Objectives
Does your math textbook provide enough word problems for students to feel confident about the subject matter? Can students relate to the problems in the text, or are they mostly artificial and contrived? In this unit, I have compiled a collection of word problems about quadratic equations. I hope they will have more appeal to today's teenagers than standard textbook collections. Also, they are organized in a way that is different from any math textbook I have seen. The premise is that by categorizing a large number of word problems and arranging them in increasing order of difficulty while only changing one aspect of the problem at a time, students will gain a better understanding of the subject matter. As students progress through the categories of word problems, their quadratic-solving skills should improve and they should gain a better understanding of how each small change affects the solution and/or the choice of solution method. These principles were suggested to me during my Yale Teacher Institute National Seminar on word problems, led by Dr. Roger Howe.
I teach at a comprehensive vocational-technical high school where students spend up to one-half of each day in their chosen career area and the remainder of their day in academic classes. The school is a "choice" public school and our students are held to the same academic standards as all public school students in the state. Our math classes are generally grouped heterogeneously and we find a wide range of abilities. In recent years I have taught primarily tenth grade students in either Level 2 or Level 3 of our integrated math program. Students choose our school for a variety of reasons. Some are focused on what they want to do when they finish high school and use the vo-tech school to get a head start; some have been moderately successful students and are looking for a route to success other than a four-year college, and some are avoiding their "feeder" school. All students ask the question, "Why do I need to learn this?"
Teaching at a vocational school offers opportunities in mathematics to find relevant problem situations. I have assembled word problems related to as many career areas as I could. However, the problems are intended to be relevant for high school students in general. For the past 10 years (of the 13 years that I've been teaching math) I have made it a personal mission to improve students' understanding of the idea that doubling both dimensions of a figure QUADRUPLES (not doubles) its area. I have used models, had them draw pictures, do the calculations, etc. to find the relationship between scale factors and area and volume. I do think I have made progress; that is, I believe most of my students understand why doubling two dimensions, in fact, quadruples the area of a figure. However, they don't "own" that concept; their automatic answer, especially on a multiple-choice-type test, would still be that the area doubles if the dimensions are doubled. To further my mission, I chose to focus this unit on quadratic word problems as yet another approach to help students internalize the scale factor relationship between changes in dimensions and changes in perimeter, area and volume.
This unit begins after students have studied the skills needed to solve quadratic equations. They should be able to find x-intercepts by factoring, using the Quadratic Formula, or examining a graph or table on a graphing calculator. They should also be familiar with finding the coordinates of the vertex of a quadratic function. While quadratic functions apply to many problem territories, including projectile motion, geometry, economics, rates, and number patterns, I chose to begin this unit with projectile motion. I selected problems that relate to sports whenever possible because most teenagers can relate to sports, either as a participant or an observer, and because the parabolic path of objects in flight as a function of time is visually represented by the graph of the quadratic function.
Once students complete the projectile motion problem suite, I switch them to the geometry problem suite where they will gain much-needed practice in setting up area and volume equations based on information given in word problems. By breaking the problems into different categories, I hope that my students will gain confidence in approaching word problems, interpreting the information that's there, and write and solve equations to answer the questions posed.
Finally, when they have mastered the art of writing area and volume equations, and they are adept at solving them, I can continue on my personal mission by having students study the effects of dilations (increasing or decreasing dimensions by some multiple) on perimeter, area, and volume. According to Magdalene Lampert, in her book Teaching Problems and the Problems of Teaching, students will see the big ideas if they are given the opportunity to analyze them in multiple situations. Then, if they can abstract a mathematical idea from those situations they should be able to apply it to new situations (Lampert (2001), p.255). By the end of this unit, students will have worked with quadratic functions in multiple situations, and should, one can hope, be successful when asked to apply their knowledge in the future.
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