Grade 3 Mathematics, Quarter 1, Unit 3.1 Applying Strategies to Multiplication and Division Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Mathematical Practices to Be Integrated Apply properties of operations to find the area of a rectangular figure (by decomposing it into non-overlapping rectangles and adding their areas). Apply properties of multiplication and the relationship between multiplication and division when solving problems. Solve division problems that can also be represented as unknown factor problems. Strengthen fluency of multiplication and division facts through 100. Recognize area as additive. 4 Model with mathematics. Use square tiles to demonstrate the distributive property of multiplication. Write an equation to match your model. 7 Look for and make use of structure. Use parenthesis to record application of associative and distributive properties. Determine different ways to decompose a larger array into smaller arrays. Represent the distributive property in a model and equation. Essential Question How can you apply properties of multiplication to solve problems? How can you use the distributive property to find the area of rectangles and rectangular figures? What strategies and patterns can you use to strengthen your fluency of multiplication and division facts through 100? Providence Public Schools D-64
Applying Strategies to Multiplication and Division (7 9 days) Grade 3 Mathematics, Quarter 1, Unit 3.1 Standards Common Core State Standards for Mathematical Content Geometry 3.G Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5 Apply properties of operations as strategies to multiply and divide. 2 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 canfind 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56. (Distributive property.) 2 Students need not use formal terms for these properties. 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. 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 arithmetic. 3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Measurement and Data 3.MD 3.MD.7 Relate area to the operations of multiplication and addition. c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a b and a c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectangular figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Providence Public Schools D-65
Grade 3 Mathematics, Quarter 1, Unit 3.1 Applying Strategies to Multiplication and Division (7 9 days) Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning Students in Grade 2 represented and solved problems involving addition and subtraction. They worked with equal groups of objects to prepare for their work with multiplication. Grade 2 students also used addition and subtraction to find the total number of objects arranged in rectangular arrays with up to five rows and five columns. They wrote equations to express the total as the sum of equal addends (repeated addition). D-66 Providence Public Schools
Applying Strategies to Multiplication and Division (7 9 days) Grade 3 Mathematics, Quarter 1, Unit 3.1 Current Learning In Grade 3, students study multiplication in terms of equal groups, arrays, and area. Students find the area of a rectangle with whole-number side lengths by tiling it. Students also multiply side lengths to find areas of rectangles. They also use the tiling method to show area models representing the distributive property decomposing the big array to find the area (For example, side lengths of 5 and 7 = 5 x 5 + 5 x 2.) Grade 3 students recognize area as an additive and find the areas of rectangular figures by decomposing them. Developing understanding of multiplication and division strategies is a critical area of focus for Grade 3. Students find products and related quotients involving single-digit whole numbers and 10. This is a required fluency for Grade 3. By the end of Grade 3, students conceptually understand and know, quickly and accurately, all products of two one-digit numbers from memory. Future Learning Students in Grade 4 will apply area formulas for rectangles in real-world situations and in mathematical problems. Students will also apply their understanding of models to multiplication (equal-sized groups, arrays, area models). Students in Grade 5 will solve for area of a rectangle with fractional side lengths by tiling (using square units) and by multiplying side lengths. Additional Findings As students learn about the meaning of multiplication and develop strategies to solve multiplication problems, they will begin to use properties such as distibutivity naturally (Schifter 1999). However, discussing about the properties themselves, as well as how they serve as tools for solving a range of problems, is important if students are to add strength to their intuitive notions and advance their understanding of multiplicative structures (Principals and Standards for School Mathematics, p.161). Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.(ccss Document, p.21) According to the PARCC Model Content Frameworks, area is a major concept within measurement, and area models must function as a support from multiplicative reasoning in grade 3 and beyond. (p. 16) Providence Public Schools D-67
