Third Grade Unit 3: Patterns in Addition and Multiplication 9 weeks In this unit students will: Apply properties of operations (commutative, associative, and distributive) as strategies to multiply and divide Fluently multiply and divide within 100, using strategies such as the patterns and relationships between multiplication and division Understand multiplication and division as inverse operations Solve problems and explain their processes of solving division problems that can also be represented as unknown factor multiplication problems.● Understand concepts of area and relate area to multiplication and addition.● Find the area of a rectangle with whole- number side lengths by tiling it.● Multiply side lengths to find areas of rectangles with whole-number side lengths in context of solving real world and mathematical problems.● Construct and analyze area models with the same product.● Describe and extend numeric patterns.● Determine addition and multiplication patterns.● Understand the commutative property’s relationship to area.● Create arrays and area models to find different ways to decompose a product.● Use arrays and area models to develop understanding of the distributive property.● Solve problems involving one and two steps and represent these problems using equations with letters such as “n” or “x” representing the unknown quantity.● Create and interpret pictographs and bar graphs.Unit Resources:Unit 3 Overview Video Parent Letter Parent Standards Clarification Number Talks Vocabulary Cards Prerequisite Skills Assessment Sample Post Assessment Student Friendly Standards Concept Map

Topic 1: Multiplication and Division Big Ideas/Enduring Understandings: Multiplication and division can be modeled with arrays. The distributive property of multiplication allows us to find partial products and then find their sum. Area models are related to addition and multiplication. Area models of rectangles and squares are directly related to the commutative property of multiplication. Area models can be used as a strategy for solving multiplication problems.Essential Questions: How does understanding the properties of operations help us multiply large numbers? What is an area model and how can it help us find the product? What is an array and how can it help us find the product or quotient? How does understanding the distributive property help us multiply large numbers?Content Standards

1Third Grade Unit 3 8/10/2016

Content standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. MGSE3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g.,

by using drawings and equations with a symbol for the unknown number to represent the problem. See Glossary: Multiplication and Division Within 100. MGSE3.OA.5 Apply properties of operations as strategies to multiply and divide.1 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.)

MGSE3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

MGSE3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Vertical Alignment Second Grade Standards MGSE2.NBT.2 Skip-count by 5s, 10s, and 100s,

10 to 100 and 100 to 1000. MGSE2.OA.3 Determine whether a group of

objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends

MGSE2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

MGSE2.G.3 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Fourth Grade Standards MGSE4.OA.1 Understand that a multiplicative

comparison is a situation in which one quantity is multiplied by a specified number to get another quantity. a. Interpret a multiplication equation as a comparison

e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.

MGSE4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

MGSE4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models

Fifth Grade Standards MGSE5.NBT. 1 Recognize that in a multi-

digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

MGSE5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Instructional StrategiesOA.3

1

2Third Grade Unit 3 8/10/2016

This standard references various strategies that can be used to solve word problems involving multiplication & division. Students should apply their skills to solve word problems. Students should use a variety of representations for creating and solving one-step word problems, such as: If you divide 4 packs of 9 brownies among 6 people, how many brownies does each person receive? (4 x 9 = 36, 36 ÷ 6 = 6).

See the table above for examples of a variety of problem solving contexts, in which students need to find the product, the group size, or the number of groups. Students should be given ample experiences to explore and make sense of ALL the different problem structures.

Examples of Multiplication:There are 24 desks in the classroom. If the teacher puts 6 desks in each row, how many rows are there? This task can be solved by drawing an array by putting 6 desks in each row. This is an array model.

This task can also be solved by drawing pictures of equal groups. 4 groups of 6 equals 24 objects

A student could also reason through the problem mentally or verbally, “I know 6 and 6 are 12. 12 and 12 are 24. Therefore, there are 4 groups of 6 giving a total of 24 desks in the classroom.”

A number line could also be used to show jumps of equal distance.

Students in third grade students should use a variety of pictures, such as stars, boxes, flowers to represent unknown numbers (variables). Letters are also introduced to represent unknowns in third grade.

Examples of Division:There are some students at recess. The teacher divides the class into 4 lines with 6 students in each line. Write a division equation for this story and determine how many students are in the class (n ÷ 4 = 6. There are 24 students in the class).

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).

3Third Grade Unit 3 8/10/2016

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:

Examples of Division Problems:Determining the number of objects in each share (partitive division, where the size of the groups is unknown):Example: 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):Example: 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?

Solution: The bananas will last for 6 days.

OA.5The focus for Q1 is the commutative and associative properties. The focus for Q2 is the distributive property. This standard references properties of multiplication. While students DO NOT need to use the formal terms of these properties, student should understand that properties are rules about how numbers work.

4Third Grade Unit 3 8/10/2016

Students do need to be flexible and fluent applying each of them. 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 how changing the order of how they multiply the numbers does not change the product. They also decompose numbers to build fluency with multiplication.

The associative property states that the sum or product stays the same when the grouping of addends or factors is changed. For example, when a student multiplies 7 x 5 x 2, a student could rearrange the numbers to first multiply 5 x 2 = 10 and then multiply 10 x 7 = 70.

The commutative property (order property) states that the order of numbers does not matter when adding or multiplying numbers. For example, if a student knows that 5 x 4 = 20, then they also know that 4 x 5 = 20.

The array below could be described as a 5 x 4 array for 5 columns and 4 rows, or a 4 x 5 array for 4 rows and 5 columns. There is no “fixed” way to write the dimensions of an array as rows x columns or columns x rows.

Students should have flexibility in being able to describe both dimensions of an array.

Students should be introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don’t know. Students would be using mental math to determine a product.

Here are ways that students could use the distributive property to determine the product of 7 x 6. Again, students should use the distributive property, but can refer to this in informal language such as “breaking numbers apart”.

5Third Grade Unit 3 8/10/2016

Another example of the distributive property helps students determine the products and factors of problems by breaking numbers apart. For example, for the problem 6 x 5= ?, students can decompose the 6 into a 4 and 2, and reach the answer by multiplying 4 x 5 = 20 and 2 x 5 =10 and adding the two products (20+10=30).

To further develop understanding of properties related to multiplication and division, students use different representations and their understanding of the relationship between multiplication and division to determine if the following types of equations are true or false. 0 x 7 = 7 x 0 = 0 (Zero Property of Multiplication) 1 x 9 = 9 x 1 = 9 (Multiplicative Identity Property of 1) 3 x 6 = 6 x 3 (Commutative Property) 8 ÷ 2 ≠ 2 ÷ 8 (Students are only to determine that these are not equal) 2 x 3 x 5 = 6 x 5 10 x 2 < 5 x 2 x 2 2 x 3 x 5 = 10 x 3 1 x 6 > 3 x 0 x 2Students 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, never by 0. They change the order of numbers to determine that the order of numbers does not make a difference in multiplication (but does make

6Third Grade Unit 3 8/10/2016

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.

Use models to 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.

Different representation:An 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.

Students need to apply properties of operations (commutative, associative and distributive) as strategies to multiply and divide. Applying the concept involved is more important than students knowing the name of the property.

