The Quadratic Formula in Bits and Pieces
Lesson 7 of 10
Objective: SWBAT solve the parts of the Quadratic Formula including the discriminant, and understand the Order of Operations when simplifying the Quadratic Formula.
To Introduce students to the Quadratic Formula, I begin with the discriminant b squared minus four times a times c. I provide each student with a copy of the worksheet titled Quadratic formula- bits and pieces that we will use to work through the lesson today. I start the lesson by asking students the number of possible solutions for a Parabola on the coordinate plane. I have students sketch the possibilities on the graphing side of their individual white boards.
As students hold up their boards, I look for a Parabola of each of the following three types:
- A Quadratic Function with the Vertex on the x-axis (one real solution)
- A Quadratic Function that intersects the x-axis at two points (two real solutions)
- A Quadratic Function that never intersects the x-axis (two imaginary solutions)
Using the three selected Parabolas from the student responses, I sketch each of the three Parabolas on the board, and instruct students to take notes on the worksheet provided. I remind students of the Standard Form of a Quadratic Equation, and how to find a, b, and c. I then write the formula for the discriminant and state that it can determine the number of solutions of a Quadratic Equation.
I ask them if they have seen this formula before. Some students do recognize it as the formula under the radical in the Quadratic Formula. Then I write out how the value of the discriminant that determines each of the three possible solutions that students found with their sketches.
If the discriminant is equal to zero, then the Quadratic Function has one real solution. If the discriminant is greater than zero, then the Quadratic Function has two real solutions. Finally, if the discriminant is less than zero, then the Quadratic Function has two imaginary solutions. Here is an example of a student's notes on discriminant.
After the Introduction, students continue evaluating the discriminant in the next section of the worksheet. I emphasize throughout this introduction that the discriminant determines the number of solutions, but it is not used to identify the solution(s).
After completing problem one through four on Part I on the worksheet, Quadratic formula- bits and pieces, I stop students to introduce the Quadratic Formula. Students have previously seen the Quadratic Formula in eighth grade, some only remember the song to Pop Goes the Weasel to remember the formula. Remembering the formula is important, but in this lesson, I emphasize that the the Quadratic Formula is used to find the x-intercepts (zeros, roots, solutions) of any Quadratic Equation.
For this lesson, I have taken the focus away from identifying a, b, and c and making the substitutions into the formula. The purpose of this lesson is how to simplify the Quadratic Formula correctly by hand without technology. By solving the Quadratic Formula in pieces, it should provide students with a better understanding of the next lesson on solving a Quadratic Equation using the Quadratic Formula. In Part II, I have made the substitutions for the students, and they are to finish simplifying the formula for the x-intercepts(solutions). Students are to state the solutions when complete.
I demonstrate how students should use Order of Operations to simplify the Quadratic Formula in the video below.
After demonstrating how to simplify the Quadratic Formula for students, I assign for them to complete problems one through six on their own. In number two, students are to state that the two solutions are both imaginary, and they do not have to complete simplifying the problem. As students are working, I walk around to monitor their progress. I expect the Independent Practice to take about 15 minutes to complete.
With about 10 minutes remaining in the period, I hand each student an Exit Slip to complete. If students have not completed the six problems in the Independent Practice, I assign it as homework.
In this Exit Slip, I provide students with a Quadratic Equation. Students are to do the following:
- Identify a, b, and c in the Quadratic Equation given
- Substitute into the Quadratic Formula
- Use the discriminant only to state the type and number of solutions
- Complete the Quadratic Formula for the x-intercepts
- Graph the Parabola to verify that their answers are correct
I use this Exit Slip as a formative assessment to check for student understanding of the difference between the information provided by the discriminant and the Quadratic Formula. Students should be able to clarify the difference between the type and number of solutions and what are the solutions.