3 grade math unit guide

3rd 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

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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 Mathematics (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. Make sense of problems and persevere in solving them. S. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. 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). Common Core State Standards for Mathematical Content

Number and Operations in Base Ten -‐ 3.NBT A. Use place value understanding and properties of operations to perform multi-‐digit arithmetic. 4 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.

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. 3.NBT.2.5 Correctly align digits according to place value, in order to add or subtract.

3.NBT.A.1 introduces the concept of rounding, which is new to students and will be revisited in unit 8 in the context of multiplication.

3.NBT.A.2 will be finalized in unit 15 in order to give students time to reach fluency in addition and subtraction within 1000 by the end of the year.

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/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/search/search.html 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/grade3.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 Essential Questions

• 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

• 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


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. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking -‐ 3.0A A. Represent and solve problems involving multiplication and division. 1. Interpret products of whole numbers, e.g., interpret 5 x 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 x 7.

2. Interpret whole-‐number quotients of whole numbers, e.g., interpret 56 x 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number

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.

In 3.0A.A.1 situations with discreet objects should be explored first when developing a conceptual understanding of multiplication, followed by measurement examples involving area models.

3.0A.A.2 will be readdressed in unit 7 in order to provide students the opportunity to develop computational strategies as they

Videos http://www.youtube.com/watch?v=llnio99_YU8 (3.OA.1) 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/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://learnzillion.com/ www.AECSD3rdGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/search/search.html

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_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.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 http://map.mathshell.org/materials/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 Essential Questions • 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

• 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?


• 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


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. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking -‐ 3.0A B. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. 2 Examples: If 6 x 4 = 24 is 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.) Note:2 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.

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 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.

3.0A.B.5 will be readdressed in unit 9 with a focus on the distributive property and in unit 12 with a focus on the associative property.

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/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/search/search.html

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.

Common Core State Standards for Mathematical Practice

2. Reason abstractly and quantitatively.

6. Attend to precision. 7. Look for and make use of structure.

3.MD.5a.1 Identify what a unit square is and know it can be used to measure area of a figure.

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.

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)

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_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.html http://www.k-‐5mathteachingresources.com/3rd-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/

Vocabulary Essential Questions • 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

• 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?


• 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


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. Common Core State Standards for Mathematical Content Geometry -‐ 3.G A. 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.

Number and Operations-‐Fractions5 -‐ 3.NF A. Develop understanding of fractions as numbers. 1. Understand a fraction 1Ib 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.

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.

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

The focus of 3.NF.A.1 and 3.NF.A.2a in this unit is on fractions between 0 and 1. Fractions greater than 1 will be introduced in unit 5.


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. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

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.

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).

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.html http://www.k-‐5mathteachingresources.com/3rd-‐grade-‐number-‐activities.html http://illuminations.nctm.org

Vocabulary Essential Questions

• Denominator • Diagram • Equal areas • Equal distance • Equal parts • Equivalence • Equivalent

• 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)?


• 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


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. Common Core State Standards for Mathematical Content

Number and Operations-‐FractionsS -‐ 3.NF A. 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.

b. Represent a fraction a/b on a number line diagram by marking off a lengths l/b from O. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Measurement and Data -‐ 3.MD B. Represent and interpret data. 4. Generate measurement data by measuring lengths using rulers marked with halves and

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.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 its location on a number line.

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.NF.A.1 is repeated here to include fractions greater than 1.

Videos http://www.engageny.org/resource/common-‐core-‐video-‐series-‐grade-‐3-‐mathematics-‐inches-‐and-‐centimeters 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/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathematicsdoc http://maccss.ncdpi.wikispaces.n

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 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically.

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.

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).

et/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.html http://www.k-‐5mathteachingresources.co

Vocabulary Essential Questions • Data • Denominator • Diagram • Fourths • Fraction • Halves • Horizontal

• 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?


• 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


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. Common Core State Standards for Mathematical Content Measurement and Data -‐ 3.MD A. 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). 6 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. 7 NOTE: 6 Excludes compound units such as cm3 and finding the geometric volume of a container. 7Excludes multiplicative comparison problems (problems involving notions of "times as much"; see Glossary, Table 2).

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. 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.A.1 is included here as an opportunity to model addition and subtraction situations with time as the context.

