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New York State Common Core 3 GRADE Mathematics Curriculum GRADE 3 MODULE 1 Table of Contents GRA...

New York State Common Core 3 GRADE Mathematics Curriculum GRADE 3 MODULE 1 Table of Contents GRADE 3 MODULE 1 Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 Module Overview........................................................................................................... i Topic A: Multiplication and the Meaning of the Factors.......................................... 1.A.1 Topic B: Division as an Unknown Factor Problem................................................... 1.B.1 Topic C: Analyze Arrays to Multiply Using Units of 2 and 3..................................... 1.C.1 Topic D: Division Using Units of 2 and 3..................................................................1.D.1 Topic E: Multiplication and Division Using Units of 4.............................................. 1.E.1 Topic F: Distributive Property and Problem Solving Using Units of 2–5 and 10....... 1.F.1 Module Assessments.............................................................................................. 1.S.1 Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 i Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Grade 3 Module 1 Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 OVERVIEW This 25-day module begins the year by building on students’ fluency with addition and knowledge of arrays. Topic A initially uses repeated addition to find the total from a number of equal groups (2.OA.4). As students notice patterns, they let go of longer addition sentences in favor of more efficient multiplication facts (3.OA.1, 3.OA.9). Lessons in Topic A move students toward understanding familiar repeated addition from Grade 2 in the form of array models, which 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. In this module the factors 2, 3, 4, 5, and 10 provide an entry point for moving into more difficult factors in later modules. Study of factors links Topics A and B; Topic B extends the study to division. Students understand division as an unknown factor problem, and relate the meaning of unknown factors to either the number or the size of groups (3.OA.2, 3.OA.6). By the end of Topic B students are aware of a fundamental connection between multiplication and division that sets the foundation for the rest of the module. In Topic C, students use the array model and familiar skip-counting strategies to solidify their understanding of multiplication and practice related facts of 2 and 3. They become fluent enough with arithmetic patterns to “add” or “subtract” groups from known products to solve more complex multiplication problems (3.OA.1, 3.OA.9). They apply their skills to word problems using drawings and equations with a symbol to find the unknown factor (3.OA.3). This culminates in students using arrays to model the distributive property as they decompose units to multiply The Distributive Property (3.OA.5). 6 × 4 = _____ In Topic D students model, write and solve partitive and measurement division problems with 2 and 3 (3.OA.2). Consistent skip-counting strategies and the continued use of array models are pathways for students to naturally relate multiplication and division. Modeling (5 × 4) = 20 advances as students use tape diagrams to represent multiplication and division. A final lesson in this topic solidifies a growing understanding of the relationship between operations (3.OA.7). (1 × 4) = 4 Topic E shifts students from simple understanding to analyzing the (6 × 4) = (5 × 4) + (1 × 4) relationship between multiplication and division. Practice of both = 20 + 4 operations is combined—this time using units of 4—and a lesson is explicitly dedicated to modeling the connection between them (3.OA.7). Skip-counting, the distributive property, arrays, number bonds and tape diagrams are tools for both operations (3.OA.1, 3.OA.2, 3.OA.9). A Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 ii Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. (1 × 4) = _ _____  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 final lesson invites students to explore their work with arrays and related facts through the lens of the commutative property as it relates to multiplication (3.OA.5). Topic F introduces the factors 5 and 10, familiar from skip-counting in Grade The Commutative Property 2. Students apply the multiplication and division strategies they have used to mixed practice with all of the factors included in Module 1 (3.OA.1, 3.OA.2, 3.OA.3). Students model relationships between factors, analyzing the arithmetic patterns that emerge to compose and decompose numbers as they further explore the relationship between multiplication and division (3.OA.3, 3.OA.5, 3.OA.7, 3.OA.9). In the final lesson of the module, students apply the tools, representations, and concepts they have learned to problem-solving with multi-step word problems using all four operations (3.OA.3, 3.OA.8). They demonstrate the flexibility of their thinking as they assess the reasonableness of their answers for a variety of problem types. The mid-module assessment follows Topic C. The end-of-module assessment follows Topic F. Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 iii Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Focus Grade Level Standards Represent and solve problems involving multiplication and division.1 3.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. 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 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.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. (See Glossary, Table 2.) 3.OA.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 = ? Understand properties of multiplication and the relationship between multiplication and division.2 3.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.) 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.)3 3.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. Multiply and divide within 100.4 3.OA.