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The Untold Tales of Long Division

An Exploration of Dividing with 2-digit Whole Number divisors

Fifth Grade TEED 521 - PSDG Unit March 15, 2007 LaNiece Fenton

-1- Thematic Framework

Central Question

Through the exploration of long division students will use both conceptual and procedural understandings of mathematics to answer the following central questions:

 Why is long division important in real-life situations?

 How does division relate to other mathematical operations?

 What does it mean to divide with two-digit divisors?

 What strategies/methods are available to compute division problems?

Rationale

The rationale behind teaching the problem solving data gathering unit on exploring long division through dividing whole numbers and decimals by 2-digit whole numbers is multi-fold.

The topic I will be teaching was determined through collaboration with my cooperating teacher.

We sat down and were able to determine where the class would be in their mathematics curriculum come springtime. Once we determined long division, I began searching for supplemental activities and curriculum to draw upon.

Through both group and independent work, this unit enables students to use mathematics to make sense of the world by putting strong emphasis on making connections of classroom learning to real-life situations. One of the dispositional learning targets of the unit actually deals directly with this notion, it is: “Students will gain an appreciation for the importance of long- division in real-world situations”. By having students relate classroom mathematical content to situations in their own life, student engagement increases. Once students are able to see the connections and appreciate the important role that long-division plays in the world, they will begin to make even more connections to classroom content.

-2- In order to take into account moral and ethical considerations, I must, as a teacher, reflect upon the unit content and my own instruction with a critical eye looking for anything that could be improved or should be taken into account. One thing I will consider while teaching the unit is students available resources. By resources, I do not mean materials or supplies, but rather human resources such as parents or older brothers and sisters available to help students with individualized instruction when needed. Unfortunately, teachers are forced to deal with large class sizes that do not allow for the individualized instruction that is often necessary. Students without this support at home may lag behind others who have it. I plan to be aware of these situations and account for them when necessary.

When planning a unit it is of utmost importance to align learning targets with national, state and school requirements as it sets a standard for education among learners. The central questions and learning targets for this unit were determined through the thorough examination of

National Standards, Essential Academic Learning Requirements (EALRS), Grade Level

Resources (GLE’s), and school-wide standards. The learning targets within this unit are clearly aligned with the aforementioned standards and within the context of the unit I have clearly linked the learning targets to the EALRs, GLEs and/or National Standards associated with them.

Learners

This unit is directed at the fifth grade classroom I will be doing my student teaching in this spring. The class is composed of 22 students; there are 14 females and 8 males. Students range in age from 10 to 11 years old. The range of abilities in the classroom is small. The majority of the students are average or middle of the road. However, there are two students with specific needs I will be addressing and keeping in mind throughout teaching. One student whose native language is Russian occasionally struggles due to a language barrier. I will have to ensure

-3- he is not isolated by fellow students or by the unit’s content. The student is involved in a pull- out class called Specific Language Development (SLD), I will work with the professional in charge of this class to develop language appropriate mathematical material when necessary (i.e. word problems); although, math is actually his strongest subject. Additionally, there is a student in my class that struggles with basic single digit multiplication facts. Unfortunately the students’ inability to compute basic multiplication facts is not due to her lack of competence in mathematics but rather her refusal to learn them. Since multiplication is foundational in the progression to long-division, I anticipate this student will struggle. It is my goal to reach this student both academically and socially. I plan on encouraging her to “brush up” on her multiplication facts in order to grasp concepts being presented in the unit and I will make myself available to her for additional individual help.

The racial/ethical composition of the classroom consists of 19 Caucasian students, one

Japanese-American student, one Hispanic student, and one Russian student (the non-native

English speaker mentioned earlier). There is a broad range of socio-economic backgrounds in the classroom. Although it is a private school some families are barely scraping by in order to keep their kids at the school. Other students may be on scholarships and if not for these, they would not be attending the private school. On the contrary, there is the other extreme where student’s come from very wealthy families.

The community the students live in is also very broad. Since the school is private, there is no specific community surrounding the school where the student population lives. Even the type of communities students live in varies within the classroom, due to the broad range of socio- economic status mentioned. Although the majority of students come from homes located in suburban neighborhoods, some may live in apartments or condos in lower-income areas.

-4- Additionally, it is hard to pin point the cultural groups represented in each of the students

communities. However, I would suppose most are surrounded by predominately white, higher

socio-economic classes, with two parent families. In considering students’ prior learning

experiences at home and in their communities I actually expect them to be very similar. Most of

the students schooling has been the same as they have been at the private school since

Kindergarten. Although, I have made no specific accommodations for students’ prior learning

experiences being insufficient to meet unit goals, I will be sensitive to the possibility of this

existing within the classroom of learners.

Assessment

Learning Targets, Evidence of Achievement, and Assessment Instrument Chart Learning Targets & EALR’s/GLE’s & National Assessment Assessment Criteria/ Type of Learning Standards Instrument Evidence of Achievement 1. Students will National Standards 3-5: 1. Pre-Assessment: 1. Student attempts to answer the question: What understand that there is a Numbers & Operations- Questionnaire is the relationship between multiplication and relationship between identify and use relationships 2. Formative division? multiplication and between operations, such as Assessment: Math 2. Journal Entry: Student responds to prompt: division. (CONCEPT) division as the inverse of Journal How do multiplication and division relate? multiplication, to solve 3. Formative Evidence: In their own words students explain how problems; Assessment: Drill & multiplication and division relate. They give three Practice Problems/ numeric examples of this in action. Worksheets 3. Students use the relationship between 4. Summative Post- multiplication and division as a technique for Assessment: The checking their division computations. Untold Tales of Long 4. Part of assessment requires that: Student answer Division the question: How do division and multiplication (see page 12 & 13 for relate? full description of Evidence: Students give a detailed description of assessment and the relationship between multiplication and division rubric) and includes at least one appropriate visual of this relationship. 2. Students will be able to Grade Level Expectation: 1. Summative- 1. Students get 80% or more of the answers multiply and divide whole Mathematics 1.5.6 - Apply Assessment: Unit correct. numbers fluently. properties to solve equations Test (taken directly (SKILL) involving multiplication and from classroom division. curriculum: Scott National Standards 3-5: Foreman-Addison Numbers & Operations- Wesley Mathematics) develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 50.

-5- Learning Targets & EALR’s/GLE’s & National Assessment Assessment Criteria/ Type of Learning Standards Instrument Evidence of Achievement 3. Students will be able to Grade Level Expectation: 1. Formative 1. Journal Entry: Students are asked to respond to use and select appropriate Mathematics - 2.2.3 Apply a Assessment: Math the following prompt: Please create a division conceptual and procedural variety of strategies and Journal Entry problem and solve it three different ways. methods/strategies approaches to construct 2. Summative-Post Evidence: A division problem consisting of a two- (calculators, paper & solutions. Assessment: The digit divisor is given and is solved using three pencil, mental math, National Standards 3-5: Untold Tales of Long different strategies. estimation) to compute Number & Operations- Division 2. Part of assessment requires that: Student creates division of whole numbers develop and use strategies to (see page 12 & 13 for procedures for the following strategies – mental and decimals by 2-digit estimate the results of whole- full description of math/estimation, paper-pencil (algorithm), and divisors. (SKILL) number computations and to assessment and calculator. judge the reasonableness of rubric) Evidence: A procedure is written out in students such results own words for all three strategies defined above. A Numbers & Operations- visual is provided for each of the strategies and an select appropriate methods and explanation of when it is best to use each strategy is tools for computing with whole given. numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools. Problem Solving- apply and adapt a variety of appropriate strategies to solve problems. 4. Students will be able to Grade Level Expectation: 1. Summative-Post 1. Part of assessment requires that: Student create clearly explain, both Mathematics - 4.2.3 Use Assessment: The a mathematical word problem involving division verbally and through mathematical language to Untold Tales of Long and solve it. written work, their process explain or describe numerical, Division (see page 12 Evidence: Word problem uses a two-digit divisor, of computing division measurement, geometric, & 13 for full problem is solved using three different strategies, with 2-digit divisors (in and/or statistical ideas and description of there is work shown for all three strategies and both word and numerical information in ways assessment and student gives a clear explanation of which strategy problems). appropriate for audience and rubric) is best for solving the problem they’ve created. (SKILL/PROCESS) purpose. National Standards 3-5: Communication -communicate their mathematical thinking coherently and clearly to peers, teachers, and others. Problem Solving - solve problems that arise in mathematics and in other contexts. 5. Students will gain an Grade Level Expectation: 1. Pre-Assessment: 1. Student attempts to answer the question: What appreciation for the Mathematics 5.3.1 - Questionnaire are some situations outside of the classroom where importance of long- Understand that 2. Formative long division is used? division in real-world mathematics is used Assessment: Math 2. Journal Entry – Student is asked to respond to situations. Journal Entry. the following prompt: Create a detailed list of (DISPOSITION) extensively in daily life 3. Summative-Post long-division in real-world situations. outside the classroom. Assessment: The Evidence: A list of at least five real-world National Standards 3-5: Untold Tales of Long situations is given and details are provided Connections- recognize and Division explaining the situation when necessary. apply mathematics in (see page 12 & 13 for 3. Part of assessment requires that: Students contexts outside of full description of respond to the question: Why is division important mathematics. assessment and outside of the classroom? rubric) Evidence: A detailed explanation is given with three appropriate examples.

-6- Learning Targets & EALR’s/GLE’s & National Assessment Assessment Criteria/ Type of Learning Standards Instrument Evidence of Achievement 6. Students will be able to Grade Level Expectation: 1. Student Self- 1. Self-Evaluation: Student will fill out a self- work cooperatively in a Communication 2.2.2. - Evaluation: Student assessment for group work (see page 14). variety of group settings. Applies skills to contribute Self-Assessment Evidence: Name and date are on the self- (SKILL) responsibly in a group setting. 2. Informal evaluation, the skills chart is filled out in its Formative entirety, student describes one thing they did well Assessment: during group work and one thing they could do Anecdotal Notes better and student explains one thing their group during group work did well and one thing their group could do better. (This form will be filled out after each group setting). 2. Assessment: During Group work I will walk around doing an informal assessment (taking anecdotal notes) of individuals working together in groups. Evidence: Notes will be based on students’ attentiveness to speakers, their body posture indicating they are engaged and listening, and whether or not student is participating when appropriate.

