IDP-F

6th Grade-Negative Numbers and Absolute Values

Katie Marie Nelson

CI 4490Fall 2012

Table Of Contents

1. Learning Goals and ObjectivesCommon Core Learning Goals and ObjectivesHorizontal AlignmentVertical Alignment21st Century Skills

2. Student Background, Knowledge, and ExperienceDevelopmental Level of InstructionDevelopmental Level of StudentsStudent DataCollaboration

3.Plan for Assessment and Evaluation of Student LearningData for LessonsObjective AlignmentFormative and Summative AssessmentBackwards Design

4.Essential Content KnowledgeEssential ContentContent vs. Common CoreGlobal Awareness

5.Resources, Materials, and PreparationCollaborationResourcesTechnologyStudent Collaboration, Cooperation, and Leadership

6.Instructional ProceduresDiversity21st Century SkillsTechnologyCritical Thinking/ Problem SolvingEffective Teaching Methods

7.Assessment/ EvaluationType of DataStudent Progress

8.Adaptations/ ModificationsAccomodations for DiversityCollaboration for DiversityTeaching Methods for Diversity

9.Classroom ManagementStudent GroupingManagement Strategies

TimeMaterialsStudent Behavior

Classroom OrganizationArrangementOrder and Timing of InstructionProcedures

10. Results and Analysis of Student LearningStudent ProgressAssessment DataImplications of Assessment for Student Learning

Using dataImplications of Assessment for Instruction

Using dataFurther Practices

11. Reflection on Teaching and LearningLearning Goals and ObjectivesStudent Background, Knowledge and ExperiencePlan for Assessment and Evaluation of Student LearningEssential Content KnowledgeResources, Materials, and PreparationInstructional ProceduresAssessment/ EvaluationAdaptations/ ModificationsClassroom ManagementResults and Analysis of Student Learning

12. Lesson Plans13. Manipulative

Student examples14. Name Value Activity

Integer War15. Worksheet

Examples of Student Work16. Summative Assessment

Blank TestAnswer KeyExamples of Student Work

1. Learning Goals and Objectives

Common Core Learning Goals and Objectives

6.NS Apply and extend previous understandings of numbers to the system of rational numbers.

5.Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7. Understand ordering and absolute value of rational numbers.a. Interpret statements of inequality as statements about the relative position of

two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 C > –7 C to express the fact that –3 C is warmer than –7 C.

c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

The goal of this unit is to address all of the above standards in some way. The goal for assessments is to evaluate how well the students have been taught the ability to locate integers on coordinate planes and number lines (6.NS.6) as well as compare the value of negative numbers, positive numbers, and absolute values(6.NS.7). The real-world context of these types of numbers will be addressed but not assessed in detail. Another main goal of the unit is to ensure that students can add and subtract negative numbers; this will be assessed heavily.

Horizontal Alignment

ELA: The students will be discovering the meaning of “negative”. The goal is to not only define negative in a mathematical context but in a worldly one so that students can make connections to other areas of the curriculum.

6.RI.4. Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of a specific word choice on meaning and tone.

-Students will be working to interpret mathematical language and meaning in a context. (I.e. the temperature rose 30 degrees and then fell 15. What was the overall change in temperature?)

6.RL.9 Compare and contrast texts in different forms or genres (e.g., stories and poems; historical novels and fantasy stories) in terms of their approaches to similar themes and topics.

-Students will be discovering how negative values effect common operations6.RI.7 Integrate information presented in different media or formats (e.g., visually,

quantitatively) as well as in words to develop a coherent understanding of a topic or issue.-Students will work with how to work with information presented on graphs and with

how to change information from words to display in other forms.6.SP.1c Pose and respond to specific questions with elaboration and detail by making

comments that contribute to the topic, text, or issue under discussion.-Students will be required to engage in class discussions and defend their arguments

intelligently.Science: Students will also be learning about negative values, which will allow for

greater understanding and experimentation in the sciences.This can be applied to almost all of the science standards for sixth grade. For example,

6.L.2.1 Summarize how energy derived from the sun is used by plant to produce sugars (photosynthesis) and is transferred within food chains and food webs (terrestrial and aquatic) from producers to consumers to decomposers.

-Students will be examining energy as it is transferred through food chains. In this process, organisms both gain and lose energy which can be shown through the use and understanding of negative numbers.

-Sub-notes represent the connection to the sixth grade curriculum.

Vertical Alignment

Expanding on Fifth Grade:5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.-Students will need to understand how to set up expressions from the negative numbers language.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.-Students are adding a new range of numbers to utilize in operations.5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.-Students will be using a new set of numbers in these operations.5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.-Students will be using a new set of numbers in these operations.5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).-Students will need to be able to identify the parts of the coordinate plane in order to graph a much larger set of points.5.G.2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.-Students will be evaluating answers in the three other quadrants.

