TITLE: Algebra 1 - Squaring Binomials

TASK DEVELOPER: Erica Sadowsky

TEXT: Algebra 1, McDougal Littell

TARGET TEACHING DATE: May 1, 2007

SCHOOL: John F. Kennedy High School

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 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

3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

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The students will analyze and discuss multiplying identical binomials using the foil method and be able to develop a shortcut pattern where (a plus b)^2 = (a^2 plus 2ab plus b^2) for sums and (a - b)^2 = (a^2 - 2ab plus b^2) for differences.

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Real World Setting: Banking

You are a banker. You are faced with issue of showing a client how much money he has gained or lost on an investment over a period of years. You must write a model showing the profit and loss and demonstrate it to your client. Once you have completed your model, you will demonstrate it to your client (students in class).  

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

• Vocabulary: binomial, multiplication, trinomial, FOIL
  

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

FOR ACTIVITY 2:

• Organizing: Construction paper to demonstrate binomial multiplication.
  
• Creating patterns: On construction paper to demonstrate the multiplication.
  

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

• Solving problems: Using the real world model to solve investment problems.

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Student Involvement - The students will complete the task:

• individually
  
• in a project group
  
• in a whole class group setting

Instruction - Activities will be organized and delivered:

• as a teacher-facilitated set of hands-on activities

Special Education Accommodations - Students with special needs will require the following electronic devices: a calculator.

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

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 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

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

• Performance assessment
  
• Portfolio
  
• Exhibition

Reporting Results - The assessment results will be reported as a score point on a rubric.

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

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Teaching for Understanding in Mathematics

Activity 1: The teacher reviews yesterday's lesson and assigns problems that were not finished
(Estimated time: 10 minute)

• Step1: Have students complete do-now (on blackboard) consisting of five problems multiplying binomials using the foil method.
• Step 2: Teacher demonstrates the foil method on the blackboard.
• Step 3: The teacher chooses children to come to the board to demonstrate their work on the five problems.

Technology: calculator (not required)
Materials: pencil or pen, paper, chalk
Student product or performance: five completed problems
Scoring: teacher determination by spot checking.

Activity 2: The students are given construction paper and rulers and asked to divide a picture of a ten x ten square into dimension on the top and right side that model the form (a plus b).  
(Estimated time: 20 minutes)

• Step 1: Students choose to divide their squares as any of the following sums: 9 plus 1, 8 plus 2, 7 plus 3, 6 plus 4 (for these purposes we do not use 5 plus 5 because a and b are different variables requiring different values.)
  
• Step 2: Students use their rulers to mark lines on their squares demonstrating the dimensions they've chosen.  They determine that the square is now divided into two squares of different dimensions and two rectangles that are identical in size.  (using (7 plus 3) as a model, they determine that they have one square which is 7 x 7, two rectangles that are each 7 x 3 and another square that is 3 x 3.  
• Step 3:  Students can continue to model these square sums using other dimensions and eventually extend their findings to develop a pattern to show that (a plus b)^2 = a^2 plus 2ab plus b^2.

Technology: none
Materials: construction paper pre-printed with 10 x 10 squares, rulers, pens/pencils
Student product or performance: students show their 10 x 10's divided and labeled. They are able to demonstrate the formula showing the shortcut to finding the square of a binomial sum.  
Scoring: none

Activity 3: The teacher presents the task for the day and asks the students to work on it independently.

(Estimated time: 20 minutes)
Students utilize real world problem "Investment Banker", using the following information. (worksheet provided)

• Step 1: Students are given the following information:  To demonstrate the profit of an investment of P dollars that gains r percent of its value in one year we use P(1 plus r) and to demonstrate an investment that loses r percent of its value after one year we use P(1 - r)
• Step 2: Students are told to find the value of an investment if it has two consecutive years of a gain and two consecutive years of a loss. P(1 plus r)(1 plus r) and P(1 - r)(1 - r)
• Step 3: Students are given actual monetary values to work with:  An investment of $1000 gains 5% the first year and 5% the second year.  How much money does the investor have at the end of the second year? 

Materials: pencil, paper, calculator (not required)
Student product or performance: completed worksheet
Scoring: teacher determination by spot checking

Activity 4: Independent assignment in textbook requiring students to square sums and differences of binomials. Text:  Basic Algebra, Chapter 5, Section 8

(Estimated time 20 minutes)

Materials: textbook, paper, pencil, calculator (not required)
Student product or performance: completed assignment
Scoring: teacher determination by spot checking

Activity 5:  Review of students work.

• Students will put problems on board and demonstrate their answers to their classmates.

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John F. Kennedy High School Mathematics Rubric: Extended Constructed Response Items

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RESULTS:

90% 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 in level 2, demonstrating incomplete understanding of the subject matter. The above results were 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 class work that was collected at the end of the 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 on in groups. This should improve comprehension of the lower level learners.

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