Pre-assessment

A pre-assessment is an important part of planning and designing a unit. It allows one to

know where there students are before the unit begins. If, for example, students are not very well

versed with a unit’s basic content, it is necessary to give students sufficient background

information and spend time going over unfamiliar vocabulary, terms and concepts at the

beginning of the unit. However, if students are extremely informed about the unit it is not

necessary to go into as much background information and thereby one is able to go into more

depth on other material. Additionally, pre-assessments allow teachers to address and become

aware of any misconceptions or habits students may have.

For my pre-assessment on this unit, I formulated a questionnaire for the students to fill

out. I selected a questionnaire format because I want it to be a relaxed way for students to reflect

and communicate what they know. I do not want the students to worry or be anxious about the

pre-assessment and I believe the questionnaire will help to ease some of their fears.

-7- Finally, in order to accurately compare my post-assessment with my pre-assessment at the end of the unit, I had to make certain the pre-assessment questionnaire and post-assessment aligned. I was able to coordinate the two assessments by ensuring that they both reflected the major learning targets of the unit. I have included a copy of the pre-assessment questionnaire on the page 11.

Post-assessment

A post-assessment is an important part of assessing student learning. Although assessment is ongoing and integrated throughout the unit, the post-assessment is imperative in order to take stock of student knowledge at the summative stage. The post-assessment reflects on an accumulation of knowledge, understandings, and applications absorbed by students throughout the entire unit. It is pertinent that the post-assessment address the learning targets for the unit. Additionally, as mentioned previously, the pre and post-assessment must align with each other in order to effectively use them as instruments of providing clear data on students learning.

Post-assessment data allows for both teacher and student growth. Teacher or professional growth occurs when teachers analyze the outcome of student learning and reflect on their own instruction. It allows teachers to know what areas they covered effectively and what areas they need to improve on instructionally. Teachers can reflect upon instruction used while covering learning targets that students need improvement on and examine what they did “right” while covering learning targets students excelled at. Furthermore, teachers can evaluate their effectiveness at addressing any misconceptions identified in the pre-assessment. Student growth occurs when assessments are clearly aligned with the learning targets for the unit. When they are

-8- aligned, accurate and authentic assessment occurs and students’ learning is expressed fully through the assessment tool.

For my post-assessment, I decided to have students create a page in a book that will be intended for next years fifth graders. In their page students become the teachers and present the tools needed to succeed at long division with two-digit divisors. Details of this assignment are located on page 12 and a rubric assessing students’ attainment of intended learning targets for can be found on page 13. This assessment aligns with the pre-assessment and learning targets 1,

3, 4, and 5, mentioned previously in the assessment chart.

Formative Assessment

In order that I may assess students’ attainment of learning targets and modify teaching instruction as needed, there will be multiple formative assessments conducted throughout the unit. Examples of these formative assessments are:

 Math Journals  Leftovers from 100 (group activity/game)  “Skills & Understanding” numerical problems on concepts presented – adapted from curriculum. Traditional drill and practice including written exercises and worksheets.  “Reasoning & Problem Solving” word problems on concepts presented – adapted from curriculum. Traditional drill and practice including written exercises and worksheets.

The formative assessments that I will explore in detail in this unit plan are entries in students’ math journals. These formative assessment entries are listed below and address learning targets 1, 3, and 5 as mentioned previously on the assessment chart.

Math Journal Entry addressing Learning Target #1: Student responds to the prompt: In your own words answer the question: How do multiplication and division relate? Use numbers to show three examples of this in action. Evidence of achievement: In their own words students explain how multiplication and division relate. They give three numeric examples of this in action.

-9- Math Journal Entry addressing Learning Target #3: Student responds to the prompt: Please create a division problem that has a two-digit divisor and solve it three different ways. Evidence of achievement: Student creates a division problem consisting of a two-digit divisor solves the problem using three different strategies (answer does not need to be correct, but rather that student understands how to use the different strategies.

Math Journal Entry addressing Learning Target #5: Student responds to the prompt: Create a detailed list of long-division in real-world situations. Give at least five examples. Evidence of achievement: Student writes out a list of at least five situations outside of the classroom where long-division is used. Details are provided explaining the situation as needed.

Self-Evaluation

Student self-evaluation form can be found on page 14 and the teacher checklist for evaluating the self-evaluation can be found on page 15. Students will evaluate themselves periodically throughout the unit after group work.

Group work throughout the unit will occur after I have given students a problem to think about (usually before each lesson). Students will be asked to work independently for a few minutes and then work with their group to collaborate their findings and come up with a joint solution. Other group work will occur informally during independent work on drill and practice problems. Students will be encouraged to ask each other for help if they stumble. Both students will benefit from this type of interaction as the student asking the question receives an explanation in a different form, possibly more closely linked to their learning style, and the student explaining reinforces their knowledge by clarifying and justifying his or her own reasoning for computing the problem.

-10- Name: ______Date:-

______Pre-Assessment 1. What is division?

2. How do division and multiplication relate?

3. What are some situations outside of the classroom where we use division?

4. Solve the following problems on the back of this sheet. Please check all answers, show your work and/or explain your answer.

a.) 560  80 b.) 6000  50 c.) 452.7  89 d.) 9152  91

5. Create a division problem and solve it. It must be different than the problems listed above. Please show your work.

6. If Matt Hasselback throws a football and it travels 120 yards in 40 seconds and Josh Brown kicks a football and it travels 70 yards in 35 seconds which football is traveling faster? (Show your work and explain your answer).