Expanding in Seventh Grade7.NS.1.Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.d. Apply properties of operations as strategies to add and subtract rational numbers.

-Students will be working with and understanding many of these things already. They will be

elevating their understanding to the next level in the seventh grade from their understanding in the sixth.7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world context.b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real- world contexts.c. Apply properties of operations as strategies to multiply and divide rational numbers.d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

-Students will be working with negative numbers and will learn to use them in new operations in the seventh.7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.-Students will have evaluated real-world problems using two of the four basic operations.

-Sub-notes represent the connection to the sixth grade curriculum.

21st Century Skills

Students will be addressing the ideas of finance in great detail especially in the context of understanding what a negative number is versus a positive number. The ideas of debt/earning, and gaining/losing money will be used to represent numerical expressions in a verbal manner.

Students will also be addressing the concepts of value- which value is worth more/worth less in a given context.

Students will be working with health literacy through the examples of eating food results in gaining calories, which must then be lost through a certain amount of exercise.

Critical thinking will be addressed through real-world examples that require the students to interpret meaning in context in order to set up expressions as well as be able to defend their position on an answer or why they may have set a problem up in a certain way.

2. Student Background, Knowledge, and Experience

Developmental Level of Instruction

In general, I noticed that at this school students are behind in the curriculum. It is strange to me that negative numbers are to be covered so late in the school year. Students are also lacking some of the basic building blocks of some of the concepts that we will be covering in my IDP such as the ability to interpret a coordinate and a general number sense and ability to quickly perform basic tasks such as simple multiplication and addition.

In this content area, the classification of different numbers has been covered thoroughly. The students in this class have mostly only covered scale models and proportions and fractions in order to be able to complete their projects for their current expedition. It seems that much of the regular curriculum has gotten pushed to the side or out of the picture completely in order to provide support for the school-wide expedition of rockets. My master teacher has especially been pushed in her curriculum because she is in charge of the expedition and the final products of the students and as a result has had to make sure that all of the students are specifically able to do the appropriate math.

According to the Common Core standards, students at this age should have a firm grasp on coordinates in the first quadrant. They should also have worked with basic operations enough to be very comfortable with them.

Developmental Level of Students

The students at my placement are used to a much different form of instruction than I am. The school is an expeditionary learning school and nearly all of the lessons that they have or activities that they take part in are somehow related to the expedition. Also, the blocks of class time are also an average of two hours typically including an exploratory course or lunch in the middle. I find this to be an extraordinary amount of time for the students to try to stay engaged. I therefore tried to plan my lessons to account for the students’ need to move around and have activities that do not necessarily feel like learning.

The class that I taught consisted of the entire sixth grade; a grand total of nineteen students. As a result, the developmental level of the students ranges from far below grade level to advanced learners. Within the class, I recognize two students that excel, at least four that truly struggle, and only a couple who can perform without a decent amount of scaffolding. I tried to choose activities that were engaging to all students and could be differentiated to different ability levels easily.

I did not necessarily develop individual plans for students whom I thought might need extra help or more challenging work. It was difficult for me to determine in advance specifically which student would fall into the various categories. Once the lessons began, I was able to assess students “on the go” and assign differentiation as needed.

One of the modifications that I was able to implement was during Integer War. The original game asks students to find the sum of the cards that they draw. A certain pair had an obvious grasp of this concept so I moved them to finding the difference of their cards in order to determine a winner.

Student Data

The school that I was working in is not culturally diverse. The main diversity is in the ability levels of the students because every student in each grade is in all of the same classes because class size is so small. Some of the students receive pull-out instruction throughout the day but for the most part all students are included in all instruction. The main classification of the students tends to be attention deficit disorders. This made it crucial for me to have engaging lessons in order to keep all levels of students learning at the same rate and involved in the learning.

The school is also economically diverse. I did not run into any issues with this disparity because I provided all materials necessary to complete the days’ lessons. Also, the students and their parents have to work to be a part of this school so all students are somewhat more committed to learning because they know that their parents care about their scores.

Collaboration

The two people that I was able to collaborate with were the EC teacher and my master teacher. The EC teacher was not always in the classroom but did an excellent job of supporting my instruction by spotting students that I did not see who were struggling. When I began my unit, she asked me what I would like from her and I suggested that she do what she normally does in the classroom in terms of monitoring struggling students. I did ask her to not interject ideas into my lessons as she does with the master teacher. In one class period, she was especially helpful to me because three boys were being very rude about having to work together and she was able to have them work together because she could sit with them the entire lesson and monitor their behavior.

