QUADRATIC FORMULA FAVORS

(QuadImage)

Goal: To learn the benefits of The Quadratic Formula and its discriminant when solving quadratic equations.

What makes this lesson Awesome:

Students learn that the Quadratic formula is highly favorable. First, we can solve any quadratic equation with it, as complicated as it may appear. Second, part of the quadratic equation helps us determine before actually solving, if the roots will be real or imaginary, rational or irrational, and even how many roots the equation has. Imagine that!

Total Time: 60min

Materials: Marker, Large Chart paper, projector

I. ACCESSING PRIOR KNOWLEDGE:

Time: 15min

As the class enters, I provide each student with an entrance slip (QuadForm_entranceSlip). The slip asks the students to answer a few questions; these APK questions target common mistakes students make and also preambles the possible single or dual solutions in quadratic equations. Once completed, ask students to compare their answers with their neighbor and correct each other’s work. Allow 4 or 5 minutes of discussion and then write the answers on the board. Students at this moment will ask questions about the work and mistakes they’ve possibly made. Tell students that a quadratic equation may appear re-arranged, and they must re-write it in standard form making one side equal to zero, before solving. (see scaffold at end of lesson) Also, it may be necessary for some students to quickly review complex numbers. The first minutes of this video will help students refresh their complex number memories. http://www.youtube.com/watch?v=zK25rGJBfB0

Optional Video Narrative: APK_narrativeVid:

https://vimeo.com/65907535

Resource: QuadForm_entranceSlip

II. New Info/Application

Time: 35 minutes

I begin this segment by grouping up students. I simply say “Raise your hand if you were born in January....” Those born in January work together. Raise your hand if you were born in February!....and I continue until every student is in a group. At the end I re-arrange I may re-arrange students such that there are no more than 4 and no less than 2 students per group. Each group receives one large sheet of paper and a marker. NO graphing calculator should given in this activity. I then ask the class to help me solve a quadratic equation applying the quadratic formula on the board. I call on students to lead me on, asking them what to write, throughout the entire process. After answering any question, I project the resource that contains 4 Quadratic equations: One with no roots, one with one (double) root, a third with two rational number roots, and a forth with two irrational roots.(see resource) Students must apply the Quadratic Formula in each of them.

The students in a group must solve each equation simultaneously in order to compare answers after each equation. Students with different answers can then check themselves and make appropriate corrections. A member must not proceed alone to the next equation. Rather, they should take on each equation as a group, and check themselves for mistakes, so it's important they collaborate with each other. The most common mistakes when using the formula is computational errors. After they’ve solved all 4, have them analyze and discuss, what, in the quadratic formula determined having one root, two roots, and no roots. Allow them to struggle with this on their own. Once the groups have pinned down that it is the expression in the square root sign (radicand), that determines the nature of the roots, I ask each group to write a set of set of rules for this expression on their chart paper. After they’re done, I address the whole class and ask one member of each group to hold up their rules for all to see. We then copy the rules on one large chart paper and pin it somewhere on the board. I then say that this expression b² – 4ac is called the Discriminant.

Optional video narrative: Application_narrativeVid

https://vimeo.com/65907534

Resource: NewInfo_QuadEqua

Generalization:

Time: 10 min

To wrap up this lesson I project the graphs of the 4 quadratic equations on the whiteboard. (resource: Generalize_4graphs) I then call on students to tell me which graph pertains to which of the 4 quadratic equations and to please explain why he/she knows this. The student should go to the board and write the corresponding equation on the image of the parabola.

(Students should point out that the equation with two imaginary roots has no x intercepts, the double root equation has one x intercept, the rational and irrational root equations both have two x intercepts, yet students should be able to match them by further examining the roots.)

At the end I ask someone to summarize this last portion of our lesson which relates the roots of a quadratic equation to its graph.

Optional Video Narrative: https://vimeo.com/65907533

Scaffolds:

Students may have trouble re-writing the quadratic equations in standard form. Let students first understand that one side must be zero and that this means y is 0. Have them see the following video and then give them the equations below to re-write in standard ax²+ bx + c form:

http://www.youtube.com/watch?v=jxRurVEiEEk

Put the following equations in standard form making one side zero:

1. 5x + x² = – 8 2. 2x² = – 3 – 5x 3. 3x = -5x² + 1

Homework: Use the discriminant to determine the nature of the roots of each quadratic equation:

1. 6x²– 2x – 3 = 0

2. 9x² – 3x – 8 = -10

3. -2x² – 8x – 14 = -6

4. -7n² +16n = 8n

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