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Third Grade

Mathematics Curriculum Guidefor the Maryland College and Career Ready Standards

2014-2015Third Grade Overview

Third Grade Mathematics Curriculum for the Maryland College and Career Ready Standards

Operations and Algebraic Thinking (OA) Represent and solve problems involving multiplication and division. Understand properties of multiplication and the relationship between multiplication and division. Multiply and divide within 100. Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Number and Operations in Base Ten (NBT) Use place value understanding and properties of operations to perform multi-digit arithmetic.

Number and Operations-Fractions (NF) Develop understanding of fractions as numbers.

Measurement and Data (MD) Solve word problems involving measurement and estimation of intervals of time, liquid volumes, and

masses of objects. Represent and interpret data. Geometric measurement: understand concepts of area and relate area to multiplication and to addition. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between

linear and area measures.

Geometry (GReason with shapes and their attributes

Grade 3 Wicomico County Public Schools 2014-2015 2



Standards for Mathematical PracticeStandards Explanations and Examples1. Make sense of problems and persevere in solving them.

In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

2. Reason abstractly and quantitatively.

Third graders should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities

3. Construct viable arguments and critique the reasoning of others.

In third grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like, “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others students’ thinking.

4. Model with mathematics. Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Third graders should evaluate their results in the context of the situation and reflect on whether the results make sense.

5. Use appropriate tools strategically.

Third graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles.

6. Attend to precision. As third graders develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle they record their answers in square units.

7. Look for and make use of structure.

In third grade, students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and distributive properties).

8. Look for and express regularity in repeated reasoning.

Students in third grade should notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve products that they don‘t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work by asking themselves, “Does this make sense?”


GRADE 3 COMMON CORE INTRODUCTION

In Grade 3, the majority of instructional time should focus on two critical areas: (1) operations of multiplication and division and (2) 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. To continue the study of geometry, students describe and analyze shapes by their sides, angles, and definitions. Students need to generalize and apply strategies for computational fluency.

1. Students develop an understanding of the operations of multiplication and division through area models, arrays, pictorial representation and equations. By the end of the year, students recall all products of two single-digit numbers.

2. 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.




Third Grade Math At-A-Glance 2014 – 2015

UNITS Important DatesUnit 1: Developing Strategies for Addition and Subtraction

(10 Days) September 1 Labor Day

Unit 2: Developing an Understanding of Multiplication and Division (24 Days) September 26 Professional Day

Unit 3: Understanding Fractions as Numbers (22 Days)

October 17 MSEA ConventionOctober 23 Interim Assessment #1 (not counted as instructional day)

November 3 Professional DayNovember 4 Election Day

Unit 4: Measurement: Mass, Time, Liquid Volume(10 Days)

November 26-28Thanksgiving

Unit 5: Understanding the Size of Fractions(18 Days)

December 10 Interim Assessment #2December 22- January 2 Winter Holiday

Unit 6: Developing Additional Strategies for Multiplication and Division(18 Days)

January 19 MLK DayJanuary 26 Professional Day

PARCC PBA/MYA Testing Window March 2-27 FYI This is a two day test and will happen during either Unit 7 or 8Unit 7: Rounding to 10 and 100, Bar and Picture Graphs, Two Step

Problems(14 Days)

February 13 Professional DayFebruary 16 President’s Day

February 19 Interim Assessment #3Unit 8: Perimeter, Shapes and their Attributes

(11 Days)Unit 9: Fractions and Line Plots

(14 Days)April 1-6 Spring HolidayApril 7 Professional Day

PARCC EOY Testing Window April 20- May 15 (two days)Unit 10: Fluency with Problem Solving and Computation

(32 days or the remainder of the year)May 25 Memorial Day


Unit One (10 Days)

Developing Strategies for Addition and Subtraction

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. Students will also work on problem solving focusing on two step problems. Introduce the writing of the equation with the unknown quantity represented with a letter.

Unit One Standards

3.NBT.A.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.OA.D.8. Solve two-step word problems using the four operations (only addition and subtraction). 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. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Crossed out parts of this standard will be addressed in later units.

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

*It is important to start using a letter (variable) to stand for the unknown quantity in the equation. In previous grades a line or box was used for the unknown.

Math Language

The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use these math terms in explanations or justifications.

add addend addition algorithm decomposingdifference digits equation estimation expressionmental computation minuend place value words rounding strategiessum subtrahend



Essential Questions

What does it mean to decompose a number? How can benchmark numbers help us add? How does using ten as a benchmark number help us add and subtract? What strategies can help us when adding and subtracting? How can strategies help us when adding and subtracting? How can addition help us know we subtracted two numbers correctly? What is a pattern?How can we use patterns to solve problems? How can you describe various patterns, (i.e. with words, as a visual pattern on a 1-100 chart, or using mathematical notations)? How do two-step word problems differ from one-step word problems? What strategies can be used to solve word problems? What symbols can be used to represent an unknown amount? How can we solve addition problems? How can we solve subtraction problems? How can we model and solve subtraction problems? How can mental math strategies, for example estimation and benchmark numbers, help us when adding and subtracting? What is a number sentence and how can I use it to solve word problems? How do we solve problems in different ways? How can we solve problems mentally? What strategies help us with this? How can problem situations and problem-solving strategies be represented? How are problem-solving strategies alike and different? How can different combinations of numbers and operations be used to represent the same quantity?



The Table below is an important resource for understanding addition and subtraction structures. Problems in this format should be used on a regular basis.

Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?

2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. Howmany bunnies hopped over to the first two?

2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?

? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now?

5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?

5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?

? – 2 = 3Total Unknown Addend Unknown Both Addends Unknown1

Put Together / Take Apart2

Three red apples and two green apples are on the table. How many apples are on the table?

3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green?

3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?

5 = 0 + 5, 5 = 5 + 05 = 1 + 4, 5 = 4 + 15 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“How many more?” version):Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

(“How many fewer?” version):Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?

2 + ? = 5, 5 – 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

(Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?

2 + 3 = ?, 3 + 2 = ?

(Versions with “more”):Julie has three more apples than Lucy. Julie has 5 apples. How many apples does Lucy have?

(Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?

5 – 3 = ?, ? + 3 = 5




Grade 3 Unit One – Developing Strategies for Addition and Subtraction (10 Days)

Connections/Notes Resources3.NBT.A.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.

Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and associative properties. Adding and subtracting fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. Students explain their thinking and show their work by using strategies and algorithms, and verify that their answer is reasonable.

Example: Mary read 573 pages during her summer reading challenge. She was only required to read 399 pages. How many extra pages did Mary read beyond the challenge requirements?

Students may use several approaches to solve the problem including the traditional algorithm. Examples of other methods students may use are listed below: 399 + 1 = 400, 400 + 100 = 500, 500 + 73 = 573, therefore 1+ 100 + 73 = 174 pages (Adding up strategy)

400 + 100 is 500; 500 + 73 is 573; 100 + 73 is 173 plus 1 (for 399, to 400) is 174 (Compensating strategy)

Take away 73 from 573 to get to 500, take away 100 to get to 400, and take away 1 to get to 399. Then 73 +100 + 1 = 174 (Subtracting to count down strategy)

399 + 1 is 400, 500 (that‘s 100 more). 510, 520, 530, 540, 550, 560, 570, (that‘s 70 more), 571, 572, 573 (that‘s 3 more) so the total is1 + 100 + 70 + 3 = 174 (Adding by tens or hundreds strategy)

Teaching Student Centered MathematicsActivities for Flexible Thinking with Whole Numbers pgs.51-55Doubles & Near-Double pgs. 80-81Make Ten on the Ten Frame pgs. 82-83Strategies for Subtraction Facts pgs. 84-91Invented Strategies for Add & Sub pgs.108-111

LessonsEstimate Sums to Solve Word ProblemsEstimate Differences to Solve Word Problems

Activities and TasksThree Digit Addition Split (K5)Doubling to 1000 (K5)Difference Add (K5)Classroom Supplies (IM)

VideosUse Addition and Subtraction Fact Families to Solve for Unknown Amounts

Templates and VisualsOperation Symbols Pinch CardsLet’s Think Place Value Task CardsSix Common Strategies for AdditionFive Common Strategies for SubtractionNumber Talks (+) pgs. 186-204Number Talks (-) pgs. 209-229

Grade 3 Unit One – Developing Strategies for Addition and Subtraction (10 Days)

Connections/Notes Resources3.OA.D.8. Solve two-step word problems using the four operations addition and subtraction. 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. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Crossed out parts of this standard will be addressed in later units in this document.Students should be exposed to multiple problem-solving strategies (using any combination of words, numbers, diagrams, physical objects or symbols) and be able to choose which ones to use.

Examples: Jerry earned 231 points at school last week. This week he earned 79 points. If he uses 60 points

Teaching Student Centered MathematicsDrawings and Diagrams for Story Problems pgs. 304-306

http://learnzillion.com/lessons/1579-use-addition-and-subtraction-fact-families-to-sove-for-unknown-amounts

http://learnzillion.com/lessons/1579-use-addition-and-subtraction-fact-families-to-sove-for-unknown-amounts


Unit Two (24 Days)

Developing an Understanding of Multiplication and DivisionIn Grade 2 students have added groups of objects by skip-counting and using repeated addition. In this unit, students connect ideas 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 2’s, 5’s and 10’s, in addition to numbers that are easily skip counted, such as 3’s and 4’s.

This unit 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. Students understand division as an unknown-factor problem. Another focus for students is to identify arithmetic patterns in order to develop a deeper understanding of number and number relationships (especially with multiplication and division).

In Unit 6, the distributive and associative properties will be added as a strategy to use to become more fluent with multiplication and division.

Unit Two Standards

3.OA.A.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 x 7.

3.OA.A.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.OA.A.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. .FYI the measurement quantities in this unit will be taught in unit 7

3.OA.B.5. Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) 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.)

3.OA.B.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

.


http://learnzillion.com/lessons/1524-solve-twostep-problems-using-parenthesis


3.OA.C.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.

3.OA.D.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.

3.MD.C.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.

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

3.MD.C.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. Parts C and D of this standard will be addressed in Unit 6 and 10.

Add 3.OA.D.8 two-step word problems with multiplication and division

3.OA.D.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. (This standard is limited to problems posed with whole numbers and having whole-number answers)

Math Language

The terms below are for teacher reference. Although these do not need to be memorized by the students, students should be encouraged to use math these terms in explanations or justifications.



area array column commutative propertydecomposing divide dividenddivisor division expression equationfactor fair share length measurement divisionmental computation multiply operation partitive divisionpatterns product quotient rowstiling unit square zero property

Essential Questions

What is a pattern?How are patterns related to multiplication?How can we use patterns to solve problems? How can you describe various patterns, (i.e. with words, as a visual pattern on a 1-100 chart, or using mathematical notations)? What does it mean to decompose a number?How can multiplication products be displayed on a 1-100 chart? Why is an area model a representation for multiplication?How can an addition table help you explain the Commutative Property of Multiplication? What strategies can be used to solve word problems?How can multiple math operations be used to solve real world problems? How can an addition table help you explain the Commutative Property of Multiplication? How do two-step word problems differ from one-step word problems?What is area? How can area be determined without counting each square?Why are square units commonly associated with finding area? How can multiplication and addition be used to determine a rectangle’s area? How is the commutative property of multiplication evident in an area model? Can the same area measurement produce different size rectangles? (Ex. 24 sq.units can produce a rectangle that is a 3 X 8, 4 X 6, 1 X24, 2 X 12) Do different factors with the same area cover the same amount of space? (Ex. Is a 3 X8 the same area as a 1 X 24?) How can the same area measure produce rectangles with different dimensions? (Ex. 24 square units can produce a rectangle that is a 3 x 8, 4 x 6, 1 x 24, 2 x 12) How do different dimensions resulting in the same area cover the same amount of space? (Ex. Is a 3 X8 the same area as a 1 X 24?) How do rectangle dimensions impact the area of the rectangle?How does knowing the area of a square or rectangle relate to knowing different multiplication facts? How does knowing the area of a square or rectangle relate to knowing multiplication facts? How does knowing the dimensions of a rectangle relate to area? How does knowing the dimensions of a rectangle relate to multiplication?



How does knowing the length and width of a rectangle relate to multiplication? How does the length and width (factors) impact the area of the rectangle? How does understanding the distributive property help us multiply large numbers? What is tiling?



The Table below is an important resource for understanding multiplication and division structures. Problems in this format should be used on a regular basis.


*The comparison structures have been omitted from this table because they are not introduced until grade 4.

The first examples in each cell are examples of discrete things. These are easier for students.

The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery windows are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.

Area involves arrays of squares that n have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations.



Grade 3 Unit Two - Developing an Understanding of Multiplication and Division (24 days)

Connections/Notes Resources3.OA.A.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.

Students recognize multiplication as a means to determine the total number of objects when there are a specific number of groups with the same number of objects in each group. Multiplication requires students to think in terms of groups of things rather than individual things. Students learn that the multiplication symbol “x” means groups of - such as 5 x 7 refer to 5 groups of 7.To further develop this understanding, students interpret a problem situation requiring multiplication using pictures, objects, words, numbers, and equations. Then, given a multiplication expression (e.g., 5 x 6) students interpret the expression using a multiplication context. They should begin to use the terms, factor and product, as they describe multiplication.

Teaching Student Centered MathematicsStrategies for Multiplication Facts pages 88-90Patterns in the Nine Facts page 90-91Using arrays to solve “hard facts” pages 92-93Area Model for Multiplication pages 129-130

LessonsEgg Carton DesignsOdd Man OutBalance Beam DiscoveriesUnderstand Equal Groups as MultiplicationRelate Multiplication to the Array ModelInterpret the Meaning of FactorsMultiplication Facts by Adding and SubtractingSkip counting Units of 4Bar Diagrams to Model Commutative Property

Activities and TasksCookie Dough Activity SheetMultiple Arrays SheetArray Picture Cards Activity (K5)Treasure Trove (AIMS)Accounting for Butterflies (AIMS)Hopping into Multiplication (AIMS)

VideosInterpret Products by Drawing Pictures


Connections/Notes Resources3.OA.A.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.

Students recognize the operation of division in two different types of situations. One situation requires determining how many groups and the other situation requires sharing (determining how many in each group). Students should be exposed to appropriate terminology (quotient, dividend, divisor, and factor).

Teaching Student Centered MathematicsStrategies for Division Facts and “Near Facts” pages 93 - 95Invented Strategies for Division page 121

http://learnzillion.com/lessons/3070-interpret-products-by-drawing-pictures

http://illuminations.nctm.org/Lesson.aspx?id=1261






Connections/Notes Resources3.OA.B.5. Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) 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.)

Students represent expressions using various objects, pictures, words and symbols in order to develop their understanding of properties. They multiply by 1 and 0 and divide by 1. They change the order of numbers to determine that the order of numbers does not make a difference in multiplication (but does make a difference in division). Given three factors, they investigate changing the order of how they multiply the numbers to determine that changing the order does not change the product. They also decompose numbers to build fluency with multiplication.

Models help build understanding of the commutative property:

Example: 3 x 6 = 6 x 3

In the following diagram it may not be obvious that 3 groups of 6 is the same as 6 groups of 3. A student may need to count to verify this.

is the same quantity as

Example: 4 X 3 = 3 X 4An array explicitly demonstrates the concept of the commutative property.

Students are introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don‘t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. Students should learn that they can decompose either of the factors. It is important to note that the students may record their thinking in different ways.

Teaching Student Centered MathematicsInvented Strategies for Division Page 121

LessonsDemonstrate CommutativityCommutativity of MultiplicationBar Diagrams to Model Commutative Property

Activities and TasksMultiples (K5)Multiples Version 2 (K5)Multiplication Bump (x2) (K5)

VideosUnderstand the Commutative Property by Naming Arrays


Connections/Notes Resources

7 x 8

5 X 8 2 X 8

5 x 8 = 40 7 x 4 = 28 7 x 4 = 28

56 2 x 8 = 16

http://learnzillion.com/lessons/963-understand-the-commutative-property-by-naming-arrays



http://learnzillion.com/lessons/1516-solve-division-problems-by-drawing-pictures


Unit Three (22 Days)

Understanding Fractions as Numbers

In previous grades students learned about partitioning shapes into fair shares using words to describe the quantity. In this unit students extend this understanding to partition both shapes and number lines, representing these fair shares using fraction notation. The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.Grade 3 expectations in this domain are limited to fractions with denominators of 2, 3, 4, 6, and 8. The focus of 3.NF.A.1 and 3.NF.A.2 in this unit is on fractions between 0 and 1. Fractions greater than 1 will be introduced in Unit 9.

Unit Three Standards

3.NF.A.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.NF.A.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.G.A.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. Students’ work with partitioning shapes (3G.2) and relates to visual fraction models (3NF)


http://learnzillion.com/lessons/1742-interpret-division--unknown-factor-problem-using-arrays

http://learnzillion.com/lessons/2999-solve-word-problems-using-the-idea-of-equal-groups


Math Language

The terms below are for teacher reference only. Although these do not need to be memorized by the students, students should be encouraged to use math terms in explanations or justifications.

congruent denominator endpoint equal parts fractionsfractional parts increment intervals number line numeratorpartition unit fraction visual fraction models

Essential Questions

How can I show that one fraction is greater (or less) than another using my Fraction Strips? How can I use fractions to name parts of a whole? How can I use pattern blocks to name fractions? How can I use pattern blocks to represent fractions? How can I write a fraction to represent a part of a group? How does the numerator impact the denominator on the number line? How is the appropriate unit for measurement determined? How is the reasonableness of a measurement determined? What are the important features of a unit fraction? What fractions are on the number line between 0 and 1? What fractions are on the number line between 0 and 3? (1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, 8/3, 9/3)What is a fraction? What is a real-life example of using fractions? What is the relationship between a unit fraction and a unit of 1? What relationships can I discover about fractions? What relationships can I discover among the pattern blocks? What represents the denominator in a set? What represents the numerator in a set? What happens to the fractional pieces of the whole when the denominator increases?Why are units important in measurement? Why is the denominator important to the unit fractions? Why is the size of the whole important? Describe what a fraction looks like in a shape?


http://www.oswego.org/ocsd-web/games/splatsquares/splatsq100.html




one square

unit

http://learnzillion.com/lessons/1464-identify-addition-and-subtraction-patterns-using-a-hundreds-chart



Grade 3 Unit Three Understanding Fractions as Numbers (22 days)Grade 3 expectations in this domain are limited to fractions with denominators of 2, 3, 4, 6, and 8.