7Third Grade Unit 3 8/10/2016

Understanding the commutative property of multiplication is developed through the use of models as basic multiplication facts are learned. For example, the result of multiplying 3 x 5 (15) is the same as the result of multiplying 5 x 3 (15).

Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty. (See example above where students split an array into smaller arrays and add the sums of the groups.

Students’ understanding of the part/whole relationships is critical in understanding the connection between multiplication and division.

OA.7This standard uses the word fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as the distributive property). “Know from memory” does not mean focusing only on timed tests and repetitive practice, but ample experiences working with manipulatives, pictures, arrays, word problems, and numbers to internalize the basic facts (up to 9 x 9).

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number. Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding. Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and experience. Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context. Fluent students:

flexibly use a combination of deep understanding, number sense, and memorization. are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and

making meaning from them. are able to articulate their reasoning. find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf and: http://joboaler.com/timed-tests-and-the-development-of-math-anxiety/By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

8Third Grade Unit 3 8/10/2016

Strategies students may use to attain fluency include: Multiplication by zeroes and ones Doubles (2s facts), Doubling twice (4s), Doubling three times (8s) Tens facts (relating to place value, 5 x 10 is 5 tens or 50) Five facts (half of tens) Skip counting (counting groups of __ and knowing how many groups have been counted) Square numbers (ex: 3 x 3) Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3) Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6) Turn-around facts (Commutative Property) Fact families (Ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24) Missing factors

General Note: Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms. (Problems presented horizontally encourage solving mentally.)

By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing 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 need to understand the part/whole relationships in order to understand the connection between multiplication and division. They need to develop efficient strategies that lead to the big ideas of multiplication and division. These big ideas include understanding the properties of operations, such as the commutative and associative properties of multiplication and the distributive

property. The naming of the property is not necessary at this stage of learning. In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 x 5, and wrote equations to represent the sum. This is called

unitizing. It requires students to count groups, not just objects. They see the whole as a number of groups of a number of objects. This strategy is a foundation for multiplication in that students should make a connection between repeated addition and multiplication.

As students create arrays for multiplication using objects or drawing on graph paper, they may discover that three groups of four and four groups of three yield the same results.

They should observe that the arrays stay the same, although how they are viewed changes. Provide numerous situations for students to develop this understanding.

9Third Grade Unit 3 8/10/2016

To develop an understanding of the distributive property, students need decompose the whole into groups. Arrays can be used to develop this understanding. To find the product of 3 × 9, students can decompose 9 into the sum of 4 and 5 and find 3 × (4 + 5).

The distributive property is the basis for the standard multiplication algorithm that students can use to fluently multiply multi-digit whole numbers in Grade 5.Once students have an understanding of multiplication using efficient strategies, they should make the connection to division.

Using various strategies to solve different contextual problems that use the same two one-digit whole numbers requiring multiplication allows for students to commit to memory all products of two one-digit numbers.

NBT.3This standard extends students’ work in multiplication by having them apply their understanding of place value. This standard expects that students go beyond tricks that hinder understanding such as “just adding zeroes” and explain and reason about their products. For example, for the problem 50 x 4, students should think of this as 4 groups of 5 tens or 20 tens. Twenty tens equals 200.

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. Students may use manipulatives, drawings, document camera, or interactive whiteboard to demonstrate their understanding.Common MisconceptionsOA.5Students may experience difficulty in determining which factor represents rows or the number of objects in a group, and which factor represents the number of groups or columns. In division there are two different situations that can cause confusion depending on which factor is the unknown—the number in the group or

10Third Grade Unit 3 8/10/2016

the number of groups.OA.7Student who struggle most likely do not have fluency for the easy numbers. The child does not understand an unknown factor (a divisor) can be found from the related multiplication. It is not a matter of instilling facts divorced from their meaning, but rather the outcome of carefully designed learning. That involves the interplay of practice and reasoning.DifferentiationIncrease the RigorOA.3 Write a word problem that the number sentence 72 ÷ 9 could be used to solve. Write a word problem with the product of 35. We need 52 juice boxes for our class party. Juice boxes come in packs of 6 or 8. How many packs of each do you need to have enough for each student? Jim purchased 5 packages of muffins. Each package contained 3 muffins. Describe another situation where there would be 5 groups of 3 or 5 x 3. The monkey keeper at the zoo needs 7 apples a day to help feed the monkeys. She has 50 apples at the start of the week. Will she have enough apples for the

entire 7 day week? Explain your reasoning. Lisa has 30 shoes and says that she owns 12 pairs. Explain why she is correct or incorrect.OA.5 Eric says he has more donuts because his mom bought six boxes of four donuts each. Samantha says that she has more donuts because her mom bought four

boxes, each with six donuts. Who is correct? Explain your thinking. Melissa needs to solve 24 x 4 in her head. What strategy should she use? Danielle is trying to multiply a strand of numbers (6 x 3 x 5) in her head but is having trouble keeping them organized in her head. Describe a strategy that she

can use to solve the problem. Malcolm multiplied 3 numbers together and got 24. What 3 numbers could he have multiplied? What strategy did you use to figure the numbers out? Solve 8 x 7 using the distributive property. Could you distribute a different factor or distribute the same factor a different way?OA.7 Sean is have difficulty when multiplying by 9. Kevin tells him if he knows x10 facts, you can quickly solve x9 facts. Do you agree with Kevin? Why or why not? Use

models, equations and/or drawings to support your answer. Explain how x2 facts relate to x4 facts. How do x4 facts help you solve x8 facts? Tara knows strategies for x2, x3, x4, x5, x6, x8, x9, and x10. She asked when she would learn strategies for x7 facts and was told she knew her x7 facts already.

Explain how that is. Heather’s teacher told her that if she knows her x10 facts then she also knows her x5 facts. Heather doesn’t understand how. Explain the relationship between

the 10’s and 5’s facts. Explain a strategy for multiplying by 3. How does knowing your x5 and x2 facts help you to learn your x7 facts?NBT.3 Lukas says that he can’t finish his homework because he doesn’t know how to multiply 5 x 70. Explain a strategy to help him solve this problem. How would adding a 0 to the end of a number affect the value of the digits? (e.g., 75 becoming 750)

11Third Grade Unit 3 8/10/2016

Describe the pattern you see when you multiply 3 x 4 and then 3 x 40. How are the products of 5 x 5 and 5 x 50 similar? How are they different? How does multiplying 10 x 8 help you to solve 9 x 8? Explain how multiplying by multiples of 10 is similar to multiplying by one-digit numbers.Evidence of LearningBy the conclusion of this unit, students should be able to demonstrate the following competencies: Understand the commutative property’s relationship to area. Create arrays and area models to find different ways to decompose a product. Use arrays and area models to develop understanding of the distributive property. Demonstrates Area models and how they are related to addition and multiplication. Use area models as a strategy for solving multiplication problems.Additional Assessment Formative Assessment Lessons - (FAL) Interpret various multiplication strategies MGSE3.OA.1, MGSE3.OA.8, MGSE3.OA.9 (Multiplication)

