## Better Way to Teach the Power of the Quadratic Formula

“This report doesn’t make any sense. It’s supposed to be a professional report, not a text message to a friend. Do it over.” That’s what a senior nuclear engineer, who happens to be my son, told a recently hired junior engineer. He was responding to a report submitted by the junior engineer. My son is a very good writer, and he tells me that he is appalled at the poor writing ability of some engineering graduates. He has learned that even if an engineering student has a high GPA, it does not necessarily follow that the student can write well. In fact, some math, science, engineering, and technical types go into a college major thinking that they will not be required to write papers and reports. Many years ago, one of my geometry students told me that he was going to be a minister, because ministers don’t have to write.

What does this have to do with teaching the quadratic formula? In my previous post, I discussed how teachers can help students understand the quadratic formula from both algebraic and geometric points of view. In this post, I’ll show you how teachers can use custom-made handouts created with computer technology not only to help students gain a deeper understanding of the quadratic formula but also learn how to be better writers. (When I use the term ‘teacher,’ I’m referring to anyone who teaches math, not just the traditional classroom teacher.) The ability to write well is a major goal of Common Core and STEM education. This lesson will not only help students better understand math, but will also help them become better communicators.

To best understand this discussion, download the student and teacher versions of the free handout Power of the Quadratic Formula from www.mathteachersresource.com/instructional-content.html. This handout provides seven ideas for teacher-guided quadratic formula learning/verification activities. Students should have basic competency using the quadratic formula. They will also need a graphing calculator, a ruler to box answers, and a teacher provided handout that gives structure to the lesson. During the lesson, students are expected to be actively engaged by calculating values of expressions, checking results, writing on the handout, and graphing equations on their graphing calculator. To make your presentations more dynamic, project graphs on a screen and use simple mouse control clicks to plot points and find x-intercepts of graphs.

The teacher version of the free handout Power of the Quadratic Formula contains all solutions to the student version of the handout. The free handout Observations About the Roots of a Polynomial gives a summary of important theorems about the roots of a polynomial. You can download it at the link mentioned above.

The graph of the equation y = x4 – 8x2 + 4 is shown below. The equation x4 – 8x2 + 4 = 0 is the first equation in the free handout Power of the Quadratic Formula. Because the ratio of the exponents is 2:1, the quadratic formula can be used to solve this equation. From the graph of the equation, we can see that there are four solutions. After solving the equation, students will see that the four solutions are irrational numbers that can be approximated to 9 decimal places. When a teacher shows them an efficient way to check solutions with their graphing calculator, some students are amazed that the value of the expression really equals zero or almost zero. Seeing the graph and checking solutions makes it more real to students.

The graph of the equation 2x2 – 3xy = 4y – 2 is shown below. This is the sixth equation in the free handout Power of the Quadratic Formula. Students will see how to use the quadratic formula to express x as an explicit function of y. By rearranging the equation, y can be expressed as a function of x, which makes it possible for students to use their graphing calculator to graph the equation. As the teacher version of the handout points out, different equation formats give us different insights about the graph of the equation. The program Basic Trig Functions, offered by Math Teacher’s Resource, can graph all three versions of the equation. I love to experiment with different equation formats. I’m still amazed that different equation formats always result in the same graph. (I must be getting old.)

Teaching Points: (Of course, teachers can modify the lesson to meet the needs of their class.)