My master teacher mainly allowed me to do whatever it was that I wanted with the lessons. If I needed anything, I knew that I could ask for her help but that she would not interfere unless she saw it necessary to fix student behavior or ensure that I was getting the respect and attention that I needed.

3. Plan for Assessment and Evaluation of Student Learning

Data for Lessons

The main data that I collected to determine how my lessons would lay out I got through observation of the students. I took special care to notice who worked well with whom, what the ability levels of individuals were and how long the attention of the class as a whole could be held.

I did not consider previous lessons very much when I was planning my lessons because we had spent the previous two weeks working on an angle and distance activity.

Objective Alignment

Objective Pre-assessment Formative Assessment

Post Assessment

Be able to plot integers on number line

Class discussion of location with laminated sheets

Using laminated sheetsWorksheet

Test/Quiz = Quest

Be able to plot coordinates on plane

Class discussion of location with laminated sheets

Using laminated sheetsWorksheet

Quest

Be able to add integers

n/a Plot answers on laminated sheetName Value ActivityInteger WarWorksheet

Quest

Be able to subtract integers

n/a Plot answer on laminated sheetInteger WarWorksheet

Quest

Be able to evaluate absolute values

Class discussion of meaning of absolute values

Plot values on laminated sheetTape number lineWorksheet

Quest

Be able to add and subtract absolute values

n/a Plot values on laminated sheetWorksheet

Quest

Be able to order integers by value

Number line discussions

Tape number line and discussionWorksheet

Quest

Be able to interpret and solve word problems

Verbal examples Worksheet Quest

Formative and Summative Assessment

Formative assessment is to occur every day in the form of monitoring the students’ progress. The students will also be surveyed as a class to see how they feel about their personal comprehension of the concepts being covered in order to determine which concepts need more class time and discussion. All of the activities that are being utilized are aimed at assessing student progress.

Summative assessment will be done in the form of a test for the students that includes all of the materials covered. For the test, I will provide the students with the same manipulatives that they have been able to use throughout the unit. Scoring of the assessment will be based on correctness of answers with considerations taken for how well the material was covered. Some answers will be based on the answers to previous problems and I will make sure to score them accordingly.

Backwards Design

The goal of the unit is to get all students to at least a basic level of understanding of all of the objectives, barring the comprehension of word problems. I would like for all of the students to pass the final summative assessment. See chart on Alignment page for the methods in which the students will be assessed. Instruction will be given in the form of hands on activities and manipulatives in order to give the students the knowledge necessary to complete the assessments with confidence.

4. Essential Content Knowledge

Common misconceptions will likely include how to plot coordinates, the idea that the absolute value of a negative is still a positive number, and a desire on the part of students to switch the integers in expressions in order to make them easier to evaluate.

Essential Content1

Remember

2Understan

d

3Apply

4Analyze

5Evaluate

6Create

-Whole numbers-Integers-Operations of addition and subtraction-Parts of a number line/ coordinate plane-Coordinate as (x,y)-Quadrants-Negative numbers-Absolute value

-Negative as opposite-Number as position on a number line-Greater than/ less than-Absolute value-Amount of negatives effect on number-What quadrant does a point lie in?

-Ordering from least to greatest/ vise versa-Be able to solve addition and subtraction equations using negatives- Be able to solve addition and subtraction equations using absolute value

-Comparison of absolute value vs. ordering values-Effect of negative on operations-Distinguish the meaning of zero in a context

-Be able to defend reasoning of answers-Why operations are effected by negative numbers.

-Design real- world problems-Design grade level appropriate problems and activities to assess and push students

Content vs. Common Core6.NS Apply and extend previous understandings of numbers to the system of rational numbers.

5.Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.-Students will discuss different ways of viewing expressions in order to understand them in a context. (example: I have a debt of $6 and I lose $3 more)

6.Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.-Students will work a great deal with number lines and coordinate planes to become fluent in plotting all integers.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.-A main discussion will include the idea of negative as opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.-Students will be asked to label the quadrants on the manipulative and after a time determine the quadrant that a point lies in without having to refer to the manipulative.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

--Students will work a great deal with number lines and coordinate planes to become fluent in plotting all integers.

7. Understand ordering and absolute value of rational numbers.a. Interpret statements of inequality as statements about the relative position of

two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.-Students will use a number line on the floor to be able to visualize the position of integers and hypothesize general rules for determining inequalities.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 C > –7 C to express the fact that –3 C is warmer than –7 C.-Students will be asked to determine inequalities in a context. (example: a debt of four dollars is less than a debt of 10)

c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30

dollars, write |–30| = 30 to describe the size of the debt in dollars.-Students will use a number line on the floor to visualize absolute value as a distance from zero.

d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.-Students will be asked to concurrently compare integer values and absolute values.

