Understanding, Ordering, and Comparing Fractions, Decimals, and Percents

Grade 5 Mathematics, Quarter 3, Unit 3.1 Understanding, Ordering, and Comparing Fractions, Decimals, and Percents Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Model positive fractions (a/2, a/3, a/4, a/5, a/6, a/8, and a/12s) and powers of ten 10, 100, and 1000, including mixed numbers and improper fractions. Represent decimals (to thousandths) and benchmark percents (10%, 25%, 50%, 75%, 100%) using pictures, words, or other representations. Identify, order, and compare equivalent positive fractions, decimals, and benchmark percents (fractions to fractions, decimals to decimals, and percent to percents) within the same number formats. Essential questions How can you show or represent the number of students absent in the class as a fraction, decimal, or percent? How would you order various fractions, decimals, or percents on a number line within the same number format? Given a list of decimals, explain how you would determine which was closest to one whole? Mathematical practices to be integrated Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Use a variety of representations and approaches when problem solving. Model with mathematics. Identify important quantities and their relationships using tools (manipulatives, drawings, tables, and graphics). Draw conclusions, interpret results, revise models if needed. Use appropriate tools strategically. Make sound decisions about when tools are appropriate and helpful. Detect possible errors strategically by using estimation and other mathematical knowledge. What are the equivalent fractions that represent a given part of a set? Using a thousandths grid, how many squares would you shade to represent 40% and.155? How are mixed numbers and improper fractions related? What is your strategy for determining if a fraction has a value greater than one whole? Cumberland, Lincoln, and Woonsocket Public Schools C-37

Grade 5 Mathematics, Quarter 3, Unit 3.1 Understanding, Ordering, and Comparing Fractions, Decimals, and Percents (15 days) Written Curriculum Grade-Level Expectations M(N&O) 5 1 Demonstrates conceptual understanding of rational numbers with respect to: whole numbers from 0 to 9,999,999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations; and positive fractional numbers (proper, mixed number, and improper) (halves, fourths, eighths, thirds, sixths, twelfths, fifths, or powers of ten (10, 100, 1000)), decimals (to thousandths), or benchmark percents (10%, 25%, 50%, 75% or 100%) as a part to whole relationship in area, set, or linear models using models, explanations, or other representations. (State) M(N&O) 5 2 Demonstrates understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive fractional numbers, decimals, or benchmark percents within number formats (fractions to fractions, decimals to decimals, or percents to percents); or integers in context using models or number lines. (State) Clarifying the Standards Prior Learning In grade 4, students learned to demonstrate a conceptual understanding of rational numbers from 0 to 999,999. They identified, described, and represented benchmark fractions (1/2, 1/4, 1/3, 1/5, 1/6, 1/8, 1/10) through area, set, and linear models. They also identified multiples or factors of the denominator, and recognized decimals as hundredths in relation to money and tenths in regards to metric measurement. They ordered and compared up to 999,999 and identified equivalent, proper positive fractions or decimals. Current Learning Grade 5 students learn to show equivalency, composition, decomposition, or place value with respect to whole numbers 0 9,999,999. Grade 5 students extend their knowledge of fractions by including 1/12 and powers of 10 (10,100,1000), decimals (to thousandths), and benchmark percents (10%, 25%, 50%, 75%, 100%) They order, compare, or identify equivalent positive fractional numbers and benchmark percents within number formats (fractions to fractions, decimals to decimals, percents to percents). The content is taught at the developmental and reinforcement level. Future Learning In sixth grade, students will compare ratios and rates. They will order and compare whole-number bases with whole number exponents. They will also order and compare across number formats. Cumberland, Lincoln, and Woonsocket Public Schools C-38

Grade 5 Mathematics, Quarter 3, Unit 3.1 Understanding, Ordering, and Comparing Fractions, Decimals, and Percents (15 days) Additional Research Findings According to Principles and Standards for School Mathematics, Through the study of various meanings and models of fractions how fractions are related to each other and to the unit whole and how they are represented students gain facility in comparing fractions, often by using benchmarks such as one-half or one. They should understand the equivalence of fractions, decimals, and percents and the information each type of representation conveys. With these understandings and skills, they should be able to develop strategies for computing with familiar fractions and decimals. Students study and use of numbers should be extended to include larger numbers, fractions, and decimals. They need to develop strategies for judging the relative size of numbers (p. 149). The book also states, Students should build their understanding of fractions as parts of a whole and division. They will need to see a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths. By using an area model in which part of a region is shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. They should develop strategies for ordering and comparing fractions, often using benchmarks such as one-half and one. They should also begin to understand that between any two fractions, there is always another fraction. They should also investigate the relationship between fractions and decimals, focusing on equivalence. Students should understand the meaning of a percent as part of a whole and use common percents such as 10%, 33 1/3%, or 50% as benchmarks in interpreting situations they encounter. By studying fractions, decimals, and percents simultaneously, students can learn to move among equivalent forms, choosing and using an appropriate and convenient form to solve problems and express quantities (p. 150). Cumberland, Lincoln, and Woonsocket Public Schools C-39

