtractionOA.3Solve Word Problems Using the Idea of Equal GroupsSolve Word Problems About Equal Groups by Drawing a ModelSolve Measurement Problems by Drawing a Model

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Solve Equal Groups Problems Using ArraysOA.4Find the Missing Quotient in a Division ProblemEstimation 180 is a website of 180 days of estimation ideas that build number sense. http://www.estimation180.com/days.htmlOther Web Resources:http://www.insidemathematics.orghttp://www. yummymath .com http://www.gregtang.comSuggested Manipulativessets of countersbase ten blocksmultiplication tableopen number lines1-inch color tileshundred chartobjects to share

Vocabulary equationexpressionmultiplicationfactorproduct multipleareadivisiondividenddivisorquotient

Suggested Literature Things that Come in 2’s, 3’s & 4’sAmanda Bean’s Amazing DreamMy Full Moon is Square Too Many Kangaroo Things to DoThe Best of TimesThe Doorbell RangEach Orange Had Eight SlicesTwo of Everything Spunky Monkeys On Parade One Hundred Hungry Ants Bats on Parade

VideosUnit 2 Overview Video Task Descriptions

Scaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State Tasks

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Task Task TypeGrouping Strategy Skills Standard(s) Brief Description

One Hundred Hungry Ants!

Scaffolding TaskIndividual/Partners Multiplication and Arrays MGSE3.OA.1

In this task, students will determine the factors of a product by creating equal groups of counters/colored tiles.

What’s My Product? Scaffolding TaskIndividual/Partners Multiplication MGSE3.OA.1

This task allows students to interpret products of whole numbers by creating equal groups with manipulatives.

The Doorbell Rang Scaffolding TaskIndividual/Partners Division MGSE3.OA.2

In this task, students will be introduced to division.

Skittles Cupcake Combos

Constructing TaskIndividual/Partners Division MGSE3.OA.3

This task assesses students’ understanding of division and their ability to organize data.

Stuck on Division Scaffolding TaskIndividual/Partners Division MGSE3.OA2

In this task, students will experiment with a set of 12 connecting cubes to determine the division patterns when the dividend is 12.

Base Ten Multiplication

Practice TaskPartners Multiplication MGSE3.OA.1

In this task students determine the factors of 100 by creating addition and/or multiplication models by using Base Ten Blocks.

What Comes First, the Chicken or the Egg?

Constructing TaskIndividual/Partners Division MGSE3.OA.4

In this task, students will use estimation skills to multiply by groups of ten.

Sharing Pumpkin Seeds Constructing TaskIndividual/Partners Division

MGSE3.OA.2MGSE3.OA.3

In this task, students will decide how to share pumpkin seeds fairly with a group of children.

Egg Tower 3-Act TaskWhole Group Multiplication and Arrays MGSE3.OA.1

In this task, students will view a picture and tell what they noticed. Next, they will be asked to discuss what they wonder about or are curious about. Students will then use mathematics to answer their own questions.

Shake, Rattle, and Roll Revisited

Practice TaskIndividual/Partners Multiplication MGSE3.OA.1

MGSE3.OA.2In this task, students will roll two cubes to get factors and use factors to find the product.

Field Day Blunder Constructing TaskPartners Multiplication MGSE3.OA.3

MGSE3.OA.4In this task, students will complete a multi-step word problem

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Topic 2: Understand Properties of Multiplication and the Relationship Between Multiplication and DivisionBig Ideas/Enduring Understandings: Multiplication is commutative, but division is not. As the divisor increases, the quotient decreases; as the divisor decreases, the quotient increases. There is a relationship between the divisor, the dividend, the quotient, and any remainder. The associative property of multiplication can be used to simplify computation. The distributive property of multiplication allows us to find partial products and then find their sum. Essential Questions: How does understanding the properties of operations help us multiply large numbers?

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How is multiplication related to addition? How can multiplication be used when dividing?Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.Understand properties of multiplication and the relationship between multiplication and division. MGSE3.OA.5 Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. 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 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)

MGSE3.OA.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8 MGSE.OA.7 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. Vertical Alignment

Second-Grade StandardsWork with equal groups of objects to gain foundations for multiplication MGSE2.OA.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

MGSE2.OA.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.

Understand place value MGSE2.NBT.2 Skip-count by 5s, 10s, and

100s, 10 to 100 and 100 to 1000. Reason with shapes and their attributes MGSE2.G.2 Partition a rectangle into rows

and columns of same-size squares and count to find the total number of them.

Fourth-Grade StandardsUse place value understanding and properties of operations to perform multi-digit arithmetic MGSE4.NBT.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.