8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.-Students will be working with coordinates in all four quadrants. They will discuss what occurs when one piece of a coordinate is made negative.

-Indicates specific connections between Common Core and the unit.

Global Awareness

Mathematics is a universal subject. No matter where you go in the world, you will add and subtract (and utilize negative numbers) in the same way. By preparing the students to be fluent in mathematics, they can speak intelligently on these subjects with people from other cultures.

5. Resources, Materials, and Preparation

Collaboration

I was able to collaborate with were the EC teacher in order to best decide how to best ensure that I met the needs of all students. The main things that she addresses in the sixth grade class are behavior issues and we were able to discuss ways in which I could make sure that these students stayed on their better behavior so that not only could they learn but also so that they would not disrupt the rest of the class. These methods included grouping particular students together. Some of the other students are better at keeping their peers on task and she remarked that it was important to pair these students but also making sure that the peers did not become frustrated or overworked. Her main concern was that the activities I presented would be fun and engaging for the students so that they would be too occupied to act out.

Resources

I chose resources that I felt could be adapted to easy use by all students. The main resources were the class set of laminated coordinate plane/number line and playing cards. I knew that all of my students had used dry erase markers and playing cards before and these materials would be familiar to them. The materials in themselves are simplistic and easy to use with proper instructions for the corresponding activities. I made sure to address concerns about usage such as returning the materials at the end of class and to use the markers for only appropriate purposes.The resources that I used allowed the students to demonstrate their strengths and weaknesses by completing the activities that went along with them. Basically, the resources were universal and were thereby able to address the strengths, weaknesses, and needs of all students.

Technology

Unfortunately the classroom that I was interning in did not have ready access to technology. The teacher has an overhead digital projector that does not work well and is really poorly located in the room. There is a class set of calculators but the unit that I taught was designed to be calulator inactive. I think that this classroom could greatly benefit from an influx in technology. The students have a class set of laptops in their humanities class but they have to stay in that classroom. I think that if the school could afford to put a Smart Board in the math classroom (or all of them) that this group if students could do far more interactive activities. I personally would have liked to utilize technology for the board work that we did instead of constantly having to mend my coordinate plane. Also, an interactive number line would likely have aided students understanding.

Student Collaboration, Cooperation, and Leadership

The activities that I presented to the students fostered collaboration, cooperation, and leadership. The students had to work together in order to complete the Name Value exercise. There also had to be partner work to play Integer War in order to practice adding integers. Students were asked to volunteer to pass out materials and explain their answers to board work every day. The students that volunteered for these roles also encouraged others to do the same.

6. Instructional Procedures

Diversity

The school that I was working in is not culturally diverse. The main diversity is in the ability levels of the students because every student in each grade is in all of the same classes because class size is so small. Some of the students receive pull-out instruction throughout the day but for the most part all students are included in all instruction. The main classification of the students tends to be attention deficit disorders. This made it crucial for me to have engaging lessons in order to keep all levels of students learning at the same rate and involved in the learning.

The school is also economically diverse. I did not run into any issues with this disparity because I provided all materials necessary to complete the days’ lessons. Also, the students and their parents have to work to be a part of this school so all students are somewhat more committed to learning because they know that their parents care about their scores.

21st Century Skills

Students will be addressing the ideas of finance in great detail especially in the context of understanding what a negative number is versus a positive number. The ideas of debt/earning, and gaining/losing money will be used to represent numerical expressions in a verbal manner.

Students will also be addressing the concepts of value- which value is worth more/worth less in a given context.

Students will be working with health literacy through the examples of eating food results in gaining calories, which must then be lost through a certain amount of exercise.

Critical thinking will be addressed through real-world examples that require the students to interpret meaning in context in order to set up expressions as well as be able to defend their position on an answer or why they may have set a problem up in a certain way.

Technology

Unfortunately the classroom that I was interning in did not have ready access to technology. The teacher has an overhead digital projector that does not work well and is really poorly located in the room. There is a class set of calculators but the unit that I taught was designed to be calulator inactive.

Critical Thinking/ Problem Solving

Rather than use a lecture format to teach the new information, the goal is to have guided discussions in which the students work together to formulate definitions for new concepts and hypothesize about new processes. For instance, the students will work together to define rules for the effect of any given number of negative signs added to an integer. The students will also need to think critically in order to interpret problems for meaning in a real-world context.