This formative assessment is designed to be implemented approximately two-thirds of the way through the instructional unit. Shared Assessments: See assessment folder.Adopted ResourcesMy Math:Reminder: The Standard Algorithms for addition and subtraction are taught in grade 4Chapter 4: Understand Multiplication1.1 Hands-On: Model Multiplication1.2 Multiplication as Repeated Addition1.3 Hands-On: Multiply with Arrays1.4 Arrays and MultiplicationChapter 5: Understand Division5.1 Hands-On: Model Division5.2 Division as Equal Sharing5.3 Relate Division and Subtraction5.4 Hands-On: Relate Division and Multiplication5.5 Inverse Operations Chapter 7: Multiplication and Division7.1 Multiply by 37.2 Divide by 37.3 Double a known fact7.4 Multiply by 47.5 Divide by 4

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath Hot Dogs for a Picnic (OA.3) Camping (OA.7) Carpet Caper (OA.7) Equal Snacks (OA.7) Filling the Pool (OA.7) Fish Dilemma (OA.7) Great Pizza Dilemma (OA.7) Harvest Dinner (OA.7) I-Did-A-Read (OA.7)

Think Math (previous adoption)Chapter 9: Multiplication Situations9.1 Practice with Multiplication and Division9.2 Connecting Multiplication and Division9.4 Combining Multiplication and Division9.5 Separating ArraysChapter 12: Multiplication Strategies12.1 Multiplication and Division12.2 Using Sums to Multiply12.3 Multiply with Base Ten Blocks12.4 Multiply with Arrays12.5 Separating Arrays to Multiply12.7 Finding Missing Factors12.8 DivisionChapter 15: Multiplication and Division15.1 Multiply by 10

12Third Grade Unit 3 8/10/2016

7.6 Problem Solve7.7 Multiply by 0 and 17.8 Divide by 0 and 1Chapter 8: Apply Multiplication and Division8.1 Multiply by 68.2 Multiply by 78.3 Divide by 6 and 78.4 Multiply by 88.5 Multiply by 98.6 Divide by 8 and 98.7 Problem SolvingChapter 9: Properties and Equations9.1 Take apart to multiply9.2 Distributive Property9.4 Associative Property9.5 Write equations

Conceptua Math:3.4-2-2 Writing Equations: Equal Shares with Remainders3.2-1.1 Modeling the Distributive Property3.2-1-2 Using Groups of 1, 2, and 5 to Solve for Groups of 3 and 63.2-1-3 Writing Equations with the Distributive Property: Groups of 3 and 63.2-1-4 Multiplying by Groups of 3 and 6: Numbers Only3.2-3-1 Modeling the Distributive Property: Groups of 7, 8 and 93.2-3-2 Using Smaller Group Sizes to Solve for Groups of 7, 8, and 93.2-3-4 Multiplying y 7, 8, and 9: Numbers Only3.4-3-1 Using Groups of 5 and 10 to Solve for Groups of 4 and 93.4-3-2 Writing Equations for Distributive Property for Groups of 4 and 9

Is Dan Losing His Marbles (OA.7) Missing Key Dilemma (OA.7) Mrs. Hasson’s Decorating Dilemma (OA.7) Portfolio Pizza Party (OA.7) Presents (OA.7) Shopping for Shoes (OA.7) Shovel, Shovel, Shovel (OA.7) Sleigh Rides (OA.7) Terrific Tiles (OA.7) Two-Inch Squares (OA.7) What is Fair (OA.7)

13Third Grade Unit 3 8/10/2016

3.4-3-3 Using 5 and 10 Groups to Solve 4 and 9 Groups3.4-3-4 Writing Equations for Distributive Property: Different Number of Groups3.4-4-2 Distributive Property: Separating Factors Into Parts3.4-4-3 Using the Distributive Property to Solve Equations3.4-4-4 Distributive Property: Numbers Only3.2-4-1 Modeling with Divisors of 6, 7, 8, and 9: Find the Number of Groups3.6-1-1 Multiplying Single-Digit Numbers by Multiples of Ten: Unit Names3.6-1-2 Multiplying Single-Digit Numbers by Multiples of Ten: Equations3.6-1-3 Using the Distributive Property with Multiples of Ten: Numbers OnlyWeb Resources: K-5 Math Teaching Resources http://www.k-5mathteachingresources.com/3rd-grade-number-activities.htmlOA.3Word Problems (Arrays) Set 1OA.5Turn Your ArrayDecompose a Factor (ver. 1)OA.7Domino MultiplicationMultiplication Bump (x2 – x5)Multiples Game (x2 – x5)Multiplication Four in a Row (x1, 2, 5, 10)I have…Who has (x2 & x5)I have…Who has (x2 & x10)I have…Who has (x3 & x5)Six SticksDivision Race 1 (Divisors 2, 5, 10)Division Squares (Divisors 2, 5, 10)NBT.3

14Third Grade Unit 3 8/10/2016

Multiples of Ten MultiplyIllustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources. https://www.illustrativemathematics.org/OA.3Gifts from Grandma (variation 1)Two Interpretations of DivisionAnalyzing Word Problems Involving MultiplicationOA.5Valid Equalities (Part 2)OA.7Kiri’s Multiplication Matching GameNBT.3How Many Colored Pencils?Learn Zillion https://learnzillion.com/resources/73932OA.3Solve Word Problems Using the Idea of Equal GroupsSolve Word Problems About Equal Groups by Drawing a ModelSolve Measurement Problems by Drawing a ModelSolve Equal Groups Problems Using ArraysOA.5Understand the Commutative Property by Naming ArraysUnderstand the Commutative Property of Multiplication in Word ProblemsUnderstand Multiplication and Division RelationshipsOA.7Multiply Using Doubles PatternMultiply Using the Half-of-Ten StrategyMultiply by Subtracting from Groups of TenMultiply by Combining Known FactsDivide Using Fact FamiliesNBT.3Multiply by Multiples of 10 with Base-Ten BlocksMultiply by Multiples of 10 Using Number LinesNational Council of Teachers of Mathematics, Illuminations: All About Multiplication – Exploring equal sets http://illuminations.nctm.org/Lesson.aspx?id=1254 Suggested Manipulativessets of countersopen number linesobjects to share

Vocabulary area modelarraycommutative property of multiplication

Suggested Literature My Full Moon is Square Stacks of TroubleToo Many KangaroosThe Best of Times

15Third Grade Unit 3 8/10/2016

multiplication chart distributive property of multiplicationdivideequationfactormultiplyoperationproductquotient

The Doorbell RangBigger, Better, Best! Sam’s Sneaker Squares Two of EverythingSea SquaresOne Hundred Hungry AntsStay in LineClean Sweep CampersThings that Come in 2’s, 3’s & 4’s Amanda Bean’s Amazing DreamBats on ParadeSpunky Monkeys on ParadeEach Orange Had Eight Slices Racing Around Sam’s Sneaker Squares Chickens on the Move

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State TasksTask Name Task Type

Grouping StrategySkills Standards Description

Skittles Cupcake Combos

Constructing TaskIndividual/Partners Division MGSE3.OA.3 This task assesses students’ understanding of division

and their ability to organize data.

Arrays on the Farm Scaffolding TaskSmall Group/Partners Multiplication and Arrays MGSE3.OA.5

In this task the students use arrays to solve multiplication problems. Farmers grow their crops in arrays to make them easier to look after and to harvest.

Seating Arrangements Constructing TaskIndividual/Partners Multiplication and Arrays MGSE3.OA.5 In this task, students will solve a word problem

requiring them to make arrays using the number 24.Making the “Hard” Constructing Task Distributive Property of MGSE3.OA.5 In this task, students will practice using the

16Third Grade Unit 3 8/10/2016

Facts Easy Small Group/Partners Multiplication MGSE3.OA.7

distributive property.

Third Grade Unit 3: Patterns in Addition and Multiplication

17Third Grade Unit 3 8/10/2016

Topic 2: Problem Solving & Arithmetic PatternsBig Ideas/Enduring Understandings: Solve two-step word problems with the four operations Describe and extend numeric patterns that involve addition and multiplicationEssential Questions: How can I use different strategies to help me solve problems with the four operations? How do we determine the sequence in an arithmetic pattern? How do we extend an arithmetic pattern that involves addition and multiplication?Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. MGSE3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown

quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

MGSE3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Vertical AlignmentSecond-Grade Standards MGSE2.NBT.2 Skip-count by 5s, 10s, and 100s,

10 to 100 and 100 to 1000. MGSE2.OA.1 Use addition and subtraction

within 100 to solve one- and two-step word problems by using drawings and equations with a symbol for the unknown number to represent the problem. Problems include contexts that involve adding to, taking from, putting together/taking apart (part/part/whole) and comparing with unknowns in all positions.

MGSE2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends

Fourth-Grade Standards MGSE4.OA.2 Multiply or divide to solve word

problems involving multiplicative comparison. Use drawings and equations with a symbol or letter for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

MGSE4.OA.3 Solve multistep word problems with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Fifth-Grade Standards MGSE5.OA.3 Generate two numerical patterns

using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Instructional Strategies

18Third Grade Unit 3 8/10/2016

OA.8This standard refers to two-step word problems using the four operations. The size of the numbers should be limited to related 3rd grade standards (e.g., 3.OA.7 and 3.NBT.2). Adding and subtracting numbers should include numbers within 1,000, and multiplying and dividing numbers should include single-digit factors and products less than 100.

This standard calls for students to represent problems using equations with a letter to represent unknown quantities.

Example: Mike runs 2 miles a day. His goal is to run 25 miles. After 5 days, how many miles does Mike have left to run in order to meet his goal? Write an equation and find the solution (2 x 5 + m = 25).

This standard refers to estimation strategies, including using compatible numbers (numbers that sum to 10, 50, or 100) or rounding. The focus in this standard is to have students use and discuss various strategies. Students should estimate during problem solving, and then revisit their estimate to check for reasonableness.

Example: Here are some typical estimation strategies for the problem: On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. How many total miles did they travel?

The assessment of estimation strategies should only have one reasonable answer (500 or 530), or a range (between 500 and 550). Problems should be structured so that all acceptable estimation strategies will arrive at a reasonable answer. Students should be expected to explain their thinking in arriving at the answer.

It is important that students 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

19Third Grade Unit 3 8/10/2016

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 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 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.

Student should use various estimation skills solve word problems. They should include: identifying when estimation is appropriate determining the level of accuracy needed selecting the appropriate method of estimation 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:

1. (using the highest place value and estimating from the front end making adjustments to the estimate by taking into account the remaining amounts) 2. rounding and adjusting (students round down or round up and then adjust their estimate depending on how much the rounding changed the original

values)

20Third Grade Unit 3 8/10/2016

Students gain a full understanding of which operation to use in any given situation through contextual problems. Number skills and concepts are developed as students solve problems. Problems should be presented on a regular basis as students work with numbers and computations.

Researchers and mathematics educators advise against providing “key words” for students to look for in problem situations because they can be misleading. Students should use various strategies to solve problems. Students should analyze the structure of the problem to make sense of it. They should think through the problem and the meaning of the answer before attempting to solve it. (M. Burns)

Encourage students to represent the problem situation in a drawing or with counters or blocks. Students should determine the reasonableness of the solution to all problems using mental computations and estimation strategies.

Students can use base–ten blocks on centimeter grid paper to construct rectangular arrays to represent problems.

Students are to identify arithmetic patterns and explain the patterns using properties of operations. They can explore patterns by determining likenesses, differences and changes. Use patterns in addition and multiplication tables.

OA.9This standard calls for students to examine arithmetic patterns involving both addition and multiplication.

Students gain a full understanding of which operation to use in any given situation through contextual problems. Number skills and concepts are developed as students solve problems. Problems should be presented on a regular basis as students work with numbers and computations.

Researchers and mathematics educators advise against providing “key words” for students to look for in problem situations because they can be misleading. Students should use various strategies to solve problems. Students should analyze the structure of the problem to make sense of it. They should think through the problem and the meaning of the answer before attempting to solve it.

Encourage students to represent the problem situation in a drawing or using manipulatives such as counters, tiles, and blocks. Students should determine the reasonableness of the solution to all problems using mental computations and estimation strategies.

Students can use base–ten blocks on centimeter grid paper to construct rectangular arrays to represent problems involving area.

Students are to identify arithmetic patterns and explain these patterns using properties of operations. They can explore patterns by determining likenesses, differences and changes. Use patterns in addition and multiplication tables.

Arithmetic patterns are patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term. This standard also mentions identifying patterns related to the properties of operations.Examples: Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends (14 = 7 + 7).

21Third Grade Unit 3 8/10/2016

Multiples of even numbers (2, 4, 6, and 8) are always even numbers. On a multiplication chart, the products in each row and column increase by the same amount (skip counting). On an addition chart, the sums in each row and column increase by the same amount. Using a multiplication table, highlight a row of numbers and ask students what they notice about the highlighted numbers.

Explain a pattern using properties of operations.

What do you notice about the numbers highlighted in pink in the multiplication table? Explain a pattern using properties of operations.

When (commutative property) one changes the order of the factors they will still get the same product, example 6 x 5 = 30 and 5 x 6 = 30.

Teacher: What pattern do you notice when 2, 4, 6, 8, or 10 are multiplied by any number (even or odd)?

Student: The product will always be an even number.Teacher: Why? In an addition table ask what patterns they notice. Explain why the pattern works this way?

Students need ample opportunities to observe and identify important numerical patterns related to operations. They should build on their previous experiences with properties related to addition and subtraction. Students investigate addition and multiplication tables in search of patterns and explain why

22Third Grade Unit 3 8/10/2016

these patterns make sense mathematically. (MPs 7&8).

All of the understandings of multiplication and division situations, of the levels of representation and solving, and of patterns need to culminate by the end of Grade 3 in fluent multiplying and dividing of all single digit numbers and 10.

