rd 3 Grade Math Unit Guide 2014-2015 Jackson County School District Year At A Glance 3rd Grade Math Unit 1 Developing strategies for addition and subtraction 20 Days Unit 2 Exploring equal groups as a foundation for multiplication and division 10 Days Unit 3 Developing conceptual understanding of area 10 Days Unit 4 Understanding unit fractions 10 Days Unit 5 Using fractions in measurement and data 10 Days Unit 6 Solving addition and subtraction problems involving measurement 10 Days Unit 7 Understanding the relationship between multiplication and division 10 Days Unit 8 Investigating patterns in number and operations 15 Days Unit 9 Developing strategies for multiplication and division 10 Days Unit 10 Understanding equivalent fractions 10 Days Unit 11 Comparing fractions 10 Days Unit 12 Solving problems involving area 10 Days Unit 13 Solving problems involving shapes 10 Days Unit 14 Using multiplication and division to solve measurement problems 10 Days Unit 15 Demonstrating computational fluency in problem solving 10 Days 3 5 3 5 # # # 44 ! # 5 # 6 , . 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·ZRUNLQJUDGH 3rd Grade Math Sequenced Units for the Common Core State Standards in Mathematics Grade 3 In the years prior to Grade 3 students gained an understanding of number and used strategies based on place value, properties of operations, and the relationship between addition and subtraction to add and subtract within 1000. They worked with standard units of measure for length and described attributes of shapes. Two major emphases of the Grade 3 year are the operations of multiplication and division and the concept of fractions. These concepts are introduced early in the year in order to build a foundation for students to revisit and extend their conceptual understanding with respect to these concepts as the year progresses. By the end of the year, students recall all products of two single-‐digit numbers. Third grade students develop understanding of fractions as numbers, and compare and reason about fraction sizes. This work with fractions is a cornerstone for developing reasoning skills and conceptual understanding of fraction size and fractions as part of the number system throughout this year and their future work with fractions and ratios. To continue the study of geometry, students describe and analyze shapes by their sides, angles, and definitions. In the final unit in this sequence of units, students generalize and apply strategies for computational fluency. This document reflects our current thinking related to the intent of the Common Core State Standards for M athematics (CCSSM) and assumes 160 days for instruction, divided among 15 units. The number of days suggested for each unit assumes 45-‐minute class periods and is included to convey how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the CCSSM; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit. Throughout Grade 3, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice: 1. 2. 3. 4. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. S. 6. 7. 8. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. When using this document to help in planning your district's instructional program, you will also need to refer to the CCSSM document, relevant progressions documents for the CCSSM, and the appropriate assessment consortium framework. Grade 3 Subject Math # of Units Timeline UNIT CURRICULUM MAP Unit 1: Developing strategies for addition and subtraction. Suggested number of days: 20 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In Grade 2 students used addition and subtraction within 1000 using concrete objects and strategies. In this unit students increase the sophistication of computation strategies for addition and subtraction that will be finalized by the end of the year. This unit introduces the concept of rounding, which provides students with another strategy to judge the reasonableness of their answers in addition and subtraction situations. Perimeter provides a context in which students can practice both rounding and addition and subtraction (e.g. estimating the perimeter of a polygon). Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations in Base Ten -‐ www.youtube.com 3.NBT 3.NBT.1.1 E xplain t he p rocess f or 3.NBT.A.1 introduces the A. Use place value understanding and rounding n umbers u sing p lace v alue. Math Fact Fluency Practice concept of rounding, which is properties of operations to perform multi-‐ 4 3.NBT.1.2 I dentify t he p lace v alue o f www.mathwire.com new to students and will be digit arithmetic. the ones, tens, and hundreds place in a revisited in unit 8 in the context www.oswego.org/ocsd-‐web/games/ 1. Use place value understanding to round whole number. http://mathfactspro.com/mathfluencygame.html#/math-‐ whole numbers to the nearest 10 or 100. of multiplication. 3.NBT.1.3 Round numbers to the facts-‐addition-‐games nearest hundred. http://jerome.northbranfordschools.org/Content/Math_F 3.NBT.1.4 Round numbers to the act_Fluency_Practice_Sheets.asp nearest ten. http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games 2. Fluently add and subtract 3.NBT.2.1 Identify and apply the https://www.illustrativemathematics.org/3 3.NBT.A.2 will be finalized in within 1000 using strategies and properties of addition to solve unit 15 in order to give students algorithms based on place problems. https://learnzillion.com/ time to reach fluency in addition value, properties of operations, 3.NBT.2.2 Identify and apply the and subtraction within 1000 by and/or the relationship between properties of subtraction to solve www.AECSD3rdGradeMathematicsdoc the end of the year. addition and subtraction. problems. 3.NBT.2.3 Check a subtraction problem http://maccss.ncdpi.wikispaces.net/Third+Grade using addition. 3.NBT.2.4 Check an addition problem www.dpi.state.nc.us using subtraction. 3.NBT.2.5 Correctly align digits http://harcourtschool.com/search/search.html according t o p lace v alue, i n o rder t o a dd or s ubtract. www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/c Measurement and Data -‐ 3.MD D. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 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 6. Attend to precision. 8. Look for and express regularity in repeated reasoning. 3.NBT.2.6 Explain and demonstrate the process of regrouping. 3.NBT.2.7 Fluently add two 2-‐digit numbers. (horizontal and vertical set up) 3.NBT.2.8 Fluently add two 3-‐digit numbers. (horizontal and vertical set up) 3.NBT.2.9 Fluently subtract two 2-‐digit numbers with and without regrouping. (horizontal and vertical set up) 3.NBT.2.10 Fluently subtract two 3-‐digit numbers with and without regrouping. (horizontal and vertical set up) 3.MD.8.1 Calculate the length of the sides when given the perimeter of an object. 3.MD.8.2 Calculate the perimeter of a polygon when given the side lengths. 3.MD.8.3 Solve mathematical problems involving rectangles with equal area and different perimeter. 3.MD.8.4 Solve mathematical problems involving rectangles with equal perimeter and different area. 3.MD.8.7 Distinguish between the area and the perimeter. 3.MD.8.8 Relate perimeter and area to the real world. 3.MD.D.8 is the first time perimeter appears in the CCSS-‐ M. Students are not expected to use formulas until Grade 4 (4.MD.A.3). 3.MD.D.8 will be addressed in full in unit 13 after students have been introduced to and worked with the concept of area. Students use precise language to make sense of their solution in the context of a problem and the magnitude of the numbers (MP.6). Students also generalize algorithms and strategies and look for shortcuts (MP.5). urriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐ grade3.html http://www.mathgoodies.com/standards/alignments/grad e3.html http://www.k-‐5mathteachingresources.com/3rd-‐grade-‐ number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐ standards Vocabulary • • • • • • • • • • • • • • • • • • • • • • Addend Area Attribute Benchmark number Compare Decomposing Estimation strategies Expanded form Inverse operations Linear Measurement Nonstandard Units Overlap Perimeter Place value Plane figures Polygon Rounding Standard form Side length Unknown quantity Variable Whole numbers Essential Questions • • • • • • • Why is the use of estimation and/or rounding important in determining if your answer is reasonable? How can you solve a three-‐digit plus a two-‐ digit addition problem in two different ways? What number patterns do you notice in the addition table? Why do these patterns make mathematical sense? Given a one-‐step word problem, what equation could represent it? How do you find the perimeter of a polygon? How is finding area different from finding perimeter? • Formative Assessment Strategies • • • • Observation – Walking around classroom and observe for understanding. Anecdotal records, conferences, checklists. 3-‐2-‐1 – 3 things you found out, 2 interesting things and 1 question you still have. Exit Cards -‐ Exit cards are written student responses to questions posed at the end of a class or learning activity or at the end of a day. Student Data Notebooks -‐ A tool for students to track their learning: Where am I going? Where am I now? How will I get there? Unit 2: Exploring equal groups as a foundation for multiplication and division. Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In Grade 2 students have added groups of objects by skip-‐counting and using repeated addition (2.0A.C.4). In this unit, students connect these concepts to multiplication and division by interpreting and representing products and quotients. Students begin developing these concepts by working with numbers with which they are more familiar, such as 2s, 5s, and 10s, in addition to numbers that are easily skip counted, such as 3s and 4s. Since multiplication is a critical area for Grade 3, students will build on these concepts throughout the year, working towards fluency by the end of the year. Videos Common Core State Standards for http://www.youtube.com/watch Mathematical Content ?v=llnio99_YU8 (3.OA.1) 0perations and Algebraic Thinking -‐ 3.0A 3.OA.1.1 Represent a situation in which a number of www.khanacademy.org In 3.0A.A.1 situations with A. Represent and solve problems involving groups can be expressed using multiplication. (MS) www.teachingchannel.org discreet objects should be multiplication and division. 3.OA.1.2 Identify a situation in which a number of www.youtube.com explored first when developing a 1. Interpret products of whole numbers, e.g., groups can be expressed using multiplication. (MS) conceptual understanding of interpret 5 x 7 as the total number of objects in 5 3.OA.1.3 Draw an array. (MS) Math Fact Fluency Practice multiplication, followed by groups of 7 objects each. For example, describe a 3.OA.1.4 Explain an array. (MS) www.mathwire.com measurement examples involving context in which a total number of objects can be 3.OA.1.5 Find the product using objects in groups. www.oswego.org/ocsd-‐ area models. expressed as 5 x 7. 3.OA.1.6 Find the product using objects in arrays. web/games/ 3.OA.1.7 Find the product using objects in area http://mathfactspro.com/mathflu models. encygame.html#/math-‐facts-‐ 3.OA.1.8 Find the product using measurement addition-‐games quantities. http://jerome.northbranfordscho 3.OA.1.9 Explain the objects in equal size groups. ols.org/Content/Math_Fact_Flue (MS) ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://learnzillion.com/ www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade 3.OA.2.1 P artition a w hole n umber i nto e qual s hares www.dpi.state.nc.us 2. Interpret whole-‐number quotients of whole 3.0A.A.2 will be readdressed in unit using a rrays. ( MS) numbers, e.g., interpret 56 x 8 as the number of 7 in order to provide students the 3.OA.2.2 P artition a w hole n umber i nto e qual p arts http://harcourtschool.com/searc objects in each share when 56 objects are opportunity to develop using a rea. h/search.html partitioned equally into 8 shares, or as a number computational strategies as they of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 3.OA.2.3 Partition a whole number into equal parts using measurement quantities. 3.OA.2.4 Identify each number in a division expression as a quotient, divisor, and/or dividend. (MS) 3.OA.2.5 Describe a situation in which a number of groups can be expressed using division. (MS) 3.OA.2.6 Identify a situation in which a number of groups can be expressed using division. (MS) 3.OA.3.1 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.2 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using drawings. 3.OA.3.3 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. 3.OA.3.4 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using equations. 3.OA.3.5 Explain that an unknown number is represented with a symbol/variable. 3.OA.3.6 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.7 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using drawings. 3.OA.3.8 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. (MS) 3.OA.3.9 Use division (quotient/divisor ≥ 5 and ≤ 10) extend the range of numbers with which they compute. 3.0A.A.3 will be readdressed in unit 7 and finalized in unit 14 to include measurement quantities in order to provide students multiple opportunities to develop and practice these concepts. www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind ex.php/common-‐core-‐standards http://map.mathshell.org/materi als/stds.php#standard1159 C. Multiply and divide within 100. 7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 x 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-‐digit numbers. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. to solve word problems that involve area and other measurement quantities other than area using equations. 3.OA.3.10 Divide an area by side length to find the unknown side length 3.OA.7.1 Fluently (accurately and quickly) divide with a dividend up to 100. 3.OA.7.2 Fluently (accurately and quickly) multiply numbers 0-‐10. 3.OA.7.3 Memorize and recall my multiples from 0-‐9. 3.OA.7.4 Recognize the relationship between multiplication and division. 3.0A.C.7 will be readdressed in unit 7 and unit 15 in order to provide students the opportunity to develop computational strategies as they extend the range of numbers with which they compute. Students use concrete objects or pictures to help conceptualize and solve problems (MP.1). They use arrays and other representations to model multiplication and division (MP.4) and contextualize given expressions (MP.2). Vocabulary • • • • • • • • • • • • • • • • • • • • Array Associative Property of Multiplication Commutative Property of Multiplication Distributive Property Division Equal Estimation Strategies Fluent Inverse Operations Length Mental Computation Strategies Multiplication Partition Product Quotient Rounding Symbol Unknown Quantity Variable Whole numbers Essential Questions • • • • • • • • • How can you use number patterns and/or models to solve multiplication problems? How is multiplication like addition? What is the advantage of using multiplication? What happens when you multiply any number by 1? By zero? How is multiplying by 1 or zero the same or different than adding by 1 or zero? What is an array? Can you use an array to show multiplication? How can you find the total number of objects in equal groups? How can you use multiplication to compare? How do you write a good mathematical explanation? Formative Assessment Strategies • • • • Take and Pass -‐ Cooperative group activity used to share or collect information from each member of the group; students write a response, then pass to the right, add their response to next paper, continue until they get their paper back, then group debriefs. Slap It -‐ Students are divided into two teams to identify correct answers to questions given by the teacher. Students use a fly swatter to slap the correct response posted on the wall. Numbered Heads Together -‐ Students sit in groups and each group member is given a number. The teacher poses a problem and all four students discuss. The teacher calls a number and that student is responsible for sharing for the group. Circle, Triangle, Square -‐ Something that is still going around in your head (Triangle) Something pointed that stood out in your mind (Square) Something that “Squared” or agreed with your thinking. Unit 3: Developing conceptual understanding of area. Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: This unit provides ample time, and should include multiple experiences, for students to explore the connections among counting tiles, skip counting the number of tiles in rows or columns, and multiplying the side lengths of a rectangle to determine area. Students' understanding of these connections is critical content at this grade, and must occur early in the school year, thereby allowing time for understanding and fluency to develop across future units. Videos Common Core State Standards for Mathematical www.khanacademy.org Content www.teachingchannel.org 0perations and Algebraic Thinking -‐ 3.0A 3.OA.5.1 Apply the properties to multiply 2 or more www.youtube.com 3.0A.B.5 will be readdressed in unit B. Understand properties of multiplication and the factors using different strategies. 9 with a focus on the distributive relationship between multiplication and division. 3.OA.5.2 Decompose an expression to represent the Math Fact Fluency Practice property and in unit 12 with a focus 5. Apply properties of operations as strategies to distributive property. www.mathwire.com on the associative property. 2 multiply and divide. Examples: If 6 x 4 = 24 is 3.OA.5.3 Justify the correctness of a problem based www.oswego.org/ocsd-‐ known, then 4 x 6 = 24 is also known. on the use of the properties (commutative, web/games/ (Commutative property of multiplication.) 3 x associative, distributive). http://mathfactspro.com/mathflu 3.OA.5.4 Use properties of operations to construct encygame.html#/math-‐facts-‐ 5 x 2 can be found by 3 x 5 and communicate a written response based on addition-‐games = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then explanation/reasoning. http://jerome.northbranfordscho 3 x 10 = 30. (Associative property of 3.OA.5.5 U se p roperties o f o perations t o c learly ols.org/Content/Math_Fact_Flue multiplication.) Knowing that 8 x 5 = 40 and construct a nd c ommunicate a c omplete w ritten ncy_Practice_Sheets.asp 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = response. http://www.mathfactcafe.com/ (8 x 5) + (8 x 2) = 40 + 16 = www.factmonster.com 56. (Distributive property.) 