MGSE4.NBT.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.

Fifth-Grade StandardsPerform operations with multi-digit whole numbers and with decimals to hundredths MGSE5.NBT.5 Fluently multiply multi-

digit whole numbers using the standard algorithm (or other strategies demonstrating understanding of multiplication) up to a 3 digit by 2 digit factor.

MGSE5.NBT.6 Fluently divide up to 4-digit dividends and 2-digit divisors by using at least one of the following methods: 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 or concrete models. (e.g., rectangular arrays, area models)

MGSE5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value,

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properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Instructional StrategiesOA.5This standard references properties of multiplication. While students DO NOT need to use the formal terms of these properties, student should understand that properties are rules about how numbers work.

Students do need to be flexible and fluent applying each of them. Students represent expressions using various objects, pictures, words and symbols in order to develop their understanding of properties. They multiply by 1 and 0 and divide by 1. They change the order of numbers to determine that the order of numbers does not make a difference in multiplication (but does make a difference in division).

Given three factors, they investigate how changing the order of how they multiply the numbers does not change the product. They also decompose numbers to build fluency with multiplication.

The associative property states that the sum or product stays the same when the grouping of addends or factors is changed. For example, when a student multiplies 7 x 5 x 2, a student could rearrange the numbers to first multiply 5 x 2 = 10 and then multiply 10 x 7 = 70.

The commutative property (order property) states that the order of numbers does not matter when adding or multiplying numbers. For example, if a student knows that 5 x 4 = 20, then they also know that 4 x 5 = 20.

The array below could be described as a 5 x 4 array for 5 columns and 4 rows, or a 4 x 5 array for 4 rows and 5 columns. There is no “fixed” way to write the dimensions of an array as rows x columns or columns x rows.

Students should have flexibility in being able to describe both dimensions of an array.

Students should be introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don’t know. Students would be using mental math to determine a product.

Here are ways that students could use the distributive property to determine the product of 7 x 6. Again, students should use the distributive property, but can refer to this in informal language such as “breaking numbers apart”.

Student 17 x 6

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7 x 5 = 357 x 1 = 735 + 7 = 42

Student 27 x 67 x 3 = 217 x 3 = 2121 + 21 = 42

Student 37 x 65 x 6 = 302 x 6 = 1230 + 12 = 42

Another example of the distributive property helps students determine the products and factors of problems by breaking numbers apart. For example, for the problem 6 x 5= ?, students can decompose the 6 into a 4 and 2, and reach the answer by multiplying 4 x 5 = 20 and 2 x 5 =10 and adding the two products (20+10=30).

To further develop understanding of properties related to multiplication and division, students use different representations and their understanding of the relationship between multiplication and division to determine if the following types of equations are true or false. 0 x 7 = 7 x 0 = 0 (Zero Property of Multiplication) 1 x 9 = 9 x 1 = 9 (Multiplicative Identity Property of 1) 3 x 6 = 6 x 3 (Commutative Property) 8 ÷ 2 ≠ 2 ÷ 8 (Students are only to determine that these are not equal) 2 x 3 x 5 = 6 x 5 10 x 2 < 5 x 2 x 2 2 x 3 x 5 = 10 x 3 1 x 6 > 3 x 0 x 2

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Students represent expressions using various objects, pictures, words and symbols in order to develop their understanding of properties. They multiply by 1 and 0 and divide by 1, never by 0. They change the order of numbers to determine that the order of numbers does not make a difference in multiplication (but does make a difference in division).

Given three factors, they investigate changing the order of how they multiply the numbers to determine that changing the order does not change the product. They also decompose numbers to build fluency with multiplication.

Use models to help build understanding of the commutative property:

Example: 3 x 6 = 6 x 3

In the following diagram it may not be obvious that 3 groups of 6 is the same as 6 groups of 3. A student may need to count to verify this.

Different representation:An array explicitly demonstrates the concept of the commutative property.

Students are introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don’t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56.

Students should learn that they can decompose either of the factors. It is important to note that the students may record their thinking in different ways.

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Students need to apply properties of operations (commutative, associative and distributive) as strategies to multiply and divide. Applying the concept involved is more important than students knowing the name of the property.

Understanding the commutative property of multiplication is developed through the use of models as basic multiplication facts are learned. For example, the result of multiplying 3 x 5 (15) is the same as the result of multiplying 5 x 3 (15).

Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty. (See example above where students split an array into smaller arrays and add the sums of the groups.

Students’ understanding of the part/whole relationships is critical in understanding the connection between multiplication and division.