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. Solve problems involving the four operations, and identify and explain patterns in 1 Limited to factors of 2–5 and 10 and the corresponding dividends in this module. 2 Limited to factors of 2–5 and 10 and the corresponding dividends in this module. 3 The Associative property is addressed in Module 3. 4 Limited to factors of 2–5 and 10 and the corresponding dividends in this module. Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 iv Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 arithmetic.5 3.OA.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, i.e., Order of Operations.) Foundational Standards 2.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. 2.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. 2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s. Focus Standards for Mathematical Practice 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. 5 In this module, problem solving is limited to multiplication and division, and limited to factors of 2–5 and 10 and the corresponding dividends. 3.OA.9 is addressed in Module 3. Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 v Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Overview of Module Topics and Lesson Objectives Standards Topics and Objectives Days 3.OA.1 A Multiplication and the Meaning of the Factors 3 3.OA.3 Lesson 1: Understand equal groups of as multiplication. Lesson 2: Relate multiplication to the array model. Lesson 3: Interpret the meaning of factors—the size of the group or the number of groups. 3.OA.2 B Division as an Unknown Factor Problem 3 3.OA.6 Lesson 4: Understand the meaning of the unknown as the size of the 3.OA.3 group in division. 3.OA.4 Lesson 5: Understand the meaning of the unknown as the number of groups in division. Lesson 6: Interpret the unknown in division using the array model. 3.OA.1 C Analyze Arrays to Multiply Using Units of 2 and 3 4 3.OA.5 Lessons 7–8: Demonstrate the commutativity of multiplication and practice 3.OA.3 related facts by skip-counting objects in array models. 3.OA.4 Lesson 9: Find related multiplication facts by adding and subtracting equal groups in array models. Lesson 10: Model the distributive property with arrays to decompose units as a strategy to multiply. Mid-Module Assessment: Topics A–C (assessment ½ day, return ½ day, 2 remediation or further applications 1 day) 3.OA.2 D Division Using Units of 2 and 3 3 3.OA.4 Lesson 11: Model division as the unknown factor in multiplication using 3.OA.6 arrays and tape diagrams. 3.OA.7 Lesson 12: Interpret the quotient as the number of groups or the number 3.OA.3 of objects in each group using units of 2. 3.OA.8 Lesson 13: Interpret the quotient as the number of groups or the number of objects in each group using units of 3. Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 vi Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Standards Topics and Objectives Days 3.OA.5 E Multiplication and Division Using Units of 4 4 3.OA.7 Lesson 14: Skip-count objects in models to build fluency with multiplication 3.OA.1 facts using units of 4. 3.OA.2 Lesson 15: Relate arrays to tape diagrams to model the commutative 3.OA.3 property of multiplication. 3.OA.4 3.OA.6 Lesson 16: Use the distributive property as a strategy to find related multiplication facts. Lesson 17: Model the relationship between multiplication and division. 3.OA.3 F Distributive Property and Problem Solving Using Units of 2–5 and 10 4 3.OA.5 Lessons 18–19: Apply the distributive property to decompose units. 3.OA.7 3.OA.8 Lesson 20: Solve two-step word problems involving multiplication and division and assess the reasonableness of answers. 3.OA.1 3.OA.2 Lesson 21: Solve two-step word problems involving all four operations and 3.OA.4 assess the reasonableness of answers. 3.OA.6 End-of-Module Assessment: Topics A–F (assessment ½ day, return ½ day, 2 remediation or further application 1 day) Total Number of Instructional Days 25 Terminology New or Recently Introduced Terms  Array (a set of numbers or objects that follow a specific pattern, a matrix)  Column (e.g., in an array)  Commutative Property/Commutative (e.g., rotate a rectangular array 90 degrees to demonstrate that factors in a multiplication sentence can switch places)  Equal groups (with reference to multiplication and division; one factor is the number of objects in a group and the other is a multiplier that indicates the number of groups)  Equation (a statement that 2 expressions are equal. E.g., 3 × 4 = 12)  Distribute (with reference to the Distributive Property; e.g. In 12 × 3 = (10 × 3) + (2 × 3) the 3 is multiplier for each part of the decomposition)  Divide/division (partitioning a total into equal groups to show how many equal groups add up to a specific number. E.g., 15 ÷ 5 = 3) Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 vii Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1  Fact (used to refer to multiplication facts, e.g., 3 × 2)  Factors (i.e., numbers that are multiplied to obtain a product)  Multiplication/multiply (an operation showing how many times a number is added to itself e.g., 5 × 3 =15)  Number of groups (factor in a multiplication problem that refers to the total equal groups)  Parentheses (e.g., ( ) used around a fact or numbers within an equation)  Quotient (the answer when one number is divided by another)  Rotate (turn, used with reference to turning arrays 90 degrees)  Row/column (in reference to rectangular arrays)  Size of groups (factor in a multiplication problem that refers to how many in a group)  Unit (i.e., one segment of a partitioned tape diagram)  Unknown (i.e., the “missing” factor or quantity in multiplication or division) Familiar Terms and Symbols6  Add 1 unit, subtract 1 unit (add or subtract a single unit of two, ten, etc.)  Number bond (shows part-part-whole relationship, shown at right)  Number sentence (similar to an equation, but not necessarily having equal sides.)  