2 Note: Students need not use formal terms for Lessons/Activities/Games these properties. https://www.illustrativemathema tics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade www.dpi.state.nc.us Measurement and Data -‐ 3.MD http://harcourtschool.com/searc C. Geometric measurement: understand concepts h/search.html of area and relate area to multiplication and to addition. 5. Recognize area as an attribute of plane figures 3.MD.5a.1 Identify what a unit square is and know it can be used to measure area of a figure. and understand concepts of area measurement. a. A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area. 3.MD.5b.1 Relate the area to real world objects. b. A plane figure which can be covered 3.MD.5b.2 Recognize area as an attribute of plane without gaps or overlaps by n unit figures with a visual model. squares is said to have an area of n 3.MD.5b.3 Explain area as an attribute of plane square units. figures. 3.MD.6.1 Determine the area of an object by 6. Measure areas by counting unit squares counting the unit squares in cm, m, in., ft., and (square cm, square m, square in, square ft, other units. and improvised units). 3.MD.6.2 Connect counting squares to multiplication when finding area. 7. Relate area to the operations of multiplication 3.MD.7a.1 Use tiles to show the area of an rectangle. and addition. 3.MD.7a.2 Multiply (b x h) or (l x w) to determine the a. Find the area of a rectangle with whole-‐ area of a rectangle. number side lengths by tiling it, and show 3.MD.7a.3 Justify that the area of a rectangle will be that the area is the same as would be the same using different methods. (Tiling and found by multiplying the side lengths. formula) Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 6. Attend to precision. 7. Look for and make use of structure. Students analyze the structure of multiplication and division (MP.7) through their work with arrays (MP.2) and work towards precisely expressing their understanding of the connection between area and multiplication (MP.6). www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Vocabulary • • • • • • • • • • • • • • • • • • • • • • Associative Property of multiplication Area Attribute Commutative Property of multiplication Distributive Property Equation Measurement Multiples Multiplication Perimeter Plane figure/figures Polygon Product Side length Square centimeter Square foot Square inch Square meter Square units Tiling Unit Square Unknown variable Whole numbers Essential Questions • • • • • • • • • How can you use number patterns and/or models to solve multiplication problems? How is multiplication like addition? What is the advantage of using multiplication? What happens when you multiply any number by 1? By zero? How is multiplying by 1 or zero the same or different than adding by 1 or zero? What does the term “square unit” represent? This rectangle has an area of square units. What does that mean? How do you find perimeter of common shapes? How do you find area of common shapes? What shapes can you create when you know the perimeter? • Formative Assessment Strategies • • • • Flag It – Students use “flags” (sticky notes) to flag important information presented in class or while working problems. Triangular Prism (Red, Yellow, Green) -‐ Students give feedback to teacher by displaying the color that corresponds to their level of understanding. Word Sort -‐ Given a set of vocabulary terms, students sort in to given categories or create their own categories for sorting. Cubing -‐ Display 6 questions from the lesson Have students in groups of 4. Each group has 1 die. Each student rolls the die and answers the question with the corresponding number. If a number is rolled more than once the student may elaborate on the previous response or roll again. Unit 4: Understanding unit fractions Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In previous grades students have had experience partitioning shapes into fair shares (1.G.A.3 and 2.G.A.3), using words to describe the quantity. In this unit students extend this understanding to partition shapes and number lines, representing these fair shares using fraction notation. Similar to how students view 1 as the building block of whole numbers, students learn to view unit fractions as building blocks-‐understanding that every fraction is a combination of unit fractions. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Geometry -‐ 3.G www.youtube.com A. Reason with shapes and their attributes. 3.G.2.1 Recognize that shapes can be divided into 2. Partition shapes into parts with equal areas. equal parts. Math Fact Fluency Practice Express the area of each part as a unit 3.G.2.2 Separate a given object into equal parts. www.mathwire.com fraction of the whole. For example, partition 3.G.2.3 Describe the area of each part as a fractional www.oswego.org/ocsd-‐ a shape into 4 parts with equal area, and part of the whole. web/games/ 3.G.2.4 Label each part as a fractional part of the http://mathfactspro.com/mathflu describe the area of each part as 1/4 of the whole. encygame.html#/math-‐facts-‐ area of the shape. 3.G.2.5 Partition shapes in multiple ways into parts addition-‐games with equal areas and express the area as a unit http://jerome.northbranfordscho fraction of the whole. ols.org/Content/Math_Fact_Flue ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ 5 Number and Operations-‐Fractions -‐ 3.NF 3.NF.1.1 Explain that the fractional pieces get smaller The focus of 3.NF.A.1 and 3.NF.A.2a www.factmonster.com in this unit is on fractions between A. Develop understanding of fractions as numbers. as the denominator gets larger. 3.NF.1.2 Explain that the denominator represents the 0 and 1. Fractions greater than 1 Lessons/Activities/Games 1. Understand a fraction 1Ib as the quantity will be introduced in unit 5. number of equal parts in the whole. (MS) https://www.illustrativemathema formed by 1 part when a whole is 3.NF.1.3 Explain that the numerator is a count of the tics.org/3 partitioned into b equal parts; understand a number of equal parts (3/4 means there are three fraction a/b as the quantity formed by a ¼’s; ¾ = ¼ + ¼ + ¼). https://learnzillion.com/ parts of size 1/b. 3.NF.1.4 Model fractions as parts of a whole or parts of a group. (MS) www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade 3.NF.2a.1 P artition ( divide) a n umber l ine i nto e qual www.dpi.state.nc.us 2. Understand a fraction as a number on the parts ( intervals). number line; represent fractions on a number 3.NF.2a.2 Identify a given fraction on a number line. http://harcourtschool.com/searc line diagram. 3.NF.2a.3 Represent and recognize a given fraction on h/search.html a. Represent a fraction 1Ib 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 1Ib and that the endpoint of the part based at 0 locates the number 1Ib on the number line. 5 NOTE: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. a number line. Students use number lines to represent fractions in a new way (MP.4). It is key for students to have meaningful conversations around this concept to develop precise language about the components of fractions and location on the number line (MP.3, MP.6). Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 6. Attend to precision. www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org Vocabulary • • • • • • • Denominator Diagram Equal areas Equal distance Equal parts Equivalence Equivalent Essential Questions • • • • • • • How do you identify and record the fraction of a given shape? How do you partition this shape so the fraction____ is represented? What does the numerator tell you about a fraction? What does the denominator tell you about a fraction? How you can represent a unit fraction using a variety of materials? How can you divide a shape in equal parts? How do you estimate parts? • • • • • • • • • • Fair Sharing Fraction Interval Number line Numerator Part Partition Reasonable Shapes Unit Fraction Whole • If a shape is divided into (4) equal pieces, what is the size of each piece? How many pieces are needed to show ____ (3/4)? • Formative Assessment Strategies • • Tic-‐Tac-‐Toe/Think-‐Tac-‐Toe -‐ A collection of activities from which students can choose to do to demonstrate their understanding. It is presented in the form of a nine square grid similar to a tic-‐tac-‐toe board and students may be expected to complete from one to “three in a row”. The activities vary in content, process, and product and can be tailored to address DOK levels. Four Corners -‐ Students choose a corner based on their level of expertise of a given subject. Based on your knowledge of _________________, which corner would you choose? Corner 1: The Dirt Road – (There’s so much dust, I can’t see where I’m going! Help!!), Corner 2: The Paved Road (It’s fairly smooth, but there are many potholes along the way.), Corner 3: The Highway (I feel fairly confident but have an occasional need to slowdown.) Corner 4: The Interstate (I ’m traveling along and could easily give directions to someone else.) Once students are in their chosen corners, allow students to discuss their progress with others. Questions may be prompted by teacher. Corner One will pair with Corner Three; Corner Two will pair with Corner four for peer tutoring. Unit 5: Using fractions in measurement and data Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In this unit students extend their work with measurement and data involving whole numbers to include fractional quantities. Measurement and data are used as a context to support students' understanding of fractions as numbers. In students' work with data, context is important, because data are not just numbers; they are numbers with meaning. Through experience with measurement, students realize fractions allow us to represent data much more accurately than just representing data with whole numbers. 3.NF.A.1 is repeated here to Videos Common Core State Standards for http://www.engageny.org/resour include fractions greater than 1. Mathematical Content ce/common-‐core-‐video-‐series-‐ S Number and Operations-‐Fractions -‐ 3.NF 3.NF.1.1 Explain that the fractional pieces get smaller grade-‐3-‐mathematics-‐inches-‐and-‐ A. Develop understanding of fractions as numbers. as the denominator gets larger. centimeters 1. Understand a fraction 1/b as the quantity 3.NF.1.2 Explain that the denominator represents the www.khanacademy.org formed by 1 part when a whole is number of equal parts in the whole. (MS) www.teachingchannel.org partitioned into b equal parts; understand a 3.NF.1.3 Explain that the numerator is a count of the www.youtube.com fraction a/b as the quantity formed by a number of equal parts (3/4 means there are three parts of size 1/b. ¼’s; ¾ = ¼ + ¼ + ¼). Math Fact Fluency Practice 3.NF.1.4 Model fractions as parts of a whole or parts www.mathwire.com of a group. (MS) www.oswego.org/ocsd-‐ web/games/ http://mathfactspro.com/mathflu encygame.html#/math-‐facts-‐ 2. Understand a fraction as a number on the 3.NF.2b.1 Recognize that a fraction a/b represents its addition-‐games number line; represent fractions on a number distance from 0 on a number line. http://jerome.northbranfordscho line diagram. 3.NF.2b.2 Recognize that a fraction a/b represents its ols.org/Content/Math_Fact_Flue b. Represent a fraction a/b on a number line location on a number line. ncy_Practice_Sheets.asp diagram by marking off a lengths l/b from O. http://www.mathfactcafe.com/ Recognize that the resulting interval has size www.factmonster.com a/b and that its endpoint locates the number a/b on the number line. 5 NOTE: Grade 3 expectations in this domain Lessons/Activities/Games are limited to fractions with denominators 2, 3, https://www.illustrativemathema 4, 6, and 8. tics.org/3 https://learnzillion.com/ 3.MD.4.1 Use a ruler to measure an object to the www.AECSD3rdGradeMathemati Measurement and Data -‐ 3.MD nearest whole, half, and quarter inch. csdoc B. Represent and interpret data. 3.MD.4.2 Collect and organize data to create a line 4. Generate measurement data by measuring plot (whole numbers, halves, and quarters). http://maccss.ncdpi.wikispaces.n lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-‐ whole numbers, halves, or quarters. Common Core State Standards for Mathematical Practice 3.MD.4.3 Create a line plot from given or collected data, where the horizontal scale is marked off in appropriate units (whole numbers, halves, and quarters). 3.MD.4.4 Label a line plot to show whole numbers, halves, and quarters. 3.MD.4.5 Use a line plot to answer questions or solve problems. 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. Data Denominator Diagram Fourths Fraction Halves Horizontal et/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/searc h/search.html www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.co Essential Questions Vocabulary • • • • • • • Students use tools to generate measurement data (MP.5) and make connections among different representations of the quantities and their relation to the given data context (MP.2). • • • • • • How do you represent a whole number as a fraction? How would you show equivalent fractions on a number line diagram? How do you know if two fractions are equivalent? How do you know if they are not equivalent? How can you write fractions in simplest form? How can you compare fractions on a number line? Where would the following fractions be located on a number line diagram? • • • • • • • • • Inch Length Line Plot Measurement Number Line Numerator Quarters Units Vertical Whole Numbers • • • • How do you represent your data on a labeled line plot diagram? What steps must you take when deciding where to place a fraction on a number line diagram? How do you measure an object in inches? How long is this item to the nearest whole number, 1/2 or 1/4 of an inch? • Formative Assessment Strategies • • • • Think-‐Write-‐Pair-‐Share -‐ Students think individually, write their thinking, pair and discuss with partner, then share with the class. Choral Response -‐ In response t o a cue, all students respond verbally at the same time. The response can be either to answer a question or to repeat something the teacher has said. Self Assessment -‐ process in which students collect information about their own learning, analyze what it reveals about their progress toward the intended learning goals and plan the next steps in their learning. Web or Concept Map -‐ Any of several forms of graphical organizers which allow learners to perceive relationships between concepts through diagramming key words representing those concepts. http://www.graphic.org/concept.html Unit 6: Solving addition and subtraction problems involving measurement Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: The focus of this unit is to develop a conceptual understanding of measuring mass, liquid volume, intervals of time, and using measurement as a context for the development of fluency in addition and subtraction. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Measurement and Data -‐ 3.MD 3.MD.1.1 Explain time intervals. www.youtube.com 3.MD.A.1 is included here as an A. Solve problems involving measurement and 3.MD.1.2 Identify minute marks on an analog clock. opportunity to model addition and estimation of intervals of time, liquid volumes, 3.MD.1.3 Identify minute position on a digital clock. Math Fact Fluency Practice subtraction situations with time as and masses of objects. 3.MD.1.4 Relate and explain a number line to the www.mathwire.com the context. 1. Tell and write time to the nearest minute minute marks on a clock. www.oswego.org/ocsd-‐ and measure time intervals in minutes. 3.MD.1.5 Use a “time” number line to measure and web/games/ solve addition or subtraction word problems to the http://mathfactspro.com/mathflu Solve word problems involving addition nearest minute. encygame.html#/math-‐facts-‐ and subtraction of time intervals in 3.MD.1.6 Use a “time” number line to measure and addition-‐games minutes, e.g., by representing the solve t wo-‐step a ddition a nd s ubtraction w ord http://jerome.northbranfordscho problem on a number line diagram. problems t o t he n earest m inute. ols.org/Content/Math_Fact_Flue 3.MD.1.7 W rite t ime t o t he n earest m inute. ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com 2. Measure and estimate liquid volumes and 3.MD.2.1 Measure liquid volume in metric units 3.MD.A.2 is addressed in full in masses of objects using standard units of grams (liters). Lessons/Activities/Games 6 unit 14 to include multiplication (g), kilograms (kg), and liters (l). Add, subtract, 3.MD.2.2 Measure mass in metric units (kilograms, https://www.illustrativemathema and division situations. multiply, or divide to solve one-‐step word grams). tics.org/3 problems involving masses or volumes that are 3.MD.2.3 Estimate liquid volume using metric units given in the same units, e.g., by using drawings (liters). https://learnzillion.com/ (such as a beaker with a measurement scale) to 3.MD.2.4 Estimate mass in metric units (kilograms, 7 6 represent the problem. NOTE: Excludes grams). www.AECSD3rdGradeMathemati compound units such as cm3 and finding the 3.MD.2.5 Use the appropriate unit to measure the csdoc 7 geometric volume of a container. Excludes mass of objects. http://maccss.ncdpi.wikispaces.n multiplicative comparison problems (problems 3.MD.2.6 Use the appropriate unit to measure the et/Third+Grade involving notions of "times as much"; see liquid volume of objects. Glossary, Table 2). www.dpi.state.nc.us 3.MD.2.7 Use the four basic operations to solve one step word problems with mass. http://harcourtschool.com/searc 3.MD.2.8 Use the four basic operations to solve one h/search.html step word problems with liquid volume. Common Core State Standards for Mathematical Practice 3.MD.2.9 Use the four basic operations to solve two step word problems with mass. 3.MD.2.10 Use the four basic operations to solve two step word problems with liquid volume. 