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Big Idea 1 – 3 rd Grade Math

This outline provides an example of how teachers will integrate performance tasks into a unit. Teachers should use this unit outline as it is described below. 3rd Grade Mathematics – Multiplication & Division within 100: The Foundation Facts Unit Topic and Length: ● In Grade 2 students have added groups of objects by skip-counting and using repeated addition (2.OA.4). In this unit students connect these concepts to multiplication and division by interpreting and representing products and quotients. Students begin developing these concepts by working with numbers with which they are more familiar, such as 2s, 5s, and 10s, in addition to numbers that are easily skip counted, such as 3s and 4s. Since multiplication is a critical area for Grade 3, students will build on these concepts throughout the year, working towards fluency by the end of the year. Another emphasis of this unit is for students to develop a solid understanding of the connection between multiplication and division. Students should recognize that multiplication strategies can be used to make sense of and solve division problems. This unit should provide students a solid foundation in solving problems with equal groups and arrays, as well as help develop a conceptual understandingof decomposing multiplication problems through the use of the distributive property. Students are not required to use the properties explicitly, but are encouraged to discuss this concept and use area diagrams to support their reasoning.

● REMEMBER: Unit 3 will include elapsed time. In order for your students to be prepared for elapsed time, you will need to review the 2nd Grade skill of telling time to the nearest 5 minutes periodically throughout Quarter 1.

● This unit should happen from August 18 - September 12 of 2014.

Common Core Content Standards: ● 3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. ● 3.OA.2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. ● 3.OA.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.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. 10 min Video - while this video is referring to 3.OA.7, many of the ideas are pertinent to the learning that will be occurring in this unit.

Posters: Kid Friendly Common Core Standards Mathematical Practice Standards

Revised 5/31/14 MP 3 – Justify Arguments & Critique the Reasoning of Others MP 7 – Look for and make use of structure. MP 8 - Look for & Express Regularity in Repeated Reasoning 1 min Video This extended example presents a sequence of eight lessons in which students 1) identify regularities they notice in pairs of related problems, 2) articulate a generalization about the behavior of an operation, 3) explore that generalization, and 4) develop arguments to prove that the generalization is true for all whole numbers. Pages 4-7 include commentary on the video Student Actions Teacher Actions Open-ended Questions ~Share strategies with each other. ~Ask students to explain and show ~What patterns are you noticing? ~Look for connections between how patterns are evident. ~What tools might you use that strategies and problems. ~Allow time daily for students to might help you organize the ~Take risks & make predictions. formulate & explain their ideas to information? ~Question if their prediction or an audience. ~How might a picture or math tool answer makes sense. ~Provide real world problems for help you recognize patterns or students to solve daily. relationships? ~How might you represent what the problem is asking? Math Practices Posters Can be posted for the class or printed on individual cards for small groups. Written in kid- friendly language.

BIG IDEAS/ENDURING UNDERSTANDINGS ESSENTIAL QUESTIONS Central concept, provides a real-world context, & continues Focuses big idea on what is relevant to the student. Provides through to reflection provocative questions that foster inquiry, understanding, and ● Understand and justify the relationship between transfer of learning. multiplication and division. ● Explain how an equation can represent a presented context. Vice versa? ● Identify number patterns in a multiplication or division ● What is the relationship between multiplication & problem. division? ● How can distributive property help you if you don’t ● Skip counting can be used as a strategy to find the remember a multiplication or division fact? total/product of “groups of something”. ● Explain how different models relate to each other.

● Multiples of 2, 5 and 10 can be used as benchmarks with the properties of operations (associative, commutative, and distributive) to solve unknown equations related to multiplication and division.

MISCONCEPTIONS: Students tend to think that they can utilize commutative property in division, such as 3 ÷ 15 = 5 and 15 ÷ 3 = 5. The use of models & discussions is essential in helping students correct this misunderstanding.

Some common misconceptions that students may have are thinking a symbol (? or ⇭ ) is always the place for the answer. This is especially true when the problem is written as 15 ÷ 3 = ? or 15 = ⇭ x 3.

