Grade level/course: 3 rd Grade/Math Trimester: 1 Unit of study number: 1.1 Unit of study title: Using Place Value Number of days for this unit: 10 Days (60 minutes per day) Common Core State Standards for Mathematical Content: Number and Operations in Base Ten 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 4 3. NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. NOTE: 4 A range of algorithms may be used. Number and Operations in Base Ten 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 4 3. NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. NOTE: 4 A range of algorithms may be used. Measurement and Data 3.MD Represent and interpret data. 3. MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step how many more and how many less problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. 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 Albuquerque Public Schools June 2012 1
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 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. 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 2 nd grade, students have extended their base-ten understanding to hundreds (2.NBT.1). They have added and subtracted within a 1,000 with understanding, but not fluency (composing and decomposing). They have become fluent with adding and subtracting within 100 (2.NBT.7). Albuquerque Public Schools June 2012 2
In 2 nd grade, students have drawn a picture graph and a bar graph to represent a data set with up to four categories. They have solved simple put-together, take-apart, and compare problems using information presented in a bar graph (2.MD.9 and 2.MD.10). Current Learning In Unit 1, the teacher will provide reinforcement activities in the area of place value by adding and subtracting within 1,000 by composing and decomposing numbers (3.NBT.2). The teacher will introduce and guide student thinking in rounding whole numbers to the nearest 10 (3.NBT.1). The teacher will continue to provide reinforcement activities in creating picture graphs and bar graphs to represent data. The teacher will provide developmental activities to have students solve one-step how many more and how many less problems using information presented in scaled bar graphs (3.MD.3). Students will use addition and subtraction skills from 3.NBT.2 to solve data problems. Students will be expected to round to the nearest 100 by Unit 3, and will be able to add and subtract fluently within 1,000 by Unit 15. Future Learning In 4 th grade, students will use place value to perform multi-digit arithmetic by fluently adding and subtracting multi-digit whole numbers using a standard algorithm (4.NBT.4). Students will make a line plot to display a data set of measurements in fractions of a unit (½, ¼, and ⅛). Students will solve problems involving addition and subtraction of problems by using information represented in line plots (4.MD.4). Additional Findings According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, Numbers and Operations, Grades 3-5, students understand the place-value structure of the base-ten number system and are able to represent and compare whole numbers and decimals. According to the PARCC Model Content Framework for Mathematics for Grade 3, Examples of Opportunities for Connections among Standards Clusters or Domains, scaled picture graphs and scaled bar graphs (3.MD.3) can be a visually appealing context for solving multiplication and division problems (p. 15). Albuquerque Public Schools June 2012 3
Content to be learned Understand place value to round whole numbers to the nearest 10 Add and subtract using strategies and algorithms based on place value Use properties of operations to add and subtract whole numbers Draw a scaled picture graph to represent a data set Draw a scaled bar graph to represent a data set Solve one-step how many more and how many less problems using information presented in scaled bar graphs Mathematical practices to be integrated 2 Reason abstractly and quantitatively. Make sense of quantities (numbers to the nearest 10) and their relationships in addition and subtraction problem situations Attend to the meaning of quantities (numbers to the nearest 10) Know and flexibly use different properties of operations for addition and subtraction problems 5 Use appropriate tools strategically. Use appropriate tools (paper and pencil, manipulatives, number line, number grid, etc.) when solving mathematical problems Interpret data to solve problems Use technological tools to explore and deepen their understanding of concepts 6 Attend to precision. Communicate precisely using strategies and algorithms based on place value Calculate accurately and efficiently Essential Questions How can you round whole numbers to the nearest 10? Albuquerque Public Schools June 2012 4
What place value is assigned to a given number? What strategies can be used to add and subtract multi-digit numbers? How can you use the relationship between addition and subtraction to solve problems? How do you draw a scaled picture graph and a scaled bar graph to represent a data set? How do you solve one-step how many more and how many less problems using information presented in scaled bar graphs? Albuquerque Public Schools June 2012 5
