Overview. Essential Questions. Grade 4 Mathematics, Quarter 3, Unit 3.3 Multiplying Fractions by a Whole Number

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1 Multiplying Fractions by a Whole Number Overview Number of instruction days: 8 10 (1 day = 90 minutes) Content to Be Learned Understand that a fraction can be written as a multiple of unit fractions with the same denominator. Understand that multiples help in multiplying a fraction by a whole number. Solve word problems involving multiplication of a fraction by a whole number. Use fraction models and equations to represent a problem involving multiplication of a fraction by a whole number. Mathematical Practices to Be Integrated 4 Model with mathematics. Represent problem situations involving fractions in multiple ways including numbers, words, visuals models, or pictures. Connect different representations (numbers, words, models) and explain the connections. 6 Attend to Precision Accurately explain and/or demonstrate the multiplication of fractions by a whole number. State the meaning of the symbols chosen to solve the problem. Essential Questions How can a fraction be written as a multiple of unit fractions with the same denominator? How does understanding multiples help in multiplying a fraction by a whole number? What strategies can you use to solve a word problem involving multiplication of a fraction by a whole number? How can you use visual representation to express multiplication of a whole number by a fraction? Providence Public Schools D-81

2 Multiplying Fractions by a Whole Number (8 10 days) Standards Common Core State Standards for Mathematical Content Number and Operations Fractions 3 3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4). b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 (2/5) as 6 (1/5), recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 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. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision D-82 Providence Public Schools

3 Multiplying Fractions by a Whole Number (8 10 days) Grade 4 Mathematics, Quarter 3, Unit 3.3 appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning Fractions were a major cluster in third grade according the PARCC framework. Students learned that a fraction has a part whole relationship and that a fraction can be represented on a number line. They generated simple equivalent fractions and expressed whole numbers as fractions. They recognized that fractions are equivalent to whole numbers (e.g., 3 = 3/1). Current Learning Fractions are a major cluster again in fourth grade according to the PARCC framework. Students learn to add and subtract fractions and mixed numbers with like denominators. They learn to recognize and generate equivalent fractions. They apply and extend their previous understandings of multiplication (repeated addition) to multiply a fraction by a whole number (e.g., 5 x ¼ = 5/4). They solve word problems involving multiplication of a fraction by a whole number. They use visual fraction models and equations to represent a problem. Students also learn to use decimal notation for fractions with denominators 10 or 100. They express fractions with denominators of 10 as an equivalent fraction with a denominator of 100. Future Learning Fractions are a major cluster in fifth grade as well according to the PARCC document. Students will expand their knowledge of equivalent fractions as a strategy for adding and subtracting fractions with different denominators. They will apply and extend their previous understanding of multiplication and division to multiply and divide fractions. They will solve real-world problems involving multiplication of fractions and mixed numbers. Additional Findings Grade 4 students learn a fundamental property of equivalent fractions: Multiplying the numerator and denominator of a fraction by the same nonzero number whole number results in a fraction that represents the same number as the original fraction. (Progressions for the Common Core State Standards in Mathematics: K 5, Number and Operations, Fractions, p. 5) Providence Public Schools D-83

4 Multiplying Fractions by a Whole Number (8 10 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. Express a fraction as a multiple of unit fractions with the same denominator. Use models, drawings and equations to multiply a fraction by a whole number. Solve word problems involving multiplication of a fraction by a whole number Use fraction models and equations to represent a problem involving multiplication of a fraction by a whole number. Instruction Learning Objectives Students will be able to: Understand a fraction a/b as a multiple of 1/b. Understand a multiple of 1/b as a fraction a/b. Use visual fraction models to represent and solve for the product of a fraction by a whole number. Use equations to represent and solve for the product of a fraction by a whole number. Solve word problems involving multiplication of a fraction by a whole number. Demonstrate understanding of the concepts and skills in this unit. Resources envision Math Grade 4, Pearson Education, Inc., 2009 See Section I, Supplemental Materials Lesson 11.5D, Fractions as Multiples of Unit Fractions: Using Models Lesson 11.5E, Multiplying a Fraction by a Whole Number: Using Models Lesson 11.5F, Multiplying a Fraction by a Whole Number: Using Symbols D-84 Providence Public Schools

5 Multiplying Fractions by a Whole Number (8 10 days) Grade 4 Mathematics, Quarter 3, Unit 3.3 Teacher Resource Masters Student Edition Investigations in Numbers, Data, and Space, Grade 4, Pearson Education, Inc., 2008 Implementing Investigations in Grade 4- Implementation Guide See Section I, Supplemental Materials Lesson 3A.1, Multiplying Whole Numbers and Fractions Lesson 3A.2, Multiplying Whole Numbers and Fractions, continued Lesson 3A.3, Assessment: Multiplying With Fractions Teacher Resources Binder Pearson Online Success Net, Implementing Investigations Site, 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 sections for ample resources to refer to when planning your unit and individual lessons. Materials Fraction bars, paper strips, scissors Instructional Considerations Key Vocabulary multiple product Planning for Effective Instructional Design and Delivery Multiplying fractions is new in grade 4. It is very important to make connections between repeated addition of unit fractions and multiplication of a unit fraction by a whole number. Students use visual fraction models and other manipulatives to build the concept and progress to expressing the products in equation form. Students need many opportunities to work with this type of problem in context to understand the connections between the models and their corresponding equations. Contexts involving a whole number times a fraction lend themselves to modeling using fraction bars and drawing and examining the patterns that they see happening while using the models. Students can apply their knowledge of the Associative Property of Multiplication to reason that 5 groups of 2 groups of 1/6 can be thought of either as 5 groups of 2/6 or 10 groups of 1/6, and therefore equal to 10/6. These connections can be made through representing them with models and drawings first, then the written equations. Incorporate Ten Minute Math Activities, the Problem of the Day, and the Daily Spiral Review that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Providence Public Schools D-85

6 Multiplying Fractions by a Whole Number (8 10 days) 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 section. Notes D-86 Providence Public Schools