Another key misconception is that the use of a symbol to represent a number once cannot be used to represent another number in a different problem/situation. Presenting students with multiple situations in which they select the symbol and explain what it represents will counter

Revised 5/31/14 this misconception.

CONTENT (noun): SKILLS (verb): ● Product ● Interpret

● Whole Number ● Describe

● Quotient ● Apply

● Properties of operations ● Multiply

● Community property of multiplication ● Divide

● Associative property of multiplication ● Justify

● Distributive property ● Prove

KEY TERMS / VOCABULARY:

● Model ● Array ● Divisor ● Diagram ● Row ● Dividend ● Multiplication ● Column ● Quotient ● Product ● Equation ● Inverse Relationship ● Repeated Addition* ● Factor ● Decomposing* ● Skip Counting ● Multiple ● Associative Property* ● Equal Groups ● Commutative Property ● Distributive Property* ● Unit ● Division * Not explicit in lessons

ASSESSMENT EVIDENCE AND ACTIVITIES:

Initial Assessment: *Initial Assessment will not directly align with summative assessment. Initial assessments are to be used to see what strategies and concepts your kids already have, so you can build off that knowledge. This also will give you a heads up to any misconceptions the students have so you can address those in your lessons. Ideally, this can be given the week prior to instruction, so that you have time to plan based on the data collected. Data from initial assessment IS NOT to be considered for grading purposes. Click here for student sheet and teacher musings FORMATIVE ASSESSMENT:

Formative assessments are built into the Conceptua lessons - Daily Lesson Skills Checks and Journal Prompts.

Summative Assessment: Summative assessments will cover all topics in Big Idea 1 and will be a paper pencil test with two components: a constructed response and a Conceptua content-based assessment.

Revised 5/31/14 Content-Based Assessment and scoring guide based upon Conceptua content for Big Idea 1 Big Idea 1 Multiplication and Division Assessment Big Idea 1 Multiplication and Division Scoring Guide

Unit 1 Constructed Response Assessment - Includes PLD on the last page Unit 1 Constructed Response Answers - some questions were slightly altered from engageNY version Students CAN NOT score higher than a “3” on this assessment, based on the factors used in comparison to the PLD

Please scan or photograph Student Samples of the Constructed Response and upload to the link below. We would like examples from ALL Proficiency Levels - Thank You. :) Dropbox For Student Samples of Summative Assessment

Please click on the link below & provide thoughts, questions, and suggestions you have regarding Unit 1. You can comment as you go, or just give an overall summary at the end of the unit. The more specific the feedback, the better we can fine tune the materials. Your feedback will be anonymous, unless you provide your information, however, if we need clarification on your comments, it would be helpful to know who provided it. Feedback Forum for Unit 1

ELD STRATEGIES & SUPPORT ● English Language Development Strategies in Mathematics from ADE site ● Use the Journal Prompts & Contextualized Problems in your grammar lessons. Identify the Parts of Speech that is the focus of the week within the mathematics context. ● Round Robin Protocols Structured discussion formats - can be used with sharing mathematical thinking & reasoning ● OEC Activity: Story Problem Partner Discussion Teacher will need to prepare a set of array cards for each group. Array Picture Cards ● OEC Activity: Story Problem Partner Discussion 2 Same activity as above, but students will use “Things That Come In Groups” posters created as a class instead of array cards. ● The Barrier Game using Array Picture Cards.

Fluency Mental Math

Spiral Review Conceptual Understanding

Word Problems Extended Constructed Response

LEARNING PLAN & ACTIVITIES: STOP - Did you look at the summative assessment before you started planning your instruction? ● CONCEPTUA Grade 3,Unit 1; 1 lesson per day as the foundation of your teaching

Big Idea 1- Multiplication & Division Within 100: The Foundation Facts MENU: Center Activities for students who finish early in Conceptua, if you are not comfortable with centers or Menus, the activities can be done individually. Some of the activities in the Menu are also included as “Additional Resources” for the lessons below.

