DUE 6-13: Facilitators Guide Template - CC 6-12.docxmc-14193-39844713.us-east-1.elb.amazonaws.com/file/13621/downloa…  · Web viewYou will have many opportunities to be the “teacher”

Module Focus: Grade 4 – Module 3 Sequence of Sessions

Overarching Objectives of this November 2013 Network Team Institute Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool

for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in order to examine the ways in which these elements contribute to and enhance conceptual understanding.

High-Level Purpose of this Session Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for

teaching these modules. Coherence: P-5.  Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that

develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.   

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.    Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Related Learning Experiences● This session is part of a sequence of Module Focus sessions examining the Grade 4 curriculum, A Story of Units.

Key Points• Number disks and area models are used heavily throughout the module to support the algorithms.

• The multiplication and division algorithms are not expected fluencies in Grade 4.

• Unit language and place value understanding drives the experience of the algorithms.

• Keep a balance of rigor by addressing each component of a lesson.

• Honor and respect the objectives.

• Find a balance between success and mastery.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for teaching these modules.

Coherence: P-5.  Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same .  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.

Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Participants will be able to articulate the key points listed above.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to the Module

8 minutesOverview of the instructional focus of Grade 1 Module 3.

Grade 4 Module 3 Grade 4 Module 3 PPT

Concept Development

110 minutes

Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

Grade 4 Module 3 Grade 4 Module 3 PPT

Module Review 9 minutesArticulate the key points of this session.

Grade 4 Module 3 Grade 4 Module 3 PPT

Session Roadmap

Section: Introduction to the Module Time: 8 minutes

[8 minutes] In this section, you will…

Provide an overview of the instructional focus of Grade 4 Module 3.

Materials used include: none

Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

1 1. NOTE THAT THIS SESSION IS DESIGNED TO BE 127 MINUTES IN LENGTH. To cut down to 120 minutes, shorten Slide 22 by 7 minutes, giving just a brief overview of Lesson 26 and not having participants work through the lesson in its entirety.Welcome! In this Module Focus Session, we will examine the second half of lessons of Grade 4 – Module 3.

1 2. Our objectives for this session are to:• Examination of the development of mathematical

understanding across the module using a focus on Concept Development within the lessons.

• Introduction to mathematical models and instructional strategies to support implementation of A Story of Units.

0 3. We will begin by exploring the module overview to understand the purpose of this module. Then we will dig in to the math of the module. We’ll lead you through the teaching sequence, one concept at a time. Along the way, we’ll also examine the other lesson components and how they function in collaboration with the concept development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole.Let’s get started with the module overview.

0 4. The third module in Grade 4 is Multi-Digit Multiplication and Division. The module includes 38 lessons and is allotted 43 instructional days. Our focus today will be Topics E-H.

3 5. Spend about 2 minutes looking at the Topic Titles for Topics A-D and the Objectives for the first half of lessons (up to Mid-Module Assessment) (Page x of the Module Overview). How do Modules 1 and 2 connect to Module 3? Gain a general understanding for where the module has taken students thus far.• What conceptual understanding have they learned? Multiplicative

Comparison (times as much); Multiplication by 10, 100, and 1000; Single-Digit by Multi-Digit (up to 4-digit) Multiplication; Application of new learning in the form of word problems;

• What models have been used? Diagrams (area and perimeter); number disks and place value charts (to establish place value when multiplying by units of 10, 100, and 1000); arrays (for pictorial representation of place value understanding); area model (to model multiplication goes back to concept of finding area);

• Why might the module begin and end with multiplication? To bring coherence to the lessons within the module; multiplication and

division at the beginning of the module build place value understanding and understanding of the area model that is needed for 2-digit×2-digit multiplication

• Why does the module begin with area and perimeter? To provide context for multiplicative comparison; to build understanding of the process of multiplication and division using the area model; to provide context for word problems

Discuss importance in understanding the flow and pace of the module, keeping the end in mind, so we don’t get “stuck” on a lesson.Finding success in lessons, not complete mastery. Mastery will be accomplished through each lesson, as they build upon themselves.

