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.)
- Teachers should read and study the teacher’s version of the Quadratic Formula handout.
- 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.