Title: PARABOLAS, FALLING BODIES, ALL THAT'S QUADRA

Grade: 10th   Subject: Mathematics   Class: Algebra II   

Purpose
By the end of this instructional unit, the students, given one representation of a quadratic function, will be able to characterize a quadratic function through the three other representations. For instance, the students, given a contextual situation, will write the appropriate quadratic equation, will graph the equation, and using a graphing calculator will answer questions relevant to the problem situation. There are a couple of sub-goals that must be achieved before achieving this goal. They include equating a graph with algebraic representation and solving quadratics using a variety of methods. By representing the function in multiple ways the students show their understanding is deeper than simply applying an algorithm to a problem.

Objective
Objective 1 - Sketching Quadratic Graphs
Given graphs of 10 different quadratic equations and graphs (without the use of a calculator), the student will match the appropriate equation to graph with at least 80% accuracy.

Objective 2 - Vocabulary/Terminology
Given ten multiple choice questions (without the use of a calculator), the student will answer quadratic vocabulary and terminology questions with at least 80% accuracy.

Objective 3 - Solve Quadratics Algebraically
Given six quadratic equations (without the use of a calculator), the student will use either factoring or the quadratic formula to solve the equations with 83% accuracy.

Objective 4
Given four quadratic equations and a coordinate graph (without the use of a calculator), the student will use the completing the square method to correctly transform a quadratic equation in standard form into vertex form and sketch the graph with 75% accuracy.

Objective 5 Solving Graphically
5.1 Given four quadratic functions and a graphing calculator, the student will correctly enter function into the calculator and find an appropriate viewing window for each equation.
5.2 Given four quadratic functions and a graphing calculator, the student will correctly enter function into the calculator and find the roots for each function.
5.3 Given four quadratic functions and a graphing calculator, the student will correctly enter function into the calculator and using the maximum or minimum utility, find the vertex for each function.

Objective 6-
Given two quadratic application problems, the student will correctly write the appropriate quadratic function, identify they the independent and dependent variables, and interpret the meaning of the roots and maximums, minimums for each function with 83% accuracy.

Materials

1. PC with PowerPoint Software connected to a projector
   
2. Computer Lab - each computer has TI Interactive Software (by Texas Instruments) and PowerPoint Software
   
3. Overhead Projector
   
4. Graphing Calculators for each student
   
5. Calculator Based Rangers
   
6. Measuring Instruments: Stop watches, rulers, measuring tape etc.
   

Procedures

1. (Day 1)-PowerPoint Presentation by instructor will introduce quadratics. Four representations of quadratics will be shown: algebraic, graphical, numerical, and contextual. Students will build the graph of the quadratic parent function y = x2 numerically with a table. Using TI-Interactive either on their own calculators or on personal computers in the computer lab, students will experiment the effects of changing signs, changing coefficients and adding and subtracting terms to the parent function. As a class, we will write the rules (or effects) that each change has on the parent graph.Students will be given and handout about everything that they will be expected to know for the unit test.
   
2. (Day 2)-Students will complete Quick Draws to practice graphing quadratic equations.Instructor will use the overhead projector to demonstrate factoring quadratic equations. Students will use worksheets/handouts to practice factoring quadratics to find the solutions and roots.
   
3. (Day 3)- Students will complete Quick Draws to practice graphing quadratic equations.Students and teacher will discuss questions from previous assignment. Problems for which students need help, will be demonstrated by students or the teacher using the overhead or white board.Teacher will model changing between standard form and vertex form of a quadratic by completing the square on the overhead and white board. Using graphing calculators and handout, students will explore and determine that the graph in standard form and vertex form are the same.Students will use worksheets/ handouts and textbook to practice completing the square and more practice factoring quadratics.A challenge problem will be written on the white board for any students desiring to solve for x in the equation ax2 + bx + c = 0.
   
4. (Day 4) - Students will complete Quick Draws to practice graphing quadratic equations.Students and teacher will discuss questions from previous assignment. Problems for which students need help, will be demonstrated by students or the teacher using the overhead or white board.Teacher will lecture to demonstrate the solution to the challenge problem from the previous day. The solution yields the quadratic formula. The teacher will teach the class the €Quadratic Formula€ Song to the tune of Pop Goes the Weasel to help them memorize the formula. Teacher will use overhead and lecture to demonstrate solving quadratics by using the quadratic formula.Students will use whiteboard, handouts and textbook to practice using the quadratic formula and factoring to solve equations and find roots of quadratic equations.
   
