Grade 4 Mathematics, Quarter 3, Unit 3.1 Adding and Subtracting Fractions and Mixed Numbers With Like Denominators. Overview

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1 Adding and Subtracting Fractions and Mixed Numbers With Like Denominators Overview Number of instruction days: 6 8 (1 day = 90 minutes) Content to Be Learned Decompose fractions into a sum of unit fractions. Add and subtract fractions and mixed numbers with like denominators. Create a line plot to display a data set in fractions of a unit. Solve problems involving addition and subtraction of fractions using the information on a line plot. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Explain to peers the meaning of a problem and look for ways to solve it. Check for reasonableness of an answer by asking Does this make sense? 4 Model with mathematics. Justify decompositions using a visual fraction model. Use data presented in line plots to solve problems involving addition and subtraction. Essential Questions How can you decompose a fraction into a sum of fractions with the same denominator? How can you add and subtract fractions and mixed numbers with like denominators? What is the relationship between addition and subtraction of fractions? How can data with fractions be represented on a line plot? Providence Public Schools D-65

2 Version 4 Adding and Subtracting Fractions and Mixed Numbers With Like Denominators (6 8 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 100. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = /8 = 8/8 + 8/8 + 1/8. c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. 4.NF Measurement and Data 4.MD Represent and interpret data. 4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. 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. D-66 Providence Public Schools

3 Adding and Subtracting Fractions and Mixed Grade 4 Mathematics, Quarter 3, Unit 3.1 Numbers With Like Denominators (6 8 days) Version 4 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 3, fractions are a major cluster according to the PARCC document. Third-grade students developed an understanding of fractions as numbers. They compared and ordered fractions using visual models. Students represented and interpreted data using scaled bar graphs and scaled picture graphs. Current Learning In Grade 4, fractions are again a major cluster according to the PARCC document. Students use visual fraction models to generate equivalent fractions of the same whole. They apply this understanding when they compare fractions with different numerators and denominators. They add and subtract fractions and mixed numbers with like denominators. They solve word problems involving adding and subtracting fractions with like denominators. Students also decompose fractions into the sum of unit fractions with like denominators (e.g., 4/3 = 1/3 + 1/3 + 1/3 + 1/3). Students make a line plot to display a data set of measurements and fractions of a unit. Future Learning In fifth grade, fractions are a major cluster according to the PARCC document. Fifth-grade students will calculate sums and differences of fractions with unlike denominators, including mixed numbers (e.g., 1/ /3 = 3/6 + 16/6 = 19/6 = 3 1/6). They will represent and solve word problems using these fractions. Students will use benchmark fractions and their number sense of fractions to estimate and check reasonableness of answers. Students will also multiply and divide fractions. Additional Findings The meaning of addition is the same for fractions and whole numbers, even though algorithms for calculating their sums can be different. Just as the sum of 4 and 7 can be seen as the length of the segment obtained by joining together two segments of lengths 4 and 7, so the sum of 2/3 and 8/5 can be seen as the length of the segment obtained joining together two segments of lengths 2/3 and 8/5. (Progressions for the Common Core State Standards in Mathematics: K 5, Number and Operations, Fractions, p. 6) Providence Public Schools D-67

4 Version 4 Adding and Subtracting Fractions and Mixed Numbers With Like Denominators (6 8 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. Explain how to decompose fractions into a sum of unit fractions. Explain how to decompose fractions into a sum of fractions in more than one way. Justify the decomposition of fractions using visual models. Add and subtract fractions and mixed numbers with like denominators using multiple strategies. Create a line plot to display a data set in fractions of a unit. Solve problems involving addition and subtraction of fractions using the information on a line plot. Learning Objectives Instruction Students will be able to: Decompose fractions into a sum of unit fractions. Decompose fractions into a sum of fractions in more than one way. Add and subtract fractions and mixed numbers with like denominators using visual models and other strategies. Create a line plot to display a data set in fractions of a unit. Solve problems involving addition and subtraction of fractions using the information on a line plot. Demonstrate understanding of the concepts and skills in this unit. D-68 Providence Public Schools

5 Adding and Subtracting Fractions and Mixed Grade 4 Mathematics, Quarter 3, Unit 3.1 Numbers With Like Denominators (6 8 days) Version 4 Resources envision Math Grade 4, Pearson Education, Inc., 2009 Topic 11, Adding and Subtracting Fractions, Teacher Edition Also see Section I, Supplemental Materials Lesson 11.1A, Decomposing and Composing Fractions Lesson 11.5A, Modeling Addition and Subtraction of Mixed Numbers Lesson 11.5B, Adding Mixed Numbers Lesson 11.5C, Subtracting Mixed Numbers Teacher Resource Masters Student Edition Investigations in Numbers, Data, and Space, Grade 4, Pearson Education, Inc., 2008 Implementing Investigations in Grade 4- Implementation Guide Unit 6, Fraction Cards and Decimal Squares, Teacher Edition Investigation 1: Parts of Rectangles Also see Section I, Supplemental Materials Lesson 2.7A, Adding and Subtracting Mixed Numbers Unit 9, Penny Jars and Plant Growth, Teacher Edition Investigation 3: Collecting and Analyzing Measurements 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 model strips Instructional Considerations Key Vocabulary mixed number Providence Public Schools D-69

6 Version 4 Adding and Subtracting Fractions and Mixed Numbers With Like Denominators (6 8 days) Planning for Effective Instructional Design and Delivery In this unit, students begin by decomposing fractions into unit fractions. Students should justify their decomposing of fractions using visual fraction models. Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions. Using visual fraction models can also be used to emphasize the concept of turning mixed numbers into an equivalent fraction. Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one way. Using a separate algorithm for mixed numbers in addition and subtraction is not necessary. Students usually add or subtract the whole numbers first and then work with the fractions using the same strategies they have applied to problems that contained only fractions. The measurement standard in this unit provides a context for students to work with fractions by measuring objects in fractions of a unit (1/2, 1/4, 1/8). Students are making a line plot of data and then adding and subtracting fractions based on data in the line plot. In Lesson 11-1 (TE, page 250B) use nonlinguistic representations for acquiring knowledge. Students use physical models such as fraction strips to help develop understanding of adding and subtracting fractions with like denominators. Students use fraction strips to model 2/10 of a mile and 5/10 of a mile to answer the question, How far does Greg walk in all? Have students continue to use fraction strips to solve problems throughout the lesson. 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 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. D-70 Providence Public Schools

7 Adding and Subtracting Fractions and Mixed Grade 4 Mathematics, Quarter 3, Unit 3.1 Numbers With Like Denominators (6 8 days) Version 4 Notes Providence Public Schools D-71

8 Version 4 Adding and Subtracting Fractions and Mixed Numbers With Like Denominators (6 8 days) D-72 Providence Public Schools