Revised 5/31/14 TOPIC 1: Multiplication: Groups of 5 & 10 (Aug 18 - 22)

“Bell” Work: post on board - 237 + 429 (use 2 different strategies to solve) Conceptua Lesson 1: Multiplying with Groups of 5 and 10: Modeling and Skip Counting Journal Response: Closure 3.1-1-1- Is skip counting a good strategy to use when you want to find the total number when you combine groups of equal size? Use words and models to explain your thinking. Additional Resource: Things that Come in Groups (Lessons for Introducing Multiplication Grade 3- Marilyn Burns) (Use 5 and 10 only) Have students (or groups) brainstorm a list of things that come in groups of 5 and things that come in groups of 10. Start a class poster(s) to be used throughout the unit. Homework: Students will look at home to find additional items that come in groups of 5 or groups of 10. (For example: Capri Sun Pouches come in boxes of 10)

“Bell” Work: Daily Journal response from 3.1-1-1 How is 3 groups of 5 similar to 5 + 5 + 5? Conceptua Lesson 2: Multiplying with Groups of 5 and 10: Modeling with Arrays Journal Response: 3.1-1-2 Daily Is making an array a good strategy to use when you want to find the total number when you combine groups of equal size? Use the words and models to explain your thinking. The word rows and columns may help you explain your thinking. Additional Resource: Making Arrays and Equations Teacher Directions Making Arrays and Equations-STUDENT activity sheet (Cut and glue activity may take 2 days. Consider having students WRITE the products instead of cutting and gluing.) Homework: Journal from Closure 3.1-1-2 After finishing her kitchen, Maria decided to buy new tile to replace the old tile in her patio. Her patio has 8 rows of 5 large tiles. How many new tiles will Maria need to buy? Draw a model and explain how Maria can find the total number of tiles she needs to buy.

“Bell” Work: Fact Fluency Practice 5 min on Math Magician (just 5’s & 10’s multiplication) ***Yes, you can use any site for fact fluency practice, however, we would recommend using a site that you can chose the factor focus.*** Conceptua Lesson 3: Multiplying with groups of 5 and 10: Writing Equations Journal Response: Closer 3.1-1-3 Marty bought large muffin pans to make muffins. Each pan has 5 rows of 5 muffin cups. How many muffins can he make if he uses 1 pan? How many if he uses 2 pans? Draw a model. Use the words rows and columns to explain your model and how you found the total number of muffins. Additional Resource: CIRCLES & STARS Homework: Teach someone at home how to play circles and stars. Bring your Four Square paper back to class. (If students don’t have dice at home, they can tear up small pieces of paper and write the numbers on them).

“Bell” Work: post on board - Beth has 8 nickels in her pocket. How much money does she have in cents? Show how you know in pictures, numbers and words. Conceptua Lesson 4: Multiplying with Groups of 5 and 10: Commutative Property Journal Response: Closure from 3.1-1-4 Draw 2 different array models for the factors 3 and 5. Using the words rows and columns, explain how the models are similar and different. Do the models show the same product? Why or Why not? Additional Resource: Daily Journal Prompt from 3.1-1-4 Emelia says that the order in which you multiply 2 factors does not change the total. Do you agree or disagree with Emelia? Use the factors 5 and 10 to explain your thinking. Draw models if you wish. Homework: Multiply by 5s Fluency Practice (link is to a fluency worksheet generator--be sure to focus on groups of 5 or 10).

Revised 5/31/14 “Bell” Work: Daily Journal 3.1-1-3 Draw 2 different models that show 4 groups of 5. Explain how your models are similar and how they are different. Conceptua Lesson 5: Multiplying with Groups of 5 and 10: Numbers Only Journal Response: Daily 3.1-1-5 Write a story problem that uses the factors 4 and 10. Write the equation for the problem. Explain how you know your equation correctly solves the problem. Additional Resource: Fact Fluency Practice Math Magician (just 5’s & 10’s multiplication), Conceptua Tool Investigation Multiplication: Groups of 5 and 10 Homework: PROBABLY A FRIDAY...NO HW

TOPIC 2: Division: Divisors of 5 and 10 (Aug 25 - 29)

“Bell” Work: post on board - 834 - 237 (Use two different strategies to solve) Conceptua Lesson 1: Dividing with Divisors of 5 and 10: Modeling Journal Response: Closer 3.1-2-1. Draw 2 different models that show how many groups of 10 are in 30. Explain how your models are similar and how they are different. Additional Resource: Commutative Property Review Homework: Draw 2 different models that show how many groups of 5 are in 30. Explain how your models are similar and how they are different.