It should be clear, this does not mean instilling facts divorced from their meanings, but rather the outcome of a carefully designed learning process that heavily involved the interplay of PRACTICE and REASONING. (Learning Progressions-Operations and Algebraic Thinking K-5).

Examples: Any sum of two even numbers is even. Any sum of two odd numbers is even. Any sum of an even number and an odd number is odd. The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups. The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in a multiplication table fall on horizontal and

vertical lines. The multiples of any number fall on a horizontal and a vertical line due to the commutative property. All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is a multiple of 10.

Students also investigate a hundreds chart in search of addition and subtraction patterns. They record and organize all the different possible sums of a number and explain why the pattern makes sense.

Common MisconceptionsOA.9The student is not able to follow the conventions of order of operations. They randomly attack pairs of numbers without regard for what the associative and distributive properties require. They do not look for and make use of structure (MP7) or they do not follow the “rules of the road”.DifferentiationIncrease the Rigor

23Third Grade Unit 3 8/10/2016

OA.8 About how many days have you been in school? Tell how you estimated and what mathematical operations you used. Ben has two cats. Each cat eats one can of wet cat food per day. Cat food is sold with 24 cans in a case. Ben wants to buy enough cat food for two weeks.

If he buys one case is this enough cat food? Explain your answer. The answer is 42. What is the question? There are 61 third-grade students in Amy’s school. 19 of them are in the library. How many are left in their classrooms? If each classroom holds about 20

students, how many classrooms are likely being used? Andrea had some markers. When she puts them in groups of 3, there was 1 left over. When she put them in groups of 4, there were 3 left over. If she had

fewer than 20 markers, how many could she have had?OA.9 What patterns do you notice on an addition table? What patterns do you notice on a multiplication table? Describe the relationship between multiplying by 2 and multiplying by 4. John says that when you multiply any number by an even number, you will always get an even product. Explain why John is or is not correct. Malik said when you multiply two odd numbers, the product is always odd. He used 3 x 7 and 7 x 7 to support his statement. Is he correct?

Acceleration Intervention

24Third Grade Unit 3 8/10/2016

The Intervention Table provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand’s Numeracy Project.

Evidence of LearningBy the conclusion of this unit, students should be able to demonstrate the following competencies: Describe and extend numeric patterns. Determine addition and multiplication patterns. Solve problems involving one and two steps and represent these problems using equations with letters such as “n” or “x” representing the unknown

quantity.Additional Assessment Shared Assessments: See assessment folder.Adopted ResourcesMy Math:Chapter 6: Multiplication and Division Patterns

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/

Think Math (previous adoption)Chapter 6: Rules and Patterns6.1 Exploring Rules

25Third Grade Unit 3 8/10/2016

Cluster of Standards Name of Intervention Snapshot of summary orStudent I can statement. . .

Materials Master

Operations and Algebraic Thinking

Solve problems involving the four operations, and identify and

explain patterns in arithmetic

MGSE3.OA.8MGSE3.OA.9

Five Sweets Per Packet Solve multiplication problems by skip counting in twos, fives, and tens.

Blank GridsStudents are encouraged to view the multiplication grid in the same way that they would view a hundreds array.

Blank Grid

Multiplication or Out Solve multiplication problems by using repeated addition.

MM 5-2MM 6-2

Twos, Fives, and Tens Solve multiplication problems by using repeated addition.

A Little Bit More/A Little Bit Less

Derive multiplication facts from 2, 5, and 10 times tables.

Fun With Fives Derive multiplication facts from 2, 5, and 10 times tables. MM 4-5

Three’s Company Solve multiplication problems by using repeated addition.

MM 5-2MM 6-2

6.1 Patterns in the Multiplication Table6.2 Multiply by 26.3 Divide by 26.4 Multiply by 56.5 Divide by 56.6 Problem Solving6.7 Multiply by 106.8 Multiples of 106.9 Divide by 10

Conceptua Math:3.2-3-3 Writing equations with the Distributive Property: Groups of 7, 8, and 93.2-4-2 Dividing with Divisors of 6, 7, 8 , and 9: Writing Equations3.2-4-4 Dividing with Divisors of 6, 7, 8, and 9: Numbers Only

login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath Hanging Airplanes (OA.8) Hot Dogs for a Picnic (OA.8) Checkerboard Investigation (OA.9)

6.2 Using Graphs to Find a Rule

Web ResourcesK-5 Math Teaching Resources http://www.k-5mathteachingresources.com/3rd-grade-number-activities.htmlOA.8Word Problems: Two Step (set 2)OA.9Roll a Rule (ver.1)Odd and Even SumsOdd and Even ProductsPatterns in the Multiplication TableIllustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources. https://www.illustrativemathematics.org/OA.8The Stamp CollectionThe Class TripOA.9Addition PatternsSymmetry of the Addition TableMaking a TenPatterns in the Multiplication TableLearn Zillion https://learnzillion.com/resources/73932

26Third Grade Unit 3 8/10/2016

OA.9Identify Patterns on an Addition ChartIdentify Patterns on a Multiplication ChartNational Council of Teachers of Mathematics, Illuminations:– Multiplication: It’s in the Cards – http://illuminations.nctm.org/Lesson.aspx?id=1267Students skip-count and examine multiplication patterns. They also explore the commutative property of multiplication. National Council of Teachers of Mathematics, Illuminations:– Multiplication: It’s in the Cards: Looking for Calculator Patterns http://illuminations.nctm.org/Lesson.aspx?id=1271Suggested Manipulativesopen number linesmultiplication chartaddition charthundred chart

Vocabularypatterndividefactormultiplyoperationproductquotient

Suggested LiteratureMy Full Moon is Square Stacks of TroubleToo Many KangaroosThe Best of TimesThe Doorbell RangBigger, Better, Best! Sam’s Sneaker Squares Two of EverythingSea SquaresOne Hundred Hungry AntsStay in LineClean Sweep CampersThings that Come in 2’s, 3’s & 4’s Amanda Bean’s Amazing DreamBats on ParadeSpunky Monkeys on ParadeEach Orange Had Eight Slices Racing Around Sam’s Sneaker Squares Chickens on the Move

VideosSEDL for OA.9

27Third Grade Unit 3 8/10/2016

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

State TasksTask Name Task Type

Grouping StrategySkills Standards Description

Skip Counting PatternsConstructing Task

Partner/Small GroupAnalyze patterns formed when skip-counting on

the 1-100 chartMGSE3.OA.9

In this task, students look for number patterns relationship to multiplication.

Take The Easy Way Out!

Practice TaskPartner/Small Group

Discovering patterns using a multiplication

chartMGSE3.OA.9

In this task, students will identify patterns and their relationship to multiplication and division.

Read All About It Constructing TaskSmall group/Partner

Applying area and problem solving MGSE3.OA.8 This task provides students with experiences solving

multistep real world problems.

It Takes Two! Constructing TaskIndividual/Partner

Write multiplication story problems MGSE3.OA.8

In this two part task, students will first work in groups to solve two-step word problems. Student groups will then create their own two-step word problems to present to the class to solve.