Grade level/course: 3 rd Grade/Math Trimester: 1 Unit of study number: 1.2 Unit of study title: Understanding Addition through Perimeter Number of days for this unit: 10 days (60 minutes per day) Common Core State Standards for Mathematical Content Measurement and Data 3.MD Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 3.MD.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. 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. Albuquerque Public Schools June 2012 1
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. 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 Starting in Kindergarten, students described attributes of objects; such as length or weight and directly compared two objects with a measurable attribute. In 1st grade, students used non-standard tools to measure length of objects as a whole number of units. In 2 nd grade, students used standard measuring tools to find lengths, solve addition and subtraction problems within 100, and begin estimating lengths. They have had experience with adding multiple addends. Albuquerque Public Schools June 2012 2
Current Learning The instructional level is developmental for this current content standard. Students will solve world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, and exhibiting rectangles with the same perimeter. In third grade, students add lengths of all the sides of a polygon in order to teach addition in real life context. They will also find unknown side lengths. According to PARCC (Partnership for Assessment of Readiness for College and Careers), the content in this standard is an additional cluster, although it makes a good review for addition. This is the first unit to address perimeter. Later units will address area. There are no fluency expectations in this content. Future Learning In 4th grade, students will know relative sizes of measurement units within one system of units. They will apply perimeter formulas for rectangles with real world and mathematical problems. Additional Findings Measurement is one of the widely used applications of mathematics. It bridges two main areas of school mathematics geometry and number. Measurement activities can simultaneously teach important everyday skills, strengthen students knowledge of other important topics in mathematics, and develop measurement concepts and processes that will be formalized and expanded in later years (Principles and Standards for School Mathematics, 2000, p. 103). Students develop concepts and skills through experiences in analyzing attributes and properties of two-dimensional objects. They form an understanding of perimeter as a measurable attribute and select appropriate units, strategies, and tools to solve problems involving perimeter (Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, p. 30). According to Piaget, understanding measurement is a developmental process which is developed through concrete experiences and leads to abstract thinking in the future (A Research Companion to Principles and Standards for School Mathematics, 2003, p. 182). Content to be learned Recognize perimeter as an attribute of plane figures. Solve real world problems and mathematical problems involving perimeters of polygons. Add to find perimeter of polygons given the side lengths. Find the unknown side length. Albuquerque Public Schools June 2012 3
Mathematical practices to be integrated 1 Make sense of problems and persevere in solving them Students might rely on using concrete objects or pictures to conceptualize and solve problems. 2 Reason abstractly and quantitatively Students can visualize the quantitative amounts of numbers, consider the units involved, and attend to their meanings. 4 Model with Mathematics Students can write an addition or subtraction equation to describe a situation and to solve problems arising in everyday life. Essential Questions What information do you need to find the perimeter of a polygon? What is your strategy for finding the perimeter of a figure or real world object? How can you find the perimeter of a rectangle when one side length is missing? What do you need to know to find an unknown side length of polygons? Albuquerque Public Schools June 2012 4
Grade level/course: Grade 3/Math Trimester: 1 Unit of study number: 1.3 Unit of study title: Applying Place Value to Addition, Subtraction and Rounding Number of days for this unit: 15 days (60 minutes per day) Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3. OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 3 NOTE: 3 This standard is limited to problems posed with whole numbers and having whole- number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Operations and Algebraic Thinking 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3. OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Number and Operations in Base Ten 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 4 3. NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. NOTE: 4 A range of algorithms may be used. Albuquerque Public Schools June 2012 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. 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. 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 7 5 + 7 3, 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 2 + 7. 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. Albuquerque Public Schools June 2012 2