“Bell” Work: Daily Journal 3.1-2-1 Explain how we can use an array model to find out how many groups of 5 are in 20. Be sure to draw a model and use the words rows and columns in your explanation. Conceptua Lesson 2: Dividing with Divisors of 5 and 10: Writing Equations Journal Response: Closer 3.1-2-2 Marissa's teacher gave the class 6 math problems for homework. Marissa thinks she can finish the homework in 1 hour (60 minutes) if she spends 10 minutes on each problem. Do you agree or disagree with Marissa? Explain your thinking. Additional Resource: Array Picture Cards You can have students focus on division, or you can have them do both multiplication and division as the directions say. Homework: Daily Journal 3.1-2-2 Sam has collected 35 bugs for a science project. If he puts 5 bugs in each jar, how many jars will he need? Draw a model and explain how Sam can use division to solve this problem.

“Bell” Work: Fact Fluency Practice 5 min on Math Magician (just 5’s & 10’s division) Conceptua Lesson 3: Modeling Inverse Relationships with Factors of 5 and 10 Journal Response: Closer 3.1-2-3 Drew says he can easily solve the problem 40 ÷ 10 because he knows 10 x 4 = 40. What does he mean? Draw models if you wish. Additional Resource: Give the answer and ask for the fact. The product is 10, what could the fact be? The answer is 30, what could a fact be? The quotient is 6, what is the question? Stick to multiples of 5 and 10 for now. Homework: Students will write two related story problems (one multiplication and one division) using things that come in groups of 5 or 10 (Teacher may want to review “Things that come in groups” posters before students go home, or you may want to assign the facts to be used).

“Bell” Work: Daily Journal 3.1-2-3 Roger is trying to solve the problem ? x 5 = 15. He says that division can be used to find an unknown. Do you agree or disagree with Roger? Explain your thinking. Draw models if you wish.

Revised 5/31/14 Conceptua Lesson 4: Dividing with Divisors of 5 and 10: Numbers Only Journal Response: Closer 3.1-2-4 Mr. James wants some students to present their science fair projects at Parent Night. He has 20 minutes planned for all of the presentations. If each student is given 5 minutes, how many students will be able to present? Write an equation for this problem and explain your thinking. Additional Resource: Solving Division Story Problems with 5 and 10 Homework: Division Fact Fluency Practice

ADDITIONAL RESOURCE on DIVISION: Story Problems Dividing by 5 and 10

TOPIC 3: Multiplication: Groups of 0, 1, 2, and 4 (Sep 2 - 5) “Bell” Work: post on board - Mr. Ruiz gave the following problem to his class: 742 - 386. Jesus said the difference was 444. Alex said the difference was 356. a) Who has the correct answer? b) Explain to the boy who got it incorrect what he did wrong. Conceptua Lesson 1: Multiplying with Groups of 0, 1, 2, and 4: Modeling Journal Response: Daily 3.1-3-1 Marta says that 4 groups of 0 is 4. Do you agree or disagree with Marta? How can you use repeated addition or a model to explain your thinking? Additional Resource: Practice with arrays *Student pages ask students to draw arrays, which will be necessary for the summative assessment. Pages 10 and 11 are intended to be used as homework (below). The entire lesson is included as reference material for teachers. Homework: Homework Lesson 1