3.MD.A.2 is addressed in full in unit 14 to include multiplication and division situations.


Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 4. Model with mathematics.

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.

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).


Vocabulary Essential Questions • Analog clock • Digital clock • Grams • Interval • Kilograms • Liter • Liquid

• 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?


• 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


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. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking -‐ 3.0A A. Represent and solve problems involving multiplication and division. 2. Interpret whole-‐number quotients of whole numbers, e.g., interpret 56 x 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. 1NOTE: 1See Glossary, Table 2.

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)





3.0A.A.2 and 3.0A.C.7 are revisited in this unit to extend the range of numbers to include all numbers within 100 when multiplying and dividing.

3.0A.A.3 includes equal groups, arrays, and area problem types. Note that multiplicative compare problems are introduced in Grade 4 (4.0A.A.2).

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/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/3 https://learnzillion.com/ www.AECSD3rdGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Third+Grade www.dpi.state.nc.us http://harcourtschool.com/searc

B. Understand properties of multiplication and the relationship between multiplication and division. 6. Understand division as an unknown-‐factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

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





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.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.

3.0A.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.

h/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.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 http://map.mathshell.org/materials/stds.php#standard1159

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 Essential Questions • Addition • Array • Bar Model • Division • Divisor • Dividend • Fluent • Multiplication • Number Line • Quotient • Subtraction • Symbol • Variable

• 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?


• 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


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. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking -‐ 3.OA D. 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.3 NOTE: 3This standard is limited to problems posed with whole numbers and having whole-‐ number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For

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 and estimation. (MS)

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

3.OA.D.8 will be revisited in unit 15 to address the use of equations and letters for unknown quantities.


example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Number and Operations in Base Ten -‐ 3.NBT A. Use place value understanding and properties of operations to perform multi-‐digit arithmetic. 4 1. Use place value understanding to round whole numbers to the nearest 10 or 100. 3. Multiply one-‐digit whole numbers by multiples of 10 in the range 10-‐90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations. NOTE: 4A range of algorithms may be used.

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.

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.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.

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.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-‐standard http://map.mathshell.org/materials/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.

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.

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.

Students examine patterns in arithmetic (MP.7) and discuss what they discover (MP.3).

Vocabulary Essential Questions • Addend • Area • Arithmetic patterns • Arrays • Decompose • Division • Factor • Improvised Units • Inverse operations • Line Plot

• 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?


• 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


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. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking -‐ 3.0A B. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. 2 Examples: If 6 x 4 = 24 is 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.) Note: 2Students 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.






3.0A.B.5 will be revisited in unit 12 to address the associative property of multiplication.


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).


Vocabulary Essential Questions • Area • Associative Property of Multiplication • Commutative Property of Multiplication • Decompose • Distributive Property • Factors • Rectangle • Rectilinear figure

• 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?


• 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


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. Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions5 -‐ 3.NF A. Develop understanding of fractions as numbers. 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. NOTE: SGrade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Common Core State Standards for Mathematical Practice 4. Model with mathematics. 6. Attend to precision.

3.NF.3a/b.1 Recognize and generate equivalent fractions. (Denominators are 2, 3, 4, 6, and 8)

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.A.3) The focus of this unit is around equivalence. Although the cluster heading includes comparison of fraction, fraction comparisons (3.NF.A.3d) will be addressed in unit 11. Students develop understanding of equivalence by modeling fractions (MP.4) and communicating their understanding of what it means for


fractions to be equivalent (MP.6). www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐three http://www.onlinemathlearning.com/common-‐core-‐grade3.html http://www.mathgoodies.com/standards/alignments/grade3.html http://www.k-‐5mathteachingresources.com/3rd-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/

Vocabulary Essential Questions • Compare • Denominator • Diagram • Equal • Equivalent • Fractions • Number Line

• 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?


• 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


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.

Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions5 -‐ 3.NF A. Develop understanding of fractions as numbers. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

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. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. S. Use appropriate tools strategically. 7. Look for and make use of structure.

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 <,>,=).

Students will use their understanding of structure (i.e., the role of the numerator and denominator) (MP.7) to reason about relative sizes of fractions (MP.3). Students use various tools to justify their comparisons, paying particular attention to the same-‐sized wholes (MP.5).



