Overview. Essential Questions. Grade 5 Mathematics, Quarter 1, Unit 1.3 Addition and Subtraction of Fractions

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1 Addition and Subtraction of Fractions Overview Number of instruction days: (1 day = 90 minutes) Content to Be Learned Add and subtract fractions and mixed numbers with unlike denominators. Use equivalent fractions to add and subtract fractions with unlike denominators. Use visual models to solve addition and subtraction problems involving fractions. Represent and solve word problems using addition and subtraction of fractions. Use benchmark fractions and number sense of fractions to estimate and check reasonableness of sums and differences. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Use benchmark fractions to check the reasonableness of answers when solving word problems involving fractions with unlike denominators. 3 Construct viable arguments and critique the reasoning of others. Construct arguments using concrete objects, drawings, or diagrams to justify sums or differences. Make conjectures about the solutions and explore the reasonableness of the solutions. Listen to or read the arguments others make to justify how they find sums and differences of fractions and decide whether the arguments make sense. Essential Questions How do you explain the process involved in adding or subtracting fractions with unlike denominators? How do you explain the process involved in adding or subtracting fractions greater than one with unlike denominators? How can you use estimation to help you determine the solutions to addition and subtraction problems involving fractions? 4 Model with mathematics. Apply knowledge of addition and subtraction of fractions to real-life situations. Draw conclusions, apply reasonableness, and interpret solutions to revise strategies to solve problems if needed. How can you add and subtract fractions with mixed numbers? How can you add and subtract fractions that include improper fractions? How are benchmark fractions used to estimate and check reasonableness of sums and differences of fractions? Providence Public Schools D-17

2 Addition and Subtraction of Fractions (12 14 days) Standards Common Core State Standards for Mathematical Content Number and Operations Fractions 5.NF Use equivalent fractions as a strategy to add and subtract fractions. 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not D-18 Providence Public Schools

3 Addition and Subtraction of Fractions (12 14 days) Grade 5 Mathematics, Quarter 1, Unit 1.3 generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 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. Clarifying the Standards Prior Learning In Grade 4, students used visual fraction models to generate equivalent fractions of the same whole. They applied this understanding when they compared fractions with different numerators and denominators. Current Learning Fifth-grade students calculate sums and differences of fractions with unlike denominators (including mixed numbers) and re-express both fractions in terms of a new denominator. They represent and solve word problems using these fractions. Students use benchmark fractions and number sense of fractions to estimate and check reasonableness of answers. Later in Grade 5, students will multiply fractions and divide a whole number by a fraction, or a fraction by a whole number. Future Learning In Grade 6, students will use their prior knowledge of fractions to continue multiplying and dividing fractions, including dividing a fraction by a fraction. Students will use the relationship between multiplication and division to help understand why procedures for dividing fractions make sense. In Grade 7, students will apply and extend their knowledge of fractions to include all operations of rational numbers. Additional Findings Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency with standard procedures for adding and subtracting fractions. They make reasonable estimates of fraction sums and differences. (Curriculum Focal Points, p. 17) Providence Public Schools D-19

4 Addition and Subtraction of Fractions (12 14 days) Developing the operations of fractions over time will help students attend carefully to the underlying unit quantities when solving problems. Students build on their knowledge of whole number operations and their knowledge of fractions to make sense of operations with fractions. (Progressions for the Common Core State Standards in Mathematics, 3 5: Number and Operations Fractions, August 12, 2011) 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. Add and subtract fractions (including mixed numbers) with unlike denominators. Use equivalent fractions to add and subtract fractions with unlike denominators. Represent and solve addition and subtraction problems involving fractions. Use benchmark fractions and number sense of fractions to estimate and check reasonableness of sums and differences. Learning Objectives Instruction Students will be able to: Add and subtract fractions (and mixed numbers) with unlike denominators. Use visual models to solve addition and subtraction problems involving fractions. Use equivalent fractions to add and subtract fractions with unlike denominators. Use benchmark fractions and number sense of fractions to estimate and check reasonableness of sums and differences. Demonstrate understanding of the concepts and skills learned in this unit. D-20 Providence Public Schools