-11- Name: ______Date:-

______The Untold Tales of Long Division Your task is to compile a page (front & back) to put in a classroom book for next years fifth graders. Your job is to become their teachers and teach them about long division.

Something to think about: If you could have had a book like this what sort of “secrets,” “helpful hints,” or tips would have helped you?

PLEASE INCLUDE…

ON THE FRONT: 1. Answers to the following:

 What is division?

 How do division and multiplication relate? (Include a written and visual answer)

 Why is division important outside of the classroom? (Answer the question and give three examples where long division can be found)

2. Create written procedures (in your own words!) explaining the following strategies:  Mental Math/Estimation  Paper & Pencil  Calculator

For each strategy include one visual and explain when it is best to use the strategy.

ON THE BACK: 1. Create a math word problem & solve it.

 The problem must have a two-digit divisor.

 You must solve the problem using three different strategies.

 Label & show work for each strategy. Draw pictures when appropriate.

 Answer this question: Which strategy is most appropriate to solve this problem? Why?

2. Help out next years students by writing down a “Top Secret Tip” or a “Helpful Hint.”

-12- See rubric for grading details!

-13- Rubric for Untold Tales Page Outstanding! Almost There Needs Improvement DEFINE DIVISION Clear and detailed Definition given but Definition not given. definition is given in not it is not in students own words. students own words or is not clear and lacking details. MULTIPLICATION & DIVISION A detailed description A description of the No description is RELATIONSHIP of the relationship relationship between given and no visual between multiplication & multiplication and used. division is given and at division is given but least 1 appropriate it is lacking detail, visual is used. at least 1 visual is Learning Target #1 used. EXAMPLES OF LONG DIVISION A detailed response A response about No response is given OUTSIDE OF THE CLASSROOM about the importance of the importance of about the Learning Target #5 division outside of the division outside of importance of classroom is given with the classroom is division outside of 3 appropriate examples. given with 2 the classroom and appropriate 0-1 appropriate examples are given. examples are given. PROCEDURES OF STRATEGIES A procedure, visual and A procedure, visual A procedure, visual explanation of when it is and explanation of and explanation of best to use the strategy when it is best to when it is best to is given for all 3 use the strategy is use the strategy is strategies. given for 2 of the given for 0-1 of the Learning Target #3 strategies. strategies. CREATED DIVISION PROBLEM *Word problem is a two- *Word problem is a *Word problem is digit divisor. two-digit divisor. not given or is not a *Problem is solved using *Problem is solved two-digit divisor. 3 different strategies. using 2-1 different *Problem is not *There is work for all 3 strategies. solved. strategies. *There is work for 2 *There is no work *A clear detailed strategies. shown. explanation of the best *An explanation of *An explanation of strategy to use for the best strategy to the best strategy to solving the problem is use for solving the use for solving the given. problem is given. problem is not given.

Learning Target #4 HELPFUL HINTS At least 1 appropriate At least 1 helpful 0 helpful hints are helpful hint is given. hint is given but it is given. not appropriate.

-14- Name: ______Date:-

______Self-Evaluation

In order to work well with others there are some important skills to consider. These skills are listed in the chart below. Please fill out this form and decide how well you worked with others.

Check the appropriate box: Skills I need to work on I do this some of I do this most or this. the time. all of the time. I respectfully listened to others.

I contributed actively to the group.

I encouraged others to participate.

I suggested solutions to problems.

I did my fair share of work.

One thing our group did well together: ______One thing our group needs to work on: ______

One thing I really did well: ______One thing I could do better: ______

-15- ______

-16- Student Name: ______Group Name: ______Date:______Teacher Checklist (for Student Self-Evaluation)

Associated Learning Target: Students will be able to work cooperatively in a variety of group settings.

Associated EALRs: Communication 2.2.2. - Applies skills to contribute responsibly in a group setting.

Criteria Checklist:

 Student puts their name and date on the self-evaluation

 Student fills out the entire skills chart

 Student describes one thing their group did well together

 Student describes one thing their group needs to work on

 Student describes one thing they did well

 Student describes one thing they could do better

-17- Communications with Families/Students

I have selected to create a website and write letters home to students as my form of communication to families. My cooperating teacher currently sends a letter home to families weekly, and I will be employing the same format she currently uses. I have attached the first letter I will send home regarding the unit in addition to hardcopies of my website I have created for my future classroom (please note: the handout is an alternative to those families who may not be able to access the class website). Please find the letter on page 17, and the website hard copies are attached to the very back of the unit. The following are Web links that support this unit:

Parent & Teacher Resources

LINK: http://www.pbs.org/teachers/math/inventory/numbersandoperations-35.html PBS Website has a wealth of information on teacher resources that support student learning.

LINK: http://illuminations.nctm.org/ActivityDetail.aspx?ID=12. This web link is a fun game for students to brush up on their factoring skills!

LINK: http://www.kcls.org If you are interested in locating additional resources for your student to supplement mathematical skills learned in classroom, this is a great place to start.

LINK: http://www.mathcantakeyouplaces.org/default.lasso. This website is a terrific resource for teachers, parents, and kids.

Teacher Resources

LINK: http://content.scholastic.com/browse/article.jsp?id=3596 This article, written by Marilyn Burns was an excellent teacher resource supplemental to Burn’s textbook we read in class.

