Completing the Square

Definition of Completing the Square

Completing the square is the process of converting a quadratic equation into a perfect square trinomial by addition or subtracting terms on both sides. 

Graphing Quadratic Functions

CA Standard 9.0: Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)2+ c.
CA Standard 10.0: Students graph quadratic functions and determine the maxima and minima of the function.

Objectives:

• Find and interpret the maximum and minimum values of a quadratic function.
• Understand patterns, functional representations, relationships and functions.
• Analyze and create multiple representations of mathematical situations and structures using algebraic symbols.
• Specify and describe locations and spatial relationships using coordinate geometry and other representational systems.

Introduction:

The general technique for graphing quadratics is the same as for graphing linear equations. However, since quadratics graph as curvy lines called "parabolas", rather than the straight lines generated by linear equations, there are some additional considerations.
The most basic quadratic is y = x². When you graph straight lines, you only needed two points to graph your line, though you generally plot three or more points just to be on the safe side. However, three points will almost certainly not be enough points for graphing a quadratic, at least not until you are very experienced.
The term quadratic comes from the word quadrate meaning square or rectangular. Similarly, one of the definitions of the term quadratic is a square. In an algebraic sense, the definition of something quadratic involves the square and no higher power of an unknown quantity; second degree. So, for our purposes, we will be working with quadratic equations which means that the highest degree we'll be encountering is a square. Normally, we see the standard quadratic equation written as the sum of three terms set equal to zero. Simply, the three terms include one that has an x², one has an x, and one term is "by itself" with no x² or x.

A quadratic function is described by an equation of the form:  f(x)=ax² + bx + c,
ax² = quadratic term
 bx = linear term
c =constant term
a ≠ 0 
Note: If a = 0, the x² term would disappear and we would have a linear equation. If the highest degree in an equation is 1, meaning that the x-term is x¹ or in the form

Ax + By = C or y = mx + b, the equation is always linear 

Lesson Description

Students will be engaged in both formal and informal approach as they are introduced to the topic of 2nd degree polynomial functions or quadratic. I will provide brief background knowledge by engaging the students in exploration activities that allow them to make connections and arrive at an understanding of quadratic functions. To enhance and sustain engagement, students will be immersed in various modalities such as multimedia, online manipulatives and group activities to achieve the learning goals objective.

Priming/Review (15-20 minutes)

Students will complete engagement and exploration activities that will allow them to get motivated and acquainted with the subject matter.

1. Watch 90 second video on graphing a quadratic function. Click the video to play..

image from www.google.com

2.  Students break into groups of 3 and build a table of values for a parent function y = x².  It must include negative and positive values of x.
3. Each group will assign different roles (in rotation) of who would investigate the function using online manipulatives (SeeingMath, Graph Sketcher, Quadratic Function Explorer,  Desmos) to generate graphs.
4. Students will then share and discuss their generated visual output among their classmates. Ask them to compare and contrast to previous knowledge of linear function's graphs and table of values.

Guided Learning

1. Students will watch video on Voicethread and then individually read/do activity on Interactive Mathematics
2. From the previous activity on SeeingMath, leave the default function as y = x² , student click on New command button and plot a quadratic functions f(x) =  x² - 4x + 6 by plugging the numeric coefficients, or  they can use Quadratic Explorer (use slider to change the values of the coefficients in a given function).

Group Activity/Assessment 

1. Working in group of 2s, generate 5 of your own quadratic function f(x) and then plot them using either SeeingMath or Quadratic Explorer.
2. Describe the behavior of the transformed equations by creating two-column notes with critical attributes of the quadratic functions such as vertex, intercepts, and line of symmetry.
3. What happens to the graph display when they change different parts of the equations?
4. Explain what condition(s) will make a quadratic a linear function, or what’s the effect of making the coefficient of x² less than 0, equal to 0, or greater than 0?
5. What causes the parabola to open upward or downward and what are the names of each openings?How do we know if the function has x-intercepts or y-intercepts?

Homework

Summarize your ideas of the different characteristics in a quadratic equation and how to find then in a graph and in the equation. Must be submitted in PowerPoint slides

Closing/Recap (3 minutes video)

Click here for a quick review on how to analyze a quadratic equation when graphing.

Profile: My name is Mr. Diaz. I teach high school math-Geometry, Algebra 2, and Trigonometry- at Frederick K. C. Price III School. This is a private Christian school in Los Angeles, California. I have been teaching there approaching 7 years, and prior to that, I taught for Los Angeles Unified School District K-5 for 6 years. I enjoy teaching future leaders math concepts and theories, because it relates to everyday life and it is a universal language. In addition, I like incorporating mathematics with technology and I’m constantly seeking out new ways to improve my skills.
I am originally from Jamaica and I went to high school in Long Island, New York. I have a BA in accounting and an MBA from CSUDH. I am scheduled to take my math CSET in the early part of April, 20014. I am currently pursuing a Master’s Degree in Educational Technology at National University.
I have been married for seven years and we are looking forward to starting a family. I have 5 sisters, no brothers and 2 sister-in-laws. Well, as you can see, I’m out numbered. On any given weekend, I cherish long drives along the California coast line-the best ocean in the world-or catching a beautiful sunset in Redondo Beach or Manhattan Beach.
In addition to being a teacher, I am also a real estate broker. Besides teaching math, real estate is my other real passion and I am a big fan of “HGTV” and “DIY” network.

CLASS OVERVIEW

         

Relevance:  This course will provide a solid foundation for further study in mathematics by helping students develop computational, procedural, and problem-solving skills. To be good at mathematics, students must learn to translate real-life situations to mathematical models and obtain solutions; and Algebra will help students develop this skill.

Course Description:  This discipline complements and expands the mathematical content and concepts of Algebra I and Geometry. Students who master Algebra II will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic and exponential functions, the binomial theorem, and the complex number system. 

Course and Homework Policy: Your success in this class is based on your understanding of the subject matter taught and the completion of assignments/homework. Therefore, you are responsible for completing homework/assignments, which are to be handed in at the beginning of the class period on the date they are due. All work must be shown in order to receive full credit. Parents will be notified via GRADELINK at the occurrence of each missed homework/ assignment and incomplete work. In addition, please contact other classmates or check GRADELINK to obtain missed homework/assignments when absent. Tests and quizzes must be completed when assigned. Excused absences result in the ability to take the missed test/quiz within two school days of the absence. Any test/quiz not completed within the two days will result in the student receiving a zero for that test/quiz. There are no make-ups for missed Pop Quizzes. Students who have excessive missed assignments of any type are absolutely unable to participate in any extra-credit work.

BASIC CLASSROOM RULES:

• School Policy: school policies will be followed closely as stated in the hand book
• Talking: Students speak "one-at-a-time", and never while another has "the floor"
• Attendance: both physical and mental
• Respect: for yourself, others (especially Mr. Díaz), and the learning materials
• Take notes. Everything written on board MUST be transcribed onto paper.
• Stay Seated: Students will remain seated unless given permission to do otherwise

                         If everyone works together, education can be enjoyable

                                                 

                                                                Expectations:

YOU are expected to:

• come prepared for class every day
• participate in class every day
• do the daily homework which is assigned
• work together cooperatively in your groups
• formulate and ask questions
• have some fun
• expect some difficulty

Mr. Díaz is expected to:

• come prepared to teach every day
• help students learn the concepts of geometry
• be available outside of class for questions
• give advice when asked
• give students timely feedback
• have some fun
• be stumped by a few questions

                              Making Math Cool: Alex Kajitani at TEDxVillageGate

   Links

 FKCP III                  Handbook                            Calendar                           Gradelink