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 4. Model with mathematics. Analog clock Digital clock Grams Interval Kilograms Liter Liquid www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Essential Questions Vocabulary • • • • • • • Students can apply the mathematics they know to persevere in solving problems arising in everyday life, society, and the workplace (MP.1, MP.4). Selecting and using appropriate tools supports the development of measurement concepts by asking students to reason about which tools are appropriate and how to use tools efficiently (MP.5). • • • • • • • How can you show time? Is there more than one way to show time? What is the difference between an analog and a digital clock? How can you measure how long an event takes from start to finish? How do we solve problems when the beginning information is unknown? How can you estimate and measure length? How can you estimate and measure capacity? • • • • • • Mass Metric Minute Number Line Scale Time Volume • • How do you differentiate between mass, weight, and capacity? Is there a way to use a pattern to solve a problem? • Formative Assessment Strategies • • • • • Index Card Summaries/Questions -‐ Periodically, distribute index cards and ask students to write on both sides, with these instructions: (Side 1) Based on our study of (unit topic), list a big idea that you understand and word it as a summary statement. (Side 2) Identify something about (unit topic) that you do not yet fully understand and word it as a statement or question. Hand Signals -‐ Ask students to display a designated hand signal to indicate their understanding of a specific concept, principal, or process: -‐ I understand____________ and can explain it (e.g., thumbs up). -‐ I do not yet understand ____________ (e.g., thumbs down). -‐ I’m not completely sure about ____________ (e.g., wave hand). One Minute Essay -‐ A one-‐minute essay question (or one-‐minute question) is a focused question with a specific goal that can, in fact, be answered within a minute or two. Analogy Prompt -‐ Present students with an analogy prompt: (A designated concept, principle, or process) is like ___________ because___________. Misconception Check -‐ Present students with common or predictable misconceptions about a designated concept, principle, or process. Ask them whether they agree or disagree and explain why. The misconception check can also be presented in the form of a multiple-‐choice or true-‐false quiz. Unit 7: Understanding the relationship between multiplication and division Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: The emphasis of this unit is for students to develop a solid understanding of the connection between multiplication and division. Students recognize that multiplication strategies can be used to make sense of and solve division problems. This unit provides students a solid foundation in solving problems with equal groups and arrays, which is necessary to support future success with measurement problems. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org 0perations and Algebraic Thinking -‐ 3.0A 3.OA.2.1 Partition a whole number into equal shares www.youtube.com 3.0A.A.2 and 3.0A.C.7 are revisited A. Represent and solve problems involving using arrays. (MS) in this unit to extend the range of multiplication and division. 3.OA.2.2 Partition a whole number into equal parts Math Fact Fluency Practice numbers to include all numbers 2. Interpret whole-‐number quotients of whole using area. www.mathwire.com within 100 when multiplying and numbers, e.g., interpret 56 x 8 as the number of 3.OA.2.3 Partition a whole number into equal parts www.oswego.org/ocsd-‐ dividing. objects in each share when 56 objects are using measurement quantities. web/games/ partitioned equally into 8 shares, or as a number 3.OA.2.4 Identify each number in a division http://mathfactspro.com/mathflu of shares when 56 objects are partitioned into expression as a quotient, divisor, and/or dividend. encygame.html#/math-‐facts-‐ equal shares of 8 objects each. For example, (MS) addition-‐games describe a context in which a number of shares or 3.OA.2.5 Describe a situation in which a number of http://jerome.northbranfordscho a number of groups can be expressed as 56 ÷ 8. groups can be expressed using division. (MS) ols.org/Content/Math_Fact_Flue 3.OA.2.6 Identify a situation in which a number of ncy_Practice_Sheets.asp groups can be expressed using division. (MS) http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games 3. Use multiplication and division within 100 to 3.OA.3.1 Use multiplication (factors ≥ 5 and ≤ 10) to https://www.illustrativemathema 3.0A.A.3 includes equal groups, solve word problems in situations involving solve word problems that involve equal groups and arrays, and area problem types. tics.org/3 equal groups, arrays, and measurement arrays using drawings. (MS) Note that multiplicative compare quantities, e.g., by using drawings and 3.OA.3.2 Use multiplication (factors ≥ 5 and ≤ 10) to https://learnzillion.com/ problems are introduced in Grade equations with a symbol for the unknown solve word problems that involve area and other 1 4 (4.0A.A.2). number to represent the problem. NOTE: measurement q uantities o ther t han a rea u sing www.AECSD3rdGradeMathemati 1 See Glossary, Table 2. drawings. csdoc 3.OA.3.3 U se m ultiplication ( factors ≥ 5 a nd ≤ 1 0) t o solve word problems that involve equal groups and http://maccss.ncdpi.wikispaces.n arrays using equations. et/Third+Grade 3.OA.3.4 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other www.dpi.state.nc.us measurement quantities other than area using equations. http://harcourtschool.com/searc 3.OA.3.5 Explain that an unknown number is represented with a symbol/variable. 3.OA.3.6 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.7 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using drawings. 3.OA.3.8 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. (MS) 3.OA.3.9 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using equations. 3.OA.3.10 Divide an area by side length to find the unknown side length. B. Understand properties of multiplication and the 3.OA.6.1 Interpret division as an unknown factor relationship between multiplication and division. problem using the fact families. 6. Understand division as an unknown-‐factor 3.OA.6.2 Interpret division as an unknown factor problem. For example, find 32 ÷ 8 by problem using a bar model. finding the number that makes 32 when 3.OA.6.3 Interpret division as an unknown factor multiplied by 8. problem using a number line. 3.OA.6.4 Interpret division as an unknown factor problem using arrays. 3.OA.6.5 Justify the correctness of a problem based on the understanding of division as an unknown factor problem. C. Multiply and divide within 100. 3.OA.7.1 Fluently (accurately and quickly) divide with 7. Fluently multiply and divide within a dividend up to 100. 100, using strategies such as the 3.OA.7.2 Fluently (accurately and quickly) multiply relationship between multiplication numbers 0-‐10. 3.OA.7.3 Memorize and recall my multiples from 0-‐9. and division (e.g., knowing that 8 x 5 = 3.OA.7.4 Recognize the relationship between 40, one knows 40 x 5 = 8) or properties multiplication and division. of operations. By the end of Grade 3, know from memory all products of two h/search.html www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org http://www.coolmath.com/ http://www.mathplayground.co m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind ex.php/common-‐core-‐standards http://map.mathshell.org/materi als/stds.php#standard1159 3.0A.C.7 is finalized in unit 15. This gives students the opportunity to develop and practice strategies in order to achieve fluency by the end of the year. one-‐digit numbers. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 7. Look for and make use of structure. Students make sense of and solve various types of multiplication and division problems (MP.1) by using the relationship between the two operations (MP.7). Vocabulary • • • • • • • • • • • • • Addition Array Bar Model Division Divisor Dividend Fluent Multiplication Number Line Quotient Subtraction Symbol Variable Essential Questions • • • • • • • • • • • • Can you use an array to solve multiplication problems? When might you need to multiply three numbers? How can you think of division as sharing? How can you think of division as repeated subtraction? What kinds of stories involve division situations? How can you use objects and draw a picture to solve a problem? What patterns develop when we multiply by multiples of 10, 100 and 1,000? What rules for multiplying can we write based on these patterns? When might it be better to estimate a product rather than determine a precise answer? How can we use what we know about basic multiplication facts and place value to multiply large numbers? How can partial products be used to simplify multiplication algorithms? How can we use regrouping to simplify multiplication algorithms? How can we use bar diagrams to solve real-‐world multiplication word problems? Formative Assessment Strategies • • • • • Journal Entry -‐ Students record in a journal their understanding of the topic, concept or lesson taught. The teacher reviews the entry to see if the student has gained an understanding of the topic, lesson or concept that was taught. Choral Response -‐ In response t o a cue, all students respond verbally at the same time. The response can be either to answer a question or to repeat something the teacher has said. A-‐B-‐C Summaries -‐ Each student in the class is assigned a different letter of the alphabet and they must select a word starting with that letter that is related to the topic being studied. Debriefing -‐ A form of reflection immediately following an activity. Idea Spinner -‐ The teacher creates a spinner marked into 4 quadrants and labeled “Predict, Explain, Summarize, Evaluate.” After new material is presented, the teacher spins the spinner and asks the students to answer a questions based on the location of the spinner. For example, if the spinner lands in the “Summarize” quadrant, the teacher might say, “List the key concepts just presented.” Unit 8: Investigating patterns in number and operations. Suggested number of days: 15 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: The focus of this unit is for students to identify arithmetic patterns in order to develop a deeper understanding of number and number relationships. In subsequent units, students will use the understanding of pattern developed in this unit to strengthen their computational strategies and skills. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Operations and Algebraic Thinking -‐ 3.OA 3.OA.8.1 Construct an equation with a letter www.youtube.com 3.OA.D.8 will be revisited in unit D. Solve problems involving the four operations, (variable) to represent the unknown quantity. 15 to address the use of equations and identify and explain patterns in arithmetic. 3.OA.8.2 Explain or demonstrate how to solve two-‐ Math Fact Fluency Practice and letters for unknown 8. Solve two-‐step word problems using the step word problems using addition and subtraction quantities. www.mathwire.com four operations. Represent these problems 3.OA.8.3 Explain or demonstrate how to solve two-‐ www.oswego.org/ocsd-‐ using equations with a letter standing for the step word problems using multiplication and web/games/ unknown quantity. Assess the division (Of single digit factors and products less http://mathfactspro.com/mathflu reasonableness of answers using mental than 100). encygame.html#/math-‐facts-‐ computation and estimation strategies 3.OA.8.4 R epresent a w ord p roblem w ith a n e quation addition-‐games 3 including rounding. using a letter to represent the unknown quantity. http://jerome.northbranfordscho 3 NOTE: This standard is limited to problems 3.OA.8.5 Solve two-‐step word problems which ols.org/Content/Math_Fact_Flue posed with whole numbers and having whole-‐ include multiple operations. ncy_Practice_Sheets.asp number answers; students should know how to 3.OA.8.6 Use mental math to estimate the answer of http://www.mathfactcafe.com/ perform operations in the conventional order a single step word problem. (MS) www.factmonster.com 3.OA.8.7 Use mental math to estimate the answer of when there are no parentheses to specify a a two-‐step word problem. Lessons/Activities/Games particular order (Order of Operations). 3.OA.8.8 Justify my answers using mental math and https://www.illustrativemathema estimation. (MS) tics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade www.dpi.state.nc.us 9. Identify arithmetic patterns (including 3.OA.9.1 Explain and model the relationship of odd patterns in the addition table or and even number patterns with addition facts. http://harcourtschool.com/searc Examples: h/search.html multiplication table), and explain them • Recognize that the sum of two even numbers is 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. even. • Recognize that the sum of two odd numbers is even. • Recognize that the sum of an even and an odd number is odd. 3.OA.9.2 Explain and model the relationship of odd and even number patterns with multiplication facts. • Recognize that if at least 1 factor is even, the product will be even. • Use divisibility rules identify arithmetic patterns. 3.OA.9.3 Use a multiplication table to locate examples of the commutative, identity, and zero properties of multiplication. 3.OA.9.4 Use an addition table to locate examples of the commutative and identity properties of addition. Number and Operations in Base Ten -‐ 3.NBT 3.NBT.1.1 Explain the process for rounding numbers A. Use place value understanding and properties 4 using place value. of operations to perform multi-‐digit arithmetic. 1. Use place value understanding to round whole 3.NBT.1.2 Identify the place value of the ones, tens, and hundreds place in a whole number. numbers to the nearest 10 or 100. 3.NBT.1.3 Round numbers to the nearest hundred. 3.NBT.1.4 Round numbers to the nearest ten. 3. Multiply one-‐digit whole numbers by 3.NBT.3.1 Correctly align digits according to place multiples of 10 in the range 10-‐90 (e.g., 9 value, in order to multiply. x 80, 5 x 60) using strategies based on 3.NBT.3.2 Explain and demonstrate the process of place value and properties of operations. multiplying a two digit number by a one digit 4 number using various algorithms. NOTE: A range of algorithms may be used. 3.NBT.3.3 Multiply 1-‐digit whole numbers by multiples of 10 in the range of 1-‐90 using different strategies. www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind ex.php/common-‐core-‐standard http://map.mathshell.org/materi als/stds.php#standard1159 Measurement and Data -‐ 3.MD B. Represent and interpret data. 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. 3.MD.3.1 Complete a scaled bar graph to represent data. 3.MD.3.2 Complete a scaled picture graph to represent data. 3.MD.3.3 Read and analyze data on horizontal and vertical scaled bar graphs. 3.MD.3.4 Read and analyze data on scaled picture graphs. 3.MD.3.5 Use information from a bar graph to solve 1-‐step “how many more” and “how many less” problems. 3.MD.3.6 Use information from a bar graph to solve 2-‐step “how many more” and “how many less” problems. 3.MD.3.7 Create problems/scenarios from information presented on a graph. Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. Students examine patterns in arithmetic (MP.7) and discuss what they discover (MP.3). Vocabulary • • • • • • • • • • Addend Area Arithmetic patterns Arrays Decompose Division Factor Improvised Units Inverse operations Line Plot Essential Questions • • • • • • What clue words help you identify which operation to use to solve word problems? Why is the use of estimation and/or rounding important in determining if your answer is reasonable? How can you solve a three-‐digit plus a two-‐ digit addition problem in two different ways? What number patterns do you notice in the addition table? Why do these patterns make mathematical sense? Given a two-‐step word problem, what equation could represent it? How can multiplication help you solve division problems? • • • • • • • • • • • • Measurement Mental Computation Multiplication Patterns Place Value Rounding Scale Scaled bar graph Scaled picture graph Unknown quantity Variable Whole Numbers Word form • • • • • • What strategies can be used to find products and/or quotients? How can you use the array model to help you solve multiplication problems? What number sentences could be used to solve this problem? How can simpler multiplication facts help you solve a more difficult fact? How do you know that your equation accurately represents this word problem? How do you know your answer is reasonable? • Formative Assessment Strategies • • • • One Sentence Summary -‐ Students are asked to write a summary sentence that answers the “who, what where, when, why, how” questions about the topic. Summary Frames -‐ Description: A ___________ is a kind of____________ that ... Compare/Contrast, Problem/Solution, Cause/Effect. One Word Summary -‐ Select (or invent) one word which best summarizes a topic. Think/Pair/Share and Turn to your partner -‐ Teacher gives direction to students. Students formulate individual response, and then turn to a partner to share their answers. Teacher calls on several random pairs to share their answers with the class. Unit 9: Developing strategies for multiplication and division Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: The focus for this unit is developing a conceptual understanding of decomposing multiplication problems through the use of the distributive property and the concept of area. Students are not required to use the properties explicitly, but are encouraged to discuss this concept and use area diagrams to support their reasoning. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org 0perations and Algebraic Thinking -‐ 3.0A 3.OA.5.1 Apply the properties to multiply 2 or more www.youtube.com 3.0A.B.5 will be revisited in unit 12 B. Understand properties of multiplication and the factors using different strategies. to address the associative property relationship between multiplication and division. 3.OA.5.2 Decompose an expression to represent the Math Fact Fluency Practice of multiplication. 5. Apply properties of operations as strategies to distributive property. www.mathwire.com 2 multiply and divide. Examples: If 6 x 4 = 24 is 3.OA.5.3 Justify the correctness of a problem based www.oswego.org/ocsd-‐ known, then 4 x 6 = 24 is also known. on the use of the properties (commutative, web/games/ (Commutative property of multiplication.) 3 x associative, distributive). http://mathfactspro.com/mathflu 3.OA.5.4 Use properties of operations to construct encygame.html#/math-‐facts-‐ 5 x 2 can be found by 3 x 5= 15, then 15 x 2 = and communicate a written response based on addition-‐games 30, or by 5 x 2 = 10, then 3 x 10 = 30. explanation/reasoning. http://jerome.northbranfordscho (Associative property of multiplication.) 