• Teachers should read and study the handouts mentioned above.
• As the teacher derives the solution, students should carefully record the steps leading to the solution. This will give students a model of how solutions should be communicated in an organized manner to a friend, parent, in a homework assignment, or on a test.
• Answers should be expressed in a manner that reflects an understanding of the results found. Example: There are four irrational roots: x ≈ + 2.70828182 or x ≈ + 5.00968842.
• Solutions should be expressed in decimal format because exact radical format is meaningless to most students. Of course, for mathematically mature students, exact radical format is fine.
• Show students an efficient way to check a solution. All solutions should be checked. Have students use a check mark to certify that they have checked solutions.
• No more than a couple exercises of this type should be assigned in a homework assignment. Homework exercises of this type should be assigned periodically throughout the course.
• From time to time, a problem of this type should appear on a test or quiz.
• Some students will claim that they can get the answer in a matter of seconds. They are right. Remind them that this is not only about getting the right answer but also about learning how to communicate ideas to another human being and gaining a deeper understanding of a math concept.
• Remind students that learning to write well is hard work. It takes time and a great deal of effort to get it right. So what’s wrong with that? It’s worth it.
• When the occasion arises, teachers should explain how Descartes’ rule of signs can be used to predict the number of positive and negative real roots. There is no reason to wait until a later chapter in the book or the next math course.
• Explain to students that many calculator outputs have a rounding error. An output like 3.08 * 10-13 = 0.000 000 000 000 308 should be treated as equal to zero. Many beginning students don’t realize that calculator outputs like 6.18 * 10-10 essentially equal zero.

The above graphics, created with the program Basic Trig Functions, are offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a textbook. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities.

The user interface for all program modules is simple and intuitive and provides numerous sample equations along with comments and suggestions for setting screen parameters to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. With simple mouse control clicks, you can find relative minimum points, relative maximum points, x-intercepts, and intersection points. A Help menu provides a quick summary of all magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

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photocredit: morguefile.com. Used by permission.

## A Different Way to Teach the Quadratic Formula

One of my core beliefs is that, whenever possible, math concepts should be understood from both an algebraic and geometric point of view. In previous posts, we looked at how René Descartes (1596 – 1650) gave us the synthesis of algebra and geometry. Now let’s look at how teachers can help students understand the quadratic formula from both an algebraic and geometric point of view by using custom made handouts created with computer technology.

To best understand this discussion, download the student and teacher versions of the free handout Quadratic Formula (teacher version) from http://www.mathteachersresource.com/instructional-content.html. This handout provides six ideas for teacher-guided quadratic formula discovery/verification activities. During the lesson, students are expected to be actively engaged calculating values of expressions and writing on the handout, so in addition to a handout, they will need a calculator and ruler. To make your presentations more dynamic, project graphs on a screen as you plot points and draw line segments with simple mouse control clicks.

The graph of the equation y = x2 + 5x – 8 is shown below. This quadratic equation is the first equation considered in the free handout Quadratic Formula. The added graphics are the graphics that students would be expected to add as the lesson progresses.

The graph of the equation h = -16t2 + 132t + 60 is shown below. This equation is the fourth equation in the free handout, Quadratic Formula (student version). The activity is about a toy rocket that is shot upward with an initial vertical velocity of 132 feet/second. The added graphics are the graphics students would be expected to add as the lesson progresses. The slope of the secant line through (6.5, 242) and (7, 200) tells us that the average vertical velocity of the toy rocket over the time interval [6.5, 7] equals -84 feet/second. The goal of this activity is to show students how mathematics can be used to extract useful information from an equation.

Teaching Points: (Depending on the class, teachers need to give appropriate coaching.)

• Students can be shown the derivation of the formula before the lesson or at a later time.
• Teachers will have to demonstrate how to enter an expression into a calculator. Students will probably make mistakes initially. Practice is the only way to improve.
• Teachers should have students use the graph to estimate answers before the actual calculation.
• Students should learn how to express answers in decimal format, because radical format is too abstract.
• Some answers require more than a simple numerical value. Teachers can dictate an English sentence that would be an appropriate way to answer the question. This is a good way for students to practice writing skills.
• The equation h = -16t2 + 132t + 60 in the toy rocket activity describes the relationship between t and h in the gravitational field in which we live. Students will learn how this equation comes about when they take a course in physics. ( -16 equals ½ of the gravitational constant for planet earth, 132 ft/sec = the initial vertical velocity, and 60 feet = the initial height above ground level.)
• Students and teachers can explore how gravity causes the average vertical velocity of an object to change over time.
• If students understand the basics of complex numbers, teachers can present activities five and six in the Quadratic Formula handout.

The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a textbook. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities.

The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. The interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition, relative minimum points, relative maximum points, x-intercepts and, intersection points can be found with simple mouse control clicks. A Help menu provides a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.