Hooked On Solutions! Constructing TaskIndividual Task

Writing two-step word problems MGSE3.OA.8 In this task, students will create word problems to

match given equations.

28Third Grade Unit 3 8/10/2016

Third Grade Unit 3: Patterns in Addition and MultiplicationTopic 3: AreaBig Ideas/Enduring Understandings: When finding the area of a rectangle, the dimensions represent the factors in a multiplication problem. Multiplication can be used to find the area of rectangles with whole numbers. Rearranging an area such as 24 sq. units based on its dimensions or factors does NOT change the amount of area being covered (Van de Walle, pg 234).

Ex. A 3 x 8 is the same area as a 4 x 6, 2 x12, and a 1 x 24. Area in measurement is equivalent to the product in multiplication. A product can have more than two factors.Essential Questions: How can area be determined without counting each square? How can the knowledge of area be used to solve real world problems? 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)Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. MGSE3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure 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

MGSE3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). MGSE3.MD.7 Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning

Vertical AlignmentSecond-Grade Standards MGSE2.OA.4 Use addition to find the total number of objects arranged in

rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Fourth-Grade Standards MGSE4.MD.3 Apply the area and perimeter formulas for rectangles in

real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

29Third Grade Unit 3 8/10/2016

MGSE4.MD.8 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.

Instructional StrategiesMD.5These standards call for students to explore the concept of covering a region with “unit squares,” which could include square tiles or shading on grid or graph paper.

Students develop understanding of using square units to measure area by: Using different sized square units Filling in an area with the same sized square units and counting the number of square units An interactive whiteboard would allow students to see that square units can be used to cover a plane figure.

Students can cover rectangular shapes with tiles and count the number of units (tiles) to begin developing the idea that area is a measure of covering. Area describes the size of an object that is two-dimensional. The formulas should not be introduced before students discover the meaning of area.

The area of a rectangle can be determined by having students lay out unit squares and count how many square units it takes to completely cover the rectangle completely without overlaps or gaps.

Students need to develop the meaning for computing the area of a rectangle. A connection needs to be made between the number of squares it takes to cover the rectangle and the dimensions of the rectangle. Ask questions such as: What does the length of a rectangle describe about the squares covering it? What does the width of a rectangle describe about the squares covering it?

The concept of multiplication can be related to the area of rectangles using arrays. Students need to discover that the length of one dimension of a rectangle tells how many squares are in each row of an array and the length of the other dimension of the rectangle tells how many squares are in each column. Ask questions about the dimensions if students do not make these discoveries. For example:

How do the squares covering a rectangle compare to an array? How is multiplication used to count the number of objects in an array?

MD.6Students should be counting the square units to find the area could be done in metric, customary, or non-standard square units. Using different sized graph paper, students can explore the areas measured in square centimeters and square inches.

An interactive whiteboard may also be used to display and count the unit squares (area) of a figure.MD.7

30Third Grade Unit 3 8/10/2016

Students should tile rectangle then multiply the side lengths to show it is the same. To find the area one could count the squares or multiply 3 x 4 = 12.

Students should solve real world and mathematical problems.Example: Drew wants to tile the bathroom floor using 1 foot tiles. How many square foot tiles will he need?

This standard extends students’ work with the distributive property. For example, in the picture below the area of a 7 x 6 figure can be determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums.

Students tile areas of rectangles, determine the area, record the length and width of the rectangle, investigate the patterns in the numbers, and discover that the area is the length times the width.

Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty. For example, students can split a 6 x 9 array into 6 groups of 5 and 6 groups of 4; then, add the sums of the groups.

The 6 groups of 5 is 30 and the 6 groups of 4 is 24. Students can write 6 x 9 as 6 x 5 + 6 x 4. 31

Third Grade Unit 3 8/10/2016

Students’ understanding of the part/whole relationships is critical in understanding the connection between multiplication and division.

Students should also make the connection of the area of a rectangle to the area model used to represent multiplication. This connection justifies the formula for the area of a rectangle.

Example: Joe and John made a poster that was 4ft. by 3ft. Melisa and Debbie made a poster that was 4ft. by 2ft. They placed their posters on the wall side-by-side so that that there was no space between them. How much area will the two posters cover?

Students use pictures, words, and numbers to explain their understanding of the distributive property in this context.

Common MisconceptionsMD.5-MD.7Students may confuse perimeter and area when they measure the sides of a rectangle and then multiply. They think the attribute they find is length, which is perimeter. Pose problems situations that require students to explain whether they are to find the perimeter or area.DifferentiationIncrease the RigorMD.5 Look at a page in the newspaper that has both advertising and news. Which area is greater- the area for the ads or the area for the news? How did

estimating area help you decide your answer? Choose a pattern block. Create a shape made of 20 of your blocks. Now choose another block. How many blocks of this type will you need to make a

shape that takes up the same space? Based on what you know, how would you describe area to a kindergartener? Describe a time when knowing the area of a figure would be important. Using color tiles, what is the area of a piece of paper? What object in the room can you find with the same area?MD.6 Give students grid paper and put them into pairs. Have Student 1 create a rectangle on their grid paper and describe the measurements to Student 2 and

see if they can draw it based on the description. Victor wants to create a 4-sided shape with the largest area possible. He has a choice of sides that are 2 units, 3 units, 4 units, or 1 unit. What

32Third Grade Unit 3 8/10/2016

measurements should Victor use? Use a representation to justify your solution.

Christine has a shape with 4 units in the first row. If she has 5 rows under the first row with 4 units in each, what is the area of her shape?

MD.7 Your grandfather is building a miniature car

out of wood. He cuts two larger rectangles for the sides and two smaller rectangles for the front and back. The total area for the car surface is 54 square units. What could the areas for the sides and front/back pieces be?

An object has an area of 12 square centimeters. What could the length and width be?

Design a shape that has a total area of 60 sq. units and is made up of both squares and rectangles (at least one of each).

Explain the relationship between the two strategies for finding area: counting square units and multiplying the side lengths together? Use an example to support your answer.

If a rectangle has an area of 36 square inches, what could the length and width be? What would the length and width be if the shape were a square? Kelly’s kitchen has a length of 10 ft. and width of 20 ft. Her living room has a length of 32 ft. and width of 40 ft. What is the total square feet of her kitchen

and living room area? Tommy’s bedroom is a rectangle with side lengths of 10 feet and 17 feet. He wanted to calculate the area by multiplying the side lengths, but he thinks it

would be easier to break the 17 into 10 and 7 before multiplying. Can Tommy do this? Explain why it would or would not work.Acceleration Intervention The Intervention Table provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand’s Numeracy Project.

33Third Grade Unit 3 8/10/2016

Cluster of Standards Name of Intervention

Snapshot of summary orStudent I can statement. . . Materials Master

Measurement and Data

Geometric Measurement:

understand concepts of area and relate area to multiplication and to

addition

MGSE3.MD.5MGSE3.MD.6MGSE3.MD.7

Animal Arrays Solve multiplication problems by using repeated addition.

MM 5-2MM 6-2

Turn Abouts Solve multiplication problems by using arrays.