Clarifying the Standards Prior Learning In both 1 st and 2 nd grades, students have had ample practice using equations to represent addition and subtraction word problems using a symbol to represent unknown quantities. In 1 st grade, students solved word problems within 20. In 2 nd grade, students solved two-step word problems within 100 (including adding to, taking from, putting together, taking apart and comparing). In 2 nd grade, students have experience adding and subtracting within 20 using mental strategies to build toward fluency. Current Learning The PARCC (Partnership for Assessment of Readiness for College and Careers), framework document identifies solving problems involving the four operations and identifying and explaining patterns in arithmetic as a major cluster. Though the standard indicates that four operations will be used to solve two-step word problems by the end of the year, this unit will focus on addition and subtraction only, as division and multiplication will be introduced in later units (units 13 and 15). During unit 15, students will be encountering two-step word problems. Although rounding and estimation is considered an additional cluster in the PARCC documents, and will not be assessed heavily, this is a new concept for 3 rd grade students and it is the foundation for other strategies students will be required to use in the future. In unit 1, students rounded to the nearest 10. In Unit 3, students will round to the nearest 10 and 100. During the problem solving process, students will create equations using letters as unknown quantities instead of symbols. Beginning in Grade 3, there is an emphasis on mental computation and reasoning. Future Learning In 4 th grade, students are solving multistep word problems (problems with two or more steps) using all four operations (including division problems with remainders). Students will also use place value understanding to round multi-digit numbers to any place. Students continue to use their mental strategies to check their answers for reasonableness. Additional Findings According to Principles and Standards for School Mathematics (2000), Estimation serves as an important companion to computation. It provides a tool for judging the Albuquerque Public Schools June 2012 3
reasonableness of calculator, mental, and paper-and-pencil computations. However, being able to compute exact answers does not automatically lead to an ability to estimate or judge the reasonableness of answers, as Reys and Yang (1998) found in their work with sixth and eighth graders. Students in grades 3-5 will need to be encouraged to routinely reflect on the size of an anticipated solution (pgs.155-156). Content to be learned Solve word problems using addition and subtraction with whole numbers Represent word problems using equations with a letter for the unknown quantity Assess reasonableness using mental computation (using estimation strategies such as rounding) Identify and explain arithmetic patterns using properties of operations Round whole numbers to the nearest 10 or 100 Use a range of algorithms (Notation 4 for 3.NMBT.1 cluster heading) Mathematical practices to be integrated 1 Make sense of problems and persevere in solving them. Explain the meaning of a problem Choose appropriate strategies or pathways to solve a problem Monitor and evaluate progress Continually ask, Does this make sense? 6 Attend to precision Communicate precisely to others Calculate accurately and efficiently 7 Look for and make use of structure Identify patterns or structure Essential Questions How do you know whether to add or subtract when solving a word problem? What strategy can you use to solve for an unknown number in an equation? What mathematical patterns can you use to solve these problems? How does rounding help you add or subtract 2 numbers more efficiently? Albuquerque Public Schools June 2012 4
Grade level/course: 3 rd Grade/Math Trimester: 1 Unit of study number: 1.4 Unit of study title: Developing Understanding of Fractions as numbers Number of days for this unit: 10 days (60 minutes per day) Common Core State Standards for Mathematical Content Number and Operations Fractions 5 3.NF 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. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8 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. Note: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8 Geometry 3.G Reason with shapes and their attributes. 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. Albuquerque Public Schools June 2012 1
Common Core State Standards for Mathematical Practice 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. 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. 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 Albuquerque Public Schools June 2012 2
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 Grades 1 and 2, students used fraction language to describe partitions of shapes into equal shares [2.G.3] (Progressions for the Common Core State Standards in Mathematics draft), 3-5 Numbers and Operations Fractions, p. 2). In the early stages of the NF Progression they use other representations such as area models, tape diagrams, and strips of paper. These can be subdivided, representing an important aspect of fractions (Progressions for the Common Core State Standards in Mathematics draft), 3-5 Numbers and Operations Fractions, p. 3). Current Learning This unit is developmental instruction as this is a critical area of instruction according to the PARCC (Partnership for Assessment of Readiness for College and Careers), framework document. The Common Core State Standards Grade 3, p. 21 introduction states: Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators to teach standards 3.NF.1 and 