Grade 5 Mathematics, Quarter 3, Unit 3.1 Understanding, Ordering, and Comparing Fractions, Decimals, and Percents (15 days) Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-40

Grade 5 Mathematics, Quarter 3, Unit 3.2 Adding and Subtracting Fractions Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Add and subtract positive proper fractions with unlike denominators. Solve problems involving the addition and subtraction of proper fractions. Apply the conventions of order of operations, with and without parentheses, when solving addition and subtraction fraction problems. Essential questions How is adding and subtracting fractions with like denominators different than adding and subtracting with unlike denominators? What strategies can be used to identify common denominators when adding and subtracting fractions with unlike denominators? How can models be used to solve problems with like and unlike denominators? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Analyze given information to develop possible strategies for solving the problem. Check answers to problems using a different method. Attend to precision. Use clear definitions and state meaning of mathematical symbols used, including using the equal sign consistently and appropriately. Strive for accuracy. How do you recognize what strategy to use for a specific word problem? How do parentheses influence how you solve an expression? Cumberland, Lincoln, and Woonsocket Public Schools C-41

Grade 5 Mathematics, Quarter 3, Unit 3.2 Adding and Subtracting Fractions (8 days) Written Curriculum Grade-Level Expectations M(N&O) 5 3 Demonstrates conceptual understanding of mathematical operations by adding and subtracting decimals and positive proper fractions with unlike denominators. (Local) M(N&O) 5 4 Accurately solves problems involving multiple operations on whole numbers or the use of the properties of factors, multiples, prime, or composite numbers; and addition or subtraction of fractions (proper) and decimals to the hundredths place. (Division of whole numbers by up to a two-digit divisor.) (State) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) Clarifying the Standards Prior Learning In grade 4, students described and illustrated the connection between repeated subtraction and division, the inverse of multiplication, dividing using whole numbers, and adding and subtracting fractions with like denominators. Fourth-grade students were able to solve problems using order of operations, including addition and subtraction with decimals and fractions. Current Learning Students in fifth grade add and subtract decimals and positive proper fractions with unlike denominators. This concept is taught at the developmental level. The multiplication and division of whole numbers is at the reinforcement level as are addition and subtraction of fractions with like denominators. Computing with multiple operations is at reinforcement level. Place value to the hundredths is taught at the developmental level in problem solving. Two-digit divisors are new. Order of operations can be used with or without parentheses. Future Learning Students in sixth grade will add and subtract positive fractions, no longer limited to proper fractions. Students will multiply and divide fractions and decimals. They will also solve problems that involve improper fractions, mixed fractions, and decimals. Sixth-grade students will subtract integers and percents of a whole. They will also solve problems involving greatest common factor and least common multiple. Order of operations is now at the reinforcement level. Additional Research Findings According to Principles and Standards for School Mathematics, As students acquire conceptual grounding related to rational numbers, they should begin to solve problems using strategies they develop or adapt from their whole-number work. At these grades, the emphasis should not be on developing general procedures to solve all decimal and fraction problems. Rather, students should generate solutions that are based on number sense and properties of the operations and that use a variety of models or representations (p. 155). Cumberland, Lincoln, and Woonsocket Public Schools C-42

Grade 5 Mathematics, Quarter 3, Unit 3.3 Adding and Subtracting Decimals Overview Number of instructional days: 7 (1 day = 45 minutes) Content to be learned Solve problems involving the addition and subtraction of decimals to the hundredths place. Apply the conventions of order of operations with and without parentheses when solving addition and subtraction decimal problems. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Explain the meaning of a problem and restate it in their words. Plan a solution pathway rather than jumping into a solution attempt. Attend to precision. Define mathematical symbols and units of measure consistently and appropriately. Use labels for clarification. Express mathematical answers with a degree of precision appropriate for the problem context. Essential questions How can models help us understand the addition and subtraction of decimals? What are some different strategies you could use to add and subtract decimals? How do you determine which strategy to use for a specific word problem? How does including parentheses change your calculation? Cumberland, Lincoln, and Woonsocket Public Schools C-43

