Comparing Fractions and Decimals


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1 Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals Overview Number of Instructional Days: 10 (1 day = minutes) Content to be Learned Explore and reason about how a number representing an amount can be written as both a fraction and a decimal. Say 32/100 as thirtytwo hundredths and rewrite this as Represent values such as 0.32 or 32/100 on a number line. Reason that comparisons are only valid when they refer to the same whole. Build area models to compare decimals. Justify decimal comparisons using visual models. Record comparisons using symbols. Essential Questions How do you write a fraction with a denominator of 10 or 100 as a decimal and show your answer using a visual model? How can you compare decimals by using their fractional equivalents? Using a visual model, how can you represent tenths and hundredths referring to the same whole? Mathematical Practices to Be Integrated Reason abstractly and quantitatively. Make sense of quantities and their relationships. Represent symbolically (i.e., equations, expressions). Look for and make use of structure. Look closely to determine a pattern or structure. Step back for an overview and shift perspective. See complicated things as being composed of single objects or several smaller objects. How do you determine where a decimal is located on a number line? When comparing decimals, why is it important for both decimals refer to the same whole? How does your visual model justify your decimal comparison? 41
2 Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals (10 days) Written Curriculum Common Core State Standards for Mathematical Content Number and Operations Fractions 3 4.NF 3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Understand decimal notation for fractions, and compare decimal fractions. 4.NF.6 4.NF.7 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Common Core 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. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 42
3 Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals (10 days) Clarifying the Standards Prior Learning In grade 3, students used visual fraction models to compare two fractions with the same numerator or same denominator. They recognized that these comparisons are valid only when two fractions refer to the same whole. Students recorded their results using the symbols <, >, and/or =. Current Learning Decimals are introduced for the first time, and this is a critical area of focus. In grade 4, students express fractions with a denominator of 10 or 100 as a decimal. They also express decimals (up to hundredths) as a fraction with a denominator of 10 or 100. Students locate decimals up to hundredths on a number line. Emphasis is placed on properly naming the decimals. Students should have ample opportunities to explore and reason about the idea that a number can be represented as both a fraction and decimal. Students use decimal notation for fractions with denominators of 10 or 100. They build area and other models to compare decimals. Visual models include area models, decimal grids, decimal circles, number lines, and meter sticks. Through these experiences, learners understand that comparisons between decimals or fractions are only valid when the whole is the same for both cases. Future Learning Fifth graders will read, write, and compare fractions to the thousandths. They will be expected to write it in expanded form. Through their understanding of place value, students will round decimals to any place. In grade 6, students will use the standard algorithm fluently to perform each operation involving multidigit decimals. Additional Findings According to Principals and Standards for School Mathematics, They (students) should also investigate the relationship between fractions and decimals focusing on equivalence. Through a variety of activities they should understand that a fraction such as 1/2 is equivalent to 5/10 and that it has a decimal representation (0.5), (p. 150) According to Progressions: 3 5 Number and Operations, The number of digits to the right of the decimal point indicates the number of zeroes in the denominator, so that 2.70 = 270/100 and 2.7 = 27/10 Students compare decimals using the meaning of a decimal as a fraction, making sure to compare fractions with the same denominator. For example, to compare 0.2 and 0.09, students think of them as 0.20 and 0.09 and see that 0.02 > 0.09 because 20/100 > 9/100. (pp. 8 and 9) 43
4 Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals (10 days) 44
5 Grade 4 Mathematics, Quarter 4, Unit 4.2 Measuring Angles and Using Angles to Solve Problems Overview Number of Instructional Days: 15 (1 day = minutes) Content to be Learned Understand that an angle is formed wherever two rays share a common endpoint. Understand that a circle has 360 onedegree turns and each degree represents 1/360 of the arc of a circle. Use a protractor to measure and sketch angles in wholenumber degrees. Recognize that an angle is made up of nonoverlapping parts and angle measures are additive. Recognize that as an angle turns n units, it measures n degrees. Essential Questions How can knowing the measurement of an angle help you determine the measurement of unknown angle(s)? How is an arc of a circle related to the degrees of an angle? Mathematical Practices to Be Integrated Use appropriate tools strategically. Decide which tools are most helpful. Detect errors by using estimation and knowledge of benchmark angles. Use technological tools to explore and deepen understanding. Attend to precision. Specify units of measure with a protractor. Accurately label each axis and angle measure in a problem. Communicate precisely with others. How do you use a protractor to draw and measure angles? Why does what you measure influence how you measure? How are angles constructed? 45
6 Grade 4 Mathematics, Quarter 4, Unit 4.2 Measuring Angles and Using Angles to Solve Problems (15 days) Written Curriculum Common Core State Standards for Mathematical Content Measurement and Data 4.MD Geometric measurement: understand concepts of angle and measure angles. 4.MD.5 4.MD.6 4.MD.7 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a onedegree angle, and can be used to measure angles. b. An angle that turns through n onedegree angles is said to have an angle measure of n degrees. Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. Recognize angle measure as additive. When an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. Common Core Standards for Mathematical Practice 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 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. 46
7 Grade 4 Mathematics, Quarter 4, Unit 4.2 Measuring Angles and Using Angles to Solve Problems (15 days) 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In grade 2, students recognized, drew, and analyzed shapes having specified attributes. They also partitioned circles into two, three, or four equal shares. In grade 3, students compared and classified shapes by their sides and the number of angles. Current Learning In grade 4, students understand that an angle is formed when two rays share a common endpoint. They recognize that angles can be measured using tools and can refer to the turn around the center of a circle. The focus is for students to conceptually understand that angles are measured in degrees and a degree consists of a turn. It takes 360 turns to complete a circle. Students recognize that angles are additive. For example, 45 onedegree angles are equal to a 45degree angle. In addition, students decompose (or divide) an angle into parts that add up to the measure of the original angle. They measure angles using a protractor and sketch angles of specified measures. This is the first time students are exposed to the idea of a turn as related to angles. Future Learning In grade 5, students will understand that attributes of twodimensional figures also belong to all subcategories. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. Sixth graders will find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. Additional Findings According to A Research Companion to Principles and Standards for School Mathematics, To understand angles, students must understand the various aspects of the angle concepts. They must overcome difficulties with orientation, discriminate angles as critical parts of geometric figures, and construct and represent the idea of turns among others. Furthermore, they must construct a high level of integration between these aspects. This difficult task is best begun in elementary and middle school years as children deal with corners of figures, comparison of angle size, and turns. (p. 164) According to Principles and Standards for School Mathematics, They (students) can begin to establish some benchmarks by which to estimate and judge the size of objects. For example, they learn that a square corner is called a right angle and establish this as a benchmark for estimating the size of other angles. (p. 172) 47
8 Grade 4 Mathematics, Quarter 4, Unit 4.2 Measuring Angles and Using Angles to Solve Problems (15 days) 48
9 Grade 4 Mathematics, Quarter 4, Unit 4.3 Solving Problems in Context Overview Number of Instructional Days: 15 (1 day = minutes) Content to be Learned Solve multistep word problems using any of the four operations involving distance, intervals of time, liquid volumes, and masses of objects or money. Represent measurement quantities using diagrams such as a number line. Fluently add and subtract any whole number using the standard algorithm. Solve problems involving simple fractions or decimals. Convert measurements from large to small units. Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Make meaning of a problem and look for entry points to its solution. Make conjectures about the meaning of the solution. Develop a plan. Monitor and evaluate progress and change course if necessary. Check answers and determine if they make sense. Reason abstractly and quantitatively. Make sense of quantities and their relationships. Represent symbolically. Understand and use different properties and operations. Construct viable arguments and critique the reasoning of others. Attempt to prove or disprove answers through examples and counterexamples. Communicate and defend mathematical reasoning. 49
10 Grade 4 Mathematics, Quarter 4, Unit 4.3 Solving Problems in Context (15 days) Essential Questions How do you recognize what operation and strategy are appropriate to solve a given problem? How can you use what you know about number relationships to develop efficient strategies for adding/subtracting multidigit numbers? Why is using a diagram to display measurement conversions helpful? Why would you want to convert differentsized standard unit measurements from larger to smaller units within a given measurement system (e.g., meters to centimeters)? What is your strategy for converting larger units to smaller units? What do making conversions with time, mass, liquid volume, distance. and money have in common? How is converting within the systems different? Written Curriculum Common Core State Standards for Mathematical Content Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Number and Operations in Base Ten 2 2 Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000, NBT Use place value understanding and properties of operations to perform multidigit arithmetic. 4.NBT.4 Fluently add and subtract multidigit whole numbers using the standard algorithm. Common Core 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 50
11 Grade 4 Mathematics, Quarter 4, Unit 4.3 Solving Problems in Context (15 days) 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. 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. 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. Clarifying the Standards Prior Learning Students solved word problems with addition and subtraction of time intervals and minutes. They added, subtracted, multiplied, and divided to solve onestep word problems involving mass or volume within the same unit. Third graders fluently added and subtracted within 1,000 using strategies based on place value. Current Learning Students must now fluently add and subtract whole numbers to a million using the standard algorithm. The concept needs to be mastered by the end of grade 4. They use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, employing the most efficient method. 51
12 Grade 4 Mathematics, Quarter 4, Unit 4.3 Solving Problems in Context (15 days) Future Learning In grade 5, students will apply the algorithm with multidigit whole numbers. They will extend understanding of volume as an attribute of solid figures and will measure by counting unit cubes. Students will convert among differentsized standard measurements and use these conversions in solving multistep, realworld problems. Additional Findings According to Principles and Standards for School Mathematics, In learning about measurement and learning how to measure, students should be actively involved drawing on familiar and accessible contexts measure objects and space in their classroom or use maps to determine locations and distances around their community. (p. 171) Students in grades 3 5 should have frequent experiences with problems that interest, challenge, and engage them in thinking about important mathematics. (p. 182) Since good problems challenge students to think, students will often struggle to arrive at solutions. It is the teacher s responsibility to know when the students need assistance and when they are able to continue working productively without help. It is essential that students have time to explore problems students need to know that a challenging problem will take some time and that perseverance is an important aspect of doing mathematics. (pp. 185 and 186) According to PARCC Model Content Frameworks, Mathematics Grades 3 11, Standard 4.MD.2 refers to using the four operations to solve word problems involving measurement, quantities such as liquid volume, mass, time and so on. Some parts of this standard could be met earlier in the year (such as using wholenumber multiplication to express measurements given in a larger unit in terms of a smaller unitsee also 4.MD.1), while others might be met only by the end of the year (such as word problems involving addition and subtraction of fractions or multiplication of a fraction by a whole number see also 4.NF.3d and 4.NF.4c). (p. 20) 52
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