LINK: http://content.scholastic.com/browse/article.jsp?id=11537. This article, also written by Marilyn Burns was directly linked to my mental math as a strategy for long division and provided very sound reasoning and tips for teachers.

-18- News from the Bowen Bunch For the week of March 26, 2007 Dear Families,

I am excited to tell you, that starting Monday, we will begin a new math unit that explores division with 2-digit whole number divisors. Throughout the unit we will be exploring long division with whole numbers, decimals, and money.

Our goals for this math unit will be for students to answer the questions: Why is long division important in real-life situations? How does division relate to other mathematical operations? What does it mean to divide with two-digit divisors? What strategies/methods are available to compute division problems?

Assessment of these goals will be through journaling, numerical and word problems, a unit test, and a final project where students create a page of a book to be given to next years fifth graders. Through the book students teach next years class about long division and the specific strategies involved in computing problems.

Below I have listed some valuable Web links that support our unit and are worth exploring. (Next to the link there is a brief description of what you find at the sight).

http://www.pbs.org/teachers/math/inventory/numbersandoperations-35.html - PBS Website has a wealth of information on teacher resources that support student learning. http://illuminations.nctm.org/ActivityDetail.aspx?ID=12 - This web link is a fun game for students to brush up on their factoring skills! http://www.kcls.org - If you are interested in locating additional resources for your student to supplement mathematical skills learned in classroom, this is a great place to start. http://www.mathcantakeyouplaces.org/default.lasso - This website is a terrific resource for teachers, parents, and kids.

As always, there will be many opportunities to get involved in this unit; I have listed some of these opportunities below: *Come in and share your knowledge on long division through a whole class discussion *Share you knowledge by providing individualized instruction/tutoring for students *Join us for a class lesson on long-division. *Come in and simply observe *Share examples of how long division is seen and used outside of the classroom

If you are interested in any of these things or have another idea of how you’d like to get involved please contact me! You can call, email, come in, or send a note with your child. If you have any questions, concerns, feedback don’t hesitate to get in touch!

Have a great week, Mrs. Fenton 206-XXX-XXXX [email protected]

-19- Unit Outline

Unit Overview

In collaboration with my cooperating teacher, we have decided I will have approximately

3 weeks to teach the unit. The tentative dates span from 3/26/07 until 4/13/07. “The Untold

Tales of Long Division: An exploration of Dividing with 2-digit Whole Number Divisors” is an interactive unit allowing students to gain a conceptual understanding of long division along with the procedural algorithms necessary to become fluent at it. Major projects in the unit include a post-assessment where students create a page in a classroom book for next years fifth graders. In their page students become teachers and present the tools needed to succeed at long division with two-digit divisors, additionally students provide one helpful hint they wish they would have had that would have helped them in the learning process. Other activities include various entries in students’ math journals, drill and practice problems and worksheets, a unit test, and “Leftovers from 100” (a long division game). Activities require both independent and group work.

Daily lessons are grouped into math understandings. Some math understandings are covered in one day while others may take up to four days. The math understandings are sequenced according to the development of conceptual understanding students need to have in order to grasp the unit’s content. Instead of simply giving students the algorithms to solve long division problems students develop a conceptual understanding that allows them to see what is happening during division and why they are doing what they are doing when it comes time to work with procedural algorithms. By first leading students through multiples of 10, mental math, estimation, and the try/check/revise strategy students begin to see what is going on during long division. Additionally, giving students examples of long division occurring outside of the classroom develops a disposition of appreciation towards it. After students get a clear picture of

-20- what long division is and why it is used students are then able to move into both standard

algorithms and more complicated algorithms dealing with decimals and money. An additional

advantage to structuring daily lessons around big picture math understandings is the ability to

sequence daily lessons appropriately and ensure they are lined up with the unit’s learning targets.

Below is a general outline of the unit’s daily lessons and what learning targets these

lessons link to.

WEEK OF: Monday Tuesday Wednesday Thursday Friday 3/26 – 3/30 Lesson: Lesson: Lesson: Lesson: Lesson: Background Number Dividing Whole Dividing Whole REVIEW Info/Intro Sense/Place Value Numbers Numbers – Multiples of 10 Conceptual – Conceptual – Estimating Try/Check/Revise Learning Target: Learning Target: Learning Target: Learning Target: Learning Target: 1, 5 1, 4 1, 4 1, 4 1, 2, 3, 4, 6 4/2 – 4/6 Lesson: Lesson: Lesson: Lesson: Lesson: Dividing Whole Dividing Whole Dividing Whole Dividing Whole REVIEW Numbers Numbers Numbers Numbers Procedural – Procedural – Procedural – Procedural – Algorithm Algorithm Choosing a Dividing with Dividing with 2- Dividing with Strategy Zeros in the digit divisors Larger Numbers Quotient Learning Target: Learning Target: Learning Target: Learning Target: Learning Target: 1, 4, 6 1, 4, 5 3, 4 3, 4, 5 2, 3, 4, 6 4/9 – 4/13 Lesson: Lesson: Lesson: Lesson: Lesson: Dividing Decimals Dividing Decimals Dividing Decimals Dividing Decimals Post-Assessment Conceptual – Conceptual – Procedural – Procedural – Work Time Multiples of 10, Dividing Money Algorithm Animal Speeds 100, 1000 Learning Target: Learning Target: Learning Target: Learning Target: Learning Target: 1, 3, 4 1, 3, 4, 5 3, 4 3, 4, 5 1, 2, 3, 4, 5 4/16 – 4/21 Lesson: Lesson: Lesson: Post Assessment REVIEW Unit Test Work Time & REVIEW Learning Target: Learning Target: Learning Target: 1, 2, 3, 4, 5 1, 2, 3, 4, 5 1, 2, 3, 4

Outlines of Lessons

Below please find a brief outline of math understandings and individual daily lessons

mentioned above. The math understandings are listed in bold before their respective daily

lessons. For the daily lessons link to the learning targets please see chart above.