Grade 5 Mathematics, Quarter 3, Unit 3.3 Adding and Subtracting Decimals (7 days) Written Curriculum Grade-Level Expectations M(N&O) 5 3 Demonstrates conceptual understanding of mathematical operations by adding and subtracting decimals and positive proper fractions with unlike denominators. (Local) M(N&O) 5 4 Accurately solves problems involving multiple operations on whole numbers or the use of the properties of factors, multiples, prime, or composite numbers; and addition or subtraction of fractions (proper) and decimals to the hundredths place. (Division of whole numbers by up to a two-digit divisor.) (State) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) Clarifying the Standards Prior Learning Grade 4 students described and illustrated the connection between repeated subtraction and division, the inverse of multiplication. They divided using whole numbers, and added and subtracted fractions with like denominators. They also solved problems using order of operations, including addition and subtraction with decimals and fractions. Current Learning Students in fifth grade add and subtract decimals to the hundredths place and positive proper fractions with unlike denominators. This concept is taught at the developmental level. The multiplication and division of whole numbers is at the reinforcement level, as well as addition and subtraction of fractions with like denominators. Multiple operations is taught at the reinforcement level. Place-value to thousandths place is taught at the developmental level in problem solving. Two-digit divisors are new. Order of operations can be used with or without parentheses. Future Learning Students in sixth grade will add and subtract positive fractions, no longer limited to proper fractions. They will multiply and divide fractions and decimals. They will also solve problems that involve improper fractions, mixed fractions, and decimals. Sixth-grade students will subtract integers and percents of a whole. They will solve problems involving greatest common factor and least common multiple. Order of operations is now reinforcement level. Additional Research Findings According to Principles and Standards for School Mathematics, As students acquire conceptual grounding related to rational numbers, they should begin to solve problems using strategies they develop or adapt from their whole-number work. At these grades, the emphasis should not be on developing general procedures to solve all decimal and fraction problems. Rather, students should generate solutions that are based on number sense and properties of the operations and that use a variety of models or representations (p. 155). Cumberland, Lincoln, and Woonsocket Public Schools C-44

Grade 5 Mathematics, Quarter 3, Unit 3.4 Organizing and Collecting Data Overview Number of instructional days: 3 (1 day = 45 minutes) Content to be learned Organize and display data using tables, bar graphs, and line graphs. Analyze data to answer questions, draw conclusions, solutions, make predictions, and solve problems. Determine the most effective method (e.g., survey, observation, experimentation) to collect data (numerical or categorical) necessary to answer teacher- or student-generated questions or hypotheses. Make connections to real-world situations and formulate new questions based on data. Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Develop questioning strategies to generate information. Understand and use prior learning in constructing arguments. Seek to understand alternative approaches suggested by others, and as a result adopt better approaches. Model with mathematics. Identify important quantities and their relationships and express these as some kind of graphic organizer. Essential questions What question(s) can be answered using your graph or table? How do you interpret the data you have collected? How does the type of data influence the choice of graph? If the trend in you data continues, what do you think your data will look like if you interview 20 more students? How is collecting, organizing, and displaying data helpful to people in the real world? How do you decide which method is effective for collecting data? How do you determine if a line graph, bar graph, and/or table best display your data? Cumberland, Lincoln, and Woonsocket Public Schools C-45

Grade 5 Mathematics, Quarter 3, Unit 3.4 Organizing and Collecting Data (3 days) Written Curriculum Grade-Level Expectations M(DSP) 5 3 Organizes and displays data using tables, bar graphs, or line graphs to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (Local) M(DSP) 5 6 In response to a teacher or student generated question or hypothesis decides the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions; and asks new questions and makes connections to real world situations. (Local) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 5 2.) Clarifying the Standards Prior Learning In grade 4, students organized and displayed data using tables, line plots, bar graphs, and pictographs to answer questions related to data. Pictographs moved from one-to-one correspondence to a key representing more than one. Students used data to formulate or justify conclusions to make predictions or to solve problems. They also decided how to collect, organize, and express data. They analyzed and drew conclusions to answer given questions. They also made predictions, asked questions, and made connections to real-world events. Current Learning In grade 5, students progress from line plots to line graphs. This concept is taught at the developmental level. The remaining concepts are taught at reinforcement level. Future Learning Students in sixth grade will organize and display data using stem-and-leaf plots. They will answer questions and justify conclusions. In grade 7, students will need to consider the limitations that could affect interpretations. Cumberland, Lincoln, and Woonsocket Public Schools C-46

Grade 5 Mathematics, Quarter 3, Unit 3.4 Organizing and Collecting Data (3 days) Additional Research Findings According to Principles and Standards for School Mathematics, Students should move toward seeing a set of data as a whole, describing its shape, and using statistical characteristics of the data such as range and measures of center to compare data sets. Much of this work emphasizes the comparison of related data sets. As students learn to describe the similarities and differences between data sets, they will have an opportunity to develop clear descriptions of the data and to formulate conclusions and arguments based on the data (p. 177). The book also states, Students should become familiar with a variety of representations such as tables, line plots, bar graphs, and line graphs by creating them, watching their teacher create them, and observing those representations found in their environment. In order to select and interpret appropriate representations, students need to understand the nature of different kinds of data: categorical data and numerical data. Students should examine classifications on categorical data that produce different views. When students are ready to present their data to an audience, they need to consider aspects of their representations that will help people understand them; the type of representation that they choose; the scales used in a graph; and headings and titles. Comparing different representations helps students learn to evaluate how well important aspects of the data are shown (p. 178). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-47

