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1st DEGREE EQUATIONS AND
INEQUALITIES IN ONE VARIABLE
Joan Koral
EDU 548
5/12/99

OVERVIEW
The topic of my unit is equations and inequalities. The purpose is to give students a solid competence in solving all types of equations and a solid understanding of how variables and equations can be utilized to solve real life problems. Equations and inequalities can be most easily related to science, and I will cover vertical motion problems, trajectory path problems, and the like. Current NCTM [National Council of Teachers of Mathematics] standards encourage teachers to make learning relevant and meaningful by showing real life connections with what is being covered in the classroom. This unit offers excellent opportunities to show students how algebra can be useful.

I also cover some language arts, by incorporating weekly [or bi-weekly] journal entries. For the authentic assessment of this unit, the students will be producing books. This project has students utilizing their writing skills, artistic skills, and research and organization skills]. I will expose the students to some history by showing a video of Andrew Wiles quest to solve Fermat’s last theorem [by NOVA].

My unit is designed for the 9th grade, course 1. In mathematics, both 7th and 8th grade cover a rather varied curriculum with little consistency from school to school. The middle school years often review previous concepts and introduce a deeper level of complexity and understanding of material. Solving equations and competently manipulating integers and variables is a very basic, crucial component for success in higher mathematics. Many schools track students by ability level; thus, students in course 1 have a wide disparity in ability and prior knowledge. This unit is important for all students and I would utilize the same instructional techniques to approach any class. I may add additional remediation work for students who don’t possess the arithmetic skills necessary to succeed.

LEARNING GOALS

• Students will be able to competently solve 1st degree equations, appropriate to course 1 learning level.
• Students will be able to decipher word problems and solve them as algebraic equations.
• Students will solve inequalities with one variable.
• Students will understand and be comfortable using terms and symbols appropriate to this material.
• Students will make connections to other subjects and real life situations where equations, variables, and/or inequalities would apply and be useful.
INSTRUCTIONAL OVERVIEW
Mathematics is a difficult subject to teach in creative ways. Math is a sequential process, in so much as, new subjects and topics must be based on previous knowledge. I chose the sequence of the activities because they make the most sense in that particular order. There is also not much opportunity for students to ‘discover’ new math concepts. Knowing that I8I stands for the absolute value of 8 cannot be arrived at logically, given any amount of time or creative learning activity. So my teaching will utilize direct instruction. However, there is room for students to make connections and discoveries about relationships. Additionally, after explaining the algorithm, there is a wide variety of interesting ways to give students both independent and guided practice.

I will begin the unit with a lesson on simple equations: how to set them up from a word problem and how to determine what information is germane to the problem and what is irrelevant. I will also review the additive inverse and multiplicative inverse properties, previous concepts the students should be familiar with. I will make connections to science, even at this early juncture. Any time we see a formula, it involves variables. I will give examples of rate, vertical motion, and acceleration problems that will incorporate equations.

I will proceed to the next step in complexity, variables on both sides of the equations.

I did not include lessons for every day of the unit, but rather I chose five sample days. There would be other lessons utilizing other topics and incorporating other subjects as they fit with this unit. However, in general, I looked at every day of the unit and tried to vary the amount and type of direct instruction, seat work, group work, etc. If one day is especially intellectually challenging, I may assign no homework. If we spend an entire period on a group activity, I may assign a more difficult cumulative review to be certain each individual student is learning. I had a difficulty truly matching the type of instruction with the content, only because this unit, like many in math, seems to focus on acquiring algorithmic skills. The students need to be able to ‘solve correctly’, not compare and contrast, defend their position, or analyze and draw conclusions.

The next topic I will address is solving equations with parentheses. For review, I will need to be certain the students remember the order of operations. I chose a fun activity since my goal is to motivate the learners to access prior knowledge. After showing examples and explaining the method, I will have students work in small groups on guided practice. In math especially, I think it helps some students to ‘talk it out’, preferably to a peer. Explaining your thought process to others, defending your position, and confronting disparities is an excellent way for students to process and internalize new information.

Absolute values and how they are handled in equations is the next topic. The next progressive step is solving equations with more than one variable. This is a discoverable concept, because variables are treated just like integers in equations. Therefore, on this day the students will work in small groups and process the information inductively.