MM 5-2

Number Strips Solve multiplication problems by skip counting in twos, fives, and tens. MM 6-1

Area and Multiplication

Provides a progression: equal groups, arrays, and area.

The Great Cover Up

Cover a shape with non-standard area units and count the number used.

The Great Cover Up PDF

The Array Game

This game allows students to practice their multiplication skills, and reinforces the ‘array’ concept of multiplication.

Third Grade Math Triumphs Intervention (came with the My Math adoption):2.4: Area ModelsEvidence of LearningBy the conclusion of this unit, students should be able to demonstrate the following competencies: When finding the area of a rectangle, understands the dimensions represent the factors in a multiplication problem. Uses multiplication to find the area of rectangles with whole numbers. Rearranges an area models and understands that the factors does NOT change the amount of area being Understands Area in measurement is equivalent to the product in multiplication.Additional Assessment Shared Assessments: See assessment folder.Adopted ResourcesMy MathChapter 13: Perimeter and Area13.3 Hands-On: Understand Area13.4 Measure Area13.5 Hands-On: Tile Rectangles to Find Area13.6 Area of Rectangles13.7 Hands-On: Area and the Distributive Property

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb Email

Think Math (previous adoption)Chapter 10: Length, Area, and Volume10.5 Area and Perimeter10.6 More Area and Perimeter

34Third Grade Unit 3 8/10/2016

Password: cobbmath Miss Guy’s Puppy Problem (MD.5 & MD.6) Carpet Caper (MD.6 & MD.7) Post Office Display (MD.6 & MD.7) Sandbox for Geoffrey (MD.6 & MD.7) Shovel, Shovel, Shovel (MD.6) Stained Glass Surprise (MD.6) Tangram Areas – Grandfather Tang’s Story

(MD.6) Tangram Money (MD.6) Terrific Tiles (MD.6) Two-Inch Squares (MD.6)

Web ResourcesK-5 Math Teaching Resources http://www.k-5mathteachingresources.com/3rd-grade-number-activities.htmlMD.5Square UnitsArea on the GeoboardFind the AreaMD.6Rectangles with Color TilesMD.7Find the Area of a RectangleJack’s RectanglesIllustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources. https://www.illustrativemathematics.org/MD.6Halves, Thirds, and SixthsFinding the Area of Polygons (MD.6 & MD.7)MD.7Three Hidden RectanglesIndia’s Bathroom TilesIntroducing the Distributive PropertyLearn Zillion https://learnzillion.com/resources/73932MD.5Cover the Area of a Shape Using Square UnitsFind the Area of a shape Using Square UnitsMD.6

35Third Grade Unit 3 8/10/2016

Find the Area of a Square or Rectangle by Counting Unit SquaresMD.7Relate Area to ArraysFind the Area of a Rectangle: Using ArraysGiven the Area, Find Missing Side Lengths of a RectangleFind Area Using Distributive PropertyApply the Distributive Property: Using Area ModelsFind Area by MultiplicationMath Playground -This website provides practice with measuring the area and perimeter of rectangles. http://www.mathplayground.com/areaSuggested Manipulativescolor tilespentominoesgeoboardsgrid paper

Vocabularyareaarea modeldistributive property of multiplicationfactormultiplyproductsquare unittilingunknown/variable

Suggested LiteratureMy Full Moon is Square Stacks of TroubleToo Many KangaroosThe Best of TimesThe Doorbell RangBigger, Better, Best! Sam’s Sneaker Squares Two of EverythingSea SquaresOne Hundred Hungry AntsStay in LineClean Sweep CampersThings that Come in 2’s, 3’s & 4’s Amanda Bean’s Amazing DreamBats on ParadeSpunky Monkeys on ParadeEach Orange Had Eight Slices Racing Around Sam’s Sneaker Squares Chickens on the Move

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications.

3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution 36

Third Grade Unit 3 8/10/2016

seeking Act Two, and a solution discussion and solution revealing Act Three. State Tasks

Task Name Task TypeGrouping Strategy

Skills Standards Description

Cover Me Scaffolding TaskPartner/Small Group Task

Analyze the concept of area MGSE3.MD.5 In this task, students investigate area using tangrams.

Fill Er’ Up Constructing TaskPartner/Small Group Task Estimating area MGSE3.MD.5

MGSE3.MD.6In this task, students practice estimating and filling the area of three different figures.

Same But DifferentConstructing Task

Partner/Small Group Task Same area, different dimensions

MGSE3.MD.5MGSE3.MD.6

In this task, students will create different area models for a given product.

Count Me In Constructing TaskPartner/Small Group Area Dimensions MGSE3.MD.5-7 In this task, students create area models and label them

with appropriate dimensions.

Paper Cut 3-Act TaskWhole Group Area Dimensions

MGSE3.MD.5MGSE3.MD.6MGSE3.MD.7

In this task, students will watch a Vimeo and tell what they noticed. Next, they will be asked to discuss what they wonder about or are curious about. Students will then use mathematics to answer their own questions.

Oops! I’m Decomposing!

Constructing TaskPartner/Small Group Task Distributive Property

of Multiplication MGSE3.MD.7

In this task, students will work through problems using area models to understand that numbers can be decomposed into “nice” numbers for multiplication and addition.

Multiplication W/ Base-Ten Blocks

Practice TaskIndividual/Partner Task One-digit by 2-digit

multiplication MGSE3.MD.7In this task, students will model multiplication of 2-digit numbers using base-ten blocks to create partial products.

Olympic Cola Display3-Act Task

Whole Group Distributive property of multiplication MGSE3.MD.7

In this task, students will use their understanding of area models to represent the distributive property to solve problems associated with an Olympic cola display.

Array ChallengePractice Task

Partner/Small Group TaskPracticing

multiplication facts using area models

MGSE3.MD.6MGSE3.MD.7

In this task, students will apply multiplication problems to the matching area model/array

Third Grade Unit 3: Patterns in Addition and MultiplicationTopic 4: Represent and Interpret DataBig Ideas/Enduring Understandings: Interpret data in picture and bar graphs Use information from a picture graph and bar graph to answer questions

37Third Grade Unit 3 8/10/2016

Essential Questions: What type of graph should I use to display data? Why do I need to ask questions and collect data?

Content StandardsContent standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. MGSE3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many

more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Vertical AlignmentSecond-Grade Standards MGSE2.MD.10 Draw a picture graph and a bar graph (with single-unit

scale) to represent a data set with up to four categories. Solve simple put-

together, take-apart, and compare problems using information presented in a bar graph.

Fourth-Grade StandardsMGSE4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1/

2,1/4,1/

8). Solve problems involving addition and subtraction of fractions with common denominators by using information presented in line plots. For example, from a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection.

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

While exploring data concepts, students should 1) Pose a question, 2) Collect data, 3) Analyze data, and 4) Interpret data (PCAI). Students should be graphing data that is relevant to their lives.

Example: Pose a question: What are some of the questions that could be asked of the date we see? Students should come up with a question. What is the typical genre read in our class? Collect and organize data: student survey.