Effective Teaching Methods

Due to the expeditionary/ exploratory ideology of this school, it is important to utilize hands on teaching methods. The students are accustomed to working with manipulatives and discovering ideas for themselves rather than obtaining knowledge in a lecture format. Also, for this age range it is important to make lessons especially engaging because they would much rather be talking to their friends. I have also found that if the students can get up and move around in a constructive way it is beneficial to the success of the lesson because many tend to be growing quickly and need to wiggle.

7. Assessment/ Evaluation

Type of Data

Formative assessment is to occur every day in the form of monitoring the students’ progress on the activities assigned. The students will also be surveyed as a class to see how they feel about their personal comprehension of the concepts being covered in order to determine which concepts need more class time and discussion. All of the activities that are being utilized are aimed at assessing student progress. The data collected will be an informal knowledge of the strengths and weaknesses of each student. This information will be used to decide which concepts need more work and to group students in order to most benefit their group work and or make sure that they can receive the individual assistance that they may need.

Summative assessment will be done in the form of a test for the students that includes all of the materials covered. For the test, I will provide the students with the same manipulatives that they have been able to use throughout the unit. Scoring of the assessment will be based on correctness of answers with considerations taken for how well the material was covered. Some answers will be based on the answers to previous problems and I will make sure to score them accordingly. The results of the test will determine the lessons that my master teacher and I will plan following my unit.

Student Progress

Student progress was to be assessed mostly informally. In this way, I was able to personally monitor their strengths and weaknesses. I do not think that the summative assessment that I created was necessarily a reflection of true student progress. The students performed well and with great accuracy on the informal assessments that took place. In general, I know that all of the students gained some knowledge from the point at which they started. This knowledge is to the success on the activities that I assigned in order to be able to assess the students.

8. Adaptations/ Modifications

Accommodations for Diversity

Due to the number of students at this school who have special educational considerations, I tried to design all of my lessons to be hands on and engaging. I also tried to make sure that the activities were simple enough to explain and perform that all of the students would be able to take part. I had special modifications for the more advanced students so that they would be challenged by all of the activities as well.

Collaboration for Diversity I worked closely with my master teacher as well as the EC teacher when she was available to help me monitor the class during activities. During planning time, my master teacher and I would discuss how the class had performed and the lesson had gone that day so that I could determine if they needed further instruction the following day. Both teachers were very honest with me about the success of the lesson and the level of student engagement that they had observed. As previously mentioned, a large part of the class has special academic considerations and rather than single out students, we would discuss the class as a whole.

Teaching Methods for Diversity

I used many different teaching strategies to try to ensure that all learner types could relate at some point. My personal favorite was the class-wide discussions. I also taught while using manipulatives and taught the students movements with their arms that they could do to remember various concepts. Students also took part in hands on activities and partner work. All of the work that was done as a class or as part of a discussion was done on the students’ copies of the coordinate plane/ number lines and copied to a coordinate plane and number line that I had drawn on the board so that students could not only see the information on the board but were actively participating as well.

9. Classroom Management

Student GroupingThe classroom that I was in is set up in cooperative pods with two to four students around

each round table. The goal is to foster cooperative learning. In general, I left the students in the seats that my master teacher had assigned because they have been designed to maximize productivity within the classroom. For instance, one student must sit next to my teacher’s desk as a part of his IEP. He is grouped with two very easy going and patient girls who also are good at redirecting him and keeping him on task.

I found that it was hard to make effective changes to the seating arrangement when I had behavioral issues because the majority of the students have been together in every single one of their classes for the last four or five years and therefore can find someone to talk to no matter where they are placed in the classroom. The round tables also makes it difficult to separate a single student who may need to spend time doing individual work.

For academic purposes, I typically allowed students to work with the other students at their table. For Integer War, I regrouped some student pairs into more homogeneous groups in order to allow for differentiation and one-on-one help. The students were able to pick their own partners for the activity in which they found the value of their name because in a way the activity dictated the groupings. As students finished the activity, they were to trade their paper with someone else that was also finished. This meant that the ability levels of the students tended to group together simply based on how long it took them to complete the activity.

Management Strategies

Time:Fortunately, I had nearly unlimited time to complete each of my lessons. My one goal for time management was to stick to a time limit for an activity when I gave one to the students. For example, I gave the students ten minutes each day to complete the bell ringer each day that we had one.

Materials:The main materials that my students had at their disposal were the laminated copies of the coordinate plane/number line, dry erase markers, playing cards, and Sharpies. My master teacher has class sets of the markers that I was able to use. For each class, I would assign a student to pass out each of the materials that were needed. At the end of the class period, I would have students pick up all of the materials especially the coordinate plane/number lines. This was because I felt that if the students were held responsible for keeping up with them, I would have to make twice as many as necessary because they would lose track of them. Before handing out the markers on the first day, I made sure to have a discussion with the students about proper usage; principally that the markers are meant for writing only on the paper and not on skin, tables, etc. I also made sure that I got all of my decks of playing cards back at the end of each class period and that the students checked the floor to make sure that all of the cards were accounted for.