5 Addition and Subtraction of Fractions (12 14 days) Grade 5 Mathematics, Quarter 1, Unit 1.3 Resources envision Math Grade 5, Pearson Education, Inc., 2009 Topic 10, Multiplying Whole Numbers, Teacher Edition Teacher Resource Masters Student Edition Also see Supplemental Insert: Lesson 10-1A Estimations Sums and Differences Lesson 10-5A, Modeling Addition and Subtraction of Mixed Numbers Lesson 10-7A More Adding and Subtraction Mixed Numbers envision Math Grade 4, Pearson Education, Inc., 2009 Topic 11:, Adding and Subtracting Fractions, Teacher Edition Teacher Resource Masters Also see Supplemental Insert Investigations in Numbers, Data, and Space Grade 5, Pearson Education, Inc., 2008 Implementing Investigations at Grade 5- Implementation Guide Unit 4, What s That Portion? Teacher Edition Investigation 3 - Adding and Subtracting Fractions, pages Teacher Resource Binder Pearson Online Success Net: Implementing Investigations Site: Connected Math 2, Grade 6, Bits and Pieces II Text Book Companion Sites: Video Tutors Homework Help Multiple Choice Skills Practice Vocabulary Quiz 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 specific recommendations. Providence Public Schools D-21

6 Addition and Subtraction of Fractions (12 14 days) Materials Fraction strips/tiles, number lines Instructional Considerations Key Vocabulary numerator denominator mixed number equivalent fraction Planning for Effective Instructional Design and Delivery In this unit, students represent fractional parts and apply knowledge of fraction equivalents when solving addition and subtraction problems involving fractions with unlike denominators (including mixed numbers). Representing fractions on a clock (see Investigations Unit 4, Investigation 3), representing fractions using visual models, and working with a number line helps students to visualize and reason about fractions and their relationships. Students may need to re-express one or more fractions with equivalent fractions in an equation based on what they have in common when adding/subtracting fractions and mixed numbers. Understanding that a fraction represents a quantity in relation to a unit whole and that the fractions in the equation must refer to the same whole is important to recognize and understand when solving problems. The CCSS does not distinguish mixed numbers and improper fractions from other types of fractions. Students understand that unit fractions are the basic building block of fractions and that every fraction is obtained by combining unit fractions. In previous grades, students built fractions from the unit fractions, seeing the numerator as the quantity of unit fractions you get by combining that many unit fractions together (for example, 5/3 is the quantity you get by combining 5 parts together when the whole is divided into 3 equal parts). Mixed numbers and improper fractions are inclusive in this understanding. Visual models will support the connection between a fraction and its corresponding mixed number as students add and subtract fractions. Using benchmark fractions to estimate sums and differences of fractions helps students to identify reasonableness of their solutions. Using their knowledge of fraction equivalents and the relationship of fractions to benchmark fractions such as ½, 1, 2, students decide which fraction is greater or less than another. Questions such as Is the fraction more or less than one whole? More/less than half? Closer to 0 or 1? should be posed throughout problem solving. Students who ask such questions of themselves will be able to make sense of and monitor their own work in this unit and in subsequent work. Throughout the unit, students can practice their estimation skills by engaging in the Ten-Minute Math Activity titled Estimation and Number Sense: Closest Estimate. (See Investigations Unit 7 TE, pp. 20, 54-56). In this activity, students make the closest estimate to a given problem and explain how they made their estimate. To support conceptual understanding, use an interactive number line. This nonlinguistic representation strategy supports learning by using a physical model. An interactive number line could look like a clothesline with benchmark fractions and whole numbers identified (folded paper hanging on the clothes line). Students can order fractions and explain their reasoning as they place and move fractions on the number line. On a number line, the whole is indicated by the unit interval (the interval between 0 and 1, for example). D-22 Providence Public Schools

7 Addition and Subtraction of Fractions (12 14 days) Grade 5 Mathematics, Quarter 1, Unit 1.3 Professional Development Support is provided in the both the envision Math and Investigations resources. Refer to the Math Background for Teachers in each Envision Topic TE listed in the unit resources section (A and B pages and additional information embedded throughout each TE). Also, in the envision Math student edition there is a Problem Solving Handbook (pages xvii-xxix). This handbook will provide support throughout the year to support students in problem solving. This handbook is also located in the Topic 1 TE, p.2g-2l. In Investigations Unit 4, see Mathematics in this Unit p , and Teacher Note, p and Descriptions of strategies (representing fractional parts and writing equivalent fractions when adding and subtracting fractions) should be considered in planning for instruction. The important concept that a fraction represents a quantity in relation to a unit whole is also presented. Incorporate Ten Minute Math Activities, the Problem of the Day, the Daily Spiral Review and Quick Checks that are aligned to the Common Core State Standards for Mathematics. EnVision Math 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. Providence Public Schools D-23

8 Addition and Subtraction of Fractions (12 14 days) Notes D-24 Providence Public Schools