5. (Day 5)- Students will complete Quick Draws to practice graphing quadratic equations.Students and teacher will discuss questions from previous assignment. Problems for which students need help, will be demonstrated by students or the teacher using the overhead or white board.Using computers, students will individually take On-Line Quiz 1 this will be on-going daily until each student masters the necessary skills.Using the PowerPoint Jeopardy game, groups of students will review and practice quadratic skills learned so far in this unit.
   
6. (Day 6)- Students will complete Quick Draws to practice graphing quadratic equations.Using the TI-83 graphing calculator plugged into overhead, the teacher demonstrate finding roots, vertices and to solve quadratics graphically.Students will use graphing calculators, handouts and worksheets to practice finding roots and vertices (maximum or minimums) and to solve quadratic equations
   
7. (Day 7) Students will complete Quick Draws to practice graphing quadratic equations.Students and teacher will discuss questions from previous assignment. Problems for which students need help, will be demonstrated by students or the teacher using the overhead or white board.Using the TI-83 graphing calculator plugged into overhead and application problems on a handout, the teacher demonstrate identifying independent and dependent variables, writing the appropriate equation, interpreting the meaning of the roots vertices.Students will use graphing calculators, handouts and worksheets to practice writing equations and finding and interpreting roots and vertices (maximum or minimums) and to solve quadratic equations.
   
8. (Day 8) - Working in groups students will use graphing calculators, handouts and worksheets to continue to practice writing equations and finding and interpreting roots and vertices (maximum or minimums) and to solve quadratic equations.
   
9. (Day 9) - Students will complete Quick Draws to practice graphing quadratic equations.Using computers, students will individually take On-Line Quiz 2 this will be on-going daily until each student masters the necessary skills.Using the PowerPoint Jeopardy game, groups of students will review and practice quadratic skills learned so far in this unit.Teacher will lecture on discuss the handout given on the first day (about what the students will be expected to know for the unit test).A handout of practice test will be available for student wishing to do it as they are studying at home.
   
10. (Day 10)- Students will be given a handout that describes the end of unit group project. Students will have the choice of an application project or a PowerPoint Presentation that reviews and discusses important concepts from the unit. Students will form working groups and brainstorm ideas.
    
11. (Days 11-13)Groups of students will use calculators, CBRs (rangers to measure speed connected to calculator), rulers, stop watches, any objects of their choosing to complete the end of unit project.
    
12. (Day 14- 15?)- Students will present projects
    
13. (Day 16) Students will take a paper and pencil assessment. There will be both a calculator and a non-calculator portion.
    

Assessments

1. Quick Draws, Online quizzes, unit project, and end of unit test.
   

Vocabulary

• Function:  A relation in which for each member of the domain there is a unique y-value. It passes the vertical line test.
  
• Quadratic Function:  A function which can be written in the form: F(x) = Ax^2 +bx + c
  
• Parabola:  The shape of the graph of a quadratic function.
  
• Vertex Form:  The form of a quadratic equation that can be given through "Completing the Square." The form is: a(x-h)^2 + k. (h,k) is the ordered pair of the vertex.
  
• Axis of Symmetry:  The equation of the line which splits the graph in half. On each side of the axis of symmetry the graph appears to be a mirror image of the other side.
  
• Root:  The x-values which yield zero as a y-value (also called zero or x-intercept). It this the x-value for which the graph crosses or touches the x-axis.
  
• Y-Intercept:  The y-value achieved when zero is plugged in for x in any function.
  
• Quadratic Formula:  This formula can be used to solve for roots. It is especially useful when the quadratic equation cannot be factored. Formula is: x equals negative b plus or minus the square root of b-squared minus 4ac, all over 2a. (If math symbols were available I would have written this symbolically.)
  
• Double Root:  If an equation, when factored, contain two of the same linear factors, a double root occurs at this x-value. Graphically, it "bounces" at this x-value
  
• Complex Root :  Also called imaginary root. Occurs when the factors include taking the square root of a negative number. Graphically, complex roots do not cross the x-axis.
  

Standards

1. Mathematics  9-12  
   Understands and applies basic and advanced properties of functions and algebra
   Uses a variety of models (e.g., written statement, algebraic formula, table of input-output values, graph) to represent functions, patterns, and relationships
   
2. Mathematics  9-12  
   Understands the general nature and uses of mathematics
   Understands that mathematics is the study of any pattern or relationship, but natural science is the study of those patterns that are relevant to the observable world