Title of Lesson: How to solve quadratic equations using completing the square and quadratic formula
Subject: Precalculus
Grade Level: 11th and 12th
Teacher: Fallon DeMonte
Objectives:
o Conditions – The students will be able to solve quadratic equations using two new methods, completing the square and quadratic formula. They must also gain an understanding about what a discriminant is and how the discriminant will help them find what type of roots they will get.
o Behavioral Verb – Formulate
o Criteria – Students will demonstrate an understanding of how to solve a quadratic equation by factoring and graphing. Students should also be able to classify what a quadratic equation is and how it relates to polynomial equations. In this lesson, students will apply their prior knowledge of factoring in order to solve by completing the square. Also, students will demonstrate how to use the quadratic formula and the relationship between the quadratic formula and finding a discriminant. They will then distinguish the differences betweens the methods for solving quadratic equations.
SCSDE Curriculum Standards Addressed:
o Math-PC-3.6 Carry out a procedure to solve polynomial equations algebraically.
NCTM National Curriculum Standards Addressed:
o Apply and adapt a variety of appropriate strategies to solve problems.
o Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
o Recognize and use connections among mathematical ideas.
Prerequisites:
o Students must be able to solve polynomial equations.
o Students must be able to articulate that a polynomial equation is an equation that is defined by a polynomial in one variable with real coefficients.
o Students must be able to apply prior knowledge of how to solve quadratic equations by factoring.
o Students must be able to understand how formulas are used and what the variables in the formulas represent, and apply this to the quadratic formula.
Materials/Preparation:
o Computer with smart notes
o Smart board
o PowerPoint
o Paper for students to take notes when doing examples
o SCLA (given during previous class)
Procedures:
o Introductory Activity: To start the class, I plan to go over the SCLA the students completed in the previous class. Normally I would just review what the students do not know from the SCLA, but most of the students had a hard time completing the entire SCLA. Most of the questions applied to factoring quadratic equations or finding the roots of a polynomial equation. These are important concepts the students need to understand before starting the new lesson.
o Main Activity: During the previous lesson the students were given a chart that they filled out with information from their textbook. This chart explained the different methods for solving quadratic equations, which included: graphing, factoring, completing the square, and the quadratic formula. To begin, I plan to go over what they put in this chart for completing the square. The first problem given on the power point is a simple quadratic equation that can be solved by factoring. Having them solve the equation by factoring first will allow them to see what the roots are so that when the method of completing the square is introduced they will know that you can get the same answer using different methods. With the first equation I plan to go over the procedure for solving a quadratic equation using completing the square (see notes sheet for work). The easiest way, in my opinion, for students to learn how to solve by completing the square is by giving them many different examples that have an increasing difficulty level. I plan to go over the examples and have them verbally explain how to solve them (examples are on the notes sheet). The lesson should be interactive so I plan to have the students come up and solve the examples on slides 4 and 6, with the help of their fellow classmates. After the students have shown an understanding of how to solve quadratic equations by completing the square I plan to introduce the quadratic formula. To start, I plan to go over what the students put on their chart for quadratic formula. I then plan to introduce them to the quadratic formula and how to use it. Similar to completing the square, I plan to go over many examples using the quadratic formula. Also, I will teach the students about finding the discriminant and how it relates to the quadratic formula (see notes sheet).
o Closure: To end the lesson, I plan to assess how well the students have learned the material. I plan to assign them 6 completing the square and 6 quadratic formula homework problems to see how well they understand the material. The homework problems will be assigned from their textbook, and will have different levels of difficulty to test students understanding of the material.
Assessment:
o The students were given an SCLA in the previous lesson in order to better construct this lesson.
o Each student will be assessed on how well they learned the new material through homework problems that they will be given some time at the end class to complete.
o There is a different level of difficulty for each homework problem.
Adaptations:
o Completing the square is very similar to factoring, so if students seem to be having a difficult time with completing the square it might be helpful to try using the factoring method first to solve the equations, assuming the equations are factorable.
o If students are having a difficult time with the quadratic formula, or finding the discriminant, then it might be helpful to teach the students the quadratic formula song. This is a method many teachers have used in the past and helps relate the formula to something they may already know.
o The most helpful thing would be to try and incorporate all of the different methods for solving quadratic equations to see that it is possible to get the same answer using different methods.
Follow-Up Lessons/Activities:
o The next lesson will be on the remainder theorem and how to use synthetic division.
o Based on the homework, I may review the different methods for solving quadratic equations if the students seem like they did not fully understand the new material.
o In the next lesson, I plan to incorporate the methods for solving quadratic equations and relate them to the new material, synthetic division.
Reflection:
When planning this lesson I decided to use a power point to present the information. The reason I chose a power point is because the classroom that I teach in has a smart board, but does not have a white board. Completing the square was a difficult topic for me to teach because I was never really taught how to use completing the square. I learned through watching many examples and trying to figure it out on my own. This is why I thought it would be best to go through many examples for completing the square. I also thought that for the quadratic formula going through examples would help students understand it better. I found that the students got bored after going through just one example, and it was difficult to keep them engaged. I do not believe my procedures were effective due to the lack of engagement from the students. Some ways I feel I could improve my lesson is to try and find a fun activity that goes along with completing the square. Also, some students mentioned after the lesson that they had learned a song to help them remember the quadratic formula, and this would have been something that might have engaged the students during the lesson.