3.G. In this unit, we are not yet comparing fractions or finding equivalent forms of fractions as this will be addressed in Unit 9. According to the Progressions for the Common Core State Standards in Mathematics (draft), 3-5 Numbers and Operations Fractions,(p.3), On the number line, the whole is the unit interval, that is, the interval from 0 to 1, measured by length. Iterating this whole to the right marks off the whole numbers, so that the intervals between consecutive whole numbers, from 0 to 1, 1 to 2, 2 to 3, etc., are all of the same length. Students might think of the number line as an infinite ruler. To construct a unit fraction on a number line diagram, e.g., ⅓, students partition the unit interval into 3 intervals of equal length and recognize that each has length ⅓. They locate the number ⅓ on the number line by marking off this length from 0, and Albuquerque Public Schools June 2012 3
locate other fractions with denominator 3 by marking off the number of lengths indicated by the numerator [3.NF.2]. Students sometimes have difficulty perceiving the unit of a number line diagram. When locating a fraction on a number line diagram, they might use as the unit the entire portion of the number line that is shown on the diagram, for example indicating the number 3 when asked to show ¾ on a number line marked from 0 to 4 number line diagrams are important representations for students as they develop an understanding of a fraction as a number. They solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators (Curriculum Focal Points for Prekindergarten through grade 8 Mathematics, p. 29). Future Learning In grade 4 students [will] learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction. This property forms the basis for much of their other work in Grade 4, including the comparison, addition, and subtraction of fractions and the introduction of finite decimals (Progressions for the Common Core State Standards in Mathematics (draft), 3-5 Numbers and Operations Fractions, p. 5). Additional Findings In grade 3, students start with unit fractions (fractions with numerator 1), which are formed by partitioning a whole into equal parts and taking one part, e.g., if a whole is partitioned into 4 equal parts then each part is ¼ of the whole, and 4 copies of that part make the whole. Next students build fractions from unit fractions, seeing the numerator 3 of ¾ as saying that ¾ is the quantity you get by putting 3 of the ¼ s together [3.NF.1]. They read any fraction this way, and in particular there is no need to introduce proper fractions and improper fractions initially; 5/3 is the quantity you get by combining 5 parts together when the whole is divided into 3 equal parts. (Progressions for the Common Core State Standards in Mathematics, 3-5 Numbers and Operations Fractions, p. 2). Content to be learned Understand a fraction from part to whole Understand a fraction as a number on a number line Represent fractions on a number line diagram Partition shapes into parts with equal parts naming each part as a unit fraction Albuquerque Public Schools June 2012 4
Mathematical practices to be integrated 3 Construct viable arguments and critique the reasoning of others Make and build a logical progression of statements to explore the truth of their conjectures and construct arguments using concrete referents such as objects, drawings, and diagrams 4 Model with mathematics Identify important quantities (halves, thirds, fourths, sixths, and eighths)in a practical situations and use relationships using such tools as diagrams 5 Use appropriate tools strategically Make sound decisions about when to use tools (a tape diagram, a number line) appropriately Detect possible errors by strategically using estimation and other mathematical knowledge. Essential Questions In a fraction, what does the numerator stand for? In a fraction, what does the denominator stand for? Where are these fractions (½, ⅓, ¼, 1 / 6, ⅛) located on the number line? Why? Given a shape, how can you partition it into the above fractions? How many equal parts are within the whole? Given a shape, how many equal parts will make the whole? How can you make a model of a given fraction? Albuquerque Public Schools June 2012 5
Grade level/course: 3 rd Grade/Math Trimester: 1 Unit of study number: 1.5 Unit of study title: Exploring Multiplication Number of days for this unit: 10 days (60 minutes a day) Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA Represent and solve problems involving multiplication and division. 3. OA.1 Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7. Operations and Algebraic Thinking 3.OA Understand properties of multiplication and the relationship between multiplication and division. 3. OA.5 Apply properties of operations as strategies to multiply and divide. 2 Examples: If 6 4 = 24 is known, then 4 6 = 24 is also known. (Commutative property of multiplication.) 3 5 2 can be found by 3 5 = 15, then 15 2 = 30, or by 5 2 = 10, then 3 10 = 30. (Associative property of multiplication.) Knowing that 8 5 = 40 and 8 2 = 16, one can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56. (Distributive property.) Note: 2 Students need not use formal terms for these properties. Operations and Algebraic Thinking 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3. OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Measurement and Data 3.MD Albuquerque Public Schools June 2012 1