Connections/Notes Resources3.NF.A.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.Some important concepts related to developing understanding of fractions include:

Understand fractional parts must be equal-sized

Example Non-example

The number of equal parts tell how many make a whole As the number of equal pieces in the whole increases, the size of the fractional pieces

decreases The size of the fractional part is relative to the whole The number of children in one-half of a classroom is different than the number of children in

one-half of a school. (the whole in each set is different therefore the half in each set will be different)

Teaching Student Centered MathematicsActivities for Flexible Thinking with Whole Numbers pg. 136

LessonsEggsactly with a Dozen EggsExploring the Value of a WholeMaking and Investigating Fraction StripsInvestigating Fraction Relationships with Relationship RodsPartition a Whole Into Equal PartsRepresent and Identify Fractional Parts of Different WholesSpecify and Partition a Whole Into Equal Parts

Activities and TasksName the Fraction (K5)Cuisenaire Fraction (K5)Fraction Barrier Game (K5)Fraction Barrier Game Grid (K5)Exploring Fraction Kits (K5)Equal Parts on the Geoboard (K5)Fractions with Color Tiles (K5)Fraction Posters (K5)Pattern Block CombinationsPicture Parts Maximizing Math (AIMS)Folding Flags (AIMS)Figuring Fractions (AIMS)Part to Whole, Whole to Part Cuisenaire RodsFraction Book

Grade 3 Unit Three Understanding Fractions as Numbers (22 days)

Connections/Notes ResourcesVideosRepresent Fractions in Different WaysWrite Fractions with Numerator and DenominatorWhy the Larger the Denominator the Smaller the Fractional Parts Using Models and Real World Examples







Unit Four (10 Days)

Measurement: Mass, Time, Liquid Volume

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. The multiplication and division components of 3.MD.A.2 will occur in Unit 7.



Third graders can learn to measure with liquid volume and to solve problems requiring the use of the four arithmetic operations, when liquid volumes are given in the same units throughout each problem. Because liquid measurement can be represented with one-dimensional scales, problems may be presented with drawings or diagrams, such as measurements on a beaker with a measurement scale in milliliters.

Unit Four Standards

3.MD.A.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.

3.MD.A.2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm3 and finding the geometric volume of a container.) 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. Excludes multiplicative comparison problems (problems involving notions of - times as much).

Math Language


elapsed time estimate gram hour container capacitykilogram liquid volume liter mass beaker tick marksmeasure metric minute millilitersnon-standard units standard units time time intervalsvolume

Essential Questions


http://learnzillion.com/lessons/3155-Count-the-unit-fractions-of-one-whole-using-fraction-strips


What does it mean to tell time to the nearest minute? What strategies can I use to help me tell and write time to the nearest minute and measure time intervals in minutes? What connections can I make between a clock and a number line? How can I use what I know about number lines to help me figure out how much time has passed between two events? How can we determine the amount of time that passes between two events? What part does elapsed time play in our daily living? How can I demonstrate my understanding of the measurement of time? How can you prove to your parents you do not spend too much time watching television? What happens when your units of measure change? Why is it important to know the mass of an object? In what ways can we determine the mass of an object? What is the difference between a standard and non-standard unit of measurement? What units are appropriate to measure mass? How are units in the same system of measurement related? What strategies could you use to figure out the mass of multiple objects? What happens to an item’s measurement when units are changed?What items in the classroom weigh close to a kilogram? How are grams and kilograms related? What everyday items weigh about a gram? About a kilogram? What is the tool best to use when measuring liquid volume? What connection can you make between the volumes and your everyday life? Does volume change when you change the measurement material? Why or why not? How can estimating help me to determine liquid volume? What are some ways I can measure the liquid volume?What are milliliters and how are they related to liters?




Grade 3 Unit Four Measurement: Mass, Time , Liquid Volume (10 days)

Connections/Notes Resources3.MD.A.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.

Students in second grade learned to tell time to the nearest five minutes. In third grade, they extend telling time and measure elapsed time both in and out of context using clocks and number lines.

Example:

Tonya wakes up at 6:45 a.m. It takes her 5 minutes to shower, 15 minutes to get dressed, and 15 minutes to eat breakfast. What time will she be ready for school?

Teaching Student Centered MathematicsElapsed Time pgs. 270-271

LessonsTelling Time Using a Number LineTelling Time to the Nearest MinuteSolve Word Problems within One Hour

Activities and TasksElapsed Time Word Problems (K5)Later and EarlierEarlier and LaterTime Riddles

VideosIdentifying the Start Time Change of Time and End Time in Real World Elapsed Time ProblemsSolving Elapsed Time Word Problems to the Nearest HourElapsed Time Word Problems to the Nearest Five MinutesSolving Elapsed Time Word Problems to the Nearest MinuteSolving Elapsed Time Word Problems Using a T-Chart

TemplateElapsed Time Ruler

Grade 3 Unit Four Measurement: Mass, Time , Liquid Volume (10 days)

Connections/Notes Resources3.MD.A.2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm3 and finding the geometric volume of a container.) 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. Excludes multiplicative comparison problems (problems involving notions of ―times as muchStudents need multiple opportunities weighing classroom objects and filling containers to help them develop a basic understanding of the size and weight of a liter, a gram, and a kilogram. Milliliters may also be used to show amounts that are less than a liter.

Teaching Student Centered MathematicsCapacity Line-Up Activity 9.9 pg.266Introducing Standard Units pgs. 274-277Estimating Measures pgs. 278-280Familiar References Activity 9.14 pg. 276


Unit Five (18 Days)

Understanding the Size of Fractions

In this unit students develop a conceptual understanding of equivalence of fractions. Multiple types of models and representations should be used to help students develop this understanding such as: fraction strips, fraction towers, fringe fractions, fractions bars, fraction circles, fraction squares. 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. This understanding of equivalence of fractions is necessary for students to reason about fraction size and structure in order to compare quantities. 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.

Unit Five Standards

3.NF.A.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.

Math Language


comparing denominator equivalent numerator unit fractionwhole greater than less than inequalities



Essential Questions

How can I compare fractions when they have the same denominators? How can I compare fractions when they have the same numerators? How can I compare fractions? How can I represent fractions of different sizes? How can I show that one fraction is greater (or less) than another using my Fraction Strips? How can I use fractions to name parts of a whole? How can I use pattern blocks to name fractions? How can I use pattern blocks to represent fractions? How can I write a fraction to represent a part of a group? How does the numerator impact the denominator on the number line? How is the appropriate unit for measurement determined? How is the reasonableness of a measurement determined? What are the important features of a unit fraction? What equivalent groups of fractions can I discover using Fraction Strips?What fractions are on the number line between 0 and 1? What is a fraction? What is a real-life example of using fractions? What is the relationship between a unit fraction and a unit of 1? What relationships can I discover about fractions? What relationships can I discover among the pattern blocks? What represents the denominator in a set? What represents the numerator in a set? When we compare two fractions, how do we know which has a greater value? Why are units important in measurement? Why is the denominator important to the unit fractions? Why is the size of the whole important?




Grade 3 – Unit Five Understanding the Size of Fractions (18 days)

Connections/Notes Resources3.NF.A.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.

