Grade 2 Mathematics, Quarter 4, Unit 4.4 Representing and Interpreting Data Using Picture and Bar Graphs Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Draw a picture graph with single-unit scale to represent a data set with up to four categories. Draw a bar graph with up to four categories. Solve simple put together, take apart, and compare problems using information presented in a bar graph. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Use picture and bar graphs to represent and solve problems. Check for reasonableness of answers and alternate solution paths. 3 Construct viable arguments and critique the reasoning of others. Construct arguments using data represented in picture and bar graphs. Justify and communicate conclusions to someone else. Essential Questions How can you use your data to make a bar graph? Picture graph? How can you use a bar graph to organize information and compare data? How can you use a picture graph to organize information and compare data? Why would you choose a bar graph over a picture graph, or vice versa, to represent your data? Providence Public Schools D-84
Representing and Interpreting Data Using Grade 2 Mathematics, Quarter 4, Unit 4.4 Picture and Bar Graphs (7 9 days) Version 5 Standards Common Core State Standards for Mathematical Content Measurement and Data 2.MD Represent and interpret data. 2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems 4 using information presented in a bar graph. 4 See Glossary, Table 1. 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 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. Providence Public Schools D-85
Grade 2 Mathematics, Quarter 4, Unit 4.4 Version 5 Representing and Interpreting Data Using Picture and Bar Graphs (7 9 days) Clarifying the Standards Prior Learning In first grade, students organized, represented, and interpreted data with up to three categories (1.MD.4). They asked and answered questions about the total number of data points. They also asked and answered questions about how many were in each category. They used data to compare and talk about how many more or how many less were in one category than in another. Current Learning Students draw a picture graph and a bar graph with single-unit scales to represent a data set with up to four categories. Earlier in the year, students solved put together problems. Students continue to solve put together, take apart, and comparison problems. See CCSS Glossary, Table 1. Future Learning In third grade, students will make picture graphs and bar graphs with multi-unit scales (3.MD.3). Using information from a scaled bar graph, students will solve one- and two-step comparison problems. Students will also use all four operations to solve problems. Additional Findings According to Adding It Up, the research shows that many elementary students have difficulty representing data sets. They sometimes have difficulty sorting, organizing and reducing the data to make it accessible or readable. Technology can help students overcome these difficulties. (p. 290) A Research Companion to Principles and Standards for School Mathematics, notes that Too often students treat data as numbers only, forgetting that those numbers have a context and that the reason for analyzing them is to learn about that context. Students seem particularly vulnerable to treating data as numbers only when they work with data they themselves have not collected. (p. 211) Collecting, organizing, representing, and interpreting data is an authentic application of mathematics. The context in a data investigation is of utmost import. Work with data and the comparison of two groups is the foundation from which further statistics will rise... Making comparisons is the heart of statistics. (A Research Companion to Principles and Standards for School Mathematics, p. 206) 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. D-86 Providence Public Schools
Representing and Interpreting Data Using Grade 2 Mathematics, Quarter 4, Unit 4.4 Picture and Bar Graphs (7 9 days) Version 5 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. Represent a data set by drawing a picture graph. (See standard for parameters.) Represent a data set by drawing a bar graph. (See standard for parameters.) Solve simple put together, take apart, and compare problems 4 using info presented in a bar graph. 4 See CCSS Glossary, Table 1. Instruction Learning Objectives Students will be able to: Draw a picture graph with single unit scale to represent a data set up to four categories. Draw a bar graph with up to four categories. Solve simple put together, take apart and compare problems 4 using info presented in a bar graph. 4 See CCSS Glossary, Table 1. Demonstrate understanding of the concepts and skills learned in this unit. Resources Investigations in Number, Data, and Space Grade 2, Pearson Education, Inc., 2008 Implementing Investigations in Grade 2 Implementation Guide Unit 4 Teacher Edition, Pockets, Teeth and Favorite Things Investigation 2, Pocket and Teeth Data Teacher Resources Binder envision Math Grade 2, Pearson Education, Inc., 2009 Topic 16 Teacher Edition, Graphs and Probability Resource Masters Student Pages See Supplemental Section in the Curriculum Framework for CCSS Aligned Lessons Lesson 1.4A, Bar Graphs, Investigations (Unit 4) Lesson 1.1A, Subtraction Facts, Investigations (Unit 9) (See Unit 1.1) Pearson Online Success Net, www.pearsonsuccessnet.com Implementing Investigations Site, http://investigations.terc.edu/ Providence Public Schools D-87
Grade 2 Mathematics, Quarter 4, Unit 4.4 Version 5 Representing and Interpreting Data Using Picture and Bar Graphs (7 9 days) 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 information and resources to refer to when planning your unit and individual lessons. Materials Counters, tiles, connecting cubes, chart paper, large grid graph paper, sticky notes. Instructional Considerations Key Vocabulary picture graph bar graph data set scale categories Planning for Effective Instructional Design and Delivery In this content standard 2.MD.10, numeracy is integrated with data and problem solving. The tasks of organizing and representing data in graphs and then using those graphs to solve story problems contain a high level of cognitive demand. Fortunately, these tasks can be embedded in a meaningful and authentic context for students. Working with data is something students enjoy for those reasons. Additionally, technology can be used to assist in these data investigations. There are two Common Core Standards for Mathematical Practice that are highlighted in this unit: 1 Make sense of problems and persevere in solving them and 3 Construct viable arguments and critique the reasoning of others. Both standards call out the importance of communication and the language involved in working in this content area. Numbers are not isolated elements here, but attached to categories, labels, amounts and judgments. Data is organized and represented in order to analyze and support or refute arguments; objective language and conjecture are equally important. In order for students to excel in type of mathematics, language needs to be explicitly modeled with time built in for meaningful discussion in pairs and small groups as well as whole group setting. Continue to do the routines, Today s Number and What Time Is It? using the parameters that are most relevant to the standards in this unit. For example, in Investigations Unit 4, Investigation 2 names the routines: Today s Number: 20 with Missing Parts and What Time Is It?: What Time Will It Be? These variations in routines contain the rigor and the content that is necessary to keep students engaged and challenged. Additionally, the different Quick Images Routines offer non-numerical and numerical fast and fun opportunities to build spatial reasoning and number sense. D-88 Providence Public Schools
Representing and Interpreting Data Using Grade 2 Mathematics, Quarter 4, Unit 4.4 Picture and Bar Graphs (7 9 days) Version 5 Notes Providence Public Schools D-89