3.OA.5.5 Use properties of operations to clearly ols.org/Content/Math_Fact_Flue Knowing that 8 x 5 = 40 and 8 x 2 = 16, one construct and communicate a complete written ncy_Practice_Sheets.asp can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = response. http://www.mathfactcafe.com/ 40 + 16 = 56. (Distributive property.) 2 www.factmonster.com Note: Students need not use formal terms for these properties. Lessons/Activities/Games https://www.illustrativemathema tics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade www.dpi.state.nc.us Measurement and Data -‐ 3.MD C. Geometric measurement: understand concepts http://harcourtschool.com/searc of area and relate area to multiplication and to h/search.html addition. 7. Relate area to the operations of multiplication and addition. c. Use tiling to show in a concrete case that the area of a rectangle with whole-‐number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-‐ overlapping rectangles and adding the areas of the non-‐overlapping parts, applying this technique to solve real world problems. Common Core State Standards for Mathematical Practice 5. Use appropriate tools strategically. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 3.MD.7c.1 Use tiling to explain the understanding of the distributive property in area problems. 3.MD.7d.1 Explain that a rectilinear figure is composed of smaller rectangles. 3.MD.7d.2 Model and separate a rectilinear figure into 2 different rectangles. 3.MD.7d.3 Determine the area of a figure by separating the figure into smaller rectangles and adding the area of each rectangle together. 3.MD.7d.4 Solve real world problems involving area of irregular shapes. Students use area diagrams and tiling (MP.5) to model the distributive property and generalize this experience to calculations (MP.7, MP.8). www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Vocabulary • • • • • • • • Area Associative Property of Multiplication Commutative Property of Multiplication Decompose Distributive Property Factors Rectangle Rectilinear figure Essential Questions • • • • • • • • • • • • What patterns develop when we multiply by multiples of 10, 100 and 1,000? What rules for multiplying can we write based on these patterns? When might it be better to estimate a product rather than determine a precise answer? How can we use what we know about basic multiplication facts and place value to multiply large numbers? How can partial products be used to simplify multiplication algorithms? How can we use regrouping to simplify multiplication algorithms? How can we use bar diagrams to solve real-‐world multiplication word problems? How do you find perimeter? How do you find the perimeter of common shapes? How do you find the perimeter of shapes? What shapes can you make when you know the perimeter? How do you find area? How do you estimate to find the area of an irregular shape? Formative Assessment Strategies • • • • Quick Write -‐ The strategy asks learners to respond in 2–10 minutes to an open-‐ended question or prompt posed by the teacher before, during, or after reading. Direct Paraphrasing -‐ Students summarize in well-‐chosen (own) words a key idea presented during the class period or the one just past. RSQC2 -‐ In two minutes, students recall and list in rank order the most important ideas from a previous day's class; in two more minutes, they summarize those points in a single sentence, then write one major question they want answered, then identify a thread or theme to connect this material to the course's major goal. I have the Question, Who has the Answer? -‐The teacher makes two sets of cards. One set contains questions related to the unit of study. The second set contains the answers to the questions. Distribute the answer cards to the students and either you or a student will read the question cards to the class. All students check their answer cards to see if they have the correct answer. A variation is to make cards into a chain activity: The student chosen to begin the chain will read the given card aloud and then wait for the next participant to read the only card that would correctly follow the progression. Play continues until all of the cards are read and the initial student is ready to read his card for the second time. Unit 10: Understanding equivalent fractions Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In this unit students develop a conceptual understanding of equivalence. Multiple types of models and representations should be used to help students develop this understanding. Students will apply their understanding of equivalence in the next unit as they learn to compare fractions. Through repeated experience locating fractions on the number line, students will recognize that many fractions label the same point and use this to support their understanding of equivalency. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org 5 Number and Operations-‐Fractions -‐ 3.NF 3.NF.3a/b.1 Recognize and generate equivalent www.youtube.com (3.NF.A.3) The focus of this unit is A. Develop understanding of fractions as numbers. fractions. (Denominators are 2, 3, 4, 6, and 8) around equivalence. Although the 3. Explain equivalence of fractions in special Math Fact Fluency Practice cluster heading includes cases, and compare fractions by reasoning www.mathwire.com comparison of fraction, fraction about their size. a. Understand two fractions www.oswego.org/ocsd-‐ comparisons (3.NF.A.3d) will be as equivalent (equal) if they are the same web/games/ addressed in unit 11. size, or the same point on a number line. http://mathfactspro.com/mathflu encygame.html#/math-‐facts-‐ b. Recognize and generate simple addition-‐games equivalent fractions, e.g., 1/2 = 2/4, http://jerome.northbranfordscho 4/6 = 2/3). Explain why the fractions ols.org/Content/Math_Fact_Flue are equivalent, e.g., by using a visual ncy_Practice_Sheets.asp fraction model. http://www.mathfactcafe.com/ www.factmonster.com 3.NF.3c.1 E xplain t hat a f raction w ith t he s ame c. Express whole numbers as fractions, and numerator a nd d enominator w ill a lways e qual 1 . Lessons/Activities/Games recognize fractions that are equivalent to 3.NF.3c.2 W rite a w hole n umber a s a f raction. https://www.illustrativemathema whole numbers. 3.NF.3c.3 R ecognize t hat s ome f ractions a re tics.org/3 Examples: Express 3 in the form 3 = 3/1; equivalent t o w hole n umbers. recognize that 6/1 = 6; locate 4/4 and 1 at https://learnzillion.com/ the same point of a number line diagram. S NOTE: Grade 3 expectations in this domain www.AECSD3rdGradeMathemati are limited to fractions with denominators csdoc 2, 3, 4, 6, and 8. http://maccss.ncdpi.wikispaces.n et/Third+Grade www.dpi.state.nc.us Common Core State Standards for Students develop understanding of Mathematical Practice http://harcourtschool.com/searc equivalence by modeling fractions h/search.html 4. Model with mathematics. (MP.4) and communicating their 6. Attend to precision. understanding of what it means for fractions to be equivalent (MP.6). Essential Questions Vocabulary • • • • • • • Compare Denominator Diagram Equal Equivalent Fractions Number Line www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ • • • • • • • How can you divide a region into equal parts? How can you show and name part of a region? How can different fractions name the same part of a whole? How can you write fractions in simplest form? How can you compare fractions? How can you locate and compare fractions and mixed numbers on a number line? How can you add fractions? • • Numerator Whole Numbers • • How can you subtract fractions? How can a fraction name a part of a group? Formative Assessment Strategies • • • • • • • Whip Around -‐ The teacher poses a question or a task. Students then individually respond on a scrap piece of paper listing at least 3 thoughts/responses/statements. When they have done so, students stand up. The teacher then randomly calls on a student to share one of his or her ideas from the paper. Students check off any items that are said by another student and sit down when all of their ideas have been shared with the group, whether or not they were the one to share them. The teacher continues to call on students until they are all seated. As the teacher listens to the ideas or information shared by the students, he or she can determine if there is a general level of understanding or if there are gaps in students’ thinking.” Word Sort -‐ Given a set of vocabulary terms, students sort in to given categories or create their own categories for sorting Triangular Prism (Red/Green/Yellow)Students give feedback to teacher by displaying the color that corresponds to their level of understanding Take and Pass -‐ Cooperative group activity used to share or collect information from each member of the group; students write a response, then pass to the right, add their response to next paper, continue until they get their paper back, then group debriefs. Student Data Notebooks -‐ A tool for students to track their learning: Where am I going? Where am I now? How will I get there? Slap It -‐ Students are divided into two teams to identify correct answers to questions given by the teacher. Students use a fly swatter to slap the correct response posted on the wall. Say Something -‐ Students take turns leading discussions in a cooperative group on sections of a reading or video Unit 11: Comparing fractions Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: In this unit students build on their prior work with fractions to reason about fraction size and structure to compare quantities. This unit focuses on a single standard to provide time for students to develop conceptual understanding of fraction comparisons and practice reasoning about size. Students defend their reasoning and critique the reasoning of others using both visual models and their understanding of the structure of fractions. This reasoning is important to develop a solid understanding of fraction magnitudes. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org 5 Number and Operations-‐Fractions -‐ 3.NF 3.NF.3d.1 Compare fractions based on the size of the www.youtube.com A. Develop understanding of fractions as numbers. numerator and denominator. 3. Explain equivalence of fractions in special 3.NF.3d.2 Compare and explain two fractions with Math Fact Fluency Practice cases, and compare fractions by reasoning the same denominator by drawing a visual model www.mathwire.com about their size. (using <,>,=). www.oswego.org/ocsd-‐ d. Compare two fractions with the same 3.NF.3d.3 Compare and explain two fractions with web/games/ the same numerator by drawing a visual model http://mathfactspro.com/mathflu numerator or the same denominator by (using <,>,=). encygame.html#/math-‐facts-‐ reasoning about their size. Recognize that addition-‐games comparisons are valid only when the two http://jerome.northbranfordscho fractions refer to the same whole. Record the ols.org/Content/Math_Fact_Flue results of comparisons with the symbols >, =, ncy_Practice_Sheets.asp or <, and justify the conclusions, e.g., by http://www.mathfactcafe.com/ using a visual fraction model. 5 www.factmonster.com NOTE: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, Lessons/Activities/Games 4, 6, and 8. https://www.illustrativemathema tics.org/3 Common Core State Standards for Students will use their Mathematical Practice https://learnzillion.com/ understanding of structure (i.e., 3. Construct viable arguments and critique the the role of the numerator and reasoning of others. S. Use appropriate tools www.AECSD3rdGradeMathemati denominator) (MP.7) to reason strategically. csdoc about relative sizes of fractions 7. Look for and make use of structure. http://maccss.ncdpi.wikispaces.n (MP.3). Students use various tools et/Third+Grade to justify their comparisons, paying particular attention to the www.dpi.state.nc.us same-‐sized wholes (MP.5). http://harcourtschool.com/searc h/search.html www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Essential Questions Vocabulary • • • • • • • Compare Denominator Diagram Equal Equivalent Fractions Number Line • • • • • How can I use fractions in real life? How can decimals be rounded to the nearest whole number? How can models be used to compute fractions with like and unlike denominators? How can models help us understand the addition and subtraction of decimals? How many ways can we use models to determine and compare equivalent fractions? • • Numerator Whole Numbers • • • • • How would you compare and order whole numbers, fractions and decimals through hundredths? How are common and decimal fractions alike and different? What strategies can be used to solve estimation problems with common and decimal fractions? How do I identify the whole? How do I use concrete materials and drawings to understand and show understanding of fractions (from 1/12ths to 1/2)? Formative Assessment Strategies • • • • Fill In Your Thoughts -‐ Written check for understanding strategy where students fill the blank. (Another term for rate of change is ____ or ____.) Circle, Triangle, Square -‐ Something that is still going around in your head (Triangle) Something pointed that stood out in your mind (Square) Something that “Squared” or agreed with your thinking. ABCD Whisper -‐ Students should get in groups of four where one student is A, the next is B, etc. Each student will be asked to reflect on a concept and draw a visual of his/her interpretation. Then they will share their answer with each other in a zigzag pattern within their group. Onion Ring -‐ Students form an inner and outer circle facing a partner. The teacher asks a question and the students are given time to respond to their partner. Next, the inner circle rotates one person to the left. The teacher asks another question and the cycle repeats itself. Unit 12: Solving problems involving area Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: The focus of this unit is to use area as a context to further develop multiplicative thinking. In this work, students bridge between concrete and abstract thinking, and use strategies to solve problems. This includes solving problems involving rectangular areas by multiplying side lengths and solving for an unknown number in related multiplication and division equations. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org 0perations and Algebraic Thinking -‐ 3.0A 3.OA.4.1 Determine the unknown number to make a www.youtube.com A. Represent and solve problems involving division equation true with both factors that are ≤ multiplication and division. 5. (MS) Math Fact Fluency Practice 4. Determine the unknown whole number in a 3.OA.4.2 Determine the unknown number to make a www.mathwire.com multiplication or division equation relating division equation true with one of the factors is ≤ 5. www.oswego.org/ocsd-‐ three whole numbers. For example, (MS) web/games/ determine the unknown number that makes 3.OA.4.3 Determine the unknown number to make a http://mathfactspro.com/mathflu division equation true. (MS) encygame.html#/math-‐facts-‐ the equation true in each of the equations 3.OA.4.4 Determine the unknown number to make a addition-‐games 8 x ? = 48, 5 = D ÷ 3, 6 x 6 = ?. multiplication equation true with both factors that http://jerome.northbranfordscho are ≤ 5. (MS) ols.org/Content/Math_Fact_Flue 3.OA.4.5 Determine the unknown number to make a ncy_Practice_Sheets.asp multiplication equation true with one of the factors http://www.mathfactcafe.com/ is ≤ 5. (MS) www.factmonster.com 3.OA.4.6 Determine the unknown number to make a multiplication equation true. (MS) Lessons/Activities/Games https://www.illustrativemathema tics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade www.dpi.state.nc.us B. Understand properties of multiplication and the 3.OA.5.1 Apply the properties to multiply 2 or more 3.0A.B.5 introduces the factors using different strategies. relationship between multiplication and division. associative property explicitly for http://harcourtschool.com/searc 5. Apply properties of operations as strategies to 3.OA.5.2 Decompose an expression to represent the the first time. This property is 2 distributive property. h/search.html multiply and divide. Examples: If 6 x 4 = 24 is fundamental for developing known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.) 2 NOTE: Students need not use formal terms for these properties. Measurement and Data -‐ 3.MD C. Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 7. Relate area to the operations of multiplication and addition. b. Multiply side lengths to find areas of rectangles with whole-‐ number side lengths in the context of solving real world and mathematical problems, and represent whole-‐number products as rectangular areas in mathematical reasoning. 3.OA.5.3 Justify the correctness of a problem based on the use of the properties (commutative, associative, distributive). 3.OA.5.4 Use properties of operations to construct and communicate a written response based on explanation/reasoning. 3.OA.5.5 Use properties of operations to clearly construct and communicate a complete written response. 3.MD.7b.1 Solve word problems using the formula (b x h) or (l x w). (real world objects) 3.MD.7b.2 Relate product and factors with area and sides of a rectangle. 2. Reason abstractly and quantitatively. 6. Attend to precision. 