3 6. Spend about 2 minutes looking at the Topic Titles for Topics E-H and the Objectives for the second half of lessons (Pages xi and xii of the Module Overview). Gain a general understanding for where the Module will take the students.• How does the first half of the Module connect to the second half? The

first half builds the foundation for the division and multiplication work of the second half (place value, area model).

• What conceptual understanding will they learn? Division with remainders; Divisibility; Division of Thousands, Hundreds, Tens, and Ones; 2-digit by 2-digit Multiplication;

• What models will be used? Array/area models; number disks; long division algorithm; place value chart; partial products (two- and four-); standard algorithm for 2-digit by 2-digit multiplication;

• What is important to note regarding the standard algorithms for multiplication and division? Students are not assessed on the standard algorithms for multiplication or division in Grade 4. The algorithm for multiplication is a Grade 5 standard. The algorithm for division is a Grade 6 standard.

Include discussion of the following:• Importance in understanding the flow and pace of the module,

keeping the end in mind; don’t get “stuck” on a lesson.• Finding success in lessons, not complete mastery. Mastery will be

accomplished through each lesson, as they build upon themselves.

Section: Concept Development Time: 111 minutes

[111 minutes] In this section, you will…

Examine of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

Materials used include:personal whiteboards and participant problem set

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

1 7. We will examine the Concept Development of the Second Half of this module by focusing on the progression and coherence of the lessons within and across each topic.We will examine the models, structure, and language used and the movement from concrete to pictorial to abstract.We will go step by step, lesson by lesson, primarily modeling the Concept Developments, but also including some Application Problems, Debriefs, and Fluencies.You will have many opportunities to be the “teacher” at your table. Try to vary who that is throughout the presentation so that everyone has the chance to be comfortable , not only practicing the mathematics, but also presenting the sequence of the lessons to the “students”.

3 8. Take 2 minutes to read Topic Opener E. Read for the standards addressed and the content students will be expected to have understanding of for moving on.Standard:

Content: “Division of tens and ones with successive remainders”• Division types (groups unknown, group size unknown)• Interpreting remainders• Division with arrays and the area model• Division algorithm as it relates to work with number disks and area

models

3 9. Complete first part with participants:10 ÷ 211 ÷ 28 ÷ 49 ÷ 4Let participants practice the rest. Keep in mind that, at this point, the word ‘remainder’ is not used. The idea is for students to gain the conceptual understanding of what is happening. ‘Remainder’ will be introduced in this lesson.

3 10. We start the Topic by connecting to the G3 models of number bonds and arrays.Discuss connection to G3 models using number bonds and arrays to find ‘groups of’.• Model array using 13 ÷6. Have participants complete 17 ÷3 on their

‘Problem Set’ (#1).• Model number bond using 13 ÷6. Have participants complete #2 on

their Problem Set.• Show how the array can be used to show a transition to a tape

diagram. Model how to box the array and to show the remainder.

Have participants complete #3 on their Problem Set.** Lesson 14 connects and advances the array to a tape diagram in order to model word problems.

5 11. Application problem asked students to draw an array to solve for 38 ÷ 4.

Show the model of an array first (groups of 4) transition to an area model, first using graph paper so students can see that they are similar transition to a rectangle, being mindful of proportions.Introduce the language of “The quotient is 9. The remainder is 2.”Have participants complete #4 on the Problem Set.Discuss Debrief questions as a whole group or in tables.• What does the quotient represent in the area model? The side length• When does the area model present a challenge in representing

division problems? When there is a remainder• How is the whole represented in the area model? By the rectangle

and square units• The quotient represents a side length. The remainder consists of

square units. Why? There are not enough units to make another

group.

6 12. Deliver the script: Model for participants how to deliver the Concept Development without reading from the script word for word; Remind them to pay attention to the progression

3 13. Lesson 17’s complexity is to regroup in the tens when there is a remaining number after division in the tens column. Problems are contrasted using remainders in the ones compared to remainders in the tens. We can keep dividing when there is a remainder in the tens.Model for participants the two examples on this slide and then have them work to complete #7 of the Problem Set.