Student Behavior:The school that I was at has a set of standards for student behavior known as SPARK. The standards are modified to fit all aspects of the school day from how to behave in class and treat others to even responsible bathroom usage. The SPARK standards outline behaviors such as Responsibility and Kindness. I used the standards when I needed to redirect students to better behavior or reward students for good behavior.One strategy that I implemented a great deal was to constantly be moving around the room so that I could easily watch all students and redirect to better behaviors without having to call students out in front of the class. When necessary I would move students to different seats near other students that I felt would influence them to work more diligently or have better behavior.

Classroom Organization

Arrangement:I did not find that the classroom arrangement was beneficial to student learning. The round tables that are used make it difficult to separate students if necessary. The students also rarely work in numbers more than pairs and the tables tend to lead to more talking on unrelated subjects. The tables also do not fit into the classroom very well impeding movement and allowing students to easily hold conversations with friends who are placed at other tables specifically to prevent this behavior.

Order and Timing of Instruction:Frankly, the scheduling of the instruction at this school is awful. Neither the students nor the teachers know where they are supposed to go most of the time because the schedule is constantly rotating. I really liked the idea of rotating when students have each class but because all of the grades see all of the teachers every day and still have to fit in their exploratory courses, the schedule also tends to have large gaps of time in the middle of regular instruction for those courses. This is not conducive to student learning or instruction. Often the teachers will have to stop an activity shortly after starting it in order to take students to an exploratory class. Many times my master teacher would not even begin an activity until students came back and instead allow for individual work or reading as a class.

Procedures:As a result of the difficulty in scheduling organization, many of the procedures may as well not be in place because although they are effective during the times that a teacher has the students for the whole class time, this rarely happens and therefore keeping the procedures constant does not happen.

10. Results and Analysis of Student Learning

Student Progress

When I interviewed the class as a whole in order to pre-assess their knowledge, approximately half of the students had a basic knowledge of where negative numbers could be found. Only one student demonstrated a working knowledge of negative numbers in operations. When assessed on their prior knowledge of absolute values, no students had any knowledge.

I was very pleased with student progress throughout the unit. Each of the assignments and activities that I assigned yielded few questions and exemplified an understanding on the part of all of the students. After reviewing the concept of coordinates, the students all demonstrated excellence in being able to plot points correctly. Students were also able to use a number line to accurately plot points and demonstrate adding and subtracting of negative numbers. In the name value activity (which did not provide a number line) on the first day of working with operations on integers, there were a good number of flaws in computation. However, when the students played integer war, there were nearly no questions or incorrect answers.

The worksheet that the students were given on the fourth day of the unit also showed great achievement and progress on the part of all students. The word problem section was the only part that students had issues with but was to be expected because it was not something that was expressly covered in the unit or assessed.

According to the summative assessment that the students received on the last day of my unit, it would seem that a majority of the students did not learn as much as previous assessments had shown. However, I know that the students have learned not only how to plot negative numbers accurately on both graphs but have learned to add with negatives very well. Two thirds of the students have demonstrated knowledge of subtracting with negative numbers. Roughly half of the students are now proficient and comfortable with absolute values. I was surprised at the number of students who were able to successfully complete the word problem at the end of the assessment.

Assessment Data

The students’ work from the name activity needed to be corrected by me on a one in three ratio. Many of the students were unable to finish in the time that was allotted in the class and were therefore unable to have a classmate check their work. However, from the mistakes that were made I was able to determine that the students would need more practice with basic operations using negative numbers (adding and subtracting).

After I looked over the score sheets from integer war and through formative assessment of the students while they were playing, I determined that all of the students had a thorough grasp of the concept and that I would be able to move on in the curriculum.

The worksheet was used solely as a formative tool. Students were to ask questions on the work as they completed it in class. From the questions that the students were asking, I hypothesized that they would do very well on the following summative assessment once completing the sheet.

I was disappointed with the results of the summative assessment. The class average was a 78. I feel that many of the students did not utilize the manipulatives that they had been using all week in the way that they were designed. The general implications are that the students may need further instruction on absolute value. There is definitely a need to cover word problems and real-life scenarios in greater depth.

Implications of Assessment for Student Learning

I feel that overall most of the learning goals were met. As mentioned above, the students need more instruction in a couple of areas. The methods in which I taught my unit appeared to be very successful in keeping all students engaged. It is beneficial to this student body to utilize activities that are more like games and allow them to work with a partner so that they may help to teach each other. The assessments that I gave showed me that the more fun that the students were having with an activity, the more that they would learn.