8. Look for and express regularity in repeated reasoning. Vocabulary Area Equation Expression Factors Multiplication Product In unit 9, students used various strategies to solve area problems. In 3.MD.C.7b students recognize that they can find area in real-‐ world situations by multiplying side lengths-‐without necessarily using a rectangular array. Students move in and out of context to solve these types of problems (MP.2) and use their repeated experience with area models to recognize that area problems can be solved using multiplication (MP.8). Students also explain precisely how an array corresponds to an expression (MP.6). Common Core State Standards for Mathematical Practice • • • • • • higher-‐level computation strategies. www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Essential Questions • • • • • What are the mathematical properties that govern addition and multiplication? How would you use them? How do you know if a number is divisible by 2, 3, 5, and 10? How can multiples be used to solve problems? What strategies aid in mastering multiplication and division facts? How can numbers be broken down into its smallest factors? • • • • Properties Unknown Variable Word Problems • • • • • How do you use weight and measurement in your life? What tools and units are used to measure the attributes of an object? How are the units of measure within a standard system related? How do you decide which unit of measurement to use? How can you apply these skills and concepts in everyday life? Formative Assessment Strategies • • • • Numbered Heads Together -‐ Students sit in groups and each group member is given a number. The teacher poses a problem and all four students discuss. The teacher calls a number and that student is responsible for sharing for the group. Gallery Walk -‐ After teams have generated ideas on a topic using a piece of chart paper, they appoint a person to stay with their work. Teams rotate around examining other team’s ideas and ask questions of the person left at the paper. Teams then meet together to discuss and add to their information so the person there also can learn from other teams. Graffiti – Groups receive a large piece of paper and felt pens of different colors. Students generate ideas in the form of graffiti. Groups can move to other papers and discuss/add to the ideas. One Question and One Comment -‐Students are assigned a chapter or passage to read and create one question and one comment generated from the reading. In class, students will meet in either small or whole class groups for discussion. Each student shares at least one comment or question. As the discussion moves student by student around the room, the next person can answer a previous question posed by another student, respond to a comment, or share their own comments and questions. As the activity builds around the room, the conversation becomes in-‐depth with opportunity for all students to learn new perspectives on the text. Unit 13: Solving problems involving shapes Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: The focus of this unit is reasoning with shapes and their attributes, including area and perimeter. The standards in this unit strongly support one another because perimeter, like area, is an attribute of shape. Prior work with area and perimeter allows students differentiate between the two measures in this unit. Videos Common Core State Standards for www.khanacademy.org Mathematical Content 3.MD.8.1 C alculate t he l ength o f t he s ides w hen g iven www.teachingchannel.org 3.MD.D.8 is addressed in full in Measurement and Data -‐ 3.MD the p erimeter o f a n o bject. www.youtube.com this unit and focuses on D. Geometric measurement: recognize perimeter 3.MD.8.2 C alculate t he p erimeter o f a p olygon w hen distinguishing between linear as an attribute of plane figures and distinguish given t he s ide l engths. Math Fact Fluency Practice and area measures and between linear and area measures. 3.MD.8.3 S olve m athematical p roblems i nvolving www.mathwire.com examining their relationship. 8. Solve real world and mathematical rectangles with equal area and different perimeter. www.oswego.org/ocsd-‐ problems involving perimeters of polygons, 3.MD.8.4 Solve mathematical problems involving web/games/ including finding the perimeter given the rectangles with equal perimeter and different area. http://mathfactspro.com/mathflu side lengths, finding an unknown side length, 3.MD.8.7 Distinguish between the area and the encygame.html#/math-‐facts-‐ and exhibiting rectangles with the same perimeter. addition-‐games perimeter and different areas or with the 3.MD.8.8 Relate perimeter and area to the real world. http://jerome.northbranfordscho same area and different perimeters. ols.org/Content/Math_Fact_Flue ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com 3.G.1.1 D efine s hapes a ccording t o t heir a ttributes. Lessons/Activities/Games Geometry -‐ 3.G 3.G.1.2 Compare and contrast quadrilaterals based https://www.illustrativemathema A. Reason with shapes and their attributes. on their attributes. tics.org/3 1. Understand that shapes in different 3 G.1.3 Sort geometric figures to identify rhombuses, categories (e.g., rhombuses, rectangles, rectangles, trapezoids, and squares as https://learnzillion.com/ and others) may share attributes (e.g., quadrilaterals. having four sides), and that the shared 3.G.1.4 Draw examples of quadrilaterals that are NOT www.AECSD3rdGradeMathemati attributes can define a larger category squares, rhombuses, or rectangles. csdoc (e.g., quadrilaterals). Recognize r hombuses, rectangles, and squares as http://maccss.ncdpi.wikispaces.n examples of quadrilaterals, and draw et/Third+Grade examples of quadrilaterals that do not belong to any of these subcategories. www.dpi.state.nc.us Common Core State Standards for Students look for and make use of http://harcourtschool.com/searc Mathematical Practice structure (MP.7) as they determine h/search.html 1. Make sense of problems and persevere in solving categories and subcategories of them. shapes by identifying and reasoning 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. about their attributes. Students make conjectures involving the attributes and measures of shapes and analyze various ways of approaching problems (MP.1, MP.3) Essential Questions Vocabulary • • • • • • • Attributes Parallelogram Quadrilaterals Rectangle Rhombus Shapes Square www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ • • • • • Do rectangles with the same area always have the same perimeter? Do rectangles with the same perimeter always have the same area? How would you explain the process for finding the area of a rectangle? What attributes do all quadrilaterals share? Given the perimeter and the length of one side of a rectangle, how can you determine the length of the other side? Formative Assessment Strategies • Summaries and Reflections -‐ Students stop and reflect, make sense of what they have heard or read, derive personal meaning from their learning experiences, and/or increase their metacognitive skills. These require that students use content-‐specific language. • Lists, Charts, and Graphic Organizers -‐ Students will organize information, make connections, and note relationships through the use of various graphic organizers. • Visual Representations of Information -‐ Students will use both words and pictures to make connections and increase memory, facilitating retrieval of information later on. This “dual coding” helps teachers address classroom diversity, preferences in learning style, and different ways of “knowing.” • Collaborative Activities -‐ Students have the opportunity to move and/or communicate with others as they develop and demonstrate their understanding of concepts. • Do’s and Don’ts -‐ List 3 Dos and 3 Don’ts when using/applying/relating to the content (e.g., 3 Dos and Don’ts for solving an equation). Example of Student Response: When adding fractions, DO find a common denominator, DO add the numerators once you’ve found a common denominators, DON’T simply add the denominators • Three Most Common Misunderstandings -‐ List what you think might be the three most common misunderstandings of a given topic based on an audience of your peers. Example of Student Response: In analyzing tone, most people probably confuse mood and tone, forget to look beyond the diction to the subtext as well, and to strongly consider the intended audience. • Yes/No Chart -‐ List what you do and don’t understand about a given topic—what you do on the left, what you don’t on the right; overly-‐vague responses don’t count. Specificity matters! Unit 14: Using multiplication and division to solve measurement problems Suggested number of days: 10 Learning Targets Notes/Comments Unit Materials and Resources Unit Overview: This unit extends students' work in unit 6 to include multiplication and division to solve problems involving measurement quantities. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking -‐ 3.0A A. Represent and solve problems involving multiplication and division. 3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 3.OA.3.1 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.2 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using drawings. 3.OA.3.3 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. 3.OA.3.4 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using equations. 3.OA.3.5 Explain that an unknown number is represented with a symbol/variable. 3.OA.3.6 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.7 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using drawings. 3.OA.3.8 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. (MS) 3.OA.3.9 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using equations. 3.0A.A.3 includes the use of all of the problem types Table 2 in ((SSM except for multiplicative compare problems-‐which will be 13 introduced in Grade 4. Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐ web/games/ http://mathfactspro.com/mathflu encygame.html#/math-‐facts-‐ addition-‐games http://jerome.northbranfordscho ols.org/Content/Math_Fact_Flue ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathema tics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathemati csdoc http://maccss.ncdpi.wikispaces.n et/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/searc h/search.html Measurement and Data -‐ 3.MD A. Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 2. Measure and estimate liquid volumes and masses of objects using standard units of 6 grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-‐step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the 7 problem. 6 NOTE: Excludes compound units such as cm3 and finding the geometric volume of a 7 container. Excludes multiplicative comparison problems (problems involving notions of "times as much"; see Glossary, Table 2). 3.OA.3.10 Divide an area by side length to find the unknown side length. 3.MD.2.1 Measure liquid volume in metric units (liters). 3.MD.2.2 Measure mass in metric units (kilograms, grams). 3.MD.2.3 Estimate liquid volume using metric units (liters). 3.MD.2.4 Estimate mass in metric units (kilograms, grams). 3.MD.2.5 Use the appropriate unit to measure the mass of objects. 3.MD.2.6 Use the appropriate unit to measure the liquid volume of objects. 3.MD.2.7 Use the four basic operations to solve one step word problems with mass. 3.MD.2.8 Use the four basic operations to solve one step word problems with liquid volume. 3.MD.2.9 Use the four basic operations to solve two step word problems with mass. 3.MD.2.10 Use the four basic operations to solve two step word problems with liquid volume. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. Vocabulary • • • • • • • Array Equation Factors Liquid Volume Mass Measure Measurement • • • • • • • Students use strategies for multiplication and division to conceptualize and solve measurement problems (MP.1, MP.2). Students select appropriate tools and justify their selection for measuring different quantities (MP.5). www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Essential Questions How can you find the total number of objects in equal groups? What are arrays, and how do they show multiplication? How can you use multiplication to compare? How can you write a story to describe a multiplication fact? How do you write a good mathematical explanation? How do you measure an object in inches? How do you measure to a fraction of an inch? • • • Metric System Quantity Volume • • How can you estimate and measure length? How can you estimate and measure capacity? Formative Assessment Strategies • • • • Anecdotal Note Cards -‐ The teacher can create a file folder with 5" x 7" note cards for each student for helpful tips and hints to guide students to remembering a process or procedure. Labels or Sticky Notes -‐Teachers can carry a clipboard with a sheet of labels or a pad of sticky notes and make observations as they circulate throughout the classroom. After the class, the labels or sticky notes can be placed in the observation notebook in the appropriate student's section and use the data collected to adjust instruction to meet student needs. Questioning -‐ Asking questions that give students opportunity for deeper thinking and provide teachers with insight into the degree and depth of student understanding. Questions should go beyond the typical factual questions requiring recall of facts or numbers. Discussion -‐ Teacher presents students with an open-‐ended question that build knowledge and develop critical and creative thinking skills. The teacher can assess student understanding by listening to responses and taking anecdotal notes. Unit 15: Demonstrating computational fluency Learning Targets in problem solving. Suggested number of days: 10 Notes/Comments Unit Materials and Resources Unit Overview: This is a culminating unit in which students focus on problem solving in order to demonstrate fluency with addition and subtraction to 1000 and demonstrate fluency for multiplication and division within 100. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Operations and Algebraic Thinking -‐ 3.OA 3.OA.7.1 Fluently (accurately and quickly) divide www.youtube.com C. Multiply and divide within 100. with a dividend up to 100. 7. Fluently multiply and divide within 100, 3.OA.7.2 Fluently (accurately and quickly) multiply Math Fact Fluency Practice using strategies such as the relationship numbers 0-‐10. www.mathwire.com between multiplication and division (e.g., 3.OA.7.3 Memorize and recall my multiples from 0-‐ www.oswego.org/ocsd-‐ knowing that 8 x 5 = 40, one knows 40 x 5 = 9. web/games/ 3.OA.7.4 Recognize the relationship between http://mathfactspro.com/mathflu 8) or properties of operations. By the end multiplication and division. encygame.html#/math-‐facts-‐ of Grade 3, know from memory all products addition-‐games of two one-‐digit numbers. http://jerome.northbranfordscho ols.org/Content/Math_Fact_Flue ncy_Practice_Sheets.asp 3.OA.D.8 was introduced in unit 8 D. Solve problems involving the four operations, 3.OA.8.1 Construct an equation with a letter http://www.mathfactcafe.com/ and is finalized in this unit to include www.factmonster.com and identify and explain patterns in arithmetic. (variable) to represent the unknown quantity. the use of letters to represent 8. Solve two-‐step word problems using the four 3.OA.8.2 Explain or demonstrate how to solve two-‐ operations. Represent these problems using unknown quantities in equations. step word problems using addition and subtraction Lessons/Activities/Games equations with a letter standing for the unknown 3.OA.8.3 Explain or demonstrate how to solve two-‐ https://www.illustrativemathema quantity. Assess the reasonableness of answers step word problems using multiplication and division tics.org/3 using mental computation and estimation (Of single digit factors and products less than 100). 3 strategies including rounding. 3.OA.8.4 Represent a word problem with an https://learnzillion.com/ 3 NOTE: This standard is limited to problems equation using a letter to represent the unknown posed with whole numbers and having whole-‐ quantity. www.AECSD3rdGradeMathemati number answers; students should know how to 3.OA.8.5 Solve two-‐step word problems which csdoc include multiple operations. http://maccss.ncdpi.wikispaces.n perform operations in the conventional order 3.OA.8.6 Use mental math to estimate the answer et/Third+Grade when there are no parentheses to specify a of a single step word problem. (MS) particular order (Order of Operations). 3.OA.8.7 Use mental math to estimate the answer www.dpi.state.nc.us of a two-‐step word problem. 3.OA.8.8 Justify my answers using mental math and http://harcourtschool.com/searc estimation. (MS) h/search.html Number and Operations in Base Ten -‐ 3.NBT A. Use place value understanding and properties 4 of operations to perform multi-‐digit arithmetic. 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. 4 NOTE: A range of algorithms may be used. 3.NBT.2.1 Identify and apply the properties of addition to solve problems. 3.NBT.2.2 Identify and apply the properties of subtraction to solve problems. 3.NBT.2.3 Check a subtraction problem using addition. 3.NBT.2.4 Check an addition problem using subtraction. 3.NBT.2.5 Correctly align digits according to place value, in order to add or subtract. 3.NBT.2.6 Explain and demonstrate the process of regrouping. 3.NBT.2.7 Fluently add two 2-‐digit numbers. (horizontal and vertical set up) 3.NBT.2.8 Fluently add two 3-‐digit numbers. (horizontal and vertical set up) 3.NBT.2.9 Fluently subtract two 2-‐digit numbers with and without regrouping. (horizontal and vertical set up) 3.NBT.2.10 Fluently subtract two 3-‐digit numbers with and without regrouping. (horizontal and vertical set up) Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 8. Look for and express regularity in repeated reasoning. Vocabulary • • • • • • • Addend Area Arithmetic patterns Array Decompose Division Factor • • • • Students demonstrate fluency in multiplication and division within 100 using various strategies and the properties of these operations (MP.5). They also represent these calculations and problem situations abstractly using letters (MP.2). www.tucerton.k12.nj.us/tes_curri culum/mathematics_2/curriculu m-‐math-‐grade-‐three http://www.onlinemathlearning.c om/common-‐core-‐grade3.html http://www.mathgoodies.com/st andards/alignments/grade3.html http://www.k-‐ 5mathteachingresources.com/3r d-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ Essential Questions What clue words help you identify which operation to use to solve word problems? Why is the use of estimation and/or rounding important in determining if your answer is reasonable? How can you solve a three-‐digit plus a two-‐ digit addition problem in two different ways? What number patterns do you notice in the addition table? Why do these patterns make mathematical sense? • • • • • • • • • • • • • Improvised Units Inverse operations Line Plot Measurement Mental Computation Place Value Scale Scaled bar graph Scaled picture graph Unknown quantity Variable Whole Numbers Word form • • • • • • • • Given a two-‐step word problem, what equation could represent it? How can multiplication help you solve division problems? What strategies can be used to find products and/or quotients? How can you use the array model to help you solve multiplication problems? What number sentences could be used to solve this problem? How can simpler multiplication facts help you solve a more difficult fact? How do you know that your equation accurately represents this word problem? How do you know your answer is reasonable? Formative Assessment Strategies • • • • Visual Representations/Drawings -‐ Graphic organizers can be used as visual representations of concepts in the content areas. Many of the graphic organizers contain a section where the student is expected to illustrate his/her idea of the concept. The Mind Map -‐ requires that students use drawings, photos or pictures from a magazine to represent a specific concept. Think/Pair/Share for Math Problem Solving -‐ Place problem on the board. Ask students to think about the steps they would use to solve the problem, but do not let them figure out the actual answer. Without telling the answer to the problem, have students discuss their strategies for solving the problem. Then let them work out the problem individually and then compare answers. Math Center Fun-‐ Practicing how to read large numbers, learning how to round numbers to various places, reviewing place value, solving word problems (as described above), recalling basic geometric terms, discussing the steps of division, discussing how to rename a fraction to lowest terms. THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #1 Developing understanding of multiplication and division and strategies for multiplication and division within 100 Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division. Operations and Algebraic Thinking 3.OA Represent and solve problems involving multiplication and division. 