4 14. Lesson 18 provides students with the opportunity to practice dividing two-digit numbers by a single-digit number, regrouping in the tens and having a quotient with a remainder. Students check their work by multiplying. Students visualize the process and work to complete the algorithm without a model.Show participants the sample problem on the slide and talk them through finding the solution.

4 15. In Lesson 20, we look at the area model and use the area model to represent division problems. Students will see that there are many ways to break apart a division problem into pieces that make it easier to solve.In the example, 48 ÷4, students see that we can break the 48 into the more manageable pieces of 40 and 8 or of 20, 20, and 8. The area models represent how we break apart the problem each time. Students will further discover that they could break 48 down to 24 and 24 or even 12, 12, 12, and 12. Each example has the consistent side length of 4.Explain this process to participants and then have them practice with #8 of the Problem Set.

4 16. We can also build from part to whole. We do this by starting with the side length of 4 and then building up to the area of 96. We first give tens to our side length. Can we give one ten? Yes! It’s 40. Can we give two tens to our side length of 4? Yes! That would be 80. Can we give three tens? No! That would be 120. Draw one length of 20. We’ve used 80 square units. Write that in the area model. How many are left? 16. Let’s give ones to our side length. Can we give one one? Yes! That would be 4. Can we give two ones? Yes! That would be 8. How many can we give? 4 because 4 fours is sixteen and that’s how many ones there are. We have a side length of 20 + 4. That equals 24.Explain and model for participants. Further show how the solution using the area model can then relate to the algorithm for division. Participants will complete #9 of the Problem Set.

4 17. Show participants how to solve by building an area model part to whole. Participants can then complete #10 of the Problem Set.

3 18. Take 2 minutes to read Topic Opener F. Read for the standards addressed and the content students will be expected to have understanding of for moving on.Standard:

Content:Find factor pairsDefine prime and compositeTest for factors and observe patternsDetermine whether a number is a multiple of anotherExplore properties of prime and composite

Why is this Topic placed between multiplication and division in the module?Knowledge of factors and multiples helps students to solve multiplication and division problems in a more efficient manner; they are able to determine unknown side lengths and to break apart problems more easily.

3 19. Lesson 22s objective is to find factor pairs. Students use their knowledge of the basic facts for multiplication in order to identify factor pairs. Further, they learn the definitions of prime and composite and identify numbers as one or the other.In Lesson 23, students test for factors and observe patterns. For example, they use division to test for factors and use the associative property as represented below. Explain both methods to participants and then have them answer the questions found on the slide.(Click to advance each question taken from the script.)

3 20. Lesson 24s objective is to determine whether a whole number is a multiple of another number. Students use division and the associative property (as demonstrated below) to determine this.In Lesson 25, students work to complete the Sieve of Eratosthenes to identify the prime numbers to 100. This is a guided lesson. If time allows, have participants complete #11 of the Problem Set.

3 21. Take 2 minutes to read Topic Opener G. Read for the standards addressed and the content students will be expected to have understanding of for moving on.Standards:

Content:

• Use unit language to divide multiples of 10, 100, and 1,000 by single-digit numbers (12 thousands ÷4 = 3 thousands).

• Represent division of up to four-digits with number disks and numerically with and without remainders and requiring decomposing remainders up to three times.

• Solve division problems with a zero in the dividend or in the quotient.

Why is division of larger dividends separated in this module?The work with factors and multiples is helpful when dividing with larger dividends. Work with larger dividends entails the same process as with smaller dividends. It is an extension of that learning.

8 22. Hand out copies of Lesson 26.Give 5 minutes to read the lesson front to back.Mark 2 ah-has relating to concepts, content, or other.Give 2-3 minute for participants to complete the Problem Set. Discuss the importance of completing the Problem Set prior to the Concept Development.Plan how to present this lesson in a shorter amount of time.Share strategies at tables and a few out loud.

4 23. Lesson 27 is an extension of the work done in Topic E in Lessons 16-18. Working with numbers disks to divide will now be familiar to students. The difference here is that this process is repeated given the larger dividend. Walk the participants through the division problem using the script. If time allows, have participants complete #12 of the Problem Set.

