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Introduction

The Teacher Writing Sample (TWS) is to be based on a specific unit or series of lessons

to shows that the student teacher has incorporated what she has learned in her education classes

based on the outcomes given to both teachers and students by the College of Education. Because

the chapter on systems of equations was lengthy and involved strengths of different learners (the

kinesthetic learners were able to draw graphs, visual learners were able to look at a graph to

understand solutions and match it to a written system of equations, and auditory learners were

able to focus on the algebraic methods of solving systems of equations), I thought it would be

best to focus on for my TWS.

The entire chapter, including the quiz and the test, took eleven school days, not all in a

row. One Friday was dedicated to practicing for the NJ ASK, another Friday was spent taking an

NJ ASK practice test, and the school was closed one day due to snow. Overall, it took about

three and a half weeks to finish the chapter.

By the time they took the test, students were expected to solve systems of equations in

multiple ways, including graphing and using algebraic methods. There were times throughout

the chapter where real-life situations were used to help students set up equations on their own.

The work done in class varied; sometimes students were expected to do it independently, other

days they were in pairs, and for the test review they worked in groups of three or four. There

were a variety of assessments including a quiz, and homework was evaluated every night to

gauge student progress. There were also at least two lunch periods each week when students

could come in for extra help.

By this time during my student teaching, I had completely taken over the class. I feel that

because of this, the results are more valid and I have a true self-evaluation.

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Philosophy Statement

Education is a key part of society. Every day, any one person learns something new: the

parts of a car, an interesting fact about a family member, or an equation in math class. In order

to learn anything, there must be a teacher.

A teacher does not necessarily need to be a person in a classroom, with a piece of chalk

and a blackboard, talking to the class about problems in a book. Instead, it is someone who is

willing to help another person learn or better understand something. Teachers do not have to be

adults, nor do they have to be older than their students.

From my experience, a person’s desire to teach is realized in elementary or middle

school, and more often than not, was because a teacher had a positive influence on them. When I

was in third grade, my teacher told the class that she knew she wanted to be a teacher when she

was in fourth grade, but most people do not know until later on in life. From that point on, I

thought about what I was best at.

As months passed, I discovered that when I understood the material, and others did not, I

was the first classmate they would turn to if the teacher was busy with someone else. I would

not only give them the answer, but also explain the process to get the answer. To my surprise,

my peers began to come to me for help, even when I did not understand something. Although

we would work on the problem together, they would thank me for the help. I knew that my

decision to be a teacher was the right choice for me.

Years later, I understand the full reason why I want to be a teacher. I enjoy explaining to

others how to get the answers to different problems. When understanding shines in their eyes,

and they are able to work out the next few problems without assistance, I feel a sense of pride

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and satisfaction; the students finally know and understand the material, and can see the

connections to what they have previously learned. Some friends of mine tell me that I help them

more than their teachers, because instead of stopping to teach, the teachers race to fulfill the

curriculum by the end of the year. There is more favoritism among those who understand the

material than with those who struggle. The one-on-one environment allows more time for

someone to learn the material, instead of rushing to understand five different concepts in a 45-

minute period.

I want to be someone who is there for the students. I do not want to hold others back, but

I want to have more available time to help those who fall behind. I think that extra time is

essential to the learning process as students cannot be expected to learn in a forty-minute class

period. They should be allowed to learn, and not just know, the material on the board. On top of

that, I want to be a teacher whom the students can trust, and one who will not judge them when

they have a problem.

My desire to be a teacher is so that students can say they learned something that year.

They will have the ability to tell their friends that they had a teacher who helped them learn the

material in multiple ways. I want others to count on me to help them when they need it, and

know that I understand their frustrations when they do not understand something. Above all, I

want the satisfaction that comes when my students show their understanding through test scores,

with the hope that some time in the future they will come back and thank me for believing in

them.

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Contextual Factors

Each school district has its unique set of contextual factors that not only attribute to the

teaching-learning process but can also affect it. These factors include, but are not limited to, the

community, school district, available technology, and student characteristics. If members of the

community are involved in supporting education, the teaching-learning process may be

positively impacted; on the other hand, if available resources are limited, the teaching-learning

process may be negatively impacted.

Bridgewater-Raritan Middle School (BRMS) is located in Bridgewater, NJ off of Foothill

Road about half a mile from Route 22 West. It also faces Route 287 North. Bridgewater

encompasses approximately 32.5 square miles of Somerset County, and is located about 25 miles

SW of Newark, NJ. As of the 2007 census, the population is at 44,408, with a total of 1,470

students enrolled in the 2008-2009 school year at BRMS and 9,135 students in the entire district.