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. 2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 ×? = 48, 5 = ÷ 3, 6 × 6 = ?. Understand properties of multiplication and the relationship between multiplication and division. 5. 6. Apply properties of operations as strategies to multiply and divide. 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.) Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Multiply and divide within 100. 7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 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. 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. Key: Major Clusters; Supporting Clusters; Additional Clusters THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF F OCUS #1, CONTINUED Number and Operations in Base Ten 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 3. Multiply one-digit whole numbers b y multiples of 10 in the range 10 –90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. Measurement and Data 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 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. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Key: Major Clusters; Supporting Clusters; Additional Clusters THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF F OCUS #2 Developing understanding of fractions, especially unit fractions (fractions with numerator 1) 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. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. 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. Number and Operations—Fractions 3.NF Develop understanding of fractions as numbers. 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. 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. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Measurement and Data 3.MD Represent and interpret data. 4. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters. Key: Major Clusters; Supporting Clusters; Additional Clusters THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF F OCUS #3 Developing understanding of the structure of rectangular arrays and of area Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape 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. Measurement and Data 3.MD Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 5. Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 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. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 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. Geometry 3.G Reason with shapes and their attributes. 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. Key: Major Clusters; Supporting Clusters; Additional Clusters THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF F OCUS #4 Describing and analyzing two-dimensional shapes Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole. Geometry 3.G Reason with shapes and their attributes. 1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Number and Operations—Fractions 3.NF Develop understanding of fractions as numbers. 1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Key: Major Clusters; Supporting Clusters; Additional Clusters THIRD GRADE CRITICAL AREAS OF FOCUS S TANDARDS AND C LUSTERS B EYOND THE C RITICAL AREAS OF F OCUS Solving multi-step problems Students apply previous understanding of addition and subtraction strategies and algorithms to solve multi-step problems. They reason abstractly and quantitatively by modeling problem situations with equations or graphs, assessing their processes and results, and justifying their answers through mental computation and estimation strategies. Students incorporate multiplication and division within 100 to solve multi-step problems with the four operations. Operations and Algebraic Thinking 3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. (Previously listed in Critical Area of Focus 1 but relates to the following.) 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. Number and Operations in Base Ten 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 1. Use place value understanding to round whole numbers to the nearest 10 or 100. 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. Measurement and Data 3.MD Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. 2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. Represent and interpret data. 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. Key: Major Clusters; Supporting Clusters; Additional Clusters Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Products and Quotients 3.OA.1 3.OA .2 3.OA .4 3.OA .6 Understands and interprets products and quotients of whole numbers. Understands and interprets products and quotients of whole numbers. Interprets products and quotients of whole numbers. Interprets products and quotients of whole numbers. Determines the unknown whole number in a multiplication or division problem by relating multiplication and division. Factors are greater than 5 and less than 10. Determines the unknown whole number in a multiplication or division problem by relating multiplication and division. Factors are greater than 5 and less than 10. Determines the unknown whole number in a multiplication or division problem by relating multiplication and division. One factor is less than or equal to 5. Determines the unknown whole number in a multiplication or division problem by relating multiplication and division. Limit to factors less than or equal to 5. Uses multiplication and division within 100 to solve word problems involving equal groups, arrays, area, and measurement quantities other than area. Factors are greater than 5 and less than 10. Uses multiplication and division within 100 to solve word problems involving equal groups and arrays. Both factors are less than or equal to 10. Given a visual aid, uses multiplication and division within 100 to solve word problems involving equal groups and arrays. Both factors are less than or equal to 10. Represents the multiplication or division situation as an equation. Multiplication and Division 3.OA.3-1 3.OA.3-2 3.OA.3-3 3.OA.3-4 Uses multiplication and division within 100 to solve word problems involving equal groups, arrays, area, and measurement quantities other than area. Factors are greater than 5 and less than 10. Identifies proper context given a numerical expression involving July 2013 Page 1 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command multiplication and division. Two-Step Problems 3.OA.8-1 3.Int.1 3.Int.2 Fraction Equivalence 3.NF.3a-1 3.NF.3a-2 3.NF.3b-1 3.NF-3c 3.NF-3d 3.NF.A.Int.1 Solves two-step unscaffolded word problems using the four operations, including rounding where appropriate, in which the unknown is in a variety of positions. Both values for each operation performed are substantial (towards the upper limits as defined by the standard assessed). Solves two-step unscaffolded word problems using the four operations, including rounding where appropriate, in which the unknown is in a variety of positions. One of the values for each operation performed is substantial (towards the upper limits as defined by the standard assessed). Solves two-step scaffolded word problems using the four operations, including rounding where appropriate, in which the unknown is in a variety of positions. One of the values for each operation performed is substantial (towards the upper limits as defined by the standard assessed). Solves two-step scaffolded word problems using the four operations and in which the sum, difference, product or quotient is always the unknown. One of the values for each operation performed is substantial (towards the upper limits as defined by the standard assessed). Understands, recognizes and generates equivalent fractions using denominators of 2, 3, 4, 6 and 8. Understands, recognizes and generates equivalent fractions using denominators of 2, 3, 4, 6 and 8. Understands, recognizes and generates equivalent fractions using denominators of 2, 4 and 8. Given visual models, understands, recognizes and generates equivalent fractions using denominators of 2, 4 and 8. Expresses whole numbers as fractions and recognize fractions that are equivalent to whole numbers. Expresses whole numbers as fractions and recognize fractions that are equivalent to whole numbers. Expresses whole numbers as fractions. Expresses whole numbers as fractions. Compares two fractions that Compares two fractions that Compares two fractions that Compares two fractions that July 2013 Page 2 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Fractions as Numbers 3.NF.1 3.NF.2 3.NF.A.Int.1 July 2013 have the same numerator or same denominator using symbols to justify conclusions. Plots the location of equivalent fractions on a number line. The student must recognize that two fractions must refer to the same whole in order to compare. have the same numerator or same denominator using symbols to justify conclusions. Plots the location of equivalent fractions on a number line. The student must recognize that two fractions must refer to the same whole in order to compare. have the same numerator or same denominator using symbols to justify conclusions (e.g., by using a visual fraction model). The student must recognize that two fractions must refer to the same whole in order to compare. have the same numerator or same denominator using symbols. The student must recognize that two fractions must refer to the same whole in order to compare. Given a whole number and two fractions in a real world situation compares the three numbers using symbols. Justifies the comparison by plotting points on a number line. Given a whole number and two fractions in a real world situation plots all three numbers on a number line and determines which fraction is closest to the whole number. Understands 1/b is equal to one whole that is partitioned into b equal parts – limiting the denominators to 2, 3, 4, 6 and 8. Understands 1/b is equal to one whole that is partitioned into b equal parts – limiting the denominators to 2, 3, 4, 6 and 8. Understands 1/b is equal to one whole that is partitioned into b equal parts – limiting the denominators to 2, 4 and 8. Understands 1/b is equal to one whole that is partitioned into b equal parts – limiting the denominators to 2 and 4. Represents 1/b on a Represents 1/b on a Represents 1/b on a Represents 1/b on a Page 3 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Time 3.MD.1-1 3.MD.1-2 July 2013 number line diagram by partitioning the number line between 0-1 into b equal parts recognizing that b is the total number of parts. number line diagram by partitioning the number line between 0-1 into b equal parts recognizing that b is the total number of parts. number line diagram by partitioning the number line between 0-1 into b equal parts recognizing that b is the total number of parts. number line diagram by partitioning the number line between 0-1 into b equal parts recognizing that b is the total number of parts. Demonstrates the understanding of the quantity a/b by marking off a parts of 1/b from 0 on the number line and states that the endpoint locates the number a/b. Demonstrates the understanding of the quantity a/b by marking off a parts of 1/b from 0 on the number line and states that the endpoint locates the number a/b. Demonstrates the understanding of the quantity a/b by marking off a parts of 1/b from 0 on the number line. Represents fractions in the form a/b using a visual model. Applies the concepts of 1/b and a/b in real world situations. Creates the number line that best fits the context. Applies the concepts of 1/b and a/b in real world situations. Tells, writes and measures time to the nearest minute. Tells, writes and measures time to the nearest minute. Tells, writes and measures time to the nearest minute. Tells, writes and measures time to the nearest minute. Creates two-step real world problems involving addition and subtraction of time intervals in minutes. Solves two–step word problems involving addition and subtraction of time intervals in minutes. Solves one-step word problems involving addition or subtraction of time intervals in minutes. Solves one-step word problems involving addition or subtraction of time intervals in minutes, with scaffolding, such as a Page 4 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command number line diagram. Volumes and Masses 3.MD.2-1 3.MD.2-2 3.MD.2-3 3.Int.5 Using grams, kilograms or liters, measures, estimates and solves two-step word problems involving liquid volumes and masses of objects using any of the four basic operations. Number values should be towards the higher end of the acceptable values for each operation. Evaluates usefulness and accuracy of estimations. Using grams, kilograms or liters, measures, estimates and solves one-step word problems involving liquid volumes and masses of objects using any of the four basic operations. Number values should be towards the higher end of the acceptable values for each operation. Geometric Measurement Recognizes area as an attribute of plane figures. Uses estimated measurements to compare answers to one-step word problems. Recognizes area as an attribute of plane figures. 3.MD.5 3.MD.6 3.MD.7b-1 Creates a visual model to show understanding that area is measured using square units and can be found by covering a plane figure without gaps or overlaps by unit squares Understands area is measured using square units. Recognizes that area can be found by covering a plane figure without gaps or overlaps by unit squares and counting them. July 2013 Using grams, kilograms or liters, measures and estimates liquid volumes and masses of objects using any of the four basic operations. Using grams, kilograms or liters, measures and estimates liquid volumes and masses of objects using concrete objects (beakers, measuring cups, scales) to develop estimates. Uses estimated measurements, when indicated, to answer onestep word problems. Recognizes area as an attribute of plane figures. Recognizes area as an attribute of plane figures. With a visual model, understands area is measured using square units. Area can be found by covering a plane figure without gaps or overlaps by unit squares and counting With a visual model, understands area is measured using square units. Area can be found by covering a plane figure without gaps or overlaps by unit squares and counting Page 5 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command and counting them. July 2013 them. Connects counting squares to multiplication when finding area. Connects counting squares to multiplication when finding area. Represents the area of a plane figure as “n” square units. Represents the area of a plane figure as “n” square units. them. Represents the area of a plane figure as “n” square units. Page 6 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Multiply One-Digit Whole Numbers 3.NBT.3 Scaled Graphs 3.MD.3-1 3.MD.3-3 3.Int.4 Measurement Data 3.MD.4 July 2013 Multiplies one-digit whole numbers by multiples of 10 in the range 10-90 using strategies based on place value. Multiplies one-digit whole numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties of operations. Uses repeated addition to multiply one-digit whole numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties of operations. Uses repeated addition to multiply one-digit whole numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties of operations with scaffolding. Completes a scaled picture graph and a scaled bar graph to represent a data set. Completes a scaled picture graph and a scaled bar graph to represent a data set. Completes a scaled picture graph and a scaled bar graph to represent a data set. Completes a scaled picture graph and a scaled bar graph to represent a data set, with scaffolding, such as using a model as a guide. Solves one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. Solves one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. Solves one-step “how many more” and “how many less” problems using information presented in scaled bar graphs. Solves one- step “how many more” and “how many less” problems using information presented in scaled bar graphs. Creates problems that provide a context for information on the graph. Generates measurement data by measuring lengths to the nearest half and fourth inch. Generates measurement data by measuring lengths to the nearest half and fourth inch. Generates measurement data by measuring lengths to the nearest half inch. Generates measurement data by measuring lengths to the nearest half inch. Shows the data by making a Shows the data by making a Shows the data by making a line plot, where the Shows the data by making a line plot, where the Page 7 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command line plot, where the horizontal scale is marked in appropriate units-whole number, halves or quarters. line plot, where the horizontal scale is marked in appropriate units-whole number, halves or quarters. horizontal scale is marked in appropriate units-whole number or halves. horizontal scale is marked in appropriate units-whole number or halves with scaffolding. Understands the properties of quadrilaterals and the subcategories of quadrilaterals. Understands the properties of quadrilaterals and the subcategories of quadrilaterals. Understands the properties of quadrilaterals and the subcategories of quadrilaterals. Identifies examples of quadrilaterals and the subcategories of quadrilaterals. Recognizes and sorts examples of quadrilaterals that have shared attributes and shows that the shared attributes can define a larger category. Recognizes that examples of quadrilaterals that have shared attributes and that the shared attributes can define a larger category. Recognizes that examples of quadrilaterals that have shared attributes and that the shared attributes can define a larger category. Recognizes that examples of quadrilaterals that have shared attributes and that the shared attributes can define a larger category. Draws examples and nonexamples of quadrilaterals with specific attributes. Draws examples and nonexamples of quadrilaterals with specific attributes. Draws examples of quadrilaterals with specific attributes. Solves real-world and mathematical problems involving perimeters of Solves real-world and mathematical problems involving perimeters of Solves mathematical problems involving perimeters of polygons, Uses the line plot to answer questions or solve problems. Understanding Shapes 3.G.1 Perimeter and Area July 2013 Solves mathematical problems involving perimeters of polygons, Page 8 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with connections to the Standards for Mathematical Practice. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command 3.G.2 3.MD.8 3.Int.3 July 2013 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. Number values should be towards the higher end of the acceptable values for each operation. 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. Number values should be towards the higher end of the acceptable values for each operation. including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same area and different perimeters. including finding the perimeter given the side lengths, and exhibiting rectangles with the same area and different perimeters. Partitions shapes in multiple ways into parts with equal areas and expresses the area as a unit fraction of the whole Partitions shapes into parts with equal areas and expresses the area as a unit fraction of the whole. Partitions shapes into parts with equal areas and expresses the area as a unit fraction of the whole. Partitions shapes into parts with equal areas and expresses the area as a unit fraction of the whole limited to halves and quarters. Page 9 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Properties of Operations 3.C.1-1 3.C.1-2 3.C.1-3 3.C.2 Clearly constructs and communicates a complete written response based on explanations/reasoning using the: Clearly constructs and communicates a complete written response based on explanations/reasoning using the: Constructs and communicates a written response based on explanations/reasoning using the: properties of operations relationship between addition and subtraction relationship between multiplication and division identification of arithmetic patterns properties of operations relationship between addition and subtraction relationship between multiplication and division identification of arithmetic patterns Response may include: Response may include: Response may include: Response may include: July 2013 properties of operations relationship between addition and subtraction relationship between multiplication and division identification of arithmetic patterns Constructs and communicates an incomplete written response based on explanations/reasoning using the: properties of operations relationship between addition and subtraction relationship between multiplication and division identification of arithmetic patterns a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) an efficient and logical progression of steps a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) a logical progression of steps a logical approach based on a conjecture and/or stated assumptions a logical, but incomplete, progression of steps minor calculation errors some use of grade-level an approach based on a conjecture and/or stated or faulty assumptions an incomplete or illogical progression of steps an intrusive calculation error Page 10 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Concrete Referents and Diagrams July 2013 with appropriate justification precision of calculation correct use of gradelevel vocabulary, symbols and labels justification of a conclusion determination of whether an argument or conclusion is generalizable evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Provides a counter-example where applicable. Clearly constructs and communicates a wellorganized and complete response based on precision of calculation correct use of gradelevel vocabulary, symbols and labels justification of a conclusion evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Clearly constructs and communicates a wellorganized and complete response based on vocabulary, symbols and labels partial justification of a conclusion based on own calculations evaluating the validity of other’s responses, approaches and conclusions. Constructs and communicates a response based on operations using concrete referents such as limited use of gradelevel vocabulary, symbols and labels partial justification of a conclusion based on own calculations Constructs and communicates an incomplete response based on operations using Page 11 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command 3.C.3-1 3.C.3-2 3.C.6-1 3.C.6-2 operations using concrete referents such as diagrams – including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include: operations using concrete referents such as diagrams – including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include: diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include: concrete referents such as diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include: July 2013 a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) an efficient and logical progression of steps with appropriate justification precision of calculation correct use of gradelevel vocabulary, symbols and labels justification of a conclusion a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) a logical progression of steps precision of calculation correct use of gradelevel vocabulary, symbols and labels justification of a conclusion a logical approach based on a conjecture and/or stated assumptions a logical, but incomplete, progression of steps minor calculation errors some use of grade-level vocabulary, symbols and labels partial justification of a conclusion based on own calculations. a conjecture and/or stated or faulty assumptions an incomplete or illogical progression of steps an intrusive calculation error limited use of gradelevel vocabulary, symbols and labels partial justification of a conclusion based on own calculations accepting the validity of other’s responses Page 12 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Distinguish Correct Explanation/ Reasoning from that which is Flawed 3.C.4-1 3.C.4-2 3.C.4-3 3.C.4-4 3.C.4-5 3.C.4-6 3.C.5-1 3.C.5-2 July 2013 determination of whether an argument or conclusion is generalizable evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning, and providing a counterexample where applicable. evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning. evaluating the validity of other’s responses, approaches and conclusions Clearly constructs and communicates a wellorganized and complete response by: Clearly constructs and communicates a wellorganized and complete response by: Constructs and communicates a complete response by: Constructs and communicates an incomplete response by: presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately evaluating explanation/reasoning; presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately distinguishing correct explanation/reasoning presenting solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately distinguishing correct explanation/reasoning from that which is presenting solutions to scaffolded two-step problems in the form of valid chains of reasoning, sometimes using symbols such as equal signs appropriately distinguishing correct Page 13 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command 3.C.7 if there is a flaw in the argument presenting and defending corrected reasoning flawed identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems presenting corrected reasoning explanation/reasoning from that which is flawed identifying an error in reasoning Response may include: Response may include: Response may include: Response may include: July 2013 from that which is flawed identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems presenting corrected reasoning a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) an efficient and logical progression of steps with appropriate justification precision of calculation correct use of gradelevel vocabulary, symbols and labels a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate) a logical progression of steps precision of calculation correct use of gradelevel vocabulary, symbols and labels justification of a conclusion a logical approach based on a conjecture and/or stated assumptions a logical, but incomplete, progression of steps minor calculation errors some use of grade-level vocabulary, symbols and labels partial justification of a conclusion based on own calculations a conjecture based on faulty assumptions an incomplete or illogical progression of steps an intrusive calculation error limited use of gradelevel vocabulary, symbols and labels partial justification of a conclusion based on own calculations accepting the validity of other’s responses. Page 14 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command July 2013 justification of a conclusion evaluation of whether an argument or conclusion is generalizable evaluating, interpreting, and critiquing the validity of other’s responses, approaches and reasoning, and providing a counterexample where applicable. evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning. evaluating the validity of other’s responses, approaches and conclusions. Page 15 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly and quantitatively, using appropriate tools strategically, looking for the making use of structure, and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Modeling 3.D.1 3.D.2 Devises a plan and applies mathematics to solve multistep, real-world contextual word problems by: Devises a plan and applies mathematics to solve multistep, real-world contextual word problems by: Devises a plan and applies mathematics to solve multistep, real-world contextual word problems by: Devises a plan and applies mathematics to solve multistep, real-world contextual word problems by: July 2013 using stated assumptions or making assumptions and using approximations to simplify a real-world situation analyzing and/or creating constraints, relationships and goals mapping relationships between important quantities by selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw conclusions using stated assumptions or making assumptions and using approximations to simplify a real-world situation mapping relationships between important quantities by selecting appropriate tools to create models analyzing relationships mathematically between important quantities to draw conclusions interpreting mathematical results in the context of the using stated assumptions and approximations to simplify a real-world situation illustrating relationships between important quantities by using provided tools to create models analyzing relationships mathematically between important quantities to draw conclusions interpreting mathematical results in a simplified context reflecting on whether using stated assumptions and approximations to simplify a real-world situation identifying important quantities by using provided tools to create models analyzing relationships mathematically to draw conclusions writing an arithmetic expression or equation to describe a situation Page 16 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly and quantitatively, using appropriate tools strategically, looking for the making use of structure, and/or looking for and expressing regularity in repeated reasoning. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command July 2013 justifying and defending models which lead to a conclusion interpreting mathematical results in the context of the situation reflecting on whether the results make sense improving the model if it has not served its purpose writing a concise arithmetic expression or equation to describe a situation situation reflecting on whether the results make sense modifying and/or improving the model if it has not served its purpose writing an arithmetic expression or equation to describe a situation the results make sense modifying the model if it has not served its purpose writing an arithmetic expression or equation to describe a situation Page 17 of 18 Performance Level Descriptors – Grade 3 Mathematics Grade 3 Math: Sub-Claim E The student demonstrates fluency in areas set forth in the Standards for Content in grades 3-6. Level 5: Distinguished Level 3: Moderate Level 4: Strong Command Level 2: Partial Command Command Command Fluency 3.NBT.2 3.OA.7 July 2013 Accurately and quickly adds and subtracts within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Accurately in a timely manner adds and subtracts within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Accurately adds and subtracts within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Adds and subtracts within 1000, using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Correctly calculates 100 percent of sums and differences in less than the allotted time on items which are timed. Correctly calculates 100 percent of sums and differences in the allotted time on items which are timed. Correctly calculates more than 75 percent and less than 100 percent of sums and differences of items which are timed. Correctly calculates at least 75 percent of the sums and differences of items which are timed. Accurately and quickly multiplies and divides within 100, using strategies relating multiplication and division or properties of operations. Accurately in a timely manner multiplies and divides within 100, using strategies relating multiplication and division or properties of operations. Accurately multiplies and divides within 100, using strategies relating multiplication and division or properties of operations. Multiplies and divides within 100, using strategies relating multiplication and division or properties of operations. Knows from memory 100 percent of the multiplication and division facts within 100 in less than the allotted time on items which are timed. Knows from memory 100 percent of the multiplication and division facts within 100 in the allotted time on items which are timed. Knows from memory more than 80 percent and less than 100 percent of the multiplication and division facts within 100 on items which are timed. Knows from memory greater than or equal to 70 percent and less than or equal to 80 percent of the multiplication and division facts within 100 on items which are timed. Page 18 of 18 Bailey●Kirkland Education Group, LLC Common Core State Standard I Can Statements 3rd Grade Mathematics CCSS Key: Operations and Algebraic Thinking (OA) Number and Operations in Base Ten (NBT) Numbers and Operations–Fractions (NF) Measurement and Data (MD) Geometry (G) PLD Key: Partial Command Moderate Command Distinguished Command Common Core State Standards for Mathematics (Outcome Based) I Can Statements Operations and Algebraic Thinking (OA) 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. I Can: 3.OA.2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. I Can: 3.OA.3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown I Can: 3.OA.1.1 Represent a situation in which a number of groups can be expressed using multiplication. (MS) 3.OA.1.2 Identify a situation in which a number of groups can be expressed using multiplication. (MS) 3.OA.1.3 Draw an array. (MS) 3.OA.1.4 Explain an array. (MS) 3.OA.1.5 Find the product using objects in groups. 3.OA.1.6 Find the product using objects in arrays. 3.OA.1.7 Find the product using objects in area models. 3.OA.1.8 Find the product using measurement quantities. 3.OA.1.9 Explain the objects in equal size groups. (MS) 3.OA.2.1 Partition a whole number into equal shares using arrays. (MS) 3.OA.2.2 Partition a whole number into equal parts using area. 3.OA.2.3 Partition a whole number into equal parts using measurement quantities. 3.OA.2.4 Identify each number in a division expression as a quotient, divisor, and/or dividend. (MS) 3.OA.2.5 Describe a situation in which a number of groups can be expressed using division. (MS) 3.OA.2.6 Identify a situation in which a number of groups can be expressed using division. (MS) 3.OA.3.1 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.2 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 1 Common Core State Standards for Mathematics (Outcome Based) I Can Statements number to represent the problem. measurement quantities other than area using drawings. 3.OA.3.3 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. 3.OA.3.4 Use multiplication (factors ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using equations. 3.OA.3.5 Explain that an unknown number is represented with a symbol/variable. 3.OA.3.6 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using drawings. (MS) 3.OA.3.7 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using drawings. 3.OA.3.8 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve equal groups and arrays using equations. (MS) 3.OA.3.9 Use division (quotient/divisor ≥ 5 and ≤ 10) to solve word problems that involve area and other measurement quantities other than area using equations. 3.OA.3.10 Divide an area by side length to find the unknown side length. 