Vocabulary Essential Questions • 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)?


• 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


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. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking -‐ 3.0A A. Represent and solve problems involving multiplication and division. 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 x ? = 48, 5 = D ÷ 3, 6 x 6 = ?.

B. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. 2 Examples: If 6 x 4 = 24 is

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)



3.0A.B.5 introduces the associative property explicitly for the first time. This property 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.) NOTE: 2Students 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.

Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively.

6. Attend to precision. 8. Look for and express regularity in repeated reasoning.




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.

higher-‐level computation strategies.

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).


Vocabulary Essential Questions • Area • Equation • Expression • Factors • Multiplication • Product

• 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?


• 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


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. Common Core State Standards for Mathematical Content 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.

Geometry -‐ 3.G A. 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 r hombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.


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.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.

3.MD.D.8 is addressed in full in this unit and focuses on distinguishing between linear and area measures and examining their relationship. Students look for and make use of structure (MP.7) as they determine categories and subcategories of 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)


Vocabulary Essential Questions • Attributes • Parallelogram • Quadrilaterals • Rectangle • Rhombus • Shapes • Square

• 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?


• 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


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.3.9 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 introduced in Grade 4.13


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 grams (g), kilograms (kg), and liters (l). 6 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. 7 NOTE: 6 Excludes compound units such as cm3 and finding the geometric volume of a container. 7Excludes multiplicative comparison problems (problems involving notions of "times as much"; see Glossary, Table 2).


2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically.

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.

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).


Vocabulary Essential Questions • Array • Equation • Factors • Liquid Volume • Mass • Measure • Measurement

• 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?


• 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 in problem solving. Suggested number of days: 10


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. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking -‐ 3.OA 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.

D. 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.3 NOTE: 3This standard is limited to problems posed with whole numbers and having whole-‐ number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

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 and estimation. (MS)

3.OA.D.8 was introduced in unit 8 and is finalized in this unit to include the use of letters to represent unknown quantities in equations.


Number and Operations in Base Ten -‐ 3.NBT A. Use place value understanding and properties of operations to perform multi-‐digit arithmetic. 4 2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. NOTE: 4A range of algorithms may be used. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 8. Look for and express regularity in repeated reasoning.

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)

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).


Vocabulary Essential Questions • Addend • Area • Arithmetic patterns • Array • Decompose • Division • Factor

• 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?


• 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.

Key: Major Clusters; Supporting Clusters; Additional Clusters

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. 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.)

6. 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.


THIRD GRADE CRITICAL AREAS OF FOCUS

CRITICAL AREA OF FOCUS #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 non- overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.


THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #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.


THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #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 non- overlapping 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.


THIRD GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #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.


THIRD GRADE CRITICAL AREAS OF FOCUS STANDARDS AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS 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.

Performance Level Descriptors – Grade 3 Mathematics

July 2013 Page 1 of 18

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

Command Level 4: Strong Command

Level 3: Moderate Command

Level 2: Partial 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. 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. Represents the multiplication or division situation as an equation.

Understands and 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.

Interprets products and quotients of whole numbers. 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.

Interprets products and quotients of whole numbers. 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.

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

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.









multiplication and division.

Two-Step Problems 3.OA.8-1 3.Int.1 3.Int.2

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).

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

Understands, recognizes and generates equivalent fractions using denominators of 2, 3, 4, 6 and 8. Expresses whole numbers as fractions and recognize fractions that are equivalent to whole numbers. Compares two fractions that

Understands, recognizes and generates equivalent fractions using denominators of 2, 3, 4, 6 and 8. Expresses whole numbers as fractions and recognize fractions that are equivalent to whole numbers. Compares two fractions that

Understands, recognizes and generates equivalent fractions using denominators of 2, 4 and 8. Expresses whole numbers as fractions. Compares two fractions that

Given visual models, understands, recognizes and generates equivalent fractions using denominators of 2, 4 and 8. Expresses whole numbers as fractions. Compares two fractions that









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. 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.

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. 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.

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.

Fractions as Numbers 3.NF.1 3.NF.2 3.NF.A.Int.1

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. Represents 1/b on a

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. Represents 1/b on a

Understands 1/b is equal to one whole that is partitioned into b equal parts – limiting the denominators to 2, 4 and 8. Represents 1/b on a

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









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. Applies the concepts of 1/b and a/b in real world situations. Creates the number line that best fits the context.