The racial breakdown of Bridgewater is as follows: White 78.3%, Asian 15.9%, Hispanic or

Latino 4.7% and African American 2.6%; 26.6% of the population speaks a language other than

English at home and 20.3% are foreign-born. Also as of 2007, 93.8% of the Bridgewater

population ages 25 and up graduated from high school; 53.4% of those earned a Bachelor’s

degree or higher. Bridgewater’s 2003 crime statistics per 1000 people were the following: Total

crime rate 17.9, violent crime rate 1.0, and non-violent crime rate 16.9. The median family

income in 2007 (with inflation-adjusted dollars) was $117,580, with 3.0% of families below the

poverty level.

Another factor that influences the learning-teaching process is the support from parents,

teachers, and the rest of the community regarding education. BRMS’ vision statement states:

“The philosophy and mission of the Bridgewater-Raritan School Counselors is to meet the

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social/emotional, academic and career needs of our students. We subscribe to the developmental

counseling approach, which involves a planned, proactive and preventative model to address the

developmental needs of the students. We serve the entire school community, including staff,

students and their families.” In previous years, the school budget was not passed, but for the

2008-2009 school year, Bridgewater’s teachers union showed how planned and proactive they

could be in order to receive the money the district needed. The union worked alongside the

superintendent and spoke at PTO meetings, polls, and other public functions in order to inform

the public why the budget needed to be passed. Teachers were calling different Bridgewater

citizens to persuade them to vote, and worked at the polls until they closed. This involvement

allowed the budget to pass, showing how supportive the community is.

Different classroom factors also affect the teaching-learning process, such as available

technology and the classroom arrangement. The classroom I am in has three computers in the

back of the room; all connected to the Internet, with one hooked up to the television in the front

of the room. This makes it easy to show the students any online applications or Microsoft Excel

or PowerPoint presentations to enhance different lessons. Students have the opportunity to be

exposed to different types of teaching styles and resources in order to help them understand the

material better. The teacher’s desk is at the back of the room which would be detrimental if the

teacher didn’t spend her time at the front of the room teaching or walking between the desks

checking student progress. The lessons are usually done on an overhead, but the whiteboard is

utilized when another example is needed or when students are called upon to answer a question.

This prevents the students from sitting at their desks and being talked to for a 42-minute period,

and it also involves them in the lesson.

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The racial breakdown of the town (mentioned above) is visible in its classrooms. Of the

twenty-two students in the class I’ve chosen to work with for the TWS, 68% of them are White,

23% are Indian, and 9% are Asian. Of the twelve females, 75% are White and 25% are Indian;

of the ten males, 60% are White, 20% are Asian, and 20% are Indian. None of the students are

English Language Learners, have special needs, nor have low math proficiency. Although the

statistics for Indian residents in Bridgewater are not mentioned above, because the classroom

ethnicities closely reflect those of the town, the students are less likely to ridicule anyone in the

class based on their background. This means that any interruptions in the classroom will most

likely revolve around everyday conversations and difficulty with the material being presented

that day.

It should be noted that the grading is as follows: 90-100 is an A, 80-89 is a B, 70-79 is a

C, and 66-69 is a D. The school system does give minus (-) and plus (+) values to the letter

grade, but for the purpose of the TWS, I mention the letter for simplicity.

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Learning Goals

Learning Goals- The students will be able to:

1. Define the following terms: system of equations, consistent, inconsistent, independent, and dependent (with respect to graphing)

2. Recognize graphs of intersecting lines, the same line and parallel lines3. Apply previous graphing abilities to solve systems of equations4. Solve systems of equations by using substitution5. Apply systems of equations to real life problems and find solutions for them6. Solve systems of equations by using elimination with addition and subtraction7. Solve systems of equations by using elimination with multiplication8. Compare the time it takes to use substitution on certain systems of equations to the time it

takes using elimination (of both kinds)9. Determine the best method to solve a system of equations10. Explain their favored method of solving a system of equations

Connection to State Standards- New Jersey Core Curriculum Content Standards (NJCCCS):

4.3 8 B.1- Graphing two equations and understanding their general behavior. Learning Goals 1, 2, 3

This is done at the beginning of the unit, when students graph a system of equations and determine what the solution is, if one exists.

4.3 8 D.4- Evaluating and simplifying algebraic expressions involving variables: including using the Order of Operations, distributive property, and substitution.Learning Goals 4, 5, 6, 7, 8

This is done throughout the entire unit, when students learn how to substitute one variable for another and use different forms of elimination in order to solve a system of equations.

4.3 9-12 B.2- Analyze and explain intersecting points as solutions of systems of equations.Learning Goals 1, 2

As mentioned in the first standard, the students are expected to graph systems of equations and explain the result.