3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ? ÷ 3, 6 × 6 = ?. I Can: 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 I Can: 3.OA.4.1 Determine the unknown number to make a division equation true with both factors that are ≤ 5. (MS) 3.OA.4.2 Determine the unknown number to make a division equation true with one of the factors is ≤ 5. (MS) 3.OA.4.3 Determine the unknown number to make a division equation true. (MS) 3.OA.4.4 Determine the unknown number to make a multiplication equation true with both factors that are ≤ 5. (MS) 3.OA.4.5 Determine the unknown number to make a multiplication equation true with one of the factors is ≤ 5. (MS) 3.OA.4.6 Determine the unknown number to make a multiplication equation true. (MS) 3.OA.5.1 Apply the properties to multiply 2 or more factors using different strategies. 3.OA.5.2 Decompose an expression to represent the distributive property. 3.OA.5.3 Justify the correctness of a problem Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 2 Common Core State Standards for Mathematics (Outcome Based) I Can Statements × 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.) based on the use of the properties (commutative, associative, distributive). 3.OA.5.4 Use properties of operations to construct and communicate a written response based on explanation/reasoning. 3.OA.5.5 Use properties of operations to clearly construct and communicate a complete written response. 3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. I Can: 3.OA.7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers. I Can: 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. I Can: 3.OA.6.1 Interpret division as an unknown factor problem using the fact families. 3.OA.6.2 Interpret division as an unknown factor problem using a bar model. 3.OA.6.3 Interpret division as an unknown factor problem using a number line. 3.OA.6.4 Interpret division as an unknown factor problem using arrays. 3.OA.6.5 Justify the correctness of a problem based on the understanding of division as an unknown factor problem. 3.OA.7.1 Fluently (accurately and quickly) divide with a dividend up to 100. 3.OA.7.2 Fluently (accurately and quickly) multiply numbers 0-10. 3.OA.7.3 Memorize and recall my multiples from 0-9. 3.OA.7.4 Recognize the relationship between multiplication and division. 3.OA.8.1 Construct an equation with a letter (variable) to represent the unknown quantity. 3.OA.8.2 Explain or demonstrate how to solve two-step word problems using addition and subtraction 3.OA.8.3 Explain or demonstrate how to solve two-step word problems using multiplication and division (Of single digit factors and products less than 100). 3.OA.8.4 Represent a word problem with an equation using a letter to represent the unknown quantity. 3.OA.8.5 Solve two-step word problems which include multiple operations. 3.OA.8.6 Use mental math to estimate the answer of a single step word problem. (MS) 3.OA.8.7 Use mental math to estimate the answer of a two-step word problem. 3.OA.8.8 Justify my answers using mental math Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 3 Common Core State Standards for Mathematics (Outcome Based) I Can Statements and estimation. (MS) 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. I Can: 3.OA.9.1 Explain and model the relationship of odd and even number patterns with addition facts. Examples: • Recognize that the sum of two even numbers is even. • Recognize that the sum of two odd numbers is even. • Recognize that the sum of an even and an odd number is odd. 3.OA.9.2 Explain and model the relationship of odd and even number patterns with multiplication facts. • Recognize that if at least 1 factor is even, the product will be even. • Use divisibility rules identify arithmetic patterns. 3.OA.9.3 Use a multiplication table to locate examples of the commutative, identity, and zero properties of multiplication. 3.OA.9.4 Use an addition table to locate examples of the commutative and identity properties of addition. Numbers and Operations–Fractions (NF) 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. I Can: 3.NF.2. I Can: 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 3.NF.1.1 Explain that the fractional pieces get smaller as the denominator gets larger. 3.NF.1.2 Explain that the denominator represents the number of equal parts in the whole. (MS) 3.NF.1.3 Explain that the numerator is a count of the number of equal parts (3/4 means there are three ¼’s; ¾ = ¼ + ¼ + ¼). 3.NF.1.4 Model fractions as parts of a whole or parts of a group. (MS) 3.NF.2a.1 Partition (divide) a number line into equal parts (intervals). 3.NF.2a.2 Identify a given fraction on a number line. 3.NF.2a.3 Represent and recognize a given fraction on a number line. 3.NF.2b.1 Recognize that a fraction a/b represents its distance from 0 on a number line. 3.NF.2b.2 Recognize that a fraction a/b represents Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 4 Common Core State Standards for Mathematics (Outcome Based) 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 3.NF.3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. I Can Statements its location on a number line. I Can: 3.NF.3a/b.1 Recognize and generate equivalent fractions. (Denominators are 2, 3, 4, 6, and 8) 3.NF.3a.2 Compare fractions using a model. 3.NF.3a.3 Compare 2 fractions that have the same numerator or denominator using a number line. 3.NF.3a.4 Plot the location of equivalent fractions on a number line. 3.NF.3c.1 Explain that a fraction with the same numerator and denominator will always equal 1. 3.NF.3c.2 Write a whole number as a fraction. 3.NF.3c.3 Recognize that some fractions are equivalent to whole numbers. 3.NF.3d.1 Compare fractions based on the size of the numerator and denominator. 3.NF.3d.2 Compare and explain two fractions with the same denominator by drawing a visual model (using <,>,=). 3.NF.3d.3 Compare and explain two fractions with the same numerator by drawing a visual model (using <,>,=). Number and Operations in Base Ten (NBT) 3.NBT.1. Use place value understanding to round whole numbers to the nearest 10 or 100. I Can: 3.NBT.2. Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. I Can: 3.NBT.1.1 Explain the process for rounding numbers using place value. 3.NBT.1.2 Identify the place value of the ones, tens, and hundreds place in a whole number. 3.NBT.1.3 Round numbers to the nearest hundred. 3.NBT.1.4 Round numbers to the nearest ten. 3.NBT.2.1 Identify and apply the properties of addition to solve problems. 3.NBT.2.2 Identify and apply the properties of subtraction to solve problems. 3.NBT.2.3 Check a subtraction problem using addition. 3.NBT.2.4 Check an addition problem using subtraction. Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 5 Common Core State Standards for Mathematics (Outcome Based) I Can Statements 3.NBT.2.5 Correctly align digits according to place value, in order to add or subtract. 3.NBT.2.6 Explain and demonstrate the process of regrouping. 3.NBT.2.7 Fluently add two 2-digit numbers. (horizontal and vertical set up) 3.NBT.2.8 Fluently add two 3-digit numbers. (horizontal and vertical set up) 3.NBT.2.9 Fluently subtract two 2-digit numbers with and without regrouping. (horizontal and vertical set up) 3.NBT.2.10 Fluently subtract two 3-digit numbers with and without regrouping. (horizontal and vertical set up) 3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. I Can: 3.NBT.3.1 Correctly align digits according to place value, in order to multiply. 3.NBT.3.2 Explain and demonstrate the process of multiplying a two digit number by a one digit number using various algorithms. 3.NBT.3.3 Multiply 1-digit whole numbers by multiples of 10 in the range of 1-90 using different strategies. Measurement and Data (MD) 3.MD.1. I Can: Tell and write time to the nearest minute, and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. 3.MD.2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. 3.MD.1.1 Explain time intervals. 3.MD.1.2 Identify minute marks on an analog clock. 3.MD.1.3 Identify minute position on a digital clock. 3.MD.1.4 Relate and explain a number line to the minute marks on a clock. 3.MD.1.5 Use a “time” number line to measure and solve addition or subtraction word problems to the nearest minute. 3.MD.1.6 Use a “time” number line to measure and solve two-step addition and subtraction word problems to the nearest minute. 3.MD.1.7 Write time to the nearest minute. I Can: 3.MD.2.1 Measure liquid volume in metric units (liters). 3.MD.2.2 Measure mass in metric units (kilograms, grams). 3.MD.2.3 Estimate liquid volume using metric units (liters). 3.MD.2.4 Estimate mass in metric units (kilograms, grams). 3.MD.2.5 Use the appropriate unit to measure the Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 6 Common Core State Standards for Mathematics (Outcome Based) I Can Statements mass of objects. 3.MD.2.6 Use the appropriate unit to measure the liquid volume of objects. 3.MD.2.7 Use the four basic operations to solve one step word problems with mass. 3.MD.2.8 Use the four basic operations to solve one step word problems with liquid volume. 3.MD.2.9 Use the four basic operations to solve two step word problems with mass. 3.MD.2.10 Use the four basic operations to solve two step word problems with liquid volume. 3.MD.3. I Can: 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. 3.MD.4. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters. I Can: 3.MD.5. Recognize area as an attribute of plane figures, and understand concepts of area measurement. I Can: a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure 3.MD.3.1 Complete a scaled bar graph to represent data. 3.MD.3.2 Complete a scaled picture graph to represent data. 3.MD.3.3 Read and analyze data on horizontal and vertical scaled bar graphs. 3.MD.3.4 Read and analyze data on scaled picture graphs. 3.MD.3.5 Use information from a bar graph to solve 1-step “how many more” and “how many less” problems. 3.MD.3.6 Use information from a bar graph to solve 2-step “how many more” and “how many less” problems. 3.MD.3.7 Create problems/scenarios from information presented on a graph. 3.MD.4.1 Use a ruler to measure an object to the nearest whole, half, and quarter inch. 3.MD.4.2 Collect and organize data to create a line plot (whole numbers, halves, and quarters). 3.MD.4.3 Create a line plot from given or collected data, where the horizontal scale is marked off in appropriate units (whole numbers, halves, and quarters). 3.MD.4.4 Label a line plot to show whole numbers, halves, and quarters. 3.MD.4.5 Use a line plot to answer questions or solve problems. 3.MD.5a.1 Identify what a unit square is and know it can be used to measure area of a figure. Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 7 Common Core State Standards for Mathematics (Outcome Based) I Can Statements area. 3.MD.5b.1 Relate the area to real world objects. 3.MD.5b.2 Recognize area as an attribute of plane figures with a visual model. 3.MD.5b.3 Explain area as an attribute of plane figures. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 3.MD.6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). I Can: 3.MD.7. Relate area to the operations of multiplication and addition. I Can: a. Find the area of a rectangle with wholenumber side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems. 3.MD.6.1 Determine the area of an object by counting the unit squares in cm, m, in., ft., and other units. 3.MD.6.2 Connect counting squares to multiplication when finding area. 3.MD.7a.1 Use tiles to show the area of an rectangle. 3.MD.7a.2 Multiply (b x h) or (l x w) to determine the area of a rectangle. 3.MD.7a.3 Justify that the area of a rectangle will be the same using different methods. (Tiling and formula) 3.MD.7b.1 Solve word problems using the formula (b x h) or (l x w). (real world objects) 3.MD.7b.2 Relate product and factors with area and sides of a rectangle. 3.MD.7c.1 Use tiling to explain the understanding of the distributive property in area problems. 3.MD.7d.1 Explain that a rectilinear figure is composed of smaller rectangles. 3.MD.7d.2 Model and separate a rectilinear figure into 2 different rectangles. 3.MD.7d.3 Determine the area of a figure by separating the figure into smaller rectangles and adding the area of each rectangle together. 3.MD.7d.4 Solve real world problems involving area of irregular shapes. Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 8 Common Core State Standards for Mathematics (Outcome Based) I Can Statements 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. I Can: 3.MD.8.1 Calculate the length of the sides when given the perimeter of an object. 3.MD.8.2 Calculate the perimeter of a polygon when given the side lengths. 3.MD.8.3 Solve mathematical problems involving rectangles with equal area and different perimeter. 3.MD.8.4 Solve mathematical problems involving rectangles with equal perimeter and different area. 3.MD.8.7 Distinguish between the area and the perimeter. 3.MD.8.8 Relate perimeter and area to the real world. Geometry (G) 3.G.1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. 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. I Can: 3.G.1.1 Define shapes according to their attributes. 3.G.1.2 Compare and contrast quadrilaterals based on their attributes. 3 G.1.3 Sort geometric figures to identify rhombuses, rectangles, trapezoids, and squares as quadrilaterals. 3.G.1.4 Draw examples of quadrilaterals that are NOT squares, rhombuses, or rectangles. I Can: 3.G.2.1 Recognize that shapes can be divided into equal parts. 3.G.2.2 Separate a given object into equal parts. 3.G.2.3 Describe the area of each part as a fractional part of the whole. 3.G.2.4 Label each part as a fractional part of the whole. 3.G.2.5 Partition shapes in multiple ways into parts with equal areas and express the area as a unit fraction of the whole. Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 9 Common Core “Shifts” in Mathematics There are six shifts in Mathematics that the Common Core requires of us if we are to be truly aligned with it in terms of curricular materials and classroom instruction. Shift 1 - Focus Teachers use the power of the eraser and significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades. Shift 2 - Coherence Principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication spiral across grade levels and students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. Shift 3 - Fluency Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions (found in the attached list of fluencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts. Shift 4 - Deep Understanding Teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations, as well as writing and speaking about their understanding. Shift 5 – Application Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content. Shift 6 - Dual Intensity Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year. Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). The Standards: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 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. 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, 1 Standards for Mathematical Practice 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 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. 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 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 x 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. 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)(x2 + x + 1), and (x – 1)(x3 + x2 + 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. 2 CCSS Standards for Mathematical Practice Questions for Teachers to Ask 1.Make sense of problems and persevere in solving them Teachers ask: • What is this problem asking? • How would you describe the problem in your own words? • Could you try this with simpler numbers? Fewer numbers? • How could you start this problem? • Would it help to create a diagram? Make a table? Draw a picture? • How is ___’s way of solving the problem like/different from yours? • Does your plan make sense? Why or why not? • What are you having trouble with? • How can you check this? 2. Reason abstractly and quantitatively Teachers ask: • What does the number ____ represent in the problem? • How can you represent the problem with symbols and numbers? • Create a representation of the problem. 3. Construct viable arguments and critique the reasoning of others Teachers ask: • How is your answer different than _____’s? • What do you think about what _____ said? • Do you agree? Why/why not? • How can you prove that your answer is correct? • What examples could prove or disprove your argument? • What do you think about _____’s argument? • Can you explain what _____ is saying? • Can you explain why his/her strategy works? • How is your strategy similar to _____? • What questions do you have for ____? • Can you convince the rest of us that your answer makes sense? 4. Model with mathematics Teachers ask: • Write a number sentence to describe this situation. • How could we use symbols to represent what is happening? • What connections do you see? • Why do the results make sense? • Is this working or do you need to change your model? *It is important that the teacher poses tasks that involve real world situations *It is important that the teacher poses tasks that involve arguments or critiques 5. Use appropriate tools strategically Teachers ask: • How could you use manipulatives or a drawing to show your thinking? • How did that tool help you solve the problem? • If we didn’t have access to that tool, what other one would you have chosen? 6. Attend to precision Teachers ask: • What does the word ____ mean? • Explain what you did to solve the problem. • Can you tell me why that is true? • How did you reach your conclusion? • Compare your answer to _____’s answer • What labels could you use? • How do you know your answer is accurate? • What new words did you use today? How did you use them? 7. Look for and make use of structure Teachers ask: • Why does this happen? • How is ____ related to ____? • Why is this important to the problem? • What do you know about ____ that you can apply to this situation? • How can you use what you know to explain why this works? • What patterns do you see? *deductive reasoning (moving from general to specific) 8. Look for and express regularity in repeated reasoning Teachers ask: • What generalizations can you make? • Can you find a shortcut to solve the problem? How would your shortcut make the problem easier? • How could this problem help you solve another problem? *inductive reasoning (moving from specific to general)
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