An important concept when comparing fractions is to look at the size of the parts and the number of the parts. For example, 1/8 is smaller than 1/2 because when 1 whole is cut into 8 pieces, the pieces are much smaller than when 1 whole is cut into 2 pieces. Students recognize when examining fractions with common denominators, the wholes have been divided into the same number of equal parts. So the fraction with the larger numerator has the larger number of equal parts.

26 < 5

6

To compare fractions that have the same numerator but different denominators, students understand that each fraction has the same number of equal parts but the size of the parts are different. They can infer that the same number of smaller pieces is less than the same number of bigger pieces.

38 < 3

4

A common misconception, the idea that the smaller the denominator, the smaller the piece or part of the set, or the larger the denominator, the larger the piece or part of the set, is based on the comparison that in whole numbers, the smaller a number, the less it is, or the larger a number, the more it is. The use of different models, such as fraction bars and number lines, allows students to compare unit fractions to reason about their sizes.As with equivalence of fractions, it is important in comparing fractions to make sure that each fraction refers to the same whole.

Teaching Student Centered MathematicsEquivalent Fraction Concepts pg.151

LessonsRecognize Equivalent Fractions Have the Same SizeShow Equivalent Fractions Refer to the Same Point on a Number LineGenerate Equivalent Fractions using Models and the Number LineRecognize Whole Numbers as Fractions and Recognize EquivalenceExplain Equivalence by Manipulating UnitsCompare Fractions with the Same Numerator

Activities and TasksComparing Fractions Comparing Fractions from with a Different Whole (IM)Comparing Fractions with the Same Denominators (IM)Comparing Fractions with the Same Numerator (IM)Comparing Fractions (IM)

Closest to One Half (IM)Fraction Comparisons with Pictures (IM)Jon and Charlie’s Run (IM)Ordering Fractions (IM)Equivalent Fractions with Fraction TilesEqual to ½ (PowerPoint)Pattern Block Fraction GameBuilding Congruent HexagonsFraction DominoesFraction Pieces (IM)Spin to Win Game


Unit Six (18 Days)

Developing Additional Strategies for Multiplication and Division

This focus for this unit is developing a conceptual understanding of decomposing multiplication problems through the use of the properties and the concept of area. Students use area as a context to further develop multiplicative thinking. 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.

Unit Six Standards

3.OA.A.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 × M = 48, 5 = M ÷ 3, 6 × 6 = M. Connections: 3.OA.A.3

3.OA.B.5. Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) 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.)

3.MD.C.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

3.OA.D.8. Solve two-step word problems using the four operations (only addition and subtraction). 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. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).


http://learnzillion.com/lessons/1735-generate-equivalent-fractions-using-a-number-linequivalent-fractions-using-a-number-line


Math Language


associative property area array commutative propertydecomposing divide dividend divisordivision expression equation factorlength mental computation multiply operationproduct quotient unit square widthzero property

Essential Questions

What does it mean to decompose a number?How can multiplication products be displayed on a 1-100 chart? Why is an area model a representation for multiplication?What strategies can be used to solve word problems?How can multiple math operations be used to solve real world problems? How can an addition table help you explain the Commutative Property of Multiplication? How do two-step word problems differ from one-step word problems?What is area? How can area be determined without counting each square?Why are square units commonly associated with finding area? How can multiplication and addition be used to determine a rectangle’s area? How is the commutative property of multiplication evident in an area model? Can the same area measurement produce different size rectangles? (Ex.24 sq.units can produce a rectangle that is a 3 X 8, 4 X 6, 1 X24, 2 X 12) Do different factors with the same area cover the same amount of space? (Ex. Is a 3 X8 the same area as a 1 X 24?) How can the same area measure produce rectangles with different dimensions? (Ex. 24 square units can produce a rectangle that is a 3 x 8, 4 x 6, 1 x 24, How do different dimensions resulting in the same area cover the same amount of space? (Ex. Is a 3 X8 the same area as a 1 X 24?) How do rectangle dimensions impact the area of the rectangle?How does knowing the area of a square or rectangle relate to knowing different multiplication facts? How does knowing the dimensions of a rectangle relate to area? How does knowing the dimensions of a rectangle relate to multiplication?



How does knowing the length and width of a rectangle relate to multiplication? How does the length and width (factors) impact the area of the rectangle? How does understanding the distributive property help us multiply large numbers?What is tiling?




Grade 3 - Unit Six - Developing Additional Strategies for Multiplication and Division (18 days)

Connections/Notes Resources3.OA.A.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 = n ÷ 3, 6 × 6 = m.

This standard is strongly connected to 3.AO.3 when students solve problems and determine unknowns in equations. Students should also experience creating story problems for given equations. When crafting story problems, they should carefully consider the question(s) to be asked and answered to write an appropriate equation. Students may approach the same story problem differently and write either a multiplication equation or division equation.Students apply their understanding of the meaning of the equal sign as ”the same as” to interpret an equation with an unknown.

When given 4 x ? = 40, they might think:o 4 groups of some number is the same as 40o 4 times some number is the same as 40o I know that 4 groups of 10 is 40 so the unknown number is 10o The missing factor is 10 because 4 times 10 equals 40.

Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.

Examples: Solve the equations below:

24 = m x 6

72 ÷ m = 9

Rachel has 3 bags. There are 4 marbles in each bag. How many marbles does Rachel have altogether? 3 x 4 = m

LessonsFind and Interpret the Unknown in Multiplication and Division

Activities and TasksMissing Numbers Multiplication (K5)What is the Missing Number? Division (K5)Division as Unknown Factor Problem (K5)

VideosFind the Missing Quotient in a Division Problem

Grade 3 - Unit Six - Developing Additional Strategies for Multiplication and Division (18 days)

Connections/Notes Resources3.OA.B.5. Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) 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.)

Students learn and use strategies for finding products and quotients that are based on the properties of operations; for example, to find 4 × 7 they may recognize that 7 = 5 +2 and compute 4 × 5 +4 ×2. This is an example of seeing and making use of structure (MP.7).

Teaching Student Centered MathematicsUseful Multiplication and Division Properties Pg. 66

http://learnzillion.com/lessons/1474-find-the-missing-quotient-in-a-division-problem


Unit Seven (14 Days)

Rounding to 10 and 100, Bar and Picture Graphs, Two-Step Problems Name???

In Grade 3, the most important development in data representation for categorical data is that students now draw picture graphs in which each picture represents more than one object, and they draw bar graphs in which the height of a given bar in tick marks must be multiplied by the scale factor in order to yield the number of objects in the given category. These developments connect with the emphasis on multiplication in this grade.

At the end of Grade 3, students can draw a scaled picture graph or a scaled bar graph to represent a data set with several categories (six or fewer categories). They can solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

This unit also extends students’ work in previous units to include multiplication and division to solve problems involving measurement quantities.

