This chapter introduces students to quadratic functions and their graphs. Students solve quadratic equations by completing the square and using the quadratic formula. They learn that the discriminant determines the number of roots of a quadratic equation. Students identify linear, exponential, and quadratic equations and finally learn to choose a model.
Graphs of Quadratic Functions -
Quadratic Equations by Graphing -
Quadratic Equations by Square Roots -
Quadratic Equations by Completing the Square -
Quadratic Equations by the Quadratic Formula -
The Discriminant -
Linear, Exponential and Quadratic Models -
Problem Solving Strategies:
Choose a Function Model -
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Problem-Solving Strand for Mathematics
In the earlier parts of this unit, Quadratic Equations and Quadratic Functions, a great deal of emphasis is placed on observing patterns that emerge when various quadratics are graphed. The classic parabolic curve, as viewed in example one of the opening lesson, alters in its width, orientation (opening up or down), and/or position on the axis in subsequent examples according to discernible differences in its equation in standard form. And yet, very importantly, it is noted that the shape of a parabola always remains symmetric.
From the beginning of the unit, the text urges students to look for patterns, compare alternative approaches, and make an informed choice between these approaches in order to solve problems efficiently. For us as teachers, taking time to examine connections between various methods and investigate both their visual interpretations and future applications is well worth class-time discussion and focus. Initially students should be encouraged to compare alternative approaches to solving quadratic equations for their relative simplicity and efficiency. Secondly, students should be urged to make explicit connections between methods and the visualization each method supports. Here again we apply the adage ascribed to Polya: “Better to solve one problem five ways than to solve five different problems.” Thirdly, students should learn that a method such as completing the square, though perhaps less mechanically simple than applying the quadratic formula, is worth learning well because it is helpful for rewriting the equations of circles, ellipses, and hyperbolas.
In the formal problem-solving lesson of the chapter, Choose a Function Model, students encounter mathematical modeling, a process that begins with real-world situations and arrives at quantitative solutions through as many refinements and adjustments as is practicable to reach acceptable results. The Review Questions offer yet another opportunity for students to work in small groups. Small group, collaborative work allows students to talk about what they do understand. If a group is confused, you might suggest that the students develop two or three clarifying questions that they could present to their classmates for discussion or clarification.
Alignment with the NCTM Process Standards
This chapter, with its extensive study of quadratic equations and functions, touches on each of the process standards of Connections, Problem Solving, and Representation. Students recognize and see connections among mathematical ideas (CON.1) and understand how mathematical ideas interconnect and build on one another to produce a coherent whole (CON.2). This is particularly evident in the development of various ways to solve quadratics, finally leading to completing the square and presenting the derivation of the quadratic formula. Furthermore, within each lesson and especially in the Problem Solving Strategies lesson at the end of the chapter, real-life applications are highlighted, applying mathematics in contexts outside of mathematics itself (CON.3).
The unit is also replete with problem-solving considerations; it builds new mathematical knowledge (PS.1), solves problems that arise in mathematics and in other contexts (PS.2), and applies and adapts a variety of appropriate strategies to solve problems (PS.3). Especially in the lesson that focuses on mathematical modeling, the unit monitors and reflects on the process of mathematical problem solving (PS.4), encouraging revisions and reformulations to be certain that the real-world situation is actualized reasonably well. Much of the unit creates and uses representations (tables, graphs, equations) to organize, record, and communicate mathematical ideas (R.1), selects, applies and translates among mathematical representations to solve problems (R.2), and uses representations to model and interpret physical, social, and mathematical phenomena (R.3).
- CON.1 - Recognize and use connections among mathematical ideas.
- CON.2 - Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
- CON.3 - Recognize and apply mathematics in contexts outside of mathematics.
- PS.1 - Build new mathematical knowledge through problem solving.
- PS.2 - Solve problems that arise in mathematics and in other contexts.
- PS.3 - Apply and adapt a variety of appropriate strategies to solve problems.
- PS.4 - Monitor and reflect on the process of mathematical problem solving.
- R.1 - Create and use representations to organize, record, and communicate mathematical ideas.
- R.2 - Select, apply, and translate among mathematical representations to solve problems.
- R.3 - Use representations to model and interpret physical, social, and mathematical phenomena.