Grade 3 Mathematics, Quarter 1, Unit 3.1 Applying Strategies to Multiplication and Division (7 9 days) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Apply properties of operations to find the area of a rectangular figure (by decomposing it into non-overlapping rectangles and adding their areas). Apply properties of multiplication and the relationship between multiplication and division when solving problems. Solve division problems that can also be represented as unknown factor problems. Monitor progress of fluency of multiplication and division facts through 100. Recognize area as additive. Represent the distributive property in a model and equation. D-68 Providence Public Schools
Applying Strategies to Multiplication and Division (7 9 days) Grade 3 Mathematics, Quarter 1, Unit 3.1 Instruction Learning Objectives Students will be able to: Resources Apply properties of operations to find the area of a rectangular figure (by decomposing it into non-overlapping rectangles and adding their areas). Apply properties of multiplication and the relationship between multiplication and division when solving problems. Solve division problems that can also be represented as unknown factor problems. Monitor progress of fluency of multiplication and division facts through 100. Recognize area as additive. Represent the distributive property in a model and equation. Demonstrate understanding of the concepts and skills learned in this unit. envision Math Grade 3, Pearson Education, Inc., 2009 Topic 6, Teacher Edition, Multiplication Fact Strategies: Use Known Facts Topic 16, Teacher Edition, Perimeter, Area, and Volume Resource Masters Student Pages Investigations in Numbers, Data, and Space Grade 3, Pearson Education, Inc., 2008 Implementing Investigations in Grade 3 Implementation Guide Unit 4 Teacher Edition Perimeter, Angles and Area Teacher Resource Binder Also see Section I, Supplemental Materials EnVision: 6-1A The Distributive Property 16-7B Area and the Distributive Property (limit to rectangular figures) 16-7C Area of Irregular Shapes (limit to rectangular figures) Providence Public Schools D-69
Grade 3 Mathematics, Quarter 1, Unit 3.1 Applying Strategies to Multiplication and Division (7 9 days) Pearson Success Net, www.pearsonsuccessnet.com Investigations (TERC), http://investigations.terc.edu/index.cfm Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment for ample resources to refer to when planning your unit and individual lessons. Materials Grid paper Square tiles Instructional Considerations Key Vocabulary Operation Multiply Divide Product Quotient additive factor Planning for Effective Instructional Design and Delivery In this unit, students apply the distributive property to find the area of a rectangular figure. Students decompose the figure into smaller non-overlapping rectangles, find the area of each part, and combine those areas to determine the area of the whole figure. Students are actually showing how the distributive property works (a and b + c is the sum of a b and a c). Students recognize area as additive by decomposing a larger rectangular array into two or more smaller arrays.. They determine the area of each array and compose those areas to find the areas of that larger array. Students can model the distributive property by composing rectangular arrays that can be decomposed into smaller arrays. An activity such as this( Decomposing an Array) is where the strategy of solving an easier known problem to help solve an unknown problem is also modeled. It is important to note that students determine Area for rectangles and rectangular arrays only..within this context, students work with array models and practice fluency of facts within 100. It is reasonable D-70 Providence Public Schools
Applying Strategies to Multiplication and Division (7 9 days) Grade 3 Mathematics, Quarter 1, Unit 3.1 (and optional) to extend the focus on facts within 100 in this unit. Arrays that are larger than 10 x 10 can be decomposed into smaller arrays. extra time, support and practice throughout the year are essential to becoming fluent. Students in third grade also use strategies based on their understanding of place value and properties of operations in problem solving. Students determine the unknown whole number in a multiplication or division problem. They solve multiplication and division problems using equal groups, arrays, and measurement. Common multiplication and division situations are detailed in Table 2 on page 89 of the Common Core State Standards for Mathematics. Grade 3 focuses on two of the three major situations: equal groups and arrays. Third-grade students apply properties of operations (commutative, associative, distributive properties of multiplication) when multiplying and dividing. They represent problems as equations, solve two-step word problems involving multiplication and division, and multiply one-digit whole numbers by multiples of 10 (10 90). Incorporate Ten Minute Math Activities, the Problem of the Day, Daily Spiral Review and Quick Checks that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction. Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources (particularly the Algebra Connections and Teacher Notes sections) Notes Providence Public Schools D-71