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. Applies the concepts of 1/b and a/b in real world situations.

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.

number line diagram by partitioning the number line between 0-1 into b equal parts recognizing that b is the total number of parts. Represents fractions in the form a/b using a visual model.

Time 3.MD.1-1 3.MD.1-2

Tells, writes and measures time to the nearest minute. Creates two-step real world problems involving addition and subtraction of time intervals in minutes.

Tells, writes and measures time to the nearest minute. Solves two–step word problems involving addition and subtraction of time intervals in minutes.

Tells, writes and measures time to the nearest minute. Solves one-step word problems involving addition or subtraction of time intervals in minutes.

Tells, writes and measures time to the nearest minute. Solves one-step word problems involving addition or subtraction of time intervals in minutes, with scaffolding, such as a









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. Uses estimated measurements to compare answers to one-step word problems.

Using grams, kilograms or liters, measures and estimates liquid volumes and masses of objects using any of the four basic operations. Uses estimated measurements, when indicated, to answer one-step word problems.

Using grams, kilograms or liters, measures and estimates liquid volumes and masses of objects using concrete objects (beakers, measuring cups, scales) to develop estimates.

Geometric Measurement 3.MD.5 3.MD.6 3.MD.7b-1

Recognizes area as an attribute of plane figures. 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

Recognizes area as an attribute of plane figures. 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.

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

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









and counting them. Connects counting squares to multiplication when finding area. Represents the area of a plane figure as “n” square units.

Connects counting squares to multiplication when finding area. Represents the area of a plane figure as “n” square units.

them. Represents the area of a plane figure as “n” square units.

them.



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.





Multiply One-Digit Whole Numbers 3.NBT.3

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.

Scaled Graphs 3.MD.3-1 3.MD.3-3 3.Int.4

Completes a scaled picture graph and a scaled bar graph to represent a data set. Solves one- and two-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.

Completes a scaled picture graph and a scaled bar graph to represent a data set. Solves one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

Completes a scaled picture graph and a scaled bar graph to represent a data set. Solves one-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

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- step “how many more” and “how many less” problems using information presented in scaled bar graphs.

Measurement Data 3.MD.4

Generates measurement data by measuring lengths to the nearest half and fourth inch. Shows the data by making a

Generates measurement data by measuring lengths to the nearest half and fourth inch. Shows the data by making a

Generates measurement data by measuring lengths to the nearest half inch. Shows the data by making a line plot, where the

Generates measurement data by measuring lengths to the nearest half inch. Shows the data by making a line plot, where the









line plot, where the horizontal scale is marked in appropriate units-whole number, halves or quarters. Uses the line plot to answer questions or solve problems.

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.

Understanding Shapes 3.G.1

Understands the properties 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. Draws examples and non-examples of quadrilaterals with specific attributes.

Understands the properties of quadrilaterals and the subcategories of quadrilaterals. Recognizes that examples of quadrilaterals that have shared attributes and that the shared attributes can define a larger category. Draws examples and non-examples of quadrilaterals with specific attributes.

Understands the properties of quadrilaterals and the subcategories of quadrilaterals. Recognizes that examples of quadrilaterals that have shared attributes and that the shared attributes can define a larger category. Draws examples of quadrilaterals with specific attributes.

Identifies examples of quadrilaterals and the subcategories of quadrilaterals. Recognizes that examples of quadrilaterals that have shared attributes and that the shared attributes can define a larger category.

Perimeter and Area

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,

Solves mathematical problems involving perimeters of polygons,








Level 2: Partial Command 3.G.2 3.MD.8 3.Int.3

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. Partitions shapes in multiple ways into parts with equal areas and expresses the area as a unit fraction of the whole

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. Partitions shapes into parts with equal areas and expresses the area as a unit fraction of the whole.

including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same area and different perimeters. Partitions shapes into parts with equal areas and expresses the area as a unit fraction of the whole.

including finding the perimeter given the side lengths, and exhibiting rectangles with the same area and different perimeters. Partitions shapes into parts with equal areas and expresses the area as a unit fraction of the whole limited to halves and quarters.