4 24. Lessons 28 and 29 build on the work of Lesson 27. Students will divide with larger numbers (up to 4-digits) using the place value chart and the standard algorithm for division side-by-side. Students will see that division with larger numbers follows the same process as that with smaller numbers.Have participants practice solving and walking through a script using unit language to solve Problem #13 on the Problem Set (The same problem that appears on this slide).Debrief: You have practiced division with 2, 3, and 4 digit numbers. Discuss what you think would be true for dividing a number with a greater number of digits.

4 25. At this point, participants have had much practice with hearing, seeing, and delivering unit language when explaining the process of division. Challenge participants to work with a partner to solve one of the problems on the slide. Each partner will solve one problem and then will explain to his/her 4partner how to teach about a zero using unit language. (Problem Set #14)

4

4 26. Lesson 31 guides students through word problems using tape diagrams to model. The model allows students to consider about the type of division and what the quotient means. The example shown in the slide models a ‘number of groups unknown’ problem.The tape diagram shows how the problem can be modeled. Notice the whole of 292 is used to label the length of the tape diagram. The size of the group is modeled on the left side of the tape diagram. The ‘?’ in the remaining section of the tape diagram indicates that we don’t know the number of groups. We can divide to show how many groups of 4 there will be.(Have participants quickly sketch a tape diagram and solve. Show result.)Participants may then solve #15 of the Problem Set. (The tape diagram and solution are modeled below.)

4 27. Larger divisors can stump some students who aren’t familiar with their division or multiplication facts. Students have been provided several lessons prior to this one to gain understanding of the method of the algorithm. Now they can be successful using these larger, and often more challenging, divisors.Participants should note that there are multiple ways to solve a problem. Two ways to solve the problem (as displayed on the slide) are outlined below. It is wise to call out and celebrate alternative solutions that are mathematically sound. Remind participants that the division algorithm is not an expected fluency until Grade 6. In Grade 4, the process is

introduced to prepare students for division work in Grade 5.

5 28. This lesson builds on the Concept Development of Topic E’s Lesson 20. In this lesson, students extend their learning of the connection of the area model to division to include larger numbers. Students will first build lengths of 1 hundred instead of lengths of ten. Students will see that there is more than one way to break apart a number.Discuss with participants both methods for decomposing the whole using number bonds and the area model. Stress that this work relates directly to earlier lessons and G3 division.

Model 672 ÷3 to find thelength of the unknown side. (See example to the right.)If time allows, have participants complete #16 of the

Problem Set.

3 29. Take 2 minutes to read Topic Opener H. Read for the standards addressed and the content students will be expected to have understanding of for moving on.Standard:

Content:• Multiply 2-digit multiples of 10 by two-digit numbers using a place

value chart and an area model.• Multiply 2-digit by 2-digit numbers using four partial products and

the standard algorithm for two-digit by two-digit multiplication.How have students prepared for this type of multiplication?Students have multiplied single-digit by multi-digit numbers. Students have modeled multiplication using place value disks, the area model, and partial products. This multiplication is simply an extension of a process with which students are already familiar.How are the students prepared to use the area model?Students have used the area model throughout the module to both multiply and divide. Earlier lessons have given a solid foundation of work with the area model to allow for a smooth transition to 2-digit by 2-digit multiplication.

2 30. This fluency activity is included within this module to show that fluencies can be used to anticipate future lessons in order to ready students for success. As we are near the end of Module 3, fluencies anticipating the work of future modules begin to appear.What is this fluency preparing students for? Geometry and FractionsWhy is it placed here? Anticipation of upcoming Lessons/Topics/ModulesHow could this fluency be revised/adapted/postponed?• If it is not new learning, it is not a fluency and should not be treated

as such.

• To revise: If students are not familiar with the geometric terms, revise to those that do have familiarity.

• To adapt: Focus on only the geometry or only the fractions, but not both.

• To postpone: Set priorities. If another fluency within the lesson is more applicable to the lesson or to future lessons, postpone this one and choose another instead.

