TITLE: Algebra I - Solving Absolute-Value Equations
TASK DEVELOPER: Donald B. Rowland
TEXT: Algebra I, McDougal Littell, Lesson 6.4
GRADE AND CONTENT AREA: 9th Grade, Mathematics, Algebra I
TARGET TEACHING DATE: 1/28/07
SCHOOL: JFK High School
STANDARDS:

PATTERNS AND ALGEBRA - GRADE 9-12

STANDARD 4.3 PATTERNS AND ALGEBRA: All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

Strand C. Modeling: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

1. Use functions to model real-world phenomena and solve problems that involve varying quantities.

  • Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)
  • Direct and inverse variation
  • Absolute value
  • Expressions, equations and inequalities
  • Same function can model variety of phenomena
  • Growth/decay and change in the natural world
  • Applications in mathematics, biology, and economics (including compound interest)

Strand D. Procedures: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

1. Evaluate and simplify expressions.

  • Add and subtract polynomials
  • Multiply a polynomial by a monomial or binomial
  • Divide a polynomial by a monomial

2. Select and use appropriate methods to solve equations and inequalities.

  • Linear equations - algebraically
  • Quadratic equations - factoring (when the coefficient of X2 is 1) and using the quadratic formula
  • All types of equations using graphing, computer, and graphing calculator techniques

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PERFORMANCES:
  • The students will takes detailed notes as the lesson is presented
  • The students will write key words in their notebook.
  • The students will complete the assigned classwork problems on the classwork sheet provided. This sheet will be collected at the end of the class period.
  • The students will complete the homework assignment related to this lesson.

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SETTING:

Real World Setting:  You are a quality control inspector for a large manufacturing company. Your job is to inspect drill bits that are produced on an assembly line. The diameter of a drill bit must be between 0.62 and 0.63 inch.

  1. Write an absolute-value inequality to represent this requirement.
  2. A bit has a diameter of 0.623 inch. Does it meet the requirement

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PREFERENCES:

Student Involvement - The students will complete the task individually, in a whole class group setting, and through independent study.

Instruction - Activities will be organized and delivered as a teacher-facilitated set of hands-on activities and in a student textbook during class time.

Special Education Accommodations - Students with special needs will require the following presentation of information: Extra processing and response time and advance notice of assignments.

Use of Resources - The school will provide classroom time to complete the task

Customer for Student Work - The student will present their work as evidence of task completion to peers and teachers.

Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: The student's teacher, the student's self evaluation, and peers.

Assessment of Student Work - The following forms of assessment will be used to determine progress and results: Performance assessment

Reporting Results - The assessment results will be reported as a letter grade and as a learning level category as detailed in the Scoring section of this document.

Timeline - The estimated time needed to plan, teach, and score this task is two planning periods and one class meeting.

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ACTIVITIES:

STANDARDS

Refer to the Standards Section above for additional details.

Alignment: Design your lesson activities below to teach the New Jersey core learning goal, expectation, indicator, and assessment limit(s) for this lesson that are identified on your TaskBuilderOnline Design Page.

PURPOSE

Focus Question: What is the purpose of learning about absolute value?

Performances: By the end of this lesson students will know and be able to solve absolute-value equations.

Lesson Outline:

- Goal: Describe the goal of the lesson - Solving Absolute-Value Equations

- Present Lesson: including sample problems as described below

- Classwork: Students will work independently and in pairs on various problems. The work will be completed on a classwork sheet. This sheet will be collected at the end of class, graded, and returned to students the following class session. The following problems will be assigned for classwork from the Algebra I book as described above.

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- Boardwork: Students will volunteer or be asked to place several problems on the blackboard. In this way, the teacher can check the student's work as well as the student's peers.

- Homework: The following problems will be assigned for homework, to be completed by the next class meeting.

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UNDERSTANDING

1. What will students say or do to show they understand (both during and at the end of the lesson)?

As the lesson is presented, the teacher will ask a number of questions to determine if the students are grasping the concepts given.

Students will be asked to complete several sample problems during the lesson. The teacher will observe student work and offer help at this time. Additional instruction will be presented if necessary.

2. What questions can I ask to uncover student thinking?

What is the purpose of learning about absolute value? How can this be applied in real life situations?  Does this apply to situations dealing with distance and length?

ENGAGEMENT

What is the "hook" that will initially engage students in the lesson?

  • How many students have ever flown on an airplane?

  • How many students have ever climbed a mountain or a large hill or a tall tree?
  • How many students have ever gone swimming where the water is really deep?
  • How many students have ever gone snorkeling?
  • How far were you able to dive below the surface?

  • Relate this discussion to absolute value and the distance from (zero) sea level.

Where students might get stuck and strategies to get them unstuck:

  • 1a. Stuck: Absolute value variables - requires two solutions

  • 1b. Unstuck: Substitute numbers to prove the point

  • 2a. Stuck: Setting up negative and positive equation

  • 2b. Unstuck: Perform several examples

LESSON AT-A-GLANCE

At the beginning of the lesson...

Describe what you and the students will do:

1. Teacher will present lesson as outlined above and assist students as necessary to ensure that the lesson is understood.

2. Student will listen, ask questions, copy down the notes, try example problems, complete assigned classwork problems, volunteer to place problems on the board, submit classwork for assessment, and copy down the homework assignment to be completed by the next class meeting.

3. Notes examples

Example 1: I x-2 I = 5

Example 2: I 2x-7 I - 5 = 4

Describe why they will complete your steps above:

Completing the steps above will ensure a successful learning session.

Source: Adapted from the BCPSS High School Mathematics Model

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SCORING:

Scoring: Students will be scored based on the rubric below. In addition to this, student understanding will be assessed based on answers to questions presented during the lesson and throughout the class period, by observing class work as it is being performed, by observing work placed on the blackboard, by assessing the class work that will be collected at the end of class, and by reviewing and assessing the homework that will be due the following class meeting.

New Jersey High School Mathematics Rubric:
Extended Constructed Response

Level 4

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

Level 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Level 2

The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Level 1

The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Level 0

The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."

Notes: Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Source: http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

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RESULTS:

Results: Most of the students were placed in the level 4 and level 3 learning categories, demonstrating a solid understanding of the subject matter. One student was placed at level 2, demonstrating incomplete understanding of the subject matter. The above results were determined by assessing answers to questions presented during the lesson and throughout the class period, by observing classwork as it is was being performed, by observing work placed on the blackboard, by assessing the classwork that was collected at the end of class, and by reviewing and assessing the homework that was completed the following class period.

Improvement: The next time this lesson is presented, I would place any level 2 or level 3 learners next to level 4 learners and assign some problems to be worked in pairs. This should improve the comprehension of the lower level learners.

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