Grade 5 Mathematics, Quarter 3, Unit 3.4 Organizing and Collecting Data (3 days) Cumberland, Lincoln, and Woonsocket Public Schools C-48

Grade 5 Mathematics, Quarter 3, Unit 3.5 Interpreting and Analyzing Data Overview Number of instructional days: 7 (1 day = 45 minutes) Content to be learned Interpret a given representation (tables, bar graph, circle graphs, or line graphs) to answer questions related to the data. Analyze data of given representation to formulate or justify conclusions to make predictions or solve problems. Determine or calculate mean, median, mode, and range for data in various contexts. Select and/or describe representations and elements that best display a set of data or a situation. Use measures of central tendency (mean, median, mode) or range to analyze situations or solve problems. Essential questions How do you interpret the data you have collected? What types of information can be best expressed through a line graph? What predictions would you make based on the data given? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Analyze givens, constraints, relationships and goals. Make conjectures about solutions. Explain relationship among equations, verbal descriptions, tables, and graphs. Model with mathematics. Identify important quantities and their relationships. Draw conclusions, interpret results, and revise models if needed. Attend to precision. Communicate formulated explanations with precision. What representations (bar graph, line graph, charts, etc.) would best represent the changes in temperature over a one-year period? How can determining the mean, median, mode, and range help you to analyze and interpret data? Cumberland, Lincoln, and Woonsocket Public Schools C-49

Grade 5 Mathematics, Quarter 3, Unit 3.5 Interpreting and Analyzing Data (7 days) Written Curriculum Grade-Level Expectations M(DSP) 5 1 Interprets a given representation (tables, bar graphs, circle graphs, or line graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (State) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 5 2.) M(DSP) 5 2 Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode) or range to analyze situations, or to solve problems. (State) M(DSP) 5 3 Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP) 5 1. (State) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 5 2.) Clarifying the Standards Prior Learning In grade 4, students interpreted pictographs or circle graphs to answer questions as new learning. While still interpreting line plots, tables, and bar graphs, they began to justify conclusions and continued to make predictions. Students in fourth grade began to use measures of central tendency (median or mode). They also continued to decide how to best organize and display data. Current Learning Students in grade 5 now use line graphs to interpret a given representation. This concept is taught at the developmental level. As they analyze the data, the mean is new to central tendency while other measures of central tendency are at the reinforcement level. Central tendency is now used to analyze situations and solve problems. This instruction is at the developmental stage. Future Learning In grade 6, students will use stem-and-leaf plots to interpret a given representation and to answer questions when organizing and displaying data. They will also analyze situations and use dispersion. Cumberland, Lincoln, and Woonsocket Public Schools C-50

Grade 5 Mathematics, Quarter 3, Unit 3.5 Interpreting and Analyzing Data (7 days) Additional Research Findings According to Principles and Standards for School Mathematics, As students construct graphs of ordered numerical data, teachers need to help them understand what the values along the horizontal and vertical axes represent. Using experience with a variety of graphs, teachers should make sure that students encounter and discuss issues such as why the scale on the horizontal axis needs to include values that are not in the data set and how to represent zero on a graph (p. 178). The book also states, A reasonable objective for upper elementary and middle grade students is that they begin to regard a set of data as a whole that can be described as a set and compared to other data sets. As students examine a set of ordered numerical data, teachers should help them learn to pay attention to important characteristics of the data set: where data are concentrated or clumped, values for which there are no data, or data points that appear to have unusual values. Building on their informal understanding of the most and the middle, students can learn about three measures of center mode, median, and, informally, the mean. Students need to learn more than simply how to identify the mode or median in a data set. They need to build an understanding of what, for example, the median tells them about the data, and they need to see this value in the context of other characteristics of the data (p. 179). Additionally, In grade 5, once students are experienced using the mode and median as part of their data descriptions, they can begin to conceptually explore the role of the mean as a balance point for the data set, using small data sets. The idea of a mean value what it is, what information it gives about the data, and how it must be interpreted in the context of other characteristics of the data is a complex one, which will continue to develop in later grades. Data can be used for developing arguments that are based on evidence and for continued problem posing. As students discuss data gathered to address a particular question, they should begin to distinguish between what the data show and what might account for the results (p. 180). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-51

Grade 5 Mathematics, Quarter 3, Unit 3.5 Interpreting and Analyzing Data (7 days) Cumberland, Lincoln, and Woonsocket Public Schools C-52