Unit Seven Standards 3.OA.A.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. Word problems involving equal groups, arrays, and measurement quantities can be used to build students’ understanding of and skill with multiplication and division, as well as to allow students to demonstrate their understanding of and skill with these operations

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

3.NBT.A.1. Use place value understanding to round whole numbers to the nearest 10 or 100.




3.NBT.A.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.

3.MD.A.2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm3 and finding the geometric volume of a container.) 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. Excludes multiplicative comparison problems (problems involving notions of ―times as much

3.MD.B.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. Scaled picture graphs and scaled bar graphs (3.MD.3) can be a visually appealing context for solving multiplication and division problems

Math Language


bar model data estimate graph hundredincrement interpret measurement patterns picture graphplace value rounding scale survey

Essential QuestionsHow are patterns related to multiplication?How can graphs be used to solve real world problems? How can multiplication products be displayed on a 1-100 chart? How can multiplication used when reading a pictograph? How can the knowledge of area be used to solve real world problems? How can we use patterns to solve problems? How can you describe various patterns, (i.e. with words, as a visual pattern on a 1-100 chart, or using mathematical notations)? How do estimation, multiplication, and division help us solve problems in everyday life? How do two-step word problems differ from one-step word problems? How is the commutative property of multiplication evident in an area model? How is the decomposition of a factor in an equation related to the distributive property of multiplication? What is a pattern? What is the connection between area models and skip counting? What strategies can be used to solve word problems? What symbols can be used to represent an unknown amount?



Why is a graph a more efficient way to view the data collected than a paragraph written describing the results? How are tables, and bar graphs, useful ways to display data? How can you use graphs to answer a question? How can surveys be used to collect data? How can graphs be used to display data gathered from a survey? How do I decide what increments to use for my scale?

Grade 3 Unit Seven– Rounding to 10 and 100, Bar and Picture Graphs, Two-Step Problems (14 days)

Connections/Notes Resources3.OA.A.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. Word problems involving equal groups, arrays, and measurement quantities can be used to build students’ understanding of and skill with multiplication and division, as well as to allow students to demonstrate their understanding of and skill with these operations

Students use a variety of representations for creating and solving one-step word problems, i.e., numbers, words, pictures, physical objects, or equations. They use multiplication and division of whole numbers up to 10 x10. Students explain their thinking, show their work by using at least one representation, and verify that their answer is reasonable.

Word problems may be represented in multiple ways:

Equations: 3 x 4 = ?, 4 x 3 = ?, 12 ÷ 4 = ? and 12 ÷ 3 = ?

Array:

Equal groups

Repeated addition: 4 + 4 + 4 or repeated subtraction

Three equal jumps forward from 0 on the number line to 12 or three equal jumps backwards from 12

Teaching Student Centered MathematicsInvented Strategies for Division pg. 121

Activities and TasksChris’s Garden Dilemma (NYC UNIT)

Videos

Solve-word-problems-using-the-idea-of-equal-groups


http://learnzillion.com/lessons/3930-find-area-using-distributive-property





Connections/Notes Resourcesto 0

Examples of division problems:

Determining the number of objects in each share (partitive division, where the size of the groups is unknown):

The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many hair clips will each person receive?

Determining the number of shares (measurement division, where the number of groups is unknown)

Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, how many days will the bananas last?





3.OA.D.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. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Students should be exposed to multiple problem-solving strategies (using any combination of words, numbers, diagrams, physical objects or symbols) and be able to choose which ones to use.

Examples:

Jerry earned 231 points at school last week. This week he earned 79 points. If he uses 60 points to earn free time on a computer, how many points will he have left?

A student may use the number line above to describe his/her thinking,

"231 + 9 = 240 so now I need to add 70 more. 240, 250 (10 more), 260 (20 more), 270, 280, 290, 300, 310 (70 more). Now I need to count back

LessonsSolve Two Step Word Problems Involving Multiplication and DivisionSolve Two Step Word Problems Involving the Four OperationsShare and Critique Peer Strategies to Varied Word ProblemsSolve Word Problems Using All Four Operations with Metric MeasurementSolve Two Step Word Problems with Multiples of Ten

Activities and TasksThe Class Trip (IM)The Stamp Collection (IM)Two Step Word Problems (Set 2) (K5)Two Step Word Problems (all four operations)




Connections/Notes Resources 60. 310, 300 (back 10), 290 (back 20), 280, 270, 260, 250 (back 60).

A student writes the equation, 231 + 79 – 60 = m and uses rounding

(230 + 80 – 60) to estimate.

A student writes the equation, 231 + 79 – 60 = m and calculates 79-60 = 19 and then calculates 231 + 19 = m.

The zoo keeper at the Salisbury Zoo saw 36 legs walk by. How many creatures do you think the zoo keeper saw? Show different possibilities with words and expressions using multiplication and addition. (Found in Tackle the Task Booklets) For example: 2 spiders, 3 butterflies, and 1 duck

2 x 8 + 3 x 6 + 1 x 2 = 36

The soccer club is going on a trip to the water park. The cost of attending the trip is $63. Included in that price is $13 for lunch and the cost of 2 wristbands, one for the morning and one for the afternoon. Write an equation representing the cost of the field trip and determine the price of one wristband.

The above diagram helps the student write the equation, w + w + 13 = 63. Using the diagram, a student might think, “I know that the two wristbands cost $50 ($63-$13) so one wristband costs $25.” To check for reasonableness, a student might use front end estimation and say 60-10 = 50 and 50 ÷ 2 = 25.

When students solve word problems, they use various estimation skills which include identifying when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate method of

Videos Solve Two Step Problems Using Parenthesis




Connections/Notes Resourcesestimation, and verifying solutions or determining the reasonableness of solutions.

Estimation strategies include, but are not limited to:

using benchmark numbers that are easy to compute front-end estimation with adjusting (using the highest place value and estimating from the front end

making adjustments to the estimate by taking into account the remaining amounts) rounding and adjusting (students round down or round up and then adjust their estimate depending

on how much the rounding changed the original values)

3.NBT.A.1. Use place value understanding to round whole numbers to the nearest 10 or 100. Connections: 3.OA.B.5

Students learn when and why to round numbers. They identify possible answers and halfway points. Then they narrow where the given number falls between the possible answers and halfway points. They also understand that by convention if a number is exactly at the halfway point of the two possible answers, the number is rounded up.

Example: Round 178 to the nearest 10.