-21- Math Understanding: Background Information/Introduction/Review (1 day) The pre-assessment will be given before lessons begin, possibly as morning work.

Day 1: Depending on student outcome on the pre-assessment we will spend approximately one day on this lesson. As a class we will review vocabulary terms pertinent to the unit. These terms will include: quotient, multiple of 10, compatible numbers, estimate, rounding, dividend, divisor, remainder, tenths, and hundredths. These vocabulary terms will be discussed and written in their math journals. Additionally, I will address any misconceptions and incorrect pre-conceived notions regarding division. I will take students through an interactive review of basic division facts, some simple division patterns, and practice finding compatible numbers.

Math Understanding: Number Sense/Place Value: (1 day) Day 2: Dividing by Multiples of 10 – Students will be introduced to the conceptual idea of what means to compute long division. Together as a class we will examine and compute division problems whose dividends and divisors are multiples of 10 where the division involves a basic fact (i.e. 240/30 = ?, basic fact is 24/3). At first we will simply explore the relationship and patterns among division problems whose dividends and divisors are multiples of 10. (I.e. 480/80; 4800/80; 48,000/80; 480,000;80). After doing this I will ask students to find the basic facts used in each of the problems we looked at. By identifying basic facts we will talk about how you can divide mentally by multiples of 10 using basic facts. Students will complete similar problems independently.

Math Understanding: Dividing Whole Numbers – Conceptual: (2 days) Day 3: I will start out with a mini-lesson where I will give examples of division problems and ask students to identify those which they can do mentally. Problems I give them will be similar to the following: 160/80; 4100/68; 4800/80. Students will be able to see that they can easily compute 160/80 and 4800/80 by taking it down to the basic facts, but they will not, presumably, know how to mentally solve 4100/68. I will explain that you can round the dividend and or divisor to a compatible number (4100/68 would change to 4100/70). I will then read the book 1,000 Miles in 12 Days, by David Hautzig, to the class. This book is about the 2,490 mile Tour de France bicycle race. The author explains the different stages within the race and presents multiple opportunities for estimating quotients using two-digit divisors. I will stop students at these opportunities and ask them to compute the problem. Students will have to estimate the dividend and quotient to mentally solve the problem, students will compute answer and then compare answers and explain reasoning to a partner. Whole class will come back together to share, discuss and work on similar problems depending on class’ understanding. Focus of discussion will include examining when estimating might be needed.

Day 4: Try/Check/Revise – Students will investigate the concept of the Try/Check/Revise strategy through a kinesthetic application where this problem is posed: “Joey bought 8 balls at Sports Authority. Footballs were $5 each and tennis balls were $3 each. If Joey spent $34, how many of each did he buy?” I will have made 8 football shaped cards with $5 written on each and 8 tennis shaped balls with $3 written on each. Students will have to manipulate cards determining how many of each ball he bought. We will make a guess,

-22- check it though multiplication and repeat (revise) until students have answered the question correctly. Further investigation of the concept will be had in the “Dog-Run” activity where students must use the strategy “Try/Check/Revise” (adapted from Scott Foresman-Addision Wesley Mathematics). In the activity students are asked to determine the size of a rectangular dog run and they only know that the width is 4ft longer than it is wide and they have 60 feet of dog fence. Throughout the activity students will inadvertently use the Try/Check/Revise method to solve the “Dog-Run.” I will ask students questions such as, “What will be longer, the length or the width? How come?” “Why will a length of 30 feet not work?” “How could you use multiplication to check the answer?” After completing the activity students will formally be introduced to the Try/Check/Revise strategy and the formal steps of using it. The procedure for this strategy will be written in their math journals. Students will complete similar word problems using this method independently. Depending on students understanding we may do more together as a class.

Day 5: Review – Students will complete a “Review” worksheet taken from their curriculum to review the unit’s content thus far. Students will be encouraged to work independently to complete the worksheet; however they are allowed to collaborate if they get stuck and are having difficulties computing or solving problems. Students will complete a math journal entry addressing learning target number 1. Students will respond to the following prompt: “In your own words answer the question: How do multiplication and division relate?”

Math Understanding: Dividing Whole Numbers – Procedural Algorithms: (2 days) Day 6: Algorithm & Leftovers from 100 – I will build conceptual understanding of dividing whole numbers by two-digit divisors with unifix cubes. I will present the problem 115/16 on the board. I will ask students what compatible numbers can be used and determine an estimate. I will then present the unifix cubes to the class (as a whole) and ask students what the first thing I must do to compute the problem. We will determine a quotient and compare the answer to our estimate. Depending on students understanding multiple examples may be introduced. I will review remainder as a vocabulary word. After conceptually presenting division of whole numbers by two-digit divisors I will present the algorithm step-by-step along with doing it with unifix cubes. Students will record this strategy’s procedure in their math journals. Students will conduct problems independently and then play the Leftovers from 100 game (adapted from Marilyn Burn’s About Teaching Mathematics book).