4.3 9-12 D.2- Select and use appropriate methods to solve [systems of] equations.Learning Goal 9

This is seen at the end of the unit, when the students are taught when the best time to use the different techniques of graphing, substitution, elimination by addition/subtraction, and elimination using multiplication. They can then look at a problem and use their own knowledge to solve different systems of equations.

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4.5 A.3- Select and apply a variety of appropriate problem-solving strategies to solve problems.Learning Goal 9

While the NJCCCS notes trying a simpler problem or making a diagram, my reasoning for this standard is the same as the previous one.

4.5 A.4- Pose problems of various types and levels of difficulty.Learning Goals 1, 2, 3, 4, 5, 6, 7, 8, 9

The homework, quizzes, and test all contain a variety of questions, some more difficult than others. For example, in the first homework only 43% of the students answered a graphing question that involved fractions correctly. Because of that, only two questions on the test had answers in the forms of decimals, and neither of them involved graphing. There are also simpler problems throughout the lesson, including reading systems of equations on a graph and determining different solutions.

4.5 B- Use communication to organize and clarify mathematical thinking; communicate mathematical thinking coherently; use the language of mathematics to express mathematical ideas precisely.Learning Goals 1, 10

This is seen throughout the entire unit.

4.5 C.2- Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).All Learning Goals

Along with the example above, the students will also be using concepts mentioned in the second standard, adding and subtracting variables, and being able to relate “infinite solutions” from equations to the coordinate plane.

4.5 D- Use reasoning to support mathematical conclusions; rely on reasoning to check the correctness of their problem solutions.All Learning Goals

This is seen throughout the entire unit.

4.5 E.1- Use symbolic and graphical representations to organize, record and communicate mathematical ideas.Learning Goals 2, 3, 4, 5, 6, 7, 9

This is the beginning of the unit, when the students study graphs before performing algebraic manipulations to find solutions of systems of equations.

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Types and Levels of Learning Goals:

The levels of the learning goals are shown in the state standards. Because it is an eighth

grade Algebra class, I incorporated standards for both eighth and ninth grade, due to Algebra

usually being taught in the latter year.

Appropriateness of Learning Goals:

These goals are appropriate because this chapter involves a variety of ways to solve

systems of equations, something the students will need to know for future lessons, tests, and

math classes. It also allows students to become more familiar with graphing equations—

something they were taught prior to the lesson—and using different algebraic manipulations to

solve for two variables, which is new material.

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Assessment Plan

Overview of Assessment Plan

The assessment of the unit had four different aspects, some of which were more formal

and can be graded, and some are more informal. Classroom observation throughout the entirety

of the unit is the least formal assessment. It includes how well students answered my questions

during a lesson, how well they were able to describe what they were doing, and how well they

performed on the warm-ups given almost daily during the first five to ten minutes of class. The

test review was also assessed through observation. The more formal assessments are the one

quiz students will take in the middle of the unit and a test they will take at the end of it.

Description of Pre- and Post- Assessment

The pre-assessment for the entire unit was done during the first lesson while the students

were working on examples in their notes. They were to graph systems of equations and explain

whether there was one solution (and name it), no solution, or infinitely many solutions. Defining

what the graphs looked like was new material, but they were taught how to graph equations prior

to that, and had a chapter test the previous day and the mid-term a few days after the leson. I

noticed that a third of the students had difficulties with algebraic manipulation to get the

equation into its slope-intercept form, and a third of the class had difficulties graphing a line

correctly.

The post-assessment was the typical end-of-unit test covering all of the material the

students were taught. It included graphing, substitution, elimination, and real-life situations for

the students to solve. A review for the test was given the day before.

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Discussion of Formative Assessment Plan

Formative assessment was given after the first lesson and continued throughout the

warm-ups, lessons, and homework assignments. In the following table, I explain that the

homework was more of a formative assessment and the daily warm-ups were considered post-

assessments. This is due to the fact that the class being used for the TWS is taught during the

last period of the day. Both my cooperating teacher and I were available during lunch and after

school a numerous times each week. Therefore, if there was a question on the homework, the

students were able ask prior to the class, and before their warm-ups. Not included in the chart is

classroom observation, which was used as a constant form of assessment throughout the entire

unit.

Learning Goals Assessments Format of Assessment1, 2, 3 Pre-Assessment

Formative Assessment

Post-Assessment

Graphing systems of equations

Homework sheet 7-1Questions on material

Warm-Up 1 (pg 372 #8-10)Quiz: Questions 1-2Test: Questions 1 & 2

4 Formative Assessment

Post-Assessment

Homework sheet 7-2Homework 2.2 in book (pg

379 # 7, 11-14, 17)Questions on material

Warm-Up: 5-Minute Check Transparency 7-3

Quiz: Questions 3-5Test: Questions 3 & 4

5 Formative Assessment

Post-Assessment

Homework 2.2 in book (pg 380 # 31-33)

Warm-Up: 5-Minute Check Transparency 7-3, 7-4

Test: Questions 9-12

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6, 8 Formative Assessment

Post-Assessment

Homework sheet 7-3Questions on material

Warm-Up: 5-Minute Check Transparency 7-4

Test: Questions 5 & 77, 8, 9, 10 Formative Assessment

Post-Assessment

Homework sheet 7-4Questions on material

Warm-Up 4 (given on transparency)

Test: Questions 6 & 8

The quiz can be seen in Appendix A and the test in Appendix B.