Ones, twos, threes, etc. (units of one, two, or three)  Repeated addition (adding equal groups together, e.g., 2 + 2 + 2 + 2)  Tape Diagram (a method for modeling problems)  Value (how much) Suggested Tools and Representations 9 × 10  18 counters per student  Tape diagram (a method for modeling problems)  Number bond (shown at right)  Array (a set of numbers or objects that follow a specific pattern, a matrix) 5 × 10 4 × 10 6 These are terms and symbols students have used or seen previously. Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 viii Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Suggested Methods of Instructional Delivery Directions for Administration of Sprints Sprints are designed to develop fluency. They should be fun, adrenaline-rich activities that intentionally build energy and excitement. A fast pace is essential. During Sprint administration, teachers assume the role of athletic coaches. A rousing routine fuels students’ motivation to do their personal best. Student recognition of increasing success is critical, and so every improvement is celebrated. One Sprint has two parts with closely related problems on each. Students complete the two parts of the Sprint in quick succession with the goal of improving on the second part, even if only by one more. With practice the following routine takes about 8 minutes. Sprint A Pass Sprint A out quickly, face down on student desks with instructions to not look at the problems until the signal is given. (Some Sprints include words. If necessary, prior to starting the Sprint quickly review the words so that reading difficulty does not slow students down.) T: You will have 60 seconds to do as many problems as you can. T: I do not expect you to finish all of them. Just do as many as you can, your personal best. (If some students are likely to finish before time is up, assign a number to count by on the back.) T: Take your mark! Get set! THINK! (When you say THINK, students turn their papers over and work furiously to finish as many problems as they can in 60 seconds. Time precisely.) After 60 seconds: T: Stop! Circle the last problem you did. I will read just the answers. If you got it right, call out “Yes!” and give a fist pump. If you made a mistake, circle it. Ready? T: (Energetically, rapid-fire call the first answer.) S: Yes! T: (Energetically, rapid-fire call the second answer.) S: Yes! Repeat to the end of Sprint A, or until no one has any more correct. If need be, read the count by answers in the same way you read Sprint answers. Each number counted by on the back is considered a correct answer. T: Fantastic! Now write the number you got correct at the top of your page. This is your personal goal for Sprint B. T: How many of you got 1 right? (All hands should go up.) T: Keep your hand up until I say the number that is 1 more than the number you got right. So, if you got 14 correct, when I say 15 your hand goes down. Ready? T: (Quickly.) How many got 2 correct? 3? 4? 5? (Continue until all hands are down.) Optional routine, depending on whether or not your class needs more practice with Sprint A: T: I’ll give you one minute to do more problems on this half of the Sprint. If you finish, stand behind your chair. (As students work you might have the person who scored highest on Sprint A pass out Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 ix Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Sprint B.) T: Stop! I will read just the answers. If you got it right, call out “Yes!” and give a fist pump. If you made a mistake, circle it. Ready? (Read the answers to the first half again as students stand.) Movement To keep the energy and fun going, always do a stretch or a movement game in between Sprint A and B. For example, the class might do jumping jacks while skip counting by 5 for about 1 minute. Feeling invigorated, students take their seats for Sprint B, ready to make every effort to complete more problems this time. Sprint B Pass Sprint B out quickly, face down on student desks with instructions to not look at the problems until the signal is given. (Repeat the procedure for Sprint A up through the show of hands for how many right.) T: Stand up if you got more correct on the second Sprint than on the first. S: (Students stand.) T: Keep standing until I say the number that tells how many more you got right on Sprint B. So if you got 3 more right on Sprint B than you did on Sprint A, when I say 3 you sit down. Ready? (Call out numbers starting with 1. Students sit as the number by which they improved is called. Celebrate the students who improved most with a cheer.) T: Well done! Now take a moment to go back and correct your mistakes. Think about what patterns you noticed in today’s Sprint. T: How did the patterns help you get better at solving the problems? T: Rally Robin your thinking with your partner for 1 minute. Go! Rally Robin is a style of sharing in which partners trade information back and forth, one statement at a time per person, for about 1 minute. This is an especially valuable part of the routine for students who benefit from their friends’ support to identify patterns and try new strategies. Students may take Sprints home. RDW or Read, Draw, Write (a Number Sentence and a Statement) Mathematicians and teachers suggest a simple process applicable to all grades: 1) Read. 2) Draw and Label. 3) Write a number sentence (equation). 4) Write a word sentence (statement). The more students participate in reasoning through problems with a systematic approach, the more they internalize those behaviors and thought processes.  What do I see?  Can I draw something?  What conclusions can I make from my drawing? Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 x Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Modeling with Interactive Guided Practice Independent Practice Questioning The teacher models the whole Each student has a copy of the The students are given a problem process with interactive question. Though guided by the to solve and possibly a questioning, some choral teacher, they work designated amount of time to response, and talk moves such as independently at times and then solve it. The teacher circulates, “What did Monique say, come together again. Timing is supports, and is thinking about everyone?” After completing the important. Students might hear, which student work to show to problem, students might reflect “You have 2 minutes to do your support the mathematical with a partner on the steps they drawing.” Or, “Put your pencils objectives of the lesson. When used to solve the problem. down. Time to work together sharing student work, students “Students, think back on what we again.” The Debrief might include are encouraged to think about did to solve this problem. What selecting different student work the work with questions such as, did we do first?” Students might to share. “What do you see Jeremy did?” then be given the same or similar “What is the same about problem to solve for homework. Jeremy’s work and Sara’s work?” “How did Jeremy show the 3/7 of the students?” “How did Sara show the 3/7 of the students?” Personal Boards Materials Needed for Personal Boards 1 High Quality Clear Sheet Protector 1 piece of stiff red tag board 11” x 8 ¼” 1 piece of stiff white tag board 11” x 8 ¼” 1 3”x 3” piece of dark synthetic cloth for an eraser 1 Low Odor Blue Dry Erase Marker: Fine Point Directions for Creating Personal Boards Cut your white and red tag to specifications. Slide into the sheet protector. Store your eraser on the red side. Store markers in a separate container to avoid stretching the sheet protector. Frequently Asked Questions About Personal Boards Why is one side red and one white? The white side of the board is the “paper.” Students generally write on it and if working individually then turn the board over to signal to the teacher they have completed their work. The teacher then says, “Show me your boards,” when most of the class is ready. What are some of the benefits of a personal board?  The teacher can respond quickly to a hole in student understandings and skills. “Let’s do some of Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 xi Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 these on our personal boards until we have more mastery.”  Student can erase quickly so that they do not have to suffer the evidence of their mistake.  They are motivating. Students love both the drill and thrill capability and the chance to do story problems with an engaging medium.  Checking work gives the teacher instant feedback about student understanding. What is the benefit of this personal board over a commercially purchased dry erase board?  It is much less expensive.  Templates such as place value charts, number bond mats, hundreds boards, and number lines can be stored between the two pieces of tag for easy access and reuse.  Worksheets, story problems, and other problem sets can be done without marking the paper so that students can work on the problems independently at another time.  Strips with story problems, number lines, and arrays can be inserted and still have a full piece of paper to write on.  The red versus white side distinction clarifies your expectations. When working collaboratively, there is no need to use the red. When working independently, the students know how to keep their work private.  The sheet protector can be removed so that student work can be projected on an overhead. Scaffolds7 The scaffolds integrated into A Story of Units give alternatives for how students access information as well as express and demonstrate their learning. Strategically placed margin notes are provided within each lesson elaborating on the use of specific scaffolds at applicable times. They address many needs presented by English language learners, students with disabilities, students performing above grade level, and students performing below grade level. Many of the suggestions are organized by Universal Design for Learning (UDL) principles and are applicable to more than one population. To read more about the approach to differentiated instruction in A Story of Units, please refer to “How to Implement A Story of Units.” 7 Students with disabilities may require Braille, large print, audio, or special digital files. Please visit the website, www.p12.nysed.gov/specialed/aim, for specific information on how to obtain student materials that satisfy the National Instructional Materials Accessibility Standard (NIMAS) format. Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 xii Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 3 1 Assessment Summary Type Administered Format Standards Addressed Mid-Module After Topic C Constructed response with rubric 3.OA.1 Assessment Task 3.OA.2 3.OA.5 3.OA.6 End-of-Module After Topic F Constructed response with rubric 3.OA.1 Assessment Task 3.OA.2 3.OA.3 3.OA.4 3.OA.5 3.OA.6 3.OA.7 3.OA.8 Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 xiii Date: 6/26/13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  New York State Common Core 3 GRADE Mathematics Curriculum GRADE 3 MODULE 1 Topic A Multiplication and the Meaning of the Factors 3.OA.1, 3.OA.3 Focus Standard: 3.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. Instructional Days: 3 Coherence -Links from: G2–M6 Foundations of Multiplication and Division -Links to: G4–M3 Multi-Digit Multiplication and Division Lesson 1 introduces students to multiplication starting with the concept of repeated addition, familiar from Grade 2. Students use repeated addition to find totals, for example from scattered stars grouped equally within larger circles. They learn to recognize each circled group of stars as a unit, and count units using the language of groups and unit form. “3 groups of 5 stars make 15 stars,” or “3 fives make 15.” By the end of Lesson 1 students use the multiplication symbol to represent these descriptions as more efficient multiplication facts. In Lesson 2, students relate the equal groups of objects in scattered configurations from Lesson 1 to the array model, exploring the correspondence between 1 equal group and 1 row. They begin to distinguish between the number of groups and the size of groups as they count rows and how many in 1 row to write multiplication facts. Students recognize the efficiency of arrays as they skip-count to find totals. By the end of Lesson 2 students use the following vocabulary: row, array, number of groups, and size of groups. Lesson 3 solidifies students’ ability to differentiate the meanings of factors. Students analyze equal groups given in scattered configurations and organized into arrays to determine whether factors represent the number of groups or the size of groups. They create pictures, number bonds, and multiplication sentences to model their understanding. In this topic students use a variety of factors since these lessons emphasize understanding the concept of multiplying rather than finding totals. Later topics limit facts to those involving one or two specific factors, allowing students to build fluency with simpler facts before moving onto more difficult ones. Topic A: Multiplication and the Meaning of the Factors Date: 6/26/13 1.A.1 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License.  