OA.6This standard refers the Table 2 in the Appendix. Since multiplication and division are inverse operations, students are expected to solve problems and explain their processes of solving division problems that can also be represented as unknown factor in multiplication problems.

Example:A student knows that 2 x 9 = 18. How can they use that fact to determine the answer to the following question: 18 people are divided into pairs in P.E. class? How many pairs are there? Write a division equation and explain your reasoning.

Multiplication and division are inverse operations and that understanding can be used to find the unknown. Fact family triangles demonstrate the inverse operations of multiplication and division by showing the two factors and how those factors relate to the product and/or quotient.

Multiplication and division are inverse operations and that understanding can be used to find the unknown. Fact family triangles demonstrate the inverse operations of multiplication and division by showing the two factors and how those factors relate to the product and/or quotient.

Students use their understanding of the meaning of the equal sign as “the same value as” to interpret an equation with an unknown. When given 32 ÷ = 4, students may think: 4 groups of some number is the same as 32 4 times some number is the same as 32 I know that 4 groups of 8 is 32 so the unknown number is 8 The missing factor is 8 because 4 times 8 is 32.

Equations in the form of a ÷ b = c and c = a ÷ b need to be used interchangeably, with the unknown in different positions.

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OA.7This standard uses the word fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as the distributive property). “Know from memory” does not mean focusing only on timed tests and repetitive practice, but ample experiences working with manipulatives, pictures, arrays, word problems, and numbers to internalize the basic facts (up to 9 x 9).

By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Strategies students may use to attain fluency include: Multiplication by zeroes and ones Doubles (2s facts), Doubling twice (4s), Doubling three times (8s) Tens facts (relating to place value, 5 x 10 is 5 tens or 50) Five facts (half of tens) Skip counting (counting groups of __ and knowing how many groups have been counted) Square numbers (ex: 3 x 3) Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3) Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6) Turn-around facts (Commutative Property) Fact families (Ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24) Missing factors

General Note: Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms. (Problems presented horizontally encourage solving mentally.)

By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Students need to understand the part/whole relationships in order to understand the connection between multiplication and division. They need to develop efficient strategies that lead to the big ideas of multiplication and division. These big ideas include understanding the properties of operations, such as the commutative and associative properties of multiplication and the

distributive property. The naming of the property is not necessary at this stage of learning. In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 x 5, and wrote equations to represent the sum. This is called

22Grade 3 Unit 2 5/15/2023

unitizing. It requires students to count groups, not just objects. They see the whole as a number of groups of a number of objects. This strategy is a foundation for multiplication in that students should make a connection between repeated addition and multiplication.

As students create arrays for multiplication using objects or drawing on graph paper, they may discover that three groups of four and four groups of three yield the same results.

They should observe that the arrays stay the same, although how they are viewed changes. Provide numerous situations for students to develop this understanding.

To develop an understanding of the distributive property, students need decompose the whole into groups. Arrays can be used to develop this understanding. To find the product of 3 × 9, students can decompose 9 into the sum of 4 and 5 and find 3 × (4 + 5).

The distributive property is the basis for the standard multiplication algorithm that students can use to fluently multiply multi-digit whole numbers in Grade 5.

Once students have an understanding of multiplication using efficient strategies, they should make the connection to division.

Using various strategies to solve different contextual problems that use the same two one-digit whole numbers requiring multiplication allows for students to commit to memory all products of two one-digit numbers.

Engage NY Lessons are included in the topic file to support instruction:Topic Overview (lesson 7-10)Lesson 7: Demonstrate the commutativity of multiplication, and practice related facts by skip-counting objects in array modelsLesson 8: Demonstrate the commutativity of multiplication, and practice related facts by skip-counting objects in array modelsLesson 9: Find related multiplication facts by adding and subtracting equal groups in array modelsLesson 10: Model the distributive property with arrays to decompose units as a strategy to multiply

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Common MisconceptionsOA.5 & OA.6Students may experience difficulty in determining which factor represents rows or the number of objects in a group, and which factor represents the number of groups or columns. In division there are two different situations that can cause confusion depending on which factor is the unknown—the number in the group or the number of groups.OA.7Student who struggle most likely do not have fluency for the easy numbers. The child does not understand an unknown factor (a divisor) can be found from the related multiplication. It is not a matter of instilling facts divorced from their meaning, but rather the outcome of carefully designed learning. That involves the interplay of practice and reasoning.DifferentiationIncrease the RigorOA.5 Eric says he has more donuts because his mom bought six boxes of four donuts each. Samantha says that she has more donuts because her mom bought