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, scale, categories, category label, and data. Students need to use both horizontal and vertical bar graphs. If you were to purchase a book for the class library which would be the best genre? Why? Example of Scaled Graph:

38Third Grade Unit 3 8/10/2016

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

Analyze and Interpret data which could include: How many more nonfiction books where read than fantasy books? Did more people read biography and mystery books or fiction and fantasy books? About how many books in all genres were read? Using the data from the graphs, what type of book was read more often than a mystery but less often than a fairytale? What interval was used for this scale? What can we say about types of books read? What is a typical type of book read? If you were to purchase a book for the class library which would be the best genre?

Representation of a data set is extended from picture graphs and bar graphs with single-unit scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate to multiplication and division with 100 (product is 100 or less and numbers used in division are 100 or less).

In picture graphs, use values for the icons in which students are having difficulty with multiplication facts. For example, represents 7 people. If there are three , there are 21 people. The intervals on the vertical scale in bar graphs should not exceed 100.

Students are to draw picture graphs in which a symbol or picture represents more than one object). Bar graphs are drawn with intervals greater than one. Ask questions that require students to compare quantities and use mathematical concepts and skills. Use symbols on picture graphs that student can easily represent half of, or know how many half of the symbol represents. Common MisconceptionsMD.3Although intervals on a bar graph are not in single units, students count each square as one. To avoid this error, have students include tick marks between each interval. Students should begin each scale with 0. They should think of skip- counting when determining the value of a bar since the scale is not in single units.Differentiation

39Third Grade Unit 3 8/10/2016

Increase the RigorMD.3 Select a graph type you would use to display the data. Why did you choose that? The set of data describes the ages of a group of people at a family party. 32, 30, 5, 2, 1, 62, 58, 28, 26, 25, 24, 2, 4, 29, 16. Create a graph to display the

data. The students asked 20 people about their favorite ice cream flavors and wanted to make a bar graph to show their data. What would be a good scale to

help organize their bar graph and why? A bar graph shows the greatest data point of 45. The lowest data point is 21. How many more votes is the greatest number than the lowest?Acceleration Intervention Third Grade Math Triumphs Intervention (came with the My Math adoption):9.1: Sort and Classify9.2: pictographs and Picture Graphs9.3: Read Tables9.4: Read Bar Graphs9.5: Make Bar GraphsEvidence of LearningBy the conclusion of this unit, students should be able to demonstrate the following competencies: Create and interpret pictographs and bar graphs.Additional Assessment Shared Assessments: See assessment folder.Adopted ResourcesMy MathChapter 12: Represent and Interpret Data12.1 Collect and record data12.2 Draw scaled picture graphs12.3 Draw scaled bar graphs

Adopted Online ResourcesMy Mathhttp://connected.mcgraw-hill.com/connected/login.doTeacher User ID: ccsde0(enumber)Password: cobbmath1Student User ID: ccsd(student ID)Password: cobbmath1

Exemplarshttp://www.exemplarslibrary.com/User: Cobb EmailPassword: cobbmath

Think Math

Web ResourcesK-5 Math Teaching Resources http://www.k-5mathteachingresources.com/3rd-grade-number-activities.htmlMD.3

40Third Grade Unit 3 8/10/2016

Represent and Interpret DataIllustrative Mathematics provides instructional and assessment tasks, lesson plans, and other resources. https://www.illustrativemathematics.org/MD.3Classroom SuppliesLearn Zillion https://learnzillion.com/resources/73932MD.3Draw Bars on a GraphNational Council of Teachers of Mathematics, Illuminations: Bar Grapher http://illuminations.nctm.org/Activity.aspx?id=4091 This is a NCTM site that contains a bar graph tool to create bar graphs. National Council of Teachers of Mathematics, Illuminations: What’s in a Name? – Creating Pictographs. Students create pictographs and answer questions about the data set. http://illuminations.nctm.org/Lesson.aspx?id=1254Suggested Manipulativestables or charts to collect/organize data

Vocabularypicture graphpictographbar graphscalekeysymbol

Suggested LiteratureMy Full Moon is Square Stacks of TroubleToo Many KangaroosThe Best of TimesThe Doorbell RangBigger, Better, Best! Sam’s Sneaker Squares Two of EverythingSea SquaresOne Hundred Hungry AntsStay in LineClean Sweep CampersThings that Come in 2’s, 3’s & 4’s Amanda Bean’s Amazing DreamBats on ParadeSpunky Monkeys on ParadeEach Orange Had Eight Slices Racing Around Sam’s Sneaker Squares Chickens on the Move

Task DescriptionsScaffolding Task Task that build up to the learning task.Constructing Task Task in which students are constructing understanding through deep/rich contextualized problem solving Practice Task Task that provide students opportunities to practice skills and concepts.Culminating Task Task designed to require students to use several concepts learned during the unit to answer a new or unique situation. Formative Assessment Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key

41Third Grade Unit 3 8/10/2016

Lesson (FAL) mathematical ideas and applications.3-Act Task Whole-group mathematical task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution

seeking Act Two, and a solution discussion and solution revealing Act Three.

State TasksTask Name Task Type

Grouping StrategySkills Standards Description

Subject To Interpretation

Constructing TaskPartner/Small Group

Creating and interpreting pictographs and bar

graphsMGSE3.MD.3

In the following task, students will organize data given to create a picture graph. Students will use the graph to answer word problems.

Watch My Garden Grow

Culminating TaskIndividual Task

Area, multiplication, problem solving, bar

graphs

MGSE3.OA.8MGSE3.MD.3MGSE3.MD.5MGSE3.MD.6MGSE3.MD.7

Students will create a flower garden representing 100 square units. The garden is composed of five rectangular regions, each with a different flower plant. A graph will be completed to represent the number of plants used in the garden. Student will compose word problems that can be answered by analyzing the data in the graph.

Unknown Product Group Size Unknown(“How many in each group? Division)

Number of Groups Unknown(“How many groups?” Division)

3 6 = ? 3 ? – 18, and 18 3 = ? ? 6 = 18, and 18 6 = ?

Equal Groups

There are 3 bags with 6 plums in each bag. How many plums are there in all?

Measurement example. You need 3

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?

Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How

If 18 plums are to be packed 6to a bag, then how many bags are needed?

Measurement example. You have 18 inches of

42Third Grade Unit 3 8/10/2016

lengths of string, each6 inches long. How much string will you need altogether?

long will each piece of string be? string, which you will cut into pieces that are6 inches long. How many pieces of string will you have?

Arrays2, Area3

There are 3 rows of apples with 6 apples in each row. How many apples are there?

Area example. What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3equal rows, how many apples will be in each row?

Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?

Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it?

Compare

A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?

Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3times as long?

A red hat costs $18 and that is3 times as much as a blue hat costs. How much does a blue hat cost?

Measurement example. A rubber band is stretched to be18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat?

Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be18 cm long. How many times as long is the rubber band now as it was at first?

General a b = ? a ? = p, and pa = ? ?b = p, and pb = ?

Word Problem Situations

2 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 window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.3Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations. The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples.

43Third Grade Unit 3 8/10/2016