3 31. The Application Problem for Lesson 34 leads directly into the lesson where students discover they are prepared to solve 40 times 22 because they know it to be the same as 4 times 10 times 22.Have participants read the Application Problem and then study the diagram. Point out that the area model shows 40×22 but that they can look at it as (4 × 10) ×22. The idea is to show students that they are able to solve more difficult problems by using what they already know as strategies. They see that in the area model. They also model it on place value charts. Deliver the script as follows:

3 32. In this lesson, students apply what they learned in Lesson 34 and extend it to the next level. To show 60 as one side length and 34 as the other, the 34 is shown as 30 + 4. Students are able to solve for each of the parts as the multiplication of units is something with which they are familiar. The expression is written in numeric form and in unit language within the model.Participants should study the model and then look to see how it is then written in the algorithm as two partial products.Debrief: When recording the partial products, must we start with the smallest product? Why or why not?We can start with either of the two partial products. The sum of the two will be the same no matter which order is used. The smaller of the two is written first in this lesson in order to allow for a smoother transition to 2-

digit by 2-digit multiplication.Participants may complete #17 of the Problem Set.

5 33. Lesson 36 builds to the next level in preparing students to multiply 2-digit by 2-digit numbers. Instead of multiplying a multiple of ten by a 2-digit number, students multiply two 2-digit numbers which are not multiples of ten.Show participants how the area model is set up for this type of problem. Within each quadrant, an expression is written and solved. We now have 4-partial products instead of two. (Students could argue that there are 4-partial products in the previous examples and that their products are simply zero.)Point out how the partial products are written in vertical format and then are added to determine the area of the large rectangle (the product of 23 and 31). The distributive property can also be used to show the breakdown of each number into partial products.Walk participants step by step through the area model, recording also within the algorithm.Participants complete #18 of the Problem Set.

4 34. Lesson 37 transitions from 4-partial products to 2-partial products, bringing students one step closer to the standard algorithm for multiplication. Using a visual of the area model, students will see that (6×30) + (6×5) is the same as 6×35. Students know how to calculate 6×35 as it was discovered in Topic C. Similarly, students will see that the same concept applies to the bottom portion of the area model.Walk students step-by-step through the example shown on the slide and then have them work to complete #19 of the Problem Set. Participants should first draw the model representing four partial products and then shade the area model to show how two partial products are also represented. Encourage participants to use the example on this slide when completing #19.

2 35. The module culminates in Lesson 38 where students see the final link from 2 partial products to solving 2-digit by 2-digit multiplication using the area model.Explain to participants the model that is shown on the slide. Students now draw just 2 partial products instead of four. An expression is written within each to show multiplication by ones and by tens. The partial products are then linked to the algorithm for multiplication. Initially, no regroupings are needed. Students are then faced with problems where regrouping becomes necessary. (See next slide.)Please note to participants that using hide zero cards can allow students to ‘see’ in the numbers in the algorithm.Use of 2 partial products encourages unit language as we multiply. We are not multiplying by ‘2’ each time, rather we are multiplying by ‘2’ and then by ‘2 tens’.

4 36. In this final problem, regrouping is necessary. Note that the regrouping is completed on the line as it has been in earlier lessons throughout this module.Show participants the regrouping within the 2nd partial product as shown above. The regrouping is completed as if there were a ‘line’ above the 2nd number. The number that is regrouped is added immediately following the multiplication by the next place value (as is typically done within the algorithm).Have participants solve both problems (#20 on the Problem Set).

Section: Module Review Time: 9 minutes

[9 minutes] In this section, you will…Faciliate as participants articulate the key points of this session and clarify as needed.

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

0 37. The Concept Development portion of the module is now complete.

5 38. Allow participants a few minutes to look at the End-of-Module Assessment. If time allows, participants may work to complete the assessment. If time does not allow, encourage a brief discussion regarding the assessment and how the work within the module prepares students for success.How did Module 1 and Module 2 prepare you for Module 3?Place value understanding, work with addition and subtraction, work with various models, and application of what was learned helped to prepare for Module 3.

Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

3 39. Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

1 40. Review the key points of the session as outlined above.

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided● Grade 4 Module 3 PPT

Additional Suggested Resources How to Implement A Story of Units A Story of Units Year Long Curriculum Overview