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Design for Instruction

Results of Pre-Assessment

The results of the pre-assessment were expected, since I worked with the students for

about a week and a half before starting this unit and I was aware of their strengths and

weaknesses. Even though they had graphed equations many times in the past and were about to

have their mid-term, about 83% of the class did not graph the equations correctly on the pre-

assessment. The problem the majority of the class had was being able to relate a negative slope

in an equation to one on a graph. These results made for constant repetition when graphing

systems of equations for the first three learning goals. However, on the homework the students

showed more of an understanding of the concept (see Appendix C). A 50 means that the student

graphed almost all of the systems of equations correctly; anything higher means they explained

the solution correctly. The students have been given a number at random and will remain at that

number for the continuation of the TWS.

Unit Overview

Day 1: Learning Goals 1, 2, 3A. Mid-Term Review for 5-10 minutesB. In notes:

a. Look at definitions and visual representations of themb. Look at four examples

i. What do we have to do to the equations in order to graph them?ii. How do we graph these equations?

iii. Do the lines intersect or not? What does your graph mean in regards to a solution of the system of equations? Look back at the first page if you are unsure.

C. Answer any questions and give out homework.

Days 2-3: Learning Goals 4, 5A. Warm Up for 5-10 minutes on graphing systems of equations

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B. Close your eyes and imagine you are looking at a graph where two lines intersect. Their point is something like (0.6, 3.45). Is it easy to see that it is the exact intersection point?

C. Before notes:a. If you were given 3x and told x = 5, what would the answer be?b. Think of the Transitive Property. If a = b and b = c, what else do you know?

D. Application to notes:a. Go through first four problems in notes as a class.b. If first equation is y = 3x + 5 and the second is y = 4x – 6, what else do you

know?E. Answer any questions and give out homeworkF. (Day 3) Warm Up for 5-10 minutes on substitution; check/ collect homeworkG. Confusion on substitution

a. Do another problem on worksheet as independent workb. Project my steps and answer a few minutes later so the students can check it

H. In notes:a. Go through two word problems as a class; first write a system of equations

then solveb. Don’t forget to check the work

I. Answer any questions and give out homework

Days 4-5: Learning Goals 2, 3, 4, 5, 6, 8 (former three on fifth day)A. Warm Up for 5-10 minutes with word problems and substitutionB. Homework:

a. Go over the one students had most difficulties withb. Answer any questions students may have

C. In notes:a. Go through first two example problems as a classb. Explain when to use addition and when to use subtraction—if you have to

subtract, rewrite the problemc. Do the next four examples as independent work

D. Answer any questions and give out homework (practice quiz in book)E. (Day 5) Quiz

a. On graphing systems of equations and using substitution to solve themb. Took about 25 minutes to do since there were two problems involving fractionsc. When finished, complete any examples not done the day before

F. I project the steps and answer to Example 3G. Answer any questions on the four problems and give out homework on elimination using

addition and subtraction

Days 6-7: Learning Goals 5, 7, 8, 9, 10 (7 and 8 only on first day)A. Warm Up for 5-10 minutes on elimination with addition and subtraction while

checking for homeworkB. Homework:

a. Display detailed answers on projector

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b. Do number 20 on worksheeti. Not given as homework

ii. A word problem, which the students needed more practice withC. In notes:

a. Do the first column of elimination using multiplication in class (2 problems)b. Work through both with detailed answers, rewriting both equations after

determining what to doD. Answer any questions and give out homeworkE. (Day 7) Warm Up for 5-10 minutes on elimination with multiplication while checking

for homeworka. No questions on Warm Upb. No questions on homework

F. In notes:a. Do second column of elimination using multiplication in class (2 problems

regarding the need to multiply both equations)b. Work through both with detailed answers, rewriting both equations after

determining what to doc. Selected textbook problems with 10-15 minutes left of class