I also learned that the students are not concerned about the grade that they end with on any given assignment because after the initial work, no one ever sees the letter/number grade that they receive. I realized this after I went over the answers to the summative assessment with the students to ensure accuracy in my grading. The idea was that if students could catch mistakes that I had made than they could receive the points back. I only had six students approach me afterward to receive their points even though I knew that at least that many more probably had a mistake. Only one student was concerned enough with their score to ask if their was a way that he could obtain additional points.

(using data)According to the final assessment, the student body as a whole did not learn as much as I had thought. However, my informal assessment of individuals revealed to me that many of them who did not do as well did in fact know the material. I think that I may have designed a bad assessment in its’ format in order for the students and myself to get as much from the resulting data as we could.

Implications of Assessment for Instruction

When I prepared to teach my unit, I had a general idea of what I was going to be teaching and the activities that I was going to be doing. I actually kind of planned one really long lesson and prepared in advance for all of my activities. However, I did not know the breakdown of when a lesson would end until the day of when I was able to assess the students formatively and determine what parts of the curriculum needed more or less time. For instance, I had originally thought that the worksheet would not take the students very long and that we could begin working with word problems that same day. The students still needed more practice and work with some of the concepts so instead that work got pushed to the next day.

The final assessment leads me to believe that the students could use more instructional time on subtraction of negative numbers and on absolute values.

(using data) Most of my assessment was done informally. I would ask the students to raise their hands and give thumbs up, thumbs down, or sideways thumb to let me know how confident they felt about a topic that was covered. I used these assessments when I needed to consider regrouping the students or with whom I needed to try to spend individual time.

In general, I do not think that my final assessment was a good measure of students’ progress and of my instructional techniques. The students themselves seemed to think that the grades that they received did not reflect the information that they had learned. Students were able to explain what they had done wrong before we even went over the test. I think that this is an example of students rushing through tests, exhibiting anxiety, or not caring in advance about the grade.

Further Practices

Given the same situation again, I would love to utilize different varieties of coteaching. Because I was doing these lessons for school, my master teacher was not able to help as much as either of us would have liked. I think that it would be effective in these lessons to use each other for more than an extra class monitor. I would have liked to be able to switch who taught the formal instruction and who led the activities in order to play to our individual strengths.

I think that it would also be very beneficial to have more modern technology in the classroom. Some of the activities that I did or could have done with a Smartboard could have really enhanced my lessons. I would have liked to have been able to have interactive number lines that could help students get a better understanding especially of absolute value.

11. Reflection on Teaching and Learning

Learning Goals and Objectives

I chose this part of the curriculum to teach because it is what my master teacher asked me to cover. I found it to be very challenging for me because I had not previously considered that I would have to teach a concept that seems so elementary to me. It is difficult for me to remember at times what curriculum is covered in middle school because I myself have studied levels of mathematics that most of my future students will probably never even dream of in my pursuit of my second degree (in mathematics). I am thankful to my cooperating teacher for making me work with a concept that I was really uncomfortable with instead of something that I know I can teach.

It was a honestly a surprise to me that my teacher had left this curriculum until so late in the first semester. To me, negative numbers are an essential building block to the middle school curriculum because they allow for applications in a realistic real-world setting instead of one in which everything is always positive. Students can also have a deeper understanding of the number sense involved in operations such as subtraction if they are able to translate to another process such as adding a negative. Understanding absolute values as a distance will also aid later as students get further into algebra and geometry and utilize concepts of distance more.

I think that I was able to do a very thorough job of covering all of the standards that I was assigned except for 6.NS.8, which refers to usage of negative integers in real-world contexts. If I had had one more day to teach I would have spent the whole time working with students on interpreting word problems and the connotations of the words that they use- whether they are referring to adding or subtracting. I also think that I was lacking when it came to teaching the significance of zero in a situation.

Student Background, Knowledge, and Experience

This school as a whole has been a very different experience for me. The math curriculum has been taught in a very strange order this semester in order to provide students with the knowledge that they need to complete their final products for the expedition that the middle school is working on this semester. Also, my master teacher has to have four complete different lesson plans every day because she teaches all grades. I do not think that this is a situation that I would be comfortable in myself. I was also surprised at how much of the basic math skills such as basic addition without a calculator that the students are lacking.

The biggest adjustment that I had to make was that I did not expect to find nearly as many students with special educational considerations. I think that I was able to change my normal teaching style to fit the situation very well with guidance from my master teacher.