Day 7: Algorithm & Dividing with Larger Numbers – For this lesson I will introduce division with two-digit whole number divisors again but this time with the dividends being much larger numbers. For example as a class we will examine the problem 3682/56. First we will find the compatible numbers (3600/50), then find the basic fact (360/5) and estimate our answer. We will put our answer into the standard algorithm and work the problem down. A “second” division problem would be formed in our algorithm and we will estimate this problem as well (322/56). First we’d find the compatible numbers (300/60) and then estimate. We will continue to fill in the standard algorithm as we go along and come up with our answer. I will ask students to check work. We will be working on graph paper for this lesson so that our algorithms remain lined up correctly. We will repeat as many more examples as is necessary.

-23- Day 8: Choosing a Strategy – Throughout this lesson we will discuss and review the various ways of solving a division problem namely mental math, paper and pencil, and calculator. Students will complete a worksheet consisting of various problems and determine the best computation method. Students will respond to the following prompt in their math journals (addressing learning target 3): Please create a division problem that has a two-digit divisor and solve it three different ways.

Day 9: Dividing with Zeros in the Quotient – I will present division problems with zeros in the quotient conceptually with math manipulatives (Place-Value Blocks). I will show why there is a zero in the tens place in problems such as 745/7. Students will work with place value blocks exploring similar problems in small groups. Students will conduct a drill & practice sheet from their math curriculum book pertaining to this lesson.

Day 10: Review – Students will complete a “Review” worksheet taken from their classroom curriculum (Scott Foresman-Addision Wesley Mathematics) to review the unit’s content thus far. Students will be encouraged to work independently to complete the worksheet; however they are allowed to collaborate if they get stuck and are having difficulties computing or solving problems.

Math Understanding: Dividing Decimals – Conceptual: (2 days) Day 11: Multiples of 10, 100, 1000 – As a class we will explore the relationship and patterns among a set of data (i.e. 245/1 = x, 245/10 = x, 245/100 = x, 245/1000 = x; students would solve these problems and look at the patter in between the quotient and divisors). After examining data I will lead students to the conclusion that when the divisor is 10, 100, 1000 you move the decimal point of the dividend left the same number of places as there are zeros in the divisor. Student will create a “cheat-sheet” in their journal reflecting this pattern. Conceptually students must understand that as you the divisor gets bigger the quotient will get smaller (when the dividend) is kept the same therefore the decimal point moves to the left and NOT to the right as it does in multiplication. Students will complete practice problems in small groups.

Day 12: Dividing Money – Students use play money as a math manipulative to conceptually understand dividing money by two-digit divisors. Students will be given a problem and have to show it with their money. They will have to trade bills for smaller ones as they compute the problem. After students have a firm grasp of the concept I will introduce them to the standard algorithm procedure for dividing money. Procedure is similar to that of dividing with whole numbers. In math journals students will write out the procedure. Students will conduct practice problems with the algorithm independently.

Math Understanding: Dividing Decimals – Procedural (2 days) Day 13: Procedural Algorithm – Students will be presented with a conceptual representation of dividing decimals with a number line. Students will be able to see the relationship between divisor and dividend. Students will learn the process of adding a zero to the end of the decimal in order to continue the division (whereas in money they rounded to the nearest

-24- cent). Students will work in groups on an assigned set of problems reviewing this and previous concepts.

Day 14: Animal Speeds – Students will get to experience real-life examples of long-division in action in regards to animal speeds. Students calculate problems dealing with animal speed out of math curriculum book. In their math journals students will respond to the following prompt (addressing learning target 5): Create a detailed list of long-division in real-world situations. Give at least five examples. Additionally, students will explore and compute problems with the book Incredible Comparisons by Russell Ash.

Math Understanding: Post-Assessment, Final Review, Unit Test (4 days) Day 15: Post-Assessment Work-Time – Students will be given their post-assessment project handout (see page 12-13) and begin work on it. Students will have all of math time to work on it.

Day 16: Post-Assessment Work-Time & Unit Review – Students will continue to work on their post-assessment project. Those who finish will complete review problems on page 240 and 241 of their math curriculum (Scott Foresman-Addision Wesley Mathematics).

Day 17: Review – Students will get into five groups of four or five and be assigned one of the following Concepts/Math Understandings to become an expert on: Number Sense/Place Value, Dividing Whole Numbers – Conceptual, Dividing Whole Numbers – Procedural Algorithms, Dividing Decimals – Conceptual, and Dividing Decimals – Procedural. A handout will be given to each group with specific items/topics they need to cover within their assigned math understanding. Students will draw upon their math books (review provided on page 244 & 245), math journals, worksheets, and returned work (problems already completed) to review content covered and prepare a mini review lesson to present to the class. Students will have approximately 30 minutes to complete their review, collaborate with group and prepare lesson. Each group presentation will be approximately 5-10 minutes. I will supplement group presentations as needed in order to have the fullest framework of knowledge covered in the review.