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 Command

Level 4: Strong Command Level 3: Moderate

Command Level 2: Partial 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:

properties of operations

relationship between addition and subtraction

relationship between multiplication and division

identification of arithmetic patterns

Response may include:

a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)

an efficient and logical progression of steps

Clearly constructs and communicates a complete written response based on explanations/reasoning using the:






a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)

a logical progression of steps

Constructs and communicates a written response based on explanations/reasoning using the:






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

Constructs and communicates an incomplete written response based on explanations/reasoning using the:






an approach based on a conjecture and/or stated or faulty assumptions

an incomplete or illogical progression of steps

an intrusive calculation error








with appropriate justification

precision of calculation

correct use of grade-level 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.




evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).

vocabulary, symbols and labels

partial justification of a conclusion based on own calculations

evaluating the validity of other’s responses, approaches and conclusions.

limited use of grade-level vocabulary, symbols and labels


Concrete Referents and Diagrams

Clearly constructs and communicates a well-organized and complete response based on

Clearly constructs and communicates a well-organized and complete response based on

Constructs and communicates a response based on operations using concrete referents such as

Constructs and communicates an incomplete response based on operations using








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:

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




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:




some use of grade-level vocabulary, symbols and labels

partial justification of a conclusion based on own calculations.

concrete referents such as diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include:

a conjecture and/or stated or faulty assumptions





accepting the validity of other’s responses








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 counter-example 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

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

Clearly constructs and communicates a well-organized and complete 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;

Clearly constructs and communicates a well-organized and complete 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

distinguishing correct explanation/reasoning

Constructs and communicates a complete response by:

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

Constructs and communicates an incomplete response by:

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








3.C.7

if there is a flaw in the argument

presenting and defending corrected reasoning



an efficient and logical progression of steps with appropriate justification



from that which is flawed

identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems

presenting corrected reasoning







flawed

identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems

presenting corrected reasoning





some use of grade-level vocabulary, symbols and labels


explanation/reasoning from that which is flawed

identifying an error in reasoning


a conjecture based on faulty assumptions





accepting the validity of other’s responses.









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 counter-example 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.



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.





Modeling 3.D.1 3.D.2

Devises a plan and applies mathematics to solve multi-step, real-world contextual word problems by:

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


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


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



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.





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


the results make sense

modifying the model if it has not served its purpose




Grade 3 Math: Sub-Claim E

The student demonstrates fluency in areas set forth in the Standards for Content in grades 3-6.




Fluency 3.NBT.2 3.OA.7

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. Correctly calculates 100 percent of sums and differences in less than the allotted time on items which are timed. Accurately and quickly 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.

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. Correctly calculates 100 percent of sums and differences in the allotted time on items which are timed. Accurately in a timely manner 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 the allotted time on items which are timed.

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. Correctly calculates more than 75 percent and less than 100 percent of sums and differences of items which are timed. Accurately multiplies and divides within 100, using strategies relating multiplication and division or properties of operations. 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.

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 at least 75 percent of the sums and differences of items which are timed. Multiplies and divides within 100, using strategies relating multiplication and division or properties of operations. 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.

Latest Revision 6/24/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 1

Bailey●Kirkland Education Group, LLC

Common Core State Standard I Can Statements 3rd Grade Mathematics

CCSS Key: PLD Key: Operations and Algebraic Thinking (OA) Partial Command Number and Operations in Base Ten (NBT) Moderate Command Numbers and Operations–Fractions (NF) Distinguished Command Measurement and Data (MD) Geometry (G)

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.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. 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.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. 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.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



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.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. 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.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



× 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.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. 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.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. 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.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



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.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.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

I Can: 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



0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

its location on a 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: 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.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. 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.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)

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. 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.

I Can: 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.

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.

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



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. 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.

I Can: 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. 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.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.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

I Can: 3.MD.5a.1 Identify what a unit square is and know it can be used to measure area of a figure.



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.

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.

3.MD.6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

I Can: 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.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 non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems.

I Can: 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.



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.

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.

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.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.

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

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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,

Standards for Mathematical Practice

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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.

CCSS Standards for Mathematical Practice

Questions for Teachers to Ask 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

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?

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.

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? *It is important that the teacher poses tasks that involve arguments or critiques

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

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

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?

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?

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)

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)

3 grade math unit guide

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