Finally, this unit will include inequalities and solution sets. Basically, this involves many definitions and properties that will be given to students via direct instruction. Students need to be able to graph the inequalities also. This is something they should know from previous classes, so the instruction will be more challenging and offer more room for choice by the student, depending on how confident they feel on this topic.

ASSESSMENT
Assessment will be on going throughout the unit. I will constantly observe the students to gauge their understanding. Informal performance assessment means I will be gathering information and assessing the comprehension as I teach and watch their responses, as the students work on guided or individual practice, and as the students work in their cooperative groups on math activities. The rationale is that I will be able to more genuinely determine the students’ progress. I am weary of my own biases and I know I can’t rely solely on something as subjective as my own opinion. However, it feels the most reliable on a day to day basis. I have to have a way to see if the students are ‘getting it’ before the first quiz or unit test. By looking only at a piece of paper, a student who does not understand may have guessed correctly or cheated; and conversely, a student who does understand may have been distracted or had problem with just that specific question. Also, by continually monitoring the students, I can adjust my teaching, incorporating more remediation and review when necessary.

I plan to use a weekly journal entry throughout the course to encourage writing about math. I will continue to use these entries to gauge student interest level, frustration level, and learning. From my own experience, I know that writing in a journal gives students more opportunity to speak and be honest than they would have in class. This gives me one more way to assess what they know and how they are feeling about the class. I also believe that in any subject it is vital to teach students how to reflect on their learning and pause to think about their progress. These journal articles will be collected weekly and worth 10% of the total grade.

Homework will be assigned on a regular basis. With math especially, it is imperative that students have mastered the information before they can move forward to cover new material. The independent practice of homework is an excellent way to reinforce the information learned that day. Especially in a direct instruction or small group situation, when a student watches someone else do it, it makes sense and they understand. However, when one gets home and has to actually do it him/herself, a new depth of knowing occurs. The homework will be collected and graded not on a right/wrong concept, but rather, if they have made an honest effort and completed 85% of the questions, they will be awarded full credit. This grade will count for 20% of the students’ class grade

Quizzes will be given every other week and will count for 30% of the grade. Unit tests will be given after a section of material is completed and will count for 40% of the students’ total grade.

AUTHENTIC ASSESSMENT
I will be having the students create a book about the subjects we have covered in this unit. I have created an accompanying rubric that will be distributed to the students at the beginning of the project. I tried to identify which components are important and what material to ‘count’ towards the assessment. I broke down the task into three areas, math content, practice exercises, and organization. I felt that those areas are the most relevant to the task being completed in a way that learning will result. Differentiating between the levels within those categories was difficult. I tried to construct the rubric in positive language that would encourage students to achieve success. I will provide guidance and be informally assessing the students as they work. Especially from students who have not had experience with writing in math class, I may encounter some resistance. I will collect each student’s chapters of the book when they complete them.
RUBRIC SCALE

LESSONS #1

TITLE: Simple equations

GOAL: Define new terms and introduce unit.

OBJ: The learner will [henceforth, abbreviated as Tlw] be able to define multiplicative and additive inverse.

Tlw know how to read, analyze, and solve word problems as equations.

Tlw distinguish useful information from the extraneous in a word problem.

NCTM STANDARDS:

#1 – Math as problem solving

#2 – Math as communication

#5 – Algebra

PROCEDURE:

Begin (10 minutes)

• present word problem for students to ponder individually
[Tom can have 500 calories for lunch. A hamburger without the roll has 320 and an average fry has 15 calories. If he eats the burger, how many fries can he eat and have a total of 500 calories?]
• On board, write what they think and how they want to set up the equations.
• Give correct labels for terms [but don’t solve yet]
Middle (20 minutes)
• In small groups, have students work on setting up equations from word problems, after determining the relevant information. (10 min.)
• In large group, give notes on multiplicative and additive inverse. (10 min)
End (15 minutes)
• Return to board, show how to use multiplicative and additive inverse to solve the original problem we set up.
• In small group, solve equations they had set up from earlier.
ASSESSMENT:
- informal observations of students working in groups. - group worksheet- homework

LESSON #2

TITLE: Variables on both sides of an equation.