NYS COMMON CORE MATHEMATICS CURRICULUM Topic A 3 A Teaching Sequence Towards Mastery of Multiplication and the Meaning of the Factors Objective 1: Understand equal groups of as multiplication. (Lesson 1) Objective 2: Relate multiplication to the array model. (Lesson 2) Objective 3: Interpret the meaning of factors—the size of the group or the number of groups. (Lesson 3) Topic A: Multiplication and the Meaning of the Factors Date: 6/26/13 1.A.2 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported.License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3 1 Lesson 1 Objective: Understand equal groups of as multiplication. Suggested Lesson Structure  Fluency Practice (10 minutes)  Application Problem (10 minutes)  Concept Development (30 minutes)  Student Debrief (10 minutes) Total Time (60 minutes) Fluency Practice (10 minutes)  Group Counting 3.OA.1 (10 minutes) Group Counting (10 minutes) Note: Basic skip-counting skills from Grade 2 shift focus in this Grade 3 activity. Group-counting lays a foundation for interpreting multiplication as repeated addition. When students count groups in this activity, they add and subtract groups of two when counting up and down. T: Let’s count to 20 forward and backward. Watch my fingers to know whether to count up or down. A closed hand means stop. (Show signals as you explain.) T: (Rhythmically point up until a change is desired. Show a closed hand then point down.) S: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0. T: Let’s count to 20 forward and backward again. This time whisper every other number. Say the other numbers in a regular voice. S: (Students whisper then speak every other number to 20 forward and backward.) T: Let’s count to 20 forward and backward again. This time, hum every other number instead of whispering. As you hum, think of the number. S: (Hum), 2, (hum), 4, (hum), 6, etc. T: Let’s count to 20 forward and backward again. This time, think every other number instead of humming. S: (Think), 2, (think), 4, (think), 6, etc. T: What did we just count by? Turn and talk to your partner. S: Twos. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.3 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3 1 T: Let’s count by twos. (Direct students to count forward to and backward from 20, changing directions at times.) Application Problem (10 minutes) There are 83 girls and 76 boys in the third grade. How many total students are in the third grade? Note: Students may choose to use a tape diagram or a number bond to model the problem. They are also likely to solve today’s application problem in less than 10 minutes. Ten minutes have been allotted in order for you to review the RDW (Read, Draw, Write) procedure for problem-solving. Directions on the Read, Draw, Write (RDW) Process: Read the problem, draw and label, write a number sentence, and write a word sentence. The more students participate in reasoning through problems with a systematic approach, the more they internalize those behaviors and thought processes. (Excerpted from “How to Implement A Story of Units.”) Concept Development (30 minutes) Materials: (S) 12 counters per student, personal white boards. Problem 1: Skip-count to find the total number of objects. T: (Select 10 students to come to the front.) At the signal, say how many arms you have. (Signal.) S: 2 arms! T: Since we each represent a group of 2 arms, let’s skip-count our volunteers by twos to find how many arms they have altogether. To keep track of our count, the students will raise up their arms when we count them. S: (Count 2, 4, 6…20.) T: How many raised arms do we have in all? S: 20. T: Arms down. How many twos did we count to find the total? Turn and whisper to your partner. S: 10 twos. T: What did you count to find the number of twos? S: I counted the number of students in the front because each person represents a group of two. T: Skip-count to find the total number of arms. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.4 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3 1 S: (Students say 2, 4, 6…. As they count, write Sample teacher board: 2 + 2 + 2….) T: Look at our addition sentence. Show thumbs up if you see the correct number of twos. S: (Show thumbs up.) T: (Under the addition sentence write 10 twos.) Clap your hands if you agree that 10 groups of two is 20. S: (Clap hands.) T: (Write ‘10 groups of two is 20’ under the other expressions.) Problem 2: Understand the relationship between repeated addition, counting groups in unit form, and multiplication equations. Seat students at tables with personal white boards and 12 counters each. NOTES ON MULTIPLE MEANS OF T: You have 12 counters. Use your counters to make equal REPRESENTATION: groups of two. How many counters will you put in each For some classes it may be necessary to group? Show with your fingers. clearly connect the word times and the S: (Students hold up 2 fingers and begin to make groups of symbol x. Have students analyze the two.) model. “How many times do you see a group of three?” Have them count the T: How many equal groups of two did you make? Tell at the groups, write the equation, and say the signal. (Signal.) words together. S: 6 groups.  “4 groups of three equals 12.” T: 6 equal groups of how many counters?  “4 times three equals 12.” S: 6 equal groups of 2 counters. T: 6 equal groups of 2 counters equal how many counters altogether? S: 12 counters. T: Write an addition sentence to show your groups on your personal white board. S: (Write 2 + 2 + 2 + 2 + 2 + 2 = 12.) T: (Record the addition sentence on the board.) How many twos did we add to make 12? S: 6 twos. Sample teacher board: T: (Record 6 twos = 12 under the addition sentence.) 6 × 2 is another way to write 2 + 2 + 2 + 2 + 2 + 2 or 6 twos. (Record 6 × 2 = 12 under 6 twos = 12 on the board.) These number sentences are all saying the same thing. Another name for number sentence is equation. T: Turn and talk to your partner. How do you think 6 × 2 = 12 relates to the other equations? Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.5 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3 1 S: They all have twos in them and the answer is 12.  I think the 6 shows how many twos there are.  You have to count two 6 times because there are 6 groups of them. That’s how you get 6 times 2.  6 × 2 might be an easier way to write a long addition sentence. T: Ways that are easier and faster are efficient. When we have equal groups, multiplication is a more efficient way of showing the total than repeated addition. Repeat the process with 4 threes, 3 fours and 2 sixes with the objective of getting students comfortable with the relationship between repeated addition, counting groups in unit form, and multiplication equations. In this lesson avoid emphasis on finding solutions. Problem 3: Write multiplication sentences from equal groups. (Draw or project the following picture.) NOTES ON MULTIPLE MEANS OF ACTION AND EXPRESSION: T: These are equal groups. Turn and tell your partner Some students may need more why they are equal. scaffolding to realize that you can’t use S: There is the same number of grey circles in each group. multiplication to find the total amount  All of the grey circles are the same size and shape, of items in groups that are not equal. You might use the following questions and there are 4 in each group. to scaffold. T: Work with your partner to write a repeated addition  “Does the drawing show groups of and a multiplication sentence for this picture. 4 modeled 3 times?” S: (Write 4 + 4 = 8, and either 2 × 4 = 8 or 4 × 2 = 8.)  “Does 4 times 3 represent this T: (Project or draw the following image.) Look at my new drawing?” drawing and the multiplication sentence I wrote to represent  “How might we redraw the picture to make it show 4 x 3?” it. Check my work by writing an addition sentence and counting to find the total number of objects.  3 x 4 = 12 MP.3 S: (Write 4 + 4 + 3 = 11.) T: Use your addition sentence as you talk in partners about why you agree or disagree with my work. S: I disagree because my addition sentence equals 11, not 12.  It’s because that last group doesn’t have 4 circles.  You can do multiplication when the groups are equal.  Here the groups aren’t equal, so the drawing doesn’t show 4 × 3. T: I hear most students disagreeing because my groups are not equal. True, to multiply you must have equal groups. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.6 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3 1 Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. Some problems do not specify a method for solving. This is an intentional reduction of scaffolding that invokes MP.5, Use Appropriate Tools Strategically. Students should solve these problems using the RDW approach used for Application Problems. For some classes, it may be appropriate to modify the assignment by specifying which problems students should work on first. With this option, let the careful sequencing of the Problem Set guide your selections so that problems continue to be scaffolded. Balance word problems with other problem types to ensure a range of practice. Assign incomplete problems for homework or at another time during the day. Student Debrief (10 minutes) Lesson Objective: Understand equal groups of as multiplication. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the ideas below to lead the discussion.  On the first page, what did you notice about the answers to your problems?  Discuss the relationship between repeated addition and the unit form 2 groups of three or 3 groups of two, depending on the drawing.  Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.7 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 3 1  Discuss the relationship between repeated addition, unit form, and the multiplication equation 3 × 2 = 6.  Review the new vocabulary presented in the lesson, including equal groups, multiply. Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.8 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 3 1 Name Date 1. Fill in the blanks to make true statements. b. 3 + 3 + 3 + 3 + 3 = _________ a. 3 groups of five = _________ 5 groups of three = _________ 3 fives = _________ 5 × 3 = _________ 3 × 5 =_________ c. 6 + 6 + 6 + 6 = ___________ _______ groups of six = __________ 4 × ______ = __________ d. 4 +____ + ____ + ____ + ____ + ____ = _________ 6 groups of ________ = ___________ 6 × ______ = __________ Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.9 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Problem Set 3 1 2. The picture below shows 2 groups of apples. Does the picture below show 2 × 3? Explain why or why not. 3. Draw a picture to show 2 × 3 = 6. 4. Caroline, Brian and Marta want to share a box of chocolates so that they each get the same amount. Circle the chocolates below to show 3 groups of 4. Then write addition and multiplication sentences to represent the problem. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.10 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Exit Ticket 3 1 Name Date 1. The picture below shows 4 groups of 2 slices of watermelon. Write repeated addition and multiplication sentences to represent the picture. 2 + ____ + ____ + ____ = ___________ 4 × ______ = __________ 2. Draw a picture to show 3 + 3 + 3 = 9. Then write a multiplication sentence to represent the picture. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.11 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Homework 3 1 Name Date 1. Fill in the blanks to make true statements. a. 4 groups of five = _________ b. 5 groups of four = _________ 4 fives = _________ 5 fours = _________ 2. 4 × 5 = _________ 5 × 4 = _________ c. 6 + 6 + 6 = ___________ _______ groups of six = __________ 3 × ______ = __________ d. 3 + ____ + ____ + ____ + ____ + ____ = _______ 6 groups of ________ = ___________ 6 × ______ = __________ Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.12 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 Homework 3 1 2. The picture below shows 3 groups of hot dogs. Does the picture below show 3 × 3? Explain why or why not. 3. Draw a picture to show 4 × 2 = 8. 4. Circle the pencils below to show 3 groups of 6. Write addition and multiplication sentences to represent the problem. Lesson 1: Understand equal groups of as multiplication. Date: 6/26/13 1.A.13 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 Lesson 2 Objective: Relate multiplication to the array model. Suggested Lesson Structure  Fluency Practice (15 minutes)  Application Problem (5 minutes)  Concept Development (30 minutes)  Student Debrief (10 minutes) Total Time (60 minutes) Fluency Practice (15 minutes)  Add and Subtract by 2 3.OA.1 (8 minutes)  Group Counting 3.OA.1 (4 minutes)  Add Equal Groups 3.OA.1 (3 minutes) Sprint: Add and Subtract by 2 (8 minutes) Materials: (S) Add and Subtract by 2 Sprint Note: This sprint supports group counting skills that are foundational to interpreting multiplication as repeated addition. Directions for Administration of Sprints One sprint has two parts with closely related problems on each. Each part is organized into 4 quadrants that move from simple to complex. This builds a challenge into each sprint for every learner. Before the lesson, cut the sprint sheet in half to create Sprint A and Sprint B. Students complete the two parts of the sprint in quick succession with the goal of improving on the second part, even if only by one more. With practice the following routine takes about 8 minutes. Sprint A (Put Sprint A face down on desks with instructions to not look at problems until signal is given.) T: You will have 60 seconds to do as many problems as you can. T: I do not expect you to finish all of them. Just do as many as you can, your personal best. T: Take your mark! Get set! THINK! (When you say THINK, students turn papers over and work furiously to finish as many problems as they can in 60 seconds. Time precisely.) Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.14 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 (After 60 seconds.) T: Stop! Circle the last problem you did. I will read just the answers. If you got it right, call out “Yes!” and give a fist pump. If you made a mistake, circle it. Ready? (Repeat to the end of Sprint A, or until no one has any more correct.) T: Now write your number correct at the top of the page. This is your personal goal for Sprint B. T: How many of you got 1 right? (All hands should go up.) T: Keep your hand up until I say a number that is 1 more than the number you got right. So, if you got 14 right, when I say 15 your hand goes down. Ready? T: (Quickly.) How many got 2 right? 3? 4? 5? (Continue until all hands are down.) (Optional routine, depending on whether or not your class needs more practice with Sprint A.) T: Take one minute to do more problems on this half of the sprint. (As students work you might have the person who scored highest on Sprint A pass out Sprint B.) T: Stop! I will read just answers. If you got it right, call out “Yes!” and give a fist pump. If you made a mistake, circle it. Ready? (Read the answers to the first half again as students stand.) Movement: To keep the energy and fun going, do a stretch or a movement game in between Sprints. Sprint B (Put Sprint B face down on desks with instructions to not look at the problems until the signal is given. Repeat the procedure for Sprint A up through the show of hands for how many right.) T: Stand up if you got more correct on the second Sprint than on the first. S: (Students stand.) T: Keep standing until I say the number that tells how many more you got right on Sprint B. If you got 3 more right on Sprint B than on Sprint A, when I say 3 you sit down. Ready? (Call out numbers starting with 1. Students sit as the number by which they improved is called.) Students may take sprints home. Group Counting (4 minutes) Note: Basic skip-counting skills from Grade 2 shift focus in this Grade 3 activity. Group-counting lays a foundation for interpreting multiplication as repeated addition. When students count groups in this activity, they add and subtract groups of three when counting up and down. T: Let’s count to 18 forward and backward. I want you to whisper, whisper, and then speak numbers. T: Watch my fingers to know whether to count up or down. A closed hand means stop. (Show signals as you explain.) T: (Rhythmically point up until a change is desired. Show a closed hand then point down.) S: (Whisper 1, whisper 2, speak 3, etc.) T: Let’s count to 18 forward and backward again. This time, think every number instead of whispering. Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.15 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 S: (Think), (think), 3, (think), (think), 6, (think), (think), 9, etc. T: What did we just count by? Turn and talk to your partner. S: Threes. T: Let’s count by threes. (Direct students to count forward and backward to 18, periodically changing directions. Emphasize the 9 to 12 transition.) Add Equal Groups (3 minutes) Materials: (S) Personal white boards Note: This activity reviews Lesson 1. Students directly relate repeated addition to multiplication. They interpret products as the number of equal groups times the number of objects in each group. T: (Project a picture array with 3 groups of 2 circled.) How many groups are circled? S: 3. T: How many are in each group? S: 2. T: Write this as an addition sentence. S: (Write 2 + 2 + 2 = 6.) T: Write a multiplication sentence for 3 twos equals 6. S: (Write 3 × 2 = 6.) Continue with possible sequence: 3 groups of 5, 5 groups of 10, and 3 groups of 4. Application Problem (5 minutes) Jordan uses 3 lemons to make 1 pitcher of lemonade. He makes 4 pitchers. How many lemons does he use altogether? Use the RDW process to show your solution. Note: This problem reviews equal groups multiplication from Lesson 1. It also leads into today’s concept development of relating multiplication to the array model. Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.16 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 Concept Development (30 minutes) Materials: (S) Personal white boards, Threes Array Template inside personal boards (pictured below), lemons image from application problem, 1 sheet of blank paper per student Problem 1: Relate equal groups to arrays. T: Look back at Jordan’s lemons. Compare the way his lemons are organized with the groups of 3 circles on NOTES ON your template. MULTIPLE MEANS OF S: The lemons in one group are touching each other, but REPRESENTATION: the circles have space between them.  Each line on As you teach the vocabulary rows, have the template shows three, like each group of lemons. students trace a row on the array with  The template is organized with everything in a finger while saying the word. You straight lines. may also have them identify other rows T: Many students are noticing straight lines on the around the room. For example, template. Let’s call a straight line going across a row. students may be seated in rows or you Use your blank paper to cover all but the top row. may have books arranged in rows. S: (Cover all but the top row.) T: Uncover 1 row at a time in the picture. As you uncover Threes Array Template (with each row, write the new total number of circles to the student work) right of it. S: (Skip-count by three using the Threes Array Template.) T: At the signal say the total number of circles you counted. (Signal.) S: 30 circles! T: Take 10 seconds to find how many rows of 3 you counted. At the signal say how many. (Signal.) S: 10 rows! T: True or false: 10 rows of 3 circles equals 30 circles? S: True! T: (Write 10 × 3 = 30 on the board.) Use the picture on your template to talk with your partner about why this equation is true. S: Yesterday we learned that we can multiply equal groups.  We skip-counted 10 rows of 3 circles each and the total is 30.  It just means 10 groups of 3 and NOTES ON when you add 10 threes, you get 30!  Yeah, but MULTIPLE MEANS OF writing 10 × 3 is a lot easier than writing out 3 + 3 + 3 + REPRESENTATION: 3… As you teach the vocabulary array, T: We call this type of organized picture an array. you may want to ask students to turn and talk, describing or defining an array for their partner. Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.17 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 T: (Project or draw the following image.) Take a look at this array. T: At the signal tell how many rectangles are in the top row. (Signal.) S: 4 rectangles. T: The size of 1 row is 4 rectangles. T: At the signal tell how many groups of four are in the array. (Signal.) S: 3 groups of four. T: To write the multiplication fact, we first write the number of groups. How many groups? S: 3 groups! T: (Write 3 × ____.) Next we write the size of the group. How many rectangles are in each group? S: 4 rectangles! T: (Fill in the fact to read 3 × 4.) Skip-count to find the total number of rectangles in the array. S: 4, 8, 12! T: (Complete the equation to read 3 × 4 = 12.) We just found the answer to the multiplication fact that represents the array. Show an array of 2 rows of 6 and repeat the process. Problem 2: Redraw equal groups as arrays. T: (Project or draw the following image.) The drawing shows 3 equal groups of 5. NOTES ON MULTIPLE MEANS OF ENGAGEMENT: T: Re-draw the picture as an array with 3 rows of 5 on Provide a challenge in this part of the your personal board. lesson by giving a multiplication sentence (e.g., 5 x 4 = _____) and no S: (Draw a 3 by 5 array.) picture. Have students draw both the T: Write a multiplication fact to describe your array. equal groups and array to represent the sentence. Then they skip-count to S: (Write 3 × 5.) find the total. T: Skip-count to find the answer to the multiplication fact. S: 5, 10, 15. (Write 3 × 5 = 15.) Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.18 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 Show 6 groups of 2 and repeat the process. Problem Set (10 minutes) Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students solve these problems using the RDW approach used for Application Problems. Student Debrief (10 minutes) Lesson Objective: Relate multiplication to the array model. The Student Debrief is intended to invite reflection and active processing of the total lesson experience. Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the ideas below to lead the discussion.  In Problems 5 and 6 how do the arrays represent equal groups?  Compare equal groups in scattered configurations and arrays.  Review the vocabulary: row, array, fact, number of groups, size of groups.  Prompt students to notice arrays around the room and possibly think of arrays in real world situations. Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.19 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 3 1 Exit Ticket (3 minutes) After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you assess the students’ understanding of the concepts that were presented in the lesson today and plan more effectively for future lessons. You may read the questions aloud to the students. Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.20 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 Sprint 3 1 Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.21 This work is licensed under a © 2013 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.  NYS COMMON CORE MATHEMATICS CURRICULUMNYS COMMON Lesson 2 Sprint 3 1 CORE MATHEMATICS CURRICULUM 3 1 Lesson 2: Relate multiplication to the array model. Date: 6/26/13 1.A.22