four boxes, each with six donuts. Who is correct? Explain your thinking. Melissa needs to solve 24 x 4 in her head. What strategy should she use? Danielle is trying to multiply a strand of numbers (6 x 3 x 5) in her head but is having trouble keeping them organized in her head. Describe a strategy that

she can use to solve the problem. Malcolm multiplied 3 numbers together and got 24. What 3 numbers could he have multiplied? What strategy did you use to figure the numbers out? Solve 8 x 7 using the distributive property. Could you distribute a different factor or distribute the same factor a different way?OA.6 Jahyden was sharing his candy with his friends. He shared 32 pieces of candy. How many friends could he have shared the candy with? Danielle has 48 lollipops to share with her friends. If she has 6 friends, how many lollipops would each friend get? If she has 8 friends, how many lollipops

would each friend get? Why does the number of lollipops change based on the number of her friends? Explain how you can use multiplication to solve a division problem. Use models, drawings and/or examples to support your answer. Rachel says that if you know 4 x 6 = 24, then you know what n equals in 24 ÷ n = 6. Why is she correct? Use the numbers 5, 6, and 30 to write a multiplication story. Write a related division story.OA.7 Sean is have difficulty when multiplying by 9. Kevin tells him if he knows x10 facts, you can quickly solve x9 facts. Do you agree with Kevin? Why or why

not? Use models, equations and/or drawings to support your answer. Explain how x2 facts relate to x4 facts. How do x4 facts help you solve x8 facts? Tara knows strategies for x2, x3, x4, x5, x6, x8, x9, and x10. She asked when she would learn strategies for x7 facts and was told she knew her x7 facts

already. Explain how that is. Heather’s teacher told her that if she knows her x10 facts then she also knows her x5 facts. Heather doesn’t understand how. Explain the relationship

between the 10’s and 5’s facts. Explain a strategy for multiplying by 3. How does knowing your x5 and x2 facts help you to learn your x7 facts?

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Accelerated InterventionThe Intervention Table provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand’s Numeracy Project.

Cluster of Standards Name of Intervention

Snapshot of summary orStudent I can statement. . . Materials Master

Operations and Algebraic Thinking

Represent and solve problems involving multiplication and

division.

MGSE3.OA.5MGSE3.OA.6

Five Sweets Per Packet

Solve multiplication problems by skip counting in twos, fives, and tens

Number Strips

Solve multiplication problems by skip counting in twos, fives, and tens MM 6-1

Animal Arrays Solve multiplication problems by using repeated additionMM 5-2MM 6-2

Multiplication or Out Solve multiplication problems by using repeated addition MM 5-2

MM 6-2

Twos, Fives, and Tens Solve multiplication problems by using repeated addition

A Little Bit More/A Little

Bit LessDerive multiplication facts from 2, 5, and 10 times tables

Fun With Fives Derive multiplication facts from 2, 5, and 10 times tables MM 4-5

Biscuit Boxes Solve Division Problems by Sharing

Pirate Crews Solve Division Problems by Sharing MM 4-6

Long Jumps Solve division by using multiplication facts MM 4-8

Sherpa (Tensing) Multiply by multiples of ten MM 6-9

Third Grade Math Triumphs for Intervention (came with the My Math adoption):2.5: Multiply by 0 and 1

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2.6: Multiply by 2 and 53.1: Relate Multiplication and Division3.2: Repeated subtraction and Skip Counting3.3: Use Arrays to Model Division3.4: Use Area Models to Show Division3.5: Divide with 0 and 13.6: Divide by 2 and 5Evidence of LearningBy the completion of this topic, students should be able to: understand how to use inverse operations to verify accuracy of computation understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. apply properties of operations (commutative, associative) as strategies to multiply and divide Additional Assessment Shared Assessments: See assessment folder.Purchased ResourcesMy Math:Chapter 5: Understanding Division5.4 Relating to multiplication5.5 Inverse operations5.6 Problem solving

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath Camping (OA.2 & OA.7) Equal Snacks (OA.2 & OA.7) Great Pizza Dilemma (OA.2 & OA.7) Mrs. Hasson’s Decorating Dilemma (OA.2 & OA.7) Filling the Pool (OA.7) Fish Dilemma (OA.7) Harvest Dinner (OA.7) I Did-A-Read (OA.7) Is Dan Losing His Marbles? (OA.7) Missing Key Dilemma (OA.7) Portfolio Pizza Party (OA.7)

Think Math (previous adoption)

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Presents (OA.7) Shopping for Shoes (OA.7) Shovel, Shovel, Shovel (OA.7)