“Bell” Work: Fact Fluency Practice 5 min on Math Magician (0, 1, 2, 4, 5, 10 multiplication) Conceptua Lesson 2: Multiplying with Groups of 0, 1, 2, and 4: Writing Equations Journal Response: Closer 3.1-3-2 Mr. Santos has a plan to build a brick wall on one side of his new patio. The wall will be 4 rows with 10 bricks in each row. How many bricks will he need to purchase at the hardware store to build the wall? Draw a model and explain your answer using the words rows and columns. Additional Resource: Practice with equal groups *Student pages (pp7-11) introduce wording/terminology that will be used on the summative assessment. Pages 10-11 are intended to be used as homework (below). The entire lesson is included as reference material for teachers. Homework: Homework Lesson 2

“Bell” Work: Daily Journal 3.1-3-2 Draw 2 different models that help you find the total in 8 groups of 2. Explain how your models are similar and how they are different. Conceptua Lesson 3: Multiplying by 1, 2, and 4: Commutative Property Journal Response: Daily 3.1-3-3 Sandra says since she knows the product for 4 x 5 is 20, she can use the commutative property to find the product for 5 x 4. What does she mean? Draw models to help you explain your thinking. Additional Resource: Fact Fluency Practice Math Magician (0, 1, 2, 4, 5, 10 multiplication) Homework: Journal Closer 3.1-3-3 Draw models on a number line for the factors 6 and 2. Do the models show the same product? Why or Why not?

“Bell” Work: Journal Daily 3.1-3-4 Mrs. Rogers bought a box of 40 crayons. She wants to give 8 crayons to each of her 4 children so they can color. Does she have enough crayons? Write an equation and explain your thinking. Conceptua Lesson 4: Multiplying with Groups of 0, 1, 2, and 4: Numbers Only Journal Response: Closer 3.1-3-4 Write a story problem that uses the factors 4 and 2. Write the equation for the problem. Explain how you know your equation correctly solves the problem. Additional Resource: Midtown Movie Schedule - Real World Investigation (Do First Problem (pg 2) with whole class) Lesson Plan Revised 5/31/14 Conceptua Tool Investigation Multiplication: Groups of 0, 1, 2, and 4 Homework: PROBABLY A FRIDAY - NO HW

TOPIC 4: Division: Groups of 0, 1, 2, & 4 (Sep - 8 - 12)

“Bell” Work: post on board - The headline of the Sunday paper said the following: 1,200 attend Wildcat Baseball Game. Mrs. Estrella said that there were one thousand two hundred people at the baseball game on Saturday. Mr. Estrella said there were twelve hundred people at the game. Is one of them correct, neither correct, or are they both correct? Why? Conceptua Lesson 1: Modeling with Divisors of 0, 1, 2, and 4: Find the Number of Groups Journal Response: Daily 3.1-4-1 Draw models to show how many groups of 4 are in 16 and how many groups of 2 are in 16. What do you notice about the models you have drawn? How do they compare to each other? Additional Resource: Things that Come in Groups (Lessons for Introducing Multiplication Grade 3- Marilyn Burns) (Add things that come in groups of 1, 2, and 4 ) Add to class poster to be used throughout the unit. Homework: Students will look at home to find additional items that come in groups of 1, 2, and 4.

“Bell” Work: Fact Fluency Practice 5 min on Math Magician (0, 1, 2, 4, 5, 10 division) Conceptua Lesson 2: Dividing with Divisors of 0, 1, 2, and 4: Writing Equations Journal Response: Closer 3.1-4-2 Miss Kim's largest class has 40 students in it. She wants only 4 students sitting at each table. How many tables does she need? She already has 8 tables in her classroom. How many more tables does she need? Additional Resource: Practice with Division *Student pages (pp 6-10) ask students to circle “groups of” or make models to represent division problems. Pages 9-10 are intended to be used as homework (below). The entire lesson is included as reference material for teachers. DON’T USE THE ‘APPLICATION’ PROBLEM FROM THE MAIN LESSON. It uses “Equal Shares Division” which will not be introduced until later. Homework: Lesson 2 Homework

