Three Methods of Solving Quadratics and Word Problems

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SWBAT decide the best method for solving a quadratic equation.

Big Idea

The structure of a quadratic equation can help students determine how to solve the equation.

Warm Up

5 minutes

For today's warm up (quadratics_solving_methods_open) I ask my students to work individually to use the quadratic formula to solve the given equation.  I want to get a sense of how students react when they start to notice that something is different about the solution. Once I form an impression, I will have students turn-and-talk with a partner to discuss what they have noticed. As students talk in pairs, I will listen for students who are finding an issue with respect to the square root of a negative number.

I expect that some of my students may just think the square root of -16 is -4.  If this is the case, I try to get this misunderstanding out for the whole class to consider. I will help guide them toward the understanding that you cannot take the square root of a negative because no number squared could ever be negative (MP2).

My students often struggle with the second part of today's warmup (MP1).  As a scaffold, I ask my students to think about what it means when they find one or two zeros (roots) for a quadratic equation.  I may ask, "What does having real zeros tell you about the graph?  And then what does having no zeros tell you about the graph?"  I plan to ask students to discuss this with their partner and verify their thoughts by graphing the associated function using technology.

Review Assignment

30 minutes

I typically implement my review assignments in a similar way. I explain to students that the goal is not to rush through and finish the assignment, but rather to better understand each question and how to solve it.  Students will be able to work with a partner. I always provide an answer key. The answer key is to be used as a supplement to their partner.  If students cannot make sense of an answer by discussing it with their partner, using the key, or checking with others nearby then they should let me know, so that I can point them in the right direction.  

The quadratic_solving_review assignment itself is straightforward.  On the answer_key I label each question with the abbreviations "F", "CTS", and "QF".  I do this to encourage my students to understand that they should be considering each of these methods, and, determining the most efficient one for each equation. Before letting them start, I will explain to students that they should set the equation equal to zero.  This first step will help them to better see the structure of the equation (MP7).  If the equation is factorable, I want them to consider this to be the best route to take.  If they have a trinomial that is not factorable, then they should look at the structure of the "a" and "b" terms.  If the leading coefficient "a" is equal to one and "b" is even, then they can use completing the square. I realize this is not a "rule" for completing the square.  Because my students struggle with fractions, I find that for many of my students it makes more sense for them to use the quadratic formula if "b/2" will result in a fraction.  Lastly, if all else fails, the quadratic formula is a tool that offers students a lot of leverage. 

The second portion of the review assignment deals with using quadratics to model mathematical situations (MP4).  Again, referring to the annotated answer_key, I encourage students to mark up the text and make notes above each important word or phrase.  Once they interpret the text and create a model, I want students to examine the structure of the quadratic equation that they write to determine the best method of solution.  

Lastly (and importantly), I want students to check their answer.  I make it a point that students should not check their answer using their equation.  I say, "What if you made a mistake with your equation?"  Then, I encourage students to go back to the problem and insert their solution everywhere necessary (for example, whenever it says "a number.").  If the solution makes sense in the word problem, then chances are they have solved it correctly.  This also forces them to make sense of the answer they have just found.


5 minutes

Today's closure activity (quadratics_solving_methods_exit) is a reflective piece for students.  I want them to consider the three methods of solving and identify which one they still find most difficult.  I also want them to try to put into writing why they are still having difficulty with it.  This metacognitive thinking will help students to identify the areas where they need to put more time in order to be as fluent as possible when working with quadratic equations.  

After they complete the reflection, I will ask students to go through the answer key and mark each problem that deals with their "trouble spot."  This way, they will have an area of emphasis when reviewing the material at a later time.