Connections: 3.OA.B.5

Teaching Student Centered MathematicsDeveloping Number RelationshipsApproximate Numbers & Rounding pg. 47

LessonsRound Two Digit Measurement on Vertical Number LineRound Two and Three Digit Numbers on a Vertical Number Line

Activities and TasksRound up or Down? (K5)Found to the Nearest TenRound to the Nearest HundredAll Aboard for Rounding (AIMS)Rounding to 50 or 500 (IM)Rounding to the Nearest 10 or 100 (IM)

Videos




Connections/Notes ResourcesUnderstand the Value of a Digit in a Multi Digit-Number

3.NBT.A.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.

Students use base ten blocks, diagrams, or hundreds charts to multiply one-digit numbers by multiples of 10 from 10-90. They apply their understanding of multiplication and the meaning of the multiples of 10. For example, 30 is 3 tens and 70 is 7 tens. They can interpret 2 x 40 as 2 groups of 4 tens or 8 groups of ten. They understand that 5 x 60 is 5 groups of 6 tens or 30 tens and know that 30 tens is 300. After developing this understanding they begin to recognize the patterns in multiplying by multiples of 10.

Teaching Student Centered MathematicsUsing Multiples of 10 & 100 pg. 116

LessonsMultiply by Multiples of Ten Using the Place Value ChartUse Place Value and the Associative Property to Multiply Multiples of Ten

Activities and TasksMultiples of Ten Multiply (K5)Multiply by Multiples of Ten Problems (K5)Multiplication Bump (X10) (K5)

VideosMultiply by Multiples of 10 with Base Ten Blocks

3.MD.A.2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm3 and finding the geometric volume of a container.) 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. Excludes multiplicative comparison problems (problems involving notions of ―times as much)

Students need multiple opportunities weighing classroom objects and filling containers to help them develop a basic understanding of the size and weight of a liter, a gram, and a kilogram. Milliliters may also be used to show amounts that are less than a liter.

Activities and TasksVolume and Mass Word Problems (K5)





3.MD.B.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

Students should have opportunities reading and solving problems using scaled graphs before being asked to draw one. The following graphs all use five as the scale interval, but students should experience different intervals to further develop their understanding of scale graphs and number facts.

Pictographs: Scaled pictographs include symbols that represent multiple units. Below is an example of a pictograph with symbols that represent multiple units. Graphs should include a title, categories, category label, key, and data.

How many more books did Juan read than Nancy?

Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs include a title, scale, scale label, categories, category label, and data.

LessonsGenerate and Organize Data

Activities and TasksButton Bar Graph (K5)Button Pictograph (K5)Jake’s Survey (K5)Collecting and Representing Data (K5)Reindeer Graphing

VideosDraw Bars on a GraphLabel and Title a GraphTitle and Label Graphs by Looking at Data CollectedDetermine Scale IncrementsDetermine an Appropriate Key for a Picture Graph





Unit Eight (11 Days)

Perimeter, Shapes and their Attributes

The focus of this unit is reasoning with shapes and their attributes. The unit especially deals with perimeter, since area was a focus in previous units. The standards in this unit strongly support one another because perimeter, like area is an attribute of shape. Prior work with area allows students to differentiate between the two measures (area and perimeter).

*Clarification- A trapezoid is a quadrilateral that has at least one set of parallel lines.

Unit Eight Standards

3.MD.D.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.



3.G.A.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

Math Language


2-dimensional 3-dimensional acute angle attributesclosed figure congruent cubes coneshexagon length obtuse angle octagonopen figure parallel parallelogram pentagonperimeter polygon properties quadrilateralrectangle rhombi/rhombus right angle rectangular prisms (as subcategory of 3-D figures)square width

Essential Questions

How do you know if a shape is a ________(square, rectangle, rhombus, trapezoid, parallelogram etc) quadrilateral? How do you know the difference between a square, a rectangle, a trapezoid, and a rhombus? How do you know the difference between shapes if several of them have the same number of sides? How does combining and breaking apart shapes affect the perimeter and area? How is a rhombus different from a square, rectangle, or trapezoid? How it is possible to have a shape that has fits into more than one category? How might finding shapes within other shapes help me in life? How might I begin at any number on ruler to measure length? How would you explain to a younger student about the different shapes and how some shapes can share attributes? In what ways can I represent data from a picture? Is a rectangle a rhombus? Why?Is it possible for a square to be a rectangle? Is a rectangle a square? Is it possible to find more than 1 way for shapes to fit together to make another shape? Is there a way to use parts of shapes to help create shapes? What are some differences between the quadrilaterals? What are some different ways to identify shapes?



What are some things you have learned about quadrilaterals? What are some ways that a hexagon (or pentagon) can look? What do know about a quadrilateral that you didn’t know at the beginning of this unit? What do you know about pattern blocks that would help me understand how to fill an area?




Grade 3 Unit Eight Perimeter, Shapes and their Attributes (11 days)

Connections/Notes Resources3.MD.D.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.Perimeter is the boundary of a two-dimensional shape. For a polygon, the length of the perimeter is the sum of the lengths of the sides. Initially, it is useful to have sides marked with unit length marks, allowing students to count the unit lengths. Later, the lengths of the sides can be labeled with numerals. As with all length tasks, students need to count the length-units and not the end-points. Next, students learn to mark off unit lengths with a ruler and label the length of each side of the polygon. For rectangles, parallelograms, and regular polygons, students can discuss and justify faster ways to find the perimeter length than just adding all of the lengths (MP3). Rectangles and parallelograms have opposite sides of equal length, so students can double the lengths of adjacent sides and add those numbers or add lengths of two adjacent sides and double that number. A regular polygon has all sides of equal length, so its perimeter length is the product of one side length and the number of sides.

Students develop an understanding of the concept of perimeter by walking around the perimeter of a room, using rubber bands to represent the perimeter of a plane figure on a geoboard, or tracing around a shape on an interactive whiteboard. They find the perimeter of objects; use addition to find perimeters; and recognize the patterns that exist when finding the sum of the lengths and widths of rectangles.

Students use geoboards, tiles, and graph paper to find all the possible rectangles that have a given perimeter (e.g., find the rectangles with a perimeter of 14 cm.) They record all the possibilities using dot or graph paper, compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles.

Given a perimeter and a length or width, students use objects or pictures to find the missing length or width. They justify and communicate their solutions using words, diagrams, pictures, and numbers.

Students use geoboards, tiles, graph paper, or technology to find all the possible rectangles with a given area (e.g. find the rectangles that have an area of 12 square units.) They record all the possibilities using dot or graph paper, compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles. Students then investigate the perimeter of the rectangles with an area of 12.