“Bell” Work: Post on Board - Beth has 45 cents ($0.45) in nickels in her pocket. How many nickels does she have? Show how you know in pictures, numbers, and words. Conceptua Lesson 3: Modeling Inverse Relationships with Factors of 1, 2 and 4 Journal Response: Closer 3.1-4-3 Andre explained that he used multiplication to help him solve a division problem. Explain how you can use multiplication to solve to find out how many groups of 4 are in 12. Additional Resource:Practice with Inverse Relationships *Student pages (pp6-10) expose students to wording/terminology that will be used on the summative assessment. Pages 9-10 are intended to be used as homework. The entire lesson is included as reference material for teachers. Homework: Lesson 3 Homework

“Bell” Work: Fact Fluency Practice 5 min on Math Magician (0, 1, 2, 4, 5, 10 division) Conceptua Lesson 4: Dividing with Divisors of 1, 2, and 4: Numbers Only Journal Response: Daily 3.1-4-4 Write a story problem that uses as starting value of 24 and equal groups of 4. Write the equation for the problem. Explain how you know your equation correctly solves the problem in your story problem. Additional Resource: Midtown Movie Schedule - Real World Investigation (Allow students to work on remaining two problems, assigning based on level of scaffolding needed).

Revised 5/31/14 Conceptua Tool Investigation Division: Divisors of 0, 1, 2, and 4 Homework: How are Division and Multiplication Alike?

GIVE FINAL ASSESSMENT ON FRIDAY, Sep 12

RESOURCES: ● Multiplication Flipbooks - Lesson Plan ● Build arrays with counters or cubes. Ask the students to give the multiplication fact and related division fact, or vice versa. ● I have Who has. (2 and 10) ● I have; Who has (2 and 5) ● Game - Multiplication Bump ● Multiplication Practice by specific number Interactive - (left hand side - can be timed) ● Math Basketball - students will practice the commutative property ● Engage NY Unit Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 ***This is one of New York’s units - it’s 25 days of lessons, drills, etc. - DO NOT PRESS PRINT! It’s 280 pages This can be used throughout Big Idea 1 & 2, just make sure you modify any materials so that it fits within the benchmark factor constraints of Conceptua.

● Georgia's Unit The Relationship Between Multiplication and Division ***This is one of Georgia’s units - it’s about a month’s worth of lessons, games, etc. - DO NOT PRESS PRINT! It’s 162 pages. This can be used throughout Big Idea 1 & 2, just make sure you modify any materials so that it fits within the benchmark factor constraints of Conceptua.

Children’s Related Literature Sampling: - One Hundred Hungry Ants By Elinor J Pinczes - Amanda Bean's Amazing Dream (Marilyn Burns Brainy Day Books)By Cindy Neuschwander - Each Orange had Eight Slices By Paul Giganti

Professional Literature Resources and Research Articles: OA Progressions Document While the entire document has valuable information, on a first read through, you might look at the overview (pgs 6-7) and the 3rd grade specific content (pgs. 22 -28). Then you might look at 2nd grade, to see what the students should be coming to you with, and then 4th grade, to see what you are preparing them for. New Twists on Old Standards This is an excerpt (1 ½ pages) from an article written by Jason Zimba, PhD (one of the lead writers of the Common Core math standards). He talks about the phrases in the new standards that distinguish them from what we have previously taught. - Arizona Common Core Coach Mathematics (Kaplan, Jerry Dr.) Mathematics Fluency: A Balanced Approach - 2 min video about the balance between fluency and conceptual understanding (the 2 gentlemen are 2 of the writers of the math standards) - Total Recall: Helping Our Students Memorize Multiplication Facts - Great Resources

Websites directly related to the unit –

For teachers: 7 min Video of sample Classroom Instruction & Number Talk Multiplication Strategies Posters

Revised 5/31/14 A Factivation! Short Video on Teacher Tube- the CONCEPT of MULTIPLICATION

You Tube Video - Division on a Number Line

For students: ● Mangahigh Once the teacher creates a class, challenges can be assigned to students where they can practice desired skills.

● Sumdog Different skills can be practiced once the teacher creates a class.

● Math Magician Good website for the students to practice all math facts. Students can select the operation they want to practice as well the fact set (5s, 10s).

● Illuminations - Concentration Game

Revised 5/31/14

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