GOAL: Students will be able to confidently and correctly solve equations

with variables on both sides.

OBJ: Tlw solve correctly equations with variables in both members.

Tlw effectively work in a small group to accomplish their task.

Tlw contribute to the class discussion their ideas on how to solve a

specific type of equation.

NCTM STANDARDS:

#1 – Math as problem solving

#2 – Math as communication #5 – Algebra

PROCEDURE:

Begin (10 minutes)

• Go over homework questions.
• Give new, challenging question to review and get their heads back in the game. [Rachel has 6 more than twice as many newspaper customers than when she started. She now has 98 customers. How many did she have when she started?]
Middle (10 minutes)
• Ask for input, how do students think we would solve equations with variables on both sides of the equal sign?
• Explain/give notes and examples
Carousel Activity (20 minutes)
• On large sheets of easel paper, I will prepare ahead of time problems to be solved.
• Students work in small groups, beginning at one paper and work the problem until I say ‘switch’.
• All groups rotate clockwise to the next sheet of paper. They review the work done by the previous group, then continue to work the problem.
• If they finish one, they sign it [all groups have different colored markers] and begin the next problem.
• One important rule is ‘no making intentional mistakes, because you never know when I will say switch. I may let you work the whole problem or maybe only a minute’.
• Each completed problem is worth 10 points per group.
Conclusion (5 minutes)

- Debrief activity.

ASSESSMENT
• Carousel activity problems.
• Informal performance of observing everyone participating.

LESSON #3

TITLE: Solving equations with parentheses.

GOAL: Students will be able to correctly solve equations containing

parentheses. OBJ: Tlw create word problems that can be turned into equations.

Tlw be able to correctly solve equations with parentheses.

Tlw be able to recite the correct order of operations. NCTM STANDARDS:

#1 – Math as problem solving

#2 – Math as communication

#4 – Mathematical connections

#5 - Algebra

PROCEDURE:

Begin (15 minutes)

• BINGO Game with flash cards to review order of operations.
Middle (10 minutes)
• Explain/give notes on how to solve equations containing parentheses.
Small Group Activity (20 minutes)
• See worksheet. [Create word problems involving equations that other students can solve].
ASSESSMENT
• Questions created by groups.
• Observe students working in groups/ see who is participating and knowledgeable.
• Cumulative review for homework.

LESSON #4

TITLE: Evaluating equations with more than one variable.

GOAL: Students will understand how to evaluate equations with more than one variable.

OBJ: Tlw be able to accurately solve equations with more than one variable.

Tlw analyze problems using inequalities.

Tlw correctly evaluate problems with inequalities. NCTM STANDARDS:

#1 – Math as problem solving

#2 – Math as communication

#5 – Algebra

PROCEDURE:

Begin (15 minutes)

• Individually, each students completes the worksheet of all the questions made by students yesterday.
Middle (20 minutes)
• Small groups are given tug-o-war problem to try to solve.
End (10 minutes)
• Debrief activity. If a group thinks they have the correct answer, have them stand up and explain how they found it. Have students tell about how they approached the problem, how to get started, etc.
ASSESSMENT

- Worksheet counts as a quiz grade.

• Informally observe and interact with students as they try to solve the problem.
• Homework.

LESSON #5
TITLE: Authentic assessment.
GOAL: Students will create one chapter for a book on equations. OBJ: Tlw correctly identify important rules and information to include in

their book.

Tlw create relevant examples and an correct answer key.

NCTM STANDARDS:

#2 – Math as communication

#4 – Mathematical connections

#5 - Algebra

PROCEDURE:

(One class period)

• See handout
ASSESSMENT - Each students’ chapter will be graded according to the rubric distributed with the assignment.

REFERENCES

Dressler, I. (1966). Algebra 1. New York, NY: Amsco.

Dressler, I. & Keenan, E. (1981). Integrated Mathematics. New York, NY: Amsco.

Foster, A., Winters, L., Rath, J., & Burrill, G. (1988). Algebra Essentails. Columbus OH: Merrill Publishing Co.

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics [Online]. Available: http://www.enc.org/reform/journals/ENC2280/280dtoc1,htm>

main page- 548 unit plan guidelines - assessment page - lessons and units