Teaching Student Centered MathematicsFixed Perimeters Activity 9.7 pg. 265

LessonsFinding Perimeter and AreaFair and SquarePerimeter ExplorationDetermine Perimeter of PolygonsExplore PerimeterDetermine Perimeter with Missing MeasurementsConstruct Rectangles from Given Areas and Find the PerimeterSolve a Variety of Word Problems with PerimeterActivities and TasksMeasuring Perimeter (K5)Perimeter with Color Tiles (K5)Designing a Rabbit Enclosure (K5)The Perimeter Stays the Same (K5)Perimeter Word Problems (K5)Constant Perimeters (AIMS)Straw Lengths and Square Units (AIMS)Mystery Perimeter

VideosFind the Perimeter of a PolygonFind the Perimeter of a Square or Rectangle by Adding Side LengthsFind the Perimeter of a Polygon in Real World ProblemsFind Perimeter with Missing Side Lengths

Grade 3 Unit Eight Perimeter, Shapes and their Attributes (11 days)


The patterns in the chart allow the students to identify the factors of 12, connect the results to the commutative property, and discuss the differences in perimeter within the same area. This chart can also be used to





Unit 9 (11 Days)

Fractions and Line Plots

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. Through experience with measurement, students realize fractions allow us to represent data much more accurately than just representing data with whole numbers. In this unit, students will revisit standard 3.NF.A.1, 3.NF.A.2, and 3.NF.A.3 which were first introduced in Unit 3 and Unit 5. These standards now include fractions greater than one.

Unit Nine Standards

3.MD.B.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

3.NF.A.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.NF.A.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.




Math Language

denominator fourths fractions halvesincrement key line plot measuremeasurement non-standard units number line numeratorstitle whole numbers

Essential QuestionsHow are units in the same system of measurement related? What strategies could you use to figure out the mass of multiple objects? What happens to an item’s measurement when units are changed?How can I determine length to the nearest ¼?How can I determine length to the nearest half?What are customary and metric units in measurement?




Grade 3 Unit 9 Fractions and Line Plots (14 days)

` Connections/Notes Resources

3.MD.B.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

Students in second grade measured length in whole units using both metric and U.S. customary systems. It‘s important to review with students how to read and use a standard ruler including details about halves and quarter marks on the ruler. Students should connect their understanding of fractions to measuring to one-half and one-quarter inch. Third graders need many opportunities measuring the length of various objects in their environment.

Some important ideas related to measuring with a ruler are: The starting point of where one places a ruler to begin measuring

Measuring is approximate. Items that students measure will not always measure exactly 14

, 12

or one

whole inch. Students will need to decide on an appropriate estimate length. Making paper rulers and folding to find the half and quarter marks will help students develop a

stronger understanding of measuring length.Example:

Students measure objects in their desk to the nearest 12

inch or 14

inch.

Display data on a line plot. How many objects measured?

Objects in My Desk

Measurement in Inches

Teaching Student Centered MathematicsMeasuring Lengths Pgs. 257-260More Than One Way Activity 9.2 pg. 259

LessonsInch by InchData With Line PlotsRepresent Data With Line PlotsAnalyze Data to Problem Solve

Activities and TasksMeasuring to the Nearest Half Inch (K5)Measuring to the Nearest Quarter Inch (K5)Measuring Strips Line Plot (K5)Hook, Line and Sticker (AIMS)Making a Transparent RulerHats off to “Ya” Line Plot TaskLeprechaun Hats for Line Plot Task

Grade 3 Unit 9 Fractions and Line Plots (14 days)

Connections/Notes Resources3.NF.A.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.

Students build fractions from unit fractions, seeing the numerator 3 of 34

as saying that 34

is the quantity you

get by putting 3 of the 14

’s together. They read any fraction this way, and in particular there is no need to

Teaching Student Centered MathematicsActivities for Flexible Thinking with Whole Numbers pg. 136



Unit 10

Fluency with Problem Solving and Computation – Remainder of the Year

This unit is designed for application and review of the problem types for which your students show the least fluency. Refer back to the previous units for comments, clarification, examples, and resources according to the needs of your students. Check data from both district and classroom assessments to provide additional lessons for the weakness. Lots of word problems are included in this unit.

Unit Ten Standards

3.OA.C.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.

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

3.MD.C.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.


3.NF.A.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.NF.A.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.




Unit 10 Fluency with Problem Solving and Computation (32 days)

Connections/Notes Resources3.OA.C.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.

LessonsSolidify Fluency of Grade 3 Standards

Activities and TasksMultiplication and Division Fluency SheetsBeaconlearningcenter.com/WebLessons Beaconlearningcenter.com/ WebLessons/CameronsTripRunning-distance-in-a-weekTriangle Towers (Math Wire)

3.NBT.A.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

LessonsDecompose to SubtractAdd Measurement Using the Standard Algorithm

Activities and TasksAddition Practice with Number Bonds

3.OA.D.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. (This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how

LessonsSolve Two Step Word ProblemsSolve Word Problems in Varied ContextsShare and Critique Peer Strategies for


http://www.khanacademy.org/math/cc-third-grade-math/cc-3rd-mult-div-topic/cc-3rd-two-step-word-problems/v/running-distance-in-a-week

http://www.beaconlearningcenter.com/WebLessons/CameronsTrip/default.htm

http://www.beaconlearningcenter.com/WebLessons/CameronsTrip/default.htm

http://www.beaconlearningcenter.com/WebLessons/Hotel6/default.htm




to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Problem Solving

Activities and TasksTwo Step Word Problems #1Squirreling Away AcornsPractice Problems (AIMS)Multi-Step ProblemsProblem Solving PowerPoint (12 problems)Two-step-word-problems

Unit 10 Fluency with Problem Solving and Computation (32 days)

Connections/Notes Resources3.MD.C.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.


LessonsDetermine Areas of Rooms in a Given Floor Plan (Area)Floor Plans (Area)

Composing and Decomposing Shapes Using Tetrominoes (Area)

Solve a Variety of Word Problems Using All Four Operations (Area and Perimeter)

Activities and TasksPerimeter and Area ChallengeFoot StuffThe Area Stays the Same


http://www.khanacademy.org/math/cc-third-grade-math/cc-3rd-mult-div-topic/cc-3rd-two-step-word-problems/v/total-seats-in-a-theater


3.NF.A.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.NF.A.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.NF.A.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.

LessonsPartition Wholes Into Equal Parts Using the Number Line Method

Activities and TasksFraction FlagPattern Blocks Fraction GameFraction Sort Lesson SeedCover Up Uncover


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