Day 18: Unit Test – Students will take their unit test. Test will be adapted from the classroom mathematics curriculum (Scott Foresman-Addision Wesley Mathematics).

Technology

Many web based resources were used throughout this unit plan and are listed in the annotated bibliography. For this section I would like to highlight the following two teacher web resources found on the scholastic website in their Instructor magazine:

-25-  http://content.scholastic.com/browse/article.jsp?id=3596 This link was to an article

written by Marilyn Burns. The article was titled 10 Big Math Ideas and was an

excellent teacher resource supplemental to Burn’s textbook we read in class.

 http://content.scholastic.com/browse/article.jsp?id=11537 This link was also an

article written by Marilyn Burns. This article was on mental math and directly linked

to my mental math section I am teaching as a strategy for long division.

Classroom Management

Good classroom management skills are necessary in order for students to work effectively as individuals and groups. Management techniques I will use throughout the unit include invoking preventative measures. The classroom seating arrangement will be set up so that collaboration with neighbors close by will be a beneficial learning tool and not a social time for students (during individual work students are allowed to ask another student for help). Clear expectations of behavior during both group and individual work time will be given to students.

Constructive talking will be allowed, however this privilege may be evoked for a period of time if it is being abused. I believe that every student is capable of succeeding at understanding the unit’s content and I will expect every student to give their best effort to succeed and attain learning targets. Additionally, as a teacher I will invoke consistency in my expectations and overall classroom management techniques. I will hold myself to the same expectations I hold my students; this includes giving students respect. I will attempt to foster a warm and caring environment and classroom community where students feel comfortable asking questions if they do not understand something. I will be patient with the development of students’ conceptual understanding of the unit and I will be flexible in order to adjust my instructional teaching to meet students’ needs. I will in every manner possible keep the curriculum engaging in order to

-26- hold students’ interest in unit content. By enacting the aforementioned classroom management techniques, many possible classroom management issues will be derailed.

Community Resources/Collaboration

Including the community into this unit will be vital in order to support and enhance student learning. Community members I will focus on involving include students’ parents and families, volunteers, and the school principal. Having these people get involved in the unit and share their knowledge with students will increase their engagement in unit content. Additionally,

I want students to incorporate their own experiences or experiences of those within their community to the unit. During the unit I will encourage students to find examples or situations requiring mathematical division outside of the classroom. Having students share these experiences with each other allows students to connect to unit content in a deeper more meaningful manner. Students are then able to compare the experiences they’ve had with mathematics outside of class to the experiences they are having in the class.

Finally, the post-assessment for this unit involves reaching out to the community. Students will make a book that aids in students’ learning for the next set of fifth graders coming through to experience the unit.

-27- Annotated Bibliography & Citations

Ash, R. (1996). Incredible Comparisons. New York: Dorling Kindersley Publishing, Inc. This book was used to integrate and incorporate literature into the unit and to supplement the lesson on dividing decimals by providing another experience for students to see division outside of the classroom.

Burns, M. 10 Big Math Ideas. Instructor. Retrieved March 13, 2007 from http://content.scholastic.com/browse/article.jsp?id=3596 This site was an excellent teacher resource supplemental to Burn’s textbook we read in class.

Burns, M. (2000). About Teaching Mathematics: A K-8 Resource (2nd edition). Sausalito: Math Solutions Publications. This textbook provided a foundation of knowledge for constructing conceptual understanding of division giving practical strategies and activities to use throughout the unit’s lessons and assessments.

Burns, M. Mental Math. Instructor. Retrieved March 13, 2007 from http://content.scholastic.com/browse/article.jsp?id=11537. This article was directly linked to my mental math as a strategy for long division and provided very sound reasoning and tips for teachers.

Charles, R., Crown, W., & Fennell, F. (2006). Scott Foresman – Addison Wesley: Mathematics (Diamond Edition). Glenview: Pearson Education Inc. This curriculum provided a basis for the content I will be required to recover in my student teaching placement; I was able to build and expand off of this curriculum.

Factor Game. Illuminations. Retrieved March 12, 2007 from http://illuminations.nctm.org/ActivityDetail.aspx?ID=12. This web link is a fun game for students to brush up on their factoring skills!

Hautzig, D. (1995). 1000 Miles in 12 Days. New York: Orchard Books. This book was used to integrate and incorporate literature into the unit and to supplement the lesson on estimating quotients.

KERA, & Travelocity. Math Can Take you Places. Retrieved March 12, 2007 from http://www.mathcantakeyouplaces.org/default.lasso. This website is a terrific resource for activities and games for teachers, parents, and kids.

Online Grade Level Resources. Washington State: Office of Superintendent of Public Instruction. Retrieved March 10, 2007 from http://www.k12.wa.us/Ealrs/default.aspx? ca=1&gl=3. I used this website as a source for locating and finding the GLEs I would address with my learning targets.

-28- PBS Teacher: Math. PBS. Retrieved March 12, 2007 from http://www.pbs.org/teachers/math/inventory/numbersandoperations-35.html. Website has a wealth of information on teacher resources that support student learning.

Principles and Standards for School Mathematics. National Council of Teachers of Mathematics. Retrieved March 10, 2007 from http://standards.nctm.org/. I used this website as a source for aligning my learning targets with national standards for mathematics in grades 3-5.

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