G. Answer any questions and give out homework

Days 8-10: All Learning GoalsA. Warm Up for 5-10 minutes on elimination with multiplication while collecting

homeworkB. Students in groups of 3 or 4 working on worksheet

a. What is the best way to solve for the problem?b. Have to do one of each methodc. Go over answers during last seven minutes of class and what methods groups

usedC. Answer any questions and give out homeworkD. (Day 9) Warm Up for 5-10 minutes on elimination while collecting homeworkE. Students in groups of 2 or 3 working on word problems

a. How do you set up the systems of equations for each problem?b. What is the best way to solve for the problem? (Graphing is not required).c. Have a student from each group go to the overhead or board to show how they

solved one of the problemsF. Answer any questions on what is on the board and give out homeworkG. (Day 10) Collect warm up sheets and check homework while students get their

textbooks outH. Students work on the even problems in the Chapter Review

a. Independent workb. If there are any questions, I will go to the student and help themc. Go over the first six to ten problems during the last five minutes of class

I. Answer any questions and tell them to study for test

Day 11: All Learning Goals: TEST

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Activities

The TWS asks for at least three activities to be done during the duration of the unit. Due

to the nature of the chapter I have chosen, only two activities took place, both of which involved

groups of two to four students. These two activities took place on days 8 and 9, and involved all

of the learning goals. The students were to look at the problems in front of them and, as a group,

decide on which method was best to solve the system of equations for each problem. The

materials and technology for both days were as follows: separate sheet(s) of paper, graph paper,

the problems to be solved, pencils, and calculators.

Activity One: Group Work Day One

On the eighth day, the students were given twelve systems of equations and were told to

solve at least four of them, in no particular order. Because there would be graphing on the test

and the students still needed practice with it, they were told that at least one of the problems had

to be solved by graphing. From walking around, I discovered that although many of the students

improved in their graphing skills, many chose more difficult problems to graph and solved them

incorrectly. On the homework from that night however, questions the students were told to

graph were 90% done correctly.

The assessment for this activity was solely based on observation as I walked across the

classroom and encountered certain questions on the material. It is also mentioned in

“Modification 2” of this TWS that the homework results of this night were better than the results

of previous nights. At the end of the lesson, I had each group tell me how they solved the

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problems they chose, and the result of their calculations. If I was given a wrong answer, the

group was told to check their work.

Activity Two: Group Work Day Two

The activity that took place on the ninth day of the chapter was similar to the one on the

eighth day except that the students were not required to graph. They were given a sheet of five

word problems and were told to set up a system of equations for each problem and then solve

them. Students who chose to graph one of the problems did so correctly.

The assessment for this activity was done during the last ten to fifteen minutes of the

class. One member from each group was asked to go to the overhead or board and write the

system of equations and its solution for a given problem. I told the groups that they should

specify what each variable was so as not to switch the answers and to again check their work.

The one group that did not label their variables had the answer “backwards,” which helped the

rest of the class understand the importance of naming variables.

Technology

The two sources of technology used throughout the unit were the TI-34 calculators each

student has Velcroed to their desks and the overhead projector. The calculators are allowed to be

used during initial instruction and during quizzes and tests. The students don’t tend to show a

reliance on them and use them only as the arithmetic becomes more difficult. The overhead was

my main teaching tool instead of using the whiteboard. This allowed me to face the students,

giving the lesson a more personal feel, and allowed me to know right away if a student was

confused or not paying attention.

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Instructional Decision-Making

Modification 1:

The first modification that I made in this unit was that I had originally planned to spend

one day on substitution and focus more time on elimination. Between student responses during

the first day of the lesson and homework results (see Appendix D) I realized that another day

needed to be spent on substitution. This helped throughout the rest of the lesson as well since

some form of substitution is used in elimination. The students also had to be proficient in it in

order to do the group work during the last few days, and for the test.

Due to the fact that some students were absent one or both of the days, or assignments not

being handed in on time, I was only able to compare the results of eleven students (half the

class). However, six of the eleven students had a percent increase while only four of them had a

decrease. Student 8 is the only one whose results remained unchanged. There were more

problems on the second homework assigned, which might have skewed the results. Yet because

more students did better, by 10% or higher for the majority of them, the extra day on substitution

created a positive outcome for the quality of the students’ work.

Modification 2:

The second modification that I made was on day 7, during the second part of elimination

using multiplication. I had thought that the students would ask a variety of questions and would

need more time to work on the examples in the notebook. Instead, the students did not have

many questions and had about a third of the period to work on a variety of problems in the

textbook.

I still did not think that the students had enough practice with the material though, and

extended the test review to cover three days instead of the two I had originally intended. As it

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can be seen in the unit overview, the first two days of the test review included group work on

different example problems and on word problems. The last day of the test review was

independent work. I did not collect the homework on day 10, but I did collect the other

elimination (of both methods) homework prior to that. The homework results shown in

Appendix E (assignments one and two given on days 6 and 7 respectively) show how useful the

group work was on day 8 (homework assignment three).

