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 9^{th} grade, course 1. In mathematics, both 7^{th} and 8^{th} 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
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.
Tlw know how to read, analyze, and solve word problems as equations.
Tlw distinguish useful information from the extraneous in a word problem.
#1 – Math as problem solving
#2 – Math as communication
#5 – Algebra
PROCEDURE:
Begin (10 minutes)
LESSON #2
TITLE: Variables on both sides of an equation.
with variables on both sides.
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
PROCEDURE:
Begin (10 minutes)
- Debrief activity.
TITLE: Solving equations with parentheses.
GOAL: Students will be able to correctly solve equations containing
Tlw be able to correctly solve equations with parentheses.
#1 – Math as problem solving
#2 – Math as communication
#4 – Mathematical connections
#5 - Algebra
PROCEDURE:
Begin (15 minutes)
TITLE: Evaluating equations with more than one variable.
OBJ: Tlw be able to accurately solve equations with more than one variable.
Tlw analyze problems using inequalities.
#1 – Math as problem solving
#2 – Math as communication
#5 – Algebra
PROCEDURE:
Begin (15 minutes)
- Worksheet counts as a quiz grade.
their book.
NCTM STANDARDS:
#2 – Math as communication
#4 – Mathematical connections
#5 - Algebra
PROCEDURE:
(One class period)
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