The support that I received constantly from the entire staff at the school was encouraging to me as a teacher candidate. The EC teacher was very helpful when I needed to make considerations for the different types of learners in the class. My master teacher also provided me with a great deal of useful ideas and feedback not only for my unit but also for my teaching career and classroom in general. Even the director and secretary of the school would ask about my progress in my unit and my school work and offer assistance and encouragement.

Plan for the Assessment and Evaluation of Student Learning

I thought that my ability to monitor student progress, strengths and weaknesses was very strong. I chose activities that I thought would be the most beneficial for the students and that would also give me plenty of time and opportunity for individual help. I wish that I had created or found another activity to help the students work with absolute values because it was the least covered in my unit. I also think that I should have made all of the students play Integer War with subtraction to get more practice so that I could have provided more support and been able to assess where the students stood in their comprehension of these topics.

I think that if I had been able to cover word problems and real-world situations with the students then I would have had much better coverage of 21st century skills.

When I made the answer key for my summative assessment, I originally had the wrong answer to the word problem that I wrote myself. This not only made me realize that it was perhaps not an appropriate assessment of the students and that I had not covered that section of the material well, it made grading the tests difficult because I had to do the work twice.

Essential Content Knowledge

It was surprising to me that I remembered as much of the essential content as I did. I had to review where the specific quadrants were. I also felt very comfortable answering students questions once I had spent time writing my lessons and reviewing all possible questions that I could come up with in advance. The most important note that I tried to stress with the students to clear up misconceptions was the idea that negative means opposite. Therefore if they were subtracting an opposite then they would perform the opposite operation (add a positive). Per example of my master teacher, I designed my lessons from the expanded Common Core charts. I feel that in this way my content and the Common Core were directly aligned.

Resources, Materials, and Preparation

I personally thought that my class set of laminated coordinate plane/ number lines was genius. I knew that the students in the middle grades at this school have the ability to lose a paper before they even leave the classroom and I did not want to waste the schools resources. The class sets also gave the students an opportunity to step into a leadership role and ensure that every one got their materials and that I got them all back. We were also able to use the materials in the seventh and eighth grade, which made them even more practical in my mind.

The integration of technology, or lack thereof, was unfortunately something that I did not have control over. I wish that I could have had a resource that allowed for interact graphs. However, I was very adamant throughout the unit that the students were not allowed to use calculators. I made sure that the problems that I created could be solved with relative ease by hand if the student understood the procedure being performed in the problem.

I was extremely pleased with the collaboration and cooperation of the students. They were very willing to participate in class discussions and build off of one another’s responses. They also were very engaged in activities even when they were allowed to choose the people that they were working with. Even when I would reorganize groupings to allow for individual attention, the students quickly adjusted and continued to work diligently.

Instructional Procedures

Although the lessons that I created did not incorporate as much technology or 21st century skills as I would have liked, they were geared specifically towards the students that I was to be working with and the resources that I had access to. In this respect, I think that I did a truly fantastic job. I was very pleased with the level of student engagement and comprehension of the content and activities by all of the sixth graders.

Assessment/ Evaluation

While I felt that I was very strong at informally assessing the students on a daily basis, I was displeased with my final test. The students did not do as well as expected but I think that this was a large part my fault. The format of the test was not very conducive to ease in taking it. It was also completely awful to grade which is what hinted to me that it was probably unpleasant to take. I also based two sets of problems off of the answers to a third part which threw many of the students off when they struggled with the original problem.

Adaptations/ Modifications

I believe that my lessons were effective in appropriately challenging all students. My goal was to design lessons that would to necessarily need to be modified and I seem to have succeeded in this regard. My concern going in to my unit was that the two more advanced students in the class would be bored but all of the students remained engaged during the majority of my unit.

Classroom Management

I was worried going into my unit that the students would see no reason to listen to me or follow my procedures. However, the very first day all of the students were on better behavior than I had ever seen from them. Admittedly they were back to their normal selves by the third day but they would still follow my directions and respect me.

The success of my management of time and materials was very encouraging to me because I am sure that this is the one of the main things that all teacher candidates worry about. I was also concerned with how I would be able to manage student behavior because I technically had no authority within the school. My main form of management was simple redirection of incorrect behaviors. I also found mixed success using the schools behavioral management techniques. I think that the students have begun to ignore some of the procedures as a whole for all of their teachers so I can honestly say that I think I did the best I could in this area.

Results and Analysis of Student Learning

I truly believe that all of the students learned from me. As previous mentioned, I think that my instruction could have been more thorough and covered all of the material that I wanted to cover if I had had five days of instruction and then the assessment instead of having to put my summative assessment on the last day. I also think that myself as well as the student body as a whole respond better to formative assessments.