The first two homework assignments were on elimination using multiplication. The first

only required one equation to be adjusted; the second required both equations in the system to be

adjusted. The third homework assignment was similar to both days’ group work in that the

students were asked to choose what they believed to be the best method to solve the system of

equations and then solve it. However, a few of the problems needed to be solved using

elimination with multiplication; the results of that last homework showed over a 10% increase in

correct answers of all the students who handed it in.

As with the first modification, due to absences and students not handing the work in,

there were 14 students who had all three assignments. With these students alone, the mean of

homework assignments 1 and 2 were both 72.62; the class mean of homework 3 was 79.50.

Although this is less than a 10% increase, it is still the difference between a C- and a B- in the

school system.

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Analysis of Student Learning

Pre-Assessment

I have used the results in Appendix C as my pre-assessment for the final results. I did not

have student work from the pre-assessment mentioned above, which was the warm-up for the

lesson, especially since the students still had many difficulties understanding how to graph lines.

After the small time of extra practice done in class, I decided to use the first homework

assignment as my pre-assessment results, which is repeated in the results of the whole class. The

formative assessment of learning goal 5 is based on selected problems from all of the homework

assignments. Any homework that was not handed in was not included in the averages I

calculated. I have not included learning goals 8 and 10 due to the fact that they were both done

within the classroom and not meant for post-assessment evaluation. There is also no chart for

learning goal 9, as I do not think that learning goal is best analyzed quantitatively.

Whole Class

The results of the pre-assessment, formative assessments, and post-assessments can be

seen in the six charts in Appendix F. Some of the results are fake negatives due to the fact that

not all of the students handed their assignments in on the days I collected them. While there

were a few students who had zero percent correct on the homework assignments, the only zeroes

I accept as “true” are those in the post-assessments.

With learning goals 1-3, even though student progress fluctuated from the pre-assessment

on, as a whole the class did well. Two of the students failed that part of the test even though

both had scored high on the pre-assessment, with student 2 scoring 100% on that part of the quiz.

77% of the class scored higher or the same from the quiz to the test; the same percentage

(although some are different students) scored higher or the same between the pre-assessment and

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the test. This shows that while student progress as a whole was not high, the majority of the

students showed some sort of improvement. With 91% passing the learning goals after the test

versus the 82% who had passed the pre-assessment (with some scores at the minimum passing

grade), I would say that the class has learned these goals.

For learning goal 4, there was yet again fluctuation seen from the first formative

assessment on. Three students failed that part of the test, only one whom did well on the first

formative assessment (student 19). 82% of the class scored the same or higher from the quiz to

the test; only 73% of the class had scored higher or the same from the first formative assessment

to the quiz. However, just over half the class (54%) scored better on the quiz than on the second

formative assessment. Even though this learning goal was incorporated into other learning goals,

these results show that many of the students had difficulties with the concept. 86% of the class

passed the learning goal on the test versus 68% passing the goal on the quiz (both with students

at the minimum passing grade), and there were three forms of assessment before the test. My

conclusion is that although as a whole the class passed, the students did not fully understand the

concept of the learning goal as well as I believe they should have.

Because the test directions were specific on the section testing for learning goal 5, the

fact that some students’ scores dropped from the formative assessment to the test is not a real

concern (save for student 14 who had more than a 20% decrease). However for the post-

assessment, all but one of the scores is at a 79.17 or higher whereas the formative assessment

results range from 0 to 100. Student 12 was the only one who had failed the learning goal on the

test, and was the only one to receive a true zero on the formative assessment. With 95% of the

class passing the section on the test versus the 59% passing the formative assessment (with some

22

students at the minimum passing grade), I believe that the students have learned this goal and

understand when systems of equations can be used in real life situations.

The results of learning goal 6 surprised me. Even though only four students showed a

decrease from the formative assessment to the test, 32% of the class failed the learning goal on

the test. However, the students who failed did so because one of the test questions was set up so

that one of the variables was equal to zero; those students assumed that it meant the system had

no solution. That misunderstanding is a result of not learning a concept before this chapter.

Therefore, I consider the chart on page 38 to be invalid and the next page shows the results I

consider valid.

The valid results on page 39 show that 14% of the class failed the learning goal on the

test, two of which were the only students who had a percent decrease from the formative

assessment. With 86% of the class passing this learning goal, most of had a large percent

increase from the formative assessment, and only one assessment check before the test, I would

say that the students learned the concept.

The results of learning goal 7 were also a surprise, this time a more pleasant one. Of the

eighteen students who passed the goal on the test, 89% received a perfect score on that section.

Three students showed a decrease from the first formative assessment to the test and only one

student showed no progress (or regression) throughout that section of the chapter. Of the four

students who failed, one showed a percent increase from formative to post-assessment. Because

82% of the class passed the learning goal on the test and 86% of the class showed no or positive

progression, I believe that as a whole the students have an adequate understanding of the learning

goal.