Represent and interpret data. 3. MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step how many more and how many less problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. Measurement and Data 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 3. MD.7 Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. Common Core State Standards for Mathematical Practice 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. 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 7 5 + 7 3, 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 2 + 7. 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 Albuquerque Public Schools June 2012 2
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. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1) (x + 1), (x 1) (x 2 + x + 1), and (x 1) (x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In 2 nd grade, students worked with equal groups of objects to gain foundations for multiplication. They determined whether a group of objects (up to 20) had an odd or even number of members, e.g., by pairing objects or counting them by 2 s and wrote equations to express an even number as a sum of two equal addends (2.OA.3). They used addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns and wrote equation to express the total as a sum of equal addends (2.OA.4). In Kindergarten through 2 nd grade, students used addition and subtraction to solve equations with an unknown number. They estimated and measured length in standard units. Current Learning The clusters in this unit (3.OA.1, 3.OA.5, 3.OA.9, 3.MD.7) are major clusters in the PARCC Content Framework. A critical area in this unit is developing an understanding of the structure of rectangular arrays and area. Students recognize area as an attribute of two-dimensional regions. They measure the area of the shape by finding the total number of same size units of area required to cover the shapes without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle. Albuquerque Public Schools June 2012 3
Developing and understanding the meaning of multiplication of whole numbers through activities and problems involving equal sized groups, arrays, and area models will build a strong foundation for continued study of multiplication. Students are to use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication involving single-digit factors. Students continue to solve for unknown numbers using multiplication. They use multiplication to solve word problems in situations involving equal groups, arrays, and measurement. Students learn the commutative property. Students identify and explain arithmetic patterns (including patterns in the addition table or multiplication table). Also in this unit, students use multiplication to interpret data with picture and bar graphs. In unit 6, students focus on developing an understanding of area. In unit 7 the focus is developing multiplication strategies. In unit 8 and 13, students understand the relationship between multiplication and division and apply it to problem solving situations. Future Learning In 4 th grade, students will develop understanding and fluency with multi-digit multiplication, and will develop an understanding of dividing to finding quotients involving multi-digit dividends. They will apply area and perimeter formulas for rectangles in real world and mathematical problems. Additional Findings Students who understand the structure of numbers and the relationships among numbers can work with them flexibly (Principles and Standards for School Mathematics,2000, p. 149). Students understand the meanings of multiplication of whole numbers through the use of representations (Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, p. 15). Measurement helps connect ideas within areas of mathematics and between mathematics and other disciplines (Principles and Standards for School Mathematics,2000, p. 171). Content to be learned Interpret products of whole numbers. Understand and apply the commutative property of multiplication as a strategy. Identify and explain patterns in arithmetic. Draw scaled picture graphs and bar graphs to represent data in several categories. Understand and show the concept of area (of rectangles) and its relationship to multiplication, including tiling. Albuquerque Public Schools June 2012 4
Mathematical practices to be integrated 3 Construct viable arguments and critique the reasoning of others. Construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 7 Look for and make use of structure Look closely to discern a pattern or structure. Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems (partition shapes). 8 Look for and express regularity in repeated reasoning. Notice if calculations are repeated. Look both for general methods and for shortcuts. Essential Questions What is multiplication? What are factors and products? What do the factors in a multiplication problem represent? How can you solve a multiplication problem? Explain your answer. What patterns of multiplication do you notice on an addition or multiplication table? How can you use multiplication to solve problems involving unknowns? How does the commutative property help you answer multiplication equation? What are some ways to represent the given data on picture and bar graphs? What information does a graph tell you? What questions can you ask or answer by looking at a graph? How do you find the area of a shape? Explain your strategy. Albuquerque Public Schools June 2012 5