TITLE: Multiplying Polynomials
TASK DEVELOPER: Nicholas Willis
CONTENT AREA AND GRADE: 9th Grade Algebra I
SCOPE AND SEQUENCE: Section 10.2 Algebra I Textbook
TARGET TEACHING DATE: May 7, 2007
SCHOOL: John F. Kennedy 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 B. Functions and Relationships: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.

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:
  1. The students will briefly review 4-5 examples of multiplying a polynomial by a monomial.
  2. The students will follow a brief lesson on how to use the distributive property and previous knowledge from performance #1 to multiply polynomials.  
  3. As a class, we will review the steps of multiplying polynomials.[Summary]
  4. The students will attempt three problems of various difficulties; easy, medium, and hard.[Guided Practice]
  5. The students will be assigned nine problems (three from each difficulty) to work on in groups of two.[Group work]
  6. The students will be assigned nine problems (three from each difficulty) to work on independently.[Independent Practice]
  7. The students will be assigned 15-20 problems for homework. 

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SETTING:

I would illustrate a real-life application of multiplying polynomials. The problem would state. The glass portion of a sliding glass door has a ratio of height to width of 2:1. The framework adds 10 inches to the width and 12 inches to the height.

  1. Write a polynomial expression that represents the total area of the window, including the framework. Answer: 2x2 32x 120

  2. Find the area when x = 8, 9, 10, and 12

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SMARTSKILLS:

Level I: Acquiring Data - Data students will acquire in this standards-based task:

  • Skills:  Students will learn the definition of binomial and trinomial.
  • Processes:  Students will be able to use foil to multiply any two binomial expressions.  

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task:

  • Arranging:  Students will be able to arrange polynomial expressions in descending order.
  • Organizing:  Students will be able to combine like terms when using the FOIL method to multiply binomial expressions.

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:

  • Solving problems:  Students will be able to multiply polynomial expressions.

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PREFERENCES:

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

Instruction - Activities will be organized and delivered by differentiating the complexity of the products and performances expected of students.

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

Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks and homework time.

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

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

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 score point on a rubric, rule, or key and as a letter grade.

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

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ACTIVITIES:

The Directed Teaching Activity (DTA)

Opening - Estimated Time: 10:00 minutes

Focusing Student Attention: Accessing prior knowledge.

Statement of objective: The students will know how to and be able to multiply polynomial expressions.

Warm-Up:

  • Step 1: The students will review 4-5 examples of multiplying a polynomial by a monomial.

  • Step 2: As a class, we will write the correct responses on the board.

Heart of the Lesson - Estimated Time: 10:00 minutes

Teacher Directed Activities - Estimated Time: 35:00 minutes

  • Step 1: The students will follow a brief lesson on how to use the distributive property to multiply polynomials.
  • Step 2: As a class, we will review the steps of multiplying polynomials.
  • Step 3:The students will attempt three problems of various difficulties; easy, medium, and hard.

    Teacher Monitored Activities - Estimated Time: 10:00 minutes

Guided Practice:

  • Step 1: The students will be assigned nine problems(three from each difficulty) to work on in groups of two.

  • Step 2: I will assess how each group is doing by performing spot checks and assisting them with any questions/concerns.

    Extension, Refinement, and Practice Activities - Estimated Time: 5:00 minutes

Closing - Estimated Time: 10:00 minutes

Independent Activities: The students will be assigned nine problems(three from each difficulty) to work on independently.

  • Step 1: Introduction to Real-life application on finding dimensions.

  • Step 2: Students will learn how to setup binomials given information regarding the dimensions of object such as window, field, or a door.
  • Step 3: Students will be given 5 problems to work on in groups.

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BENCHMARKING:
  1. Assessment Benchmarking Example:  Multiplying binomials
  2. Real World Benchmarking Example:  Multiplying polynomials to find area.

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SCORING:

John F. Kennedy High School Mathematics Rubric: Brief Constructed Response
Level 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially 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 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It 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 attempt to apply a reasonable strategy or applies an inappropriate 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 0The 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: Adapted from: http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

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METACOGNITION:

Cognitive Information: I will collect the following information in a survey at the end of the unit.

  1. Describe what skills you needed to complete this task.
  2. Explain how you solved the goal, problem, or issue in this task.
  3. Draw a picture that shows how you solved the goal, problem, or issue in this task.
  4. Explain why you completed the task your way.

Attitude Information: I will collect the following information in a survey at the end of the unit.

  1. Do you feel that you are good in algebra?
  2. Did you find this task to be difficult?
  3. Did you see the usefulness of what you were asked to do in real life?
  4. Did you enjoy the task?

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RESULTS:

Data: 75% of the students placed in the level 3 learning categories, demonstrating a solid understanding of the subject matter.  Three students placed in 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 student demonstration of work at the blackboard and by assessing the homework that was collected the next day.

Improvement: The next time I present this lesson, I would group any level 2 and level 3 learners with a level 1 learner and assign a set of problems to worked on in groups.  This should help improve comprehension for all levels. 

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