Understanding Fractions on the Number Line

Transcription

1 Understanding Fractions on the Number Line Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Mathematical Practices to Be Integrated Develop understanding of fractions as rational numbers. Understand that the unit fraction 1/b is one part of a whole that has been partitioned into equal parts. Understand and represent a fraction as a number on the number line. Define the interval from 0 to 1 as the whole and partition it into b equal parts. Understand that the endpoint of the fraction s interval locates the fraction on the number line. 2 Reason abstractly and quantitatively. Justify conclusions regarding how parts are equal. Analyze and compare arguments, using concrete representations to identify flawed logic versus correct logic and making improvements as needed. 5 Use appropriate tools strategically. Apply mathematics to solve real-world problems. Use diagrams, tables, and graphs to draw mathematical conclusions. Interpret results and reflect on them to make improvements. Essential Questions What is a fraction? What does the denominator in a fraction tell you? What does the numerator tell you? What is a unit fraction? How do you use a number line to model a fraction? To locate a fraction? How do you define the interval that represents the whole on a number line? Providence Public Schools D-57

2 Understanding Fractions on the Number Line (7 9 days) Standards Common Core State Standards for Mathematical Content Number and Operations Fractions 5 3.NF 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Develop understanding of fractions as numbers. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Common Core State Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient D-58 Providence Public Schools

3 Understanding Fractions on the Number Line (7 9 days) Grade 3 Mathematics, Quarter 2, Unit 2.4 students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning In previous grades, students partitioned shapes into equal shares using language such as halves, thirds, fourths, and quarters. Students also worked with rulers to measure to the nearest whole unit. They created bar graphs with single-unit scale, so the concept of equal intervals will be familiar to them. Current Learning Students continue to develop their understanding of fractions as rational numbers using the number line model. Students continue to practice dividing the whole into equi-distant intervals. They also continue to build their understanding of unit fractions (e.g. 1/a where a whole numbers, and a 0). Denominators are limited to 2, 3, 4, 6, and 8 in Grade 3. Students also measure to the nearest ½, ¼ and 1/8 and create graphs based on their measurement data. Future Learning Students in Grade 4 will develop understanding of fraction equivalence and operations with fractions. They will compose and decompose fractions using unit fractions (e.g. 3/8 = 1/8 + 1/8 + 1/8). Grade 5 students will apply their understanding of fractions and fraction models to add and subtract fractions with unlike denominators. An in-depth understanding of the number line model will help students understand the 4 th and 5 th grade concepts. They will become fluent in calculating sums and differences of fractions and will provide reasonable estimations. Additional Findings Adding It Up brings up an important point: Students often view the study of whole numbers, decimal fractions, common fractions and integers as disconnected topics. One tool that we believe may be useful in developing numerical understanding and in making connections across number systems is the number line. The number line givese a unified geometric representation [of rational numbers]. it may help Providence Public Schools D-59

4 Understanding Fractions on the Number Line (7 9 days) students develop a sense of the magnitudes and relationships of those numbers in a way that is les clear in other representations. (p. 418) 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. Place and locate fractions on a number line. Partition 0 to 1 into b equal parts. Explain what a nd b are and their relationship to each other in the fractions a/b and 1/b. Explain or show what a unit fraction is. (1/b etc ) Model a/b and 1/b fractions using the number line. D-60 Providence Public Schools

5 Understanding Fractions on the Number Line (7 9 days) Grade 3 Mathematics, Quarter 2, Unit 2.4 Instruction Learning Objectives Students will be able to: Develop understanding of fractions as numbers by placing and locating them on a number line. Define the interval from 0 to 1 as the whole and partition it into b equal parts. Demonstrate understanding of a/b and 1/b fractions by explaining what a nd b are and their relationship to each other. Model a/b and 1/b fractions using the number line. Demonstrate understanding of the concepts and skills learned in this unit. Resources envision Math Grade 3, Pearson Education, Inc., 2009 Topic 12 Teacher Edition, Understanding Fractions Resource Masters Student Pages Also see Section I, Supplemental Materials 12-2a Fractions and Regions Investigations in Numbers, Data, and Space Grade 3, Pearson Education, Inc., 2008 Implementing Investigations in Grade 3 Implementation Guide Unit 7 Teacher Edition Finding Fair Shares Investigation 1: Sharing Brownies Investigation 2: Many ways to Make a Share Also see Section I, Supplemental Materials 1.4A Fractions on a Number Line 1.4B Comparing Fractions on Number Lines Providence Public Schools D-61

6 Understanding Fractions on the Number Line (7 9 days) Teacher Resources Binder Pearson Success Net Investigations (TERC) 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 1-inch grid paper, Spaghetti and Meatballs for All by Marilyn Burns Instructional Considerations Key Vocabulary Fraction part whole unit fraction numerator denominator equi-distant interval Planning for Effective Instructional Design and Delivery Students continue to develop their understanding of fractions as rational numbers; and as such, all rational numbers have a place on the number line. The number line is an important model for fractions in terms of their value (or distance from 0 on the number line) as well as their location on the number line. Students should have multiple opportunities to partition the unit interval of 0 to 1 into equidistant intervals (e.g. ¼, ¼, ¼, ¼ or 1/3, 1/3, 1/3). A number line model that can be folded, bent and/or manipulated is helpful, so that students can prove to themselves that they have created equal intervals, which can then represent unit fractions in terms of value. When these unit fractions are labeled on equidistant points, they will also represent that number as having a place on the number line. Connecting their interactive models to real-life fractional models with equal intervals (i.e. rulers, thermometers, measuring cups etc) will reinforce the concept of unit fractions and their functionality. D-62 Providence Public Schools

7 Understanding Fractions on the Number Line (7 9 days) Grade 3 Mathematics, Quarter 2, Unit 2.4 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-63