Web ResourcesK-5 Math Teaching Resources http://www.k-5mathteachingresources.com/3rd-grade-number-activities.htmlOA.5Turn Your ArrayDecompose a Factor (ver. 1)OA.6Division as an Unknown Factor (x5 & x 10)OA.7Domino MultiplicationMultiplication Bump (x2 – x5)Multiples Game (x2 – x5)Multiplication Four in a Row (x1, 2, 5, 10)I have…Who has (x2 & x5)I have…Who has (x2 & x10)I have…Who has (x3 & x5)Six SticksDivision Race 1 (Divisors 2, 5, 10)Division Squares (Divisors 2, 5, 10)Illustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources. https://www.illustrativemathematics.org/OA.5Valid Equalities (Part 2)OA.7Kiri’s Multiplication Matching GameLearn Zillion: https://learnzillion.com/resources/73932OA.5Understand the Commutative Property by Naming ArraysUnderstand the Commutative Property of Multiplication in Word ProblemsUnderstand Multiplication and Division RelationshipsOA.6Interpret Division as an Unknown Factor Problem Using ArraysInterpret Division as an Unknown Factor Problem Using Fact FamiliesInterpret Division as an Unknown Factor Problem Using a Bar ModelOA.7Multiply Using Doubles Pattern

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Multiply Using the Half-of-Ten StrategyMultiply by Subtracting from Groups of TenMultiply by Combining Known FactsDivide Using Fact FamiliesOther Web Resources:http://www.insidemathematics.orghttp://www. yummymath .com http://www.gregtang.comSuggested Manipulativessets of countersbase ten blocksmultiplication tableopen number lines1-inch color tileshundred chartobjects to share

Vocabulary equationexpressionmultiplicationfactorproduct multipleareadivisiondividenddivisorquotient

Suggested Literature Things that Come in 2’s, 3’s & 4’sAmanda Bean’s Amazing DreamMy Full Moon is Square Too Many Kangaroo Things to DoThe Best of TimesThe Doorbell RangEach Orange Had Eight SlicesTwo of Everything Spunky Monkeys On Parade One Hundred Hungry Ants Bats on Parade

VideosSEDL for OA.7 Part 1sets of countersopen number lines1-inch color tilesTask Descriptions

Scaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

28Grade 3 Unit 2 5/15/2023

Task Task TypeGrouping Strategy Skills Standard(s) Brief Description

Arrays on the Farm Scaffolding TaskSmall Group/Partners Multiplication and Arrays MGSE3.OA.5

In this task the students use arrays to solve multiplication problems. Farmers grow their crops in arrays to make them easier to look after and to harvest.

Family Reunion Constructing TaskIndividual/Partners Multiplication and Division MGSE3.OA.5

MGSE3.OA.6

This task is a two part task. Part I introduces and scaffolds learning of the associative property. In Part II students will be constructing the associative property.

Seating Arrangements Constructing TaskIndividual/Partners Multiplication and Arrays MGSE3.OA.5

In this task, students will solve a word problem requiring them to make arrays using the number 24.

Array-nging Fact Families

Practice TaskIndividual/Partners Multiplication and Division MGSE3.OA.5

MGSE3.OA.6

In this task, students will make models on grid paper of arrays that show both multiplication and division number sentences.

Finding FactorsConstructing/Practice

TaskSmall Group/Partners

Multiplication and DivisionMGSE3.OA.5MGSE3.OA.6MGSE3.OA.7

This is a several part task in which students find factors through work with multiplication and division.

Use What You KnowPractice Task

Individual/Partner Unknown Factors MGSE3.OA.6In this task, students use multiplication facts and strategies to identify the unknown factor in a division problem.

Multiplication Chart Mastery

Practice TaskIndividual/Small Group Multiplication Chart

MGSE3.OA.5MGSE3.OA.6MGSE3.OA.7

In this task, students will explain and describe the patterns they find in the multiplication chart.This task would work well as a math conference interview.

Making the “Hard” Facts Easy

Constructing TaskSmall Group/Partners

Distributive Property of Multiplication

MGSE3.OA.5MGSE3.OA.7

In this task, students will practice using the distributive property.

Find the Unknown Number

Practice TaskIndividual/Partners Unknown Factors

MGSE3.OA.5MGSE3.OA.6MGSE3.OA.7

In this task, students will complete division equations by finding the unknown factors.

Making Up Multiplication

Constructing TaskIndividual/Partners Multiplication Stories MGSE3.OA.5

MGSE3.OA.6

In this task, students will learn three different ways a multiplication problem can be written. They will then practice writing their own problems.

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