23

Subgroups

Because the ratio of female students to male students in the class is just over 1:1 (1.1: 1 to

be exact) I have chosen the two genders to be the subgroups I am looking at. I have used the

valid results of learning goal 6 to compare the two subgroups for two reasons. The first is that

there was only one type of formative assessment before the test, meaning that even though

learning goal 7 incorporates learning goal 6 and it was reviewed before the test, students did not

get as much practice with the material as they did with learning goal 4. The second reason is that

it is an important skill to have for the upcoming chapter, and I would like to know how well each

subgroup understands the material.

Appendix G shows the results for the female students. It can be seen on the first graph

that all of the females showed a percent increase between the formative assessment and the test.

It should also be noted that while student 1 seems to have received a low score on this section of

the test, it is still a passing mark. Students 9 and 20 were absent the day I collected the

homework, but since they both received a 100% on that section of the test, I know they

understood the material. The second chart shows the percent increase of each female student

from the formative assessment to the test. Because eleven of the students did show a percent

increase, I can conclude that all of the females learned the content.

Appendix H shows the results for the male students. It can be seen on the first graph that

four-fifths of the males showed a percent increase between the formative assessment and the test.

For the 20% who showed a percent decrease, student 11 received no credit on the test due to the

fact that he did not show any work; student 17 did show work but did not get the correct answer.

As with the females, students 10, 19, and 21 were absent the day I collected the homework; that

they all received a 100% on that section of the test shows their understanding. The second chart

24

shows the percent change of each male student from the formative assessment to the test. Seven

of the male students had a 40% increase or higher; only five of the female students also had a

40% increase or higher. Overall the male students showed more of a percent increase than the

females did. However, because two of them had a percent decrease, my conclusion is that the

females learned more by the end of the chapter than the males did.

Individuals

The two students I have chosen to observe are of opposite genders to show how they

differ from one another. Student 9 began with a 66.67% on the pre-assessment and ended with

an 88.75% on the test; student 12 began with a 61.11% and ended with a 56.75%. However, I

believe that both students showed an increase in knowledge by the time they took the test,

especially when looking at specific learning goal performance. Student 9 was absent for a few

days, more towards the end of the chapter. Student 12 was present in class every day during the

chapter, and at the end worked with a tutor.

For the first three learning goals, student 9 fluctuated in scores on the assessments. Her

percent correct on the test for those learning goals was higher than what she received on the quiz

and pre-assessment. I feel that she became more confident with graphing as time went on and

was competent enough to draw certain conclusions from the graphs. For learning goal 4, she

again fluctuated in scores on the assessments; yet she received a perfect score on that section on

the test compared to the 66.67% she received on the quiz. Because of this, I feel student 9 did

learn that goal. Learning goals 5, 6, and 7 all show significant increase (25% or higher) between

the formative assessment and the test for student 9.

While she did not earn a perfect score on every section of the test, looking at the percent

increase for each goal shows that student 9 was able to learn and understand all of the goals. I

25

believe this student is comfortable with learning in the classroom and, even when absent, can

catch up and understand the work adequately.

Student 12 also fluctuated for the first three learning goals, albeit less than student 9 did.

He did better on the test than on the previous assessments, which shows that at the end of the

lesson he was more comfortable with graphing and reading the results. For learning goal 4, he

showed little progress throughout all of the assessments, and the percent correct on that section

of the test allowed me to know that he did not understand the material on that learning goal. As

with student 9, student 12 showed a percent increase in learning goals 5, 6, and 7 of 33.33% or

higher (up to a 100% increase with learning goal 7).

The zero percent on learning goal 4 brought this student’s grade down significantly, but

the increase for the other learning goals leads me to believe that he learned and understood most

of the material. It was most likely a mixture of help from the tutor more than the two-day test

review, but it is still comprehension of the chapter. In the end, that is more important than how

he learned it.

26

Reflection and Self-Evaluation

In order to find which learning goal the students were least and most successful in, I took

the difference between the grade on each given section on the test and the first form of

assessment the students were given for each one. I was not surprised to see that learning goals 1-

3 were least successful and learning goal 7 was most successful as I feel that is easily discernable

from the graphs, but I was surprised to see that overall the students increased for each learning

goal.

I find it surprising too that learning goals 1-3 showed the least amount of success even

though the students had been graphing before I started teaching their class. And yet the 8.51%

increase from the pre-assessment and the test shows that the majority of the class did not achieve

that skill. One reason for this is lack of understanding; if a student does not understand graphing

before a chapter, he or she is not likely to magically understand it by the end of the chapter. I

think that a few students did gain some more knowledge, but at thirteen- or fourteen-years-old,

some of the students might not be cognitively developed to understand the connection between

the graph of two lines and their solution(s). Another reason for the minute success might be that

since after the quiz graphing wasn’t used until the first test review, the students were busy

worrying about other methods of solving systems of equations. I have found that about 95% of

the students in this class prefer solving problems algebraically and show an aversion to graphing.

The fact that learning goal 7 showed the most success with an overall increase of 35.22%

was again a surprise. I have found through my own studies and watching others that solving

systems of equations using multiplication is sometimes the most difficult concept to grasp in this

chapter. I would have thought that substitution (learning goal 4) would show the most amount of

increase due to the fact that it was incorporated in all of the following lessons. I think the reason

27

students did so well on learning goal 7 was because it was the most recent topic learned, and

therefore fresh in their minds. Another reason related to the above paragraph, is that this group

of students prefers to solve problems algebraically. It might even be because the students enjoy

cutting corners, and substitution involved too many steps for them, whereas all they had to do

with multiplication was distribute, rewrite, and solve.

I believe that the students need to be proficient in graphing before starting this lesson. If

I were to do this lesson again in my own classroom, I would do my best to be sure they

understood how to graph equations before moving on to other material. I think that the students

were not exposed to many fractions, so that when fractions were involved in the graphing they

became confused; therefore more problems with fractions would have to be incorporated. Using

graphing calculators would probably help in this aspect since it allows students to physically

maneuver a point on a line.

Another way I can improve performance in all of the learning goals and better myself as a

teacher is to be less traditional. For the test review I did have two days of group work where the

students helped each other out as well as received assistance from me, but two days out of eleven

is not going to help the students in the end. As I said above, I can include graphing calculators in

the future; I can also create a type of game for the quiz review. In a class such as this one, a

game where the teams are boys against girls would have worked well. There might also be an

applet or two online to help guide the students through some problems rather than have me at the

front of the classroom lecturing. I hope that if I do teach another Algebra class in the future, I

am able to incorporate these ideas and sources.

28

Appendix A

Name:____________________________________ Chapter 7 Quiz

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

1. x - y = 9x + y = 11

2. 2x – 3y = 46y = 4x - 8

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

3. 2m + n = 1 4. 3x – y = 1 5. x = 3 - 2y m - n = 8 2x + 4y = 3 2x + 4y = 6

29

Appendix B

Name:_____________________________________ TB:_____ Date:___________

Chapter 7 Test: Systems of Equations

For # 1 and 2, graph the systems of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

1. y = 2x – 1y = x - 2

2. 2x + y = 62x – y = -2

30

For # 3 & 4, use substitution to solve the system of equations.

3. y = x + 38x – 7y = 12

4. 9 = x – 2y3x + 5 = 2y

For # 5-8, use elimination to solve the system of equations.

5. -2x – 2y = 4

-x + 2y = -1

6. 2x – 3y = 104x + 9y = -10

7. 3x + 3y = 93x + 6y = 9

8. 2x – 6y = 6-9x + 9y = 9

31

For # 9-12, write a system of equations for each word problem and solve. Show all work and express your answer with “words.”

9. The girl’s basketball team is having a bake sale for a fundraiser. They are selling brownies and cookies. The brownies cost $0.75 more than the cookies. Two brownies and two cookies cost $6.50. What are the prices of each?

10. The length of a rectangular poster is 10 inches longer than the width. If the perimeter of the poster is 124 inches, what is the width?

11. At the store, Luke bought two DVDs and four CDs for $61.00. Ben bought two DVDs and two CDs for $46.00. What were the prices for the DVDs and the CDs?

12. The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the numbers?

32

Appendix C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.00

20.00

40.00

60.00

80.00

100.00

120.00

Student Post-Assessment (Learning Goals 1-3)

Percentage

Students

Perc

ent C

orre

ct

33

Appendix D

3 5 6 7 8 9 11 14 15 16 220

10

20

30

40

50

60

70

80

90

100

Subsitution Homework: Comparing Day One and Day Two

Day OneDay Two

Students

Perc

et C

orre

ct

34

Appendix E

35

Appendix F

36

37

38

39

Appendix G

40

Appendix H

1 2 3 4 5 7 8 9 13 16 18 200.00

20.00

40.00

60.00

80.00

100.00

120.00

Female Students' Learning Goal 6 Percent Increase

Percent Increase

Female Student

Perc

ent I

ncre

ase

6 10 11 12 14 15 17 19 21 220

20

40

60

80

100

120

Male Students' Learning Goal 6 Results

Formative As-sessment

Post Assessment: Test Questions

Male Student

Perc

enta

ge C

orre

ct

41

6 10 11 12 14 15 17 19 21 22

-100.00

-50.00

0.00

50.00

100.00

150.00

Male Students' Learning Goal 6 Percent Change

Percent Change

Male Student

Perc

ent C

hang

e