• 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

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:

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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Real World Setting: Education

The Students will be able to solve the quadratic equations of the form: ax2 +bx +c = 0. The students will be able to make connections between these equations and the parabolic equations/functions. Different types of parabolic antennas' receivers can be used as examples for the real world application of such equations.

The main goal is to help the student to master different methods of solving the quadratic equations, and to discover their real world applications. the students will use different examples to model and solve the quadratic equations.

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

Vocabulary:  x = (-b ± √(b≤ -4ac))/2a.  The expression (b≤ -4ac) under the radical sign is called the discriminant. The solutions can be real or imaginary. If the discriminant is negative, the solutions are imaginary or complex. Otherwise the solutions are real. these words will be defined using mathematical examples in class.

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

Solving problems: The method of discriminant will be used to solve the applied problems. Students will be able to identify the solutions as real or imaginary based on the sign of the discriminant.

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Student Involvement - The students will complete the task:

Each student will try to solve some assigned problems individually, using the quadratic formula. Then as cooperative groups, they will work collaboratively to solve different problems assigned to each group. At the end, a selected student from each group will present the collaborative work to the class by showing the process and the solutions. 

Instruction - Activities will be organized and delivered by differentiating the complexity  of the quadratic equations and their solutions.  If the class is inclusive, students with special needs will be given special consideration and attention. A one-to-one session could be provided.

Calculator - Students will use calculators or other technologic tools that might be available.

Customer for Student Work - The student will present their work as evidence of task completion to their peers as class presentation, and to teacher as a hard copy from each group. Their work will be evaluated scored.

Reporting Results - The assessment results will be reported as a score point on a percentile rubric that is equivalent to the New Jersey rubric but in a different scale.The score rubric meets the Paterson Public Schools grading criteria.  

Timeline - The estimated time needed to plan, and teach this lesson is one to two class periods. During the second class period, the students will be assessed.

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Alignment: The lesson activities will be designed to meet the New Jersey core curriculum standards in alignment with Paterson Schools District mathematics curriculum. The students will be able to solve different type of problems involving the quadratic equations. However, the students will be limited to use strictly the quadratic formula to solve these problems.

Focus Question: I will ask the following questions to the students. How can you identify a quadratic equation? What the coefficients and the constant term mean?

Performances: By the end of this lesson students will be able to solve quadratic equations using the quadratic formula. They will understand how to identify the real roots and the imaginary roots (also called complex numbers). I will set higher expectation for students' performance. I will measure their performance through assessment test/quiz, and their class participation.

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

The students are expected to be able to identified the quadratic equations from a set of different equations. They must be able to differentiate the coefficients a, b of the x-variable  and the constant term c.They must show that they can use the quadratic formula to solve the quadratic equations.  

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

I will give them a set of equations including exponential equations, linear equations, quadratic equations, and so on, and I will ask the to identify the quadratic equations. I will also give them some quadratic equations and their solutions, and I will ask their to identify the sign of the discriminant in each case. I will also ask them to make connection between quadratic equations and some objects that they might use.

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

I will ask if anyone has ever use the quadratic formula to solve equations in their previous classes. If yes, how difficult or easy they might find this formula to be. I will engage them in a short class discussion before starting to teach.

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

• Stuck: Some students might make sign mistakes, while others might have difficulty to identify the values of a, b, and c using in the quadratic formula.
  
• Unstuck: I will help these students to use the sign properly and to perform the operations correctly.
  

Questions I can ask to push student thinking (without directly leading their thinking):

• Can you find a real life application to model a quadratic equation?
  
• How can you solve the real world problem by using the approach of solving quadratic equations?
  
• Can you apply the quadratic formula to solve problems in Finance, Engineering and any other fields?
  

At the beginning of the lesson...

Describe what you and the students will do:

1. I will have the students read the definitions.
   
2. I will have them understand the quadratic formula
3. I will use examples

Describe why they will complete your steps above:
These steps will help the students to meet the challenge of solving the quadratic equations using the quadratic formula. Their problems solving skills might be increased.

In the middle of the lesson...

Describe what you and the students will do:

1. The students will be engaged in practicing exercises from their book.
   
2. They will engage in group discussion
3. They will ask and answer questions.

Describe why they will complete your steps above:
In mathematics, practice remains one of the most important tools for students' success. I often told my students that a day without practicing mathematics is like a day without sunshine.

At the end of the lesson:

Describe what you and the students will do:

1. I will assign homework to my students.
2. I will explain the homework.
3. I will set the requirement for this homework, and the deadline to return it.

Describe why they will complete your steps above:

I will be able to measure the students' ability to use the quadratic formula before I give them the assessment test/quiz.

Source: Adapted from the BCPSS High School Mathematics Model

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Student Performance One:

Assessment Benchmarking Example: The actual benchmark for students performance for my first data driven lesson is set at 75/100. Students who achieve below this grade need more practice to increase their performance for the next assessment. 

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New Jersey High School Proficiency Test (HSPT)

Analyze: I recorded the students' grades for the assessment of my first data-driven lesson and I will examine the data to look for trends, contributing factors over a series of assessments of the same learning standard.

• Trends: During a test for the 3rd. making period I have included 2 quadratic equation problems and asked them to use any method to solve one of them, and to use the quadratic formula for the other one. Most of the students solved both problems using the quadratic formula. This trend indicates that most of them are comfortable to use the quadratic formula.
  
  The students performed well. Some students showed a 5%-10% increase in their test scores, and 82.1% of the students passed this test. This represents 8.62% increase from the previous test. Even the students who failed the test showed some improvement. I expect that this trend will continue.
  
  Recall that students can make real life connection between the quadratic equations and the parabolic antenna, some  roadway/highway curves, and even some brides. The quadratic equations have daily applications in our lives. 
  
  
• Contributing factors: I believe that peer tutoring was a high contributing factor. Group discussion helps more students to understand how to use the quadratic formula.
  
  My students will have the opportunity to be exposed to more data driven lessons.

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