Illuminations: Building Connections

# Building Connections

 This lesson focuses on having students make connections among different classes of polynomial functions by exploring the graphs of the functions. The questions in the activity sheets allow students to make connections between the x-intercepts of the graph of a polynomial and the polynomial's factors. This activity is designed for students who already have a strong understanding of linear functions, some knowledge of quadratic functions, and what is meant by a polynomial function.

### Learning Objectives

 Students will: Explain the relationship between linear factors of a polynomial function and the graph of the function Based on the graph of two lines, sketch the parabola that is the product of the two linear expressions Given the graph of a polynomial, find the equations of lines that could be components of the polynomial

### Materials

 Colored pencils Strips of paper or rulers Graphing calculators (optional) Building Polynomial Functions Activity Sheet Working Backwards Activity Sheet Higher Degree Polynomials Activity Sheet

### Assessment Options

 A suggestion for assessment is to have students summarize the activity by writing about what they have learned and discussing their understanding of the relationships among linear factors, polynomial functions, and the graphs of these functions. Their comments shed insights into their thinking and help suggest improvements in the activity. A quiz might consist of having the students graph two lines and then sketch the parabola that represents the product of the two linear expressions. Make sure that students explain their reasoning. They could also be instructed to work backward - given the graph of a parabola, they could sketch and find the equations of possible lines that are components of that parabola. I have found that students tend to blur the terms factor and x-intercept. In assessing their writing and oral communication, the teacher should insist that students use the proper terminology and should clarify any "fuzziness" that may exist between these two concepts.

### Teacher Reflection

 What activities would (a) foster connections among the classes of polynomial functions, that is, linear functions, quadratic functions, and polynomial functions of degree greater than two, and (b) foster connections between the graphical and algebraic representations of these functions? What connections do the students make between their study of the graphs of linear and quadratic functions and their study of the graphs of polynomial functions of degree greater than two? Does their understanding of polynomial functions of degree greater than two build on their understanding of the graphs of linear and quadratic functions?

### NCTM Standards and Expectations

 Algebra 9-12Understand relations and functions and select, convert flexibly among, and use various representations for them. Interpret representations of functions of two variables.

### References

 Buck, Judy Curran, October 2000 edition of Mathematics Teacher Journal. Curran, Judy. "An Investigation into StudentsÃ§ Conceptual Understanding of the Graphical Representation of Polynomial Functions." Ph.S. diss., University of New Hampshire, 1995.National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.Schwartz, Judah L., Michal Yerushalmy, and Educational Development Center. The Function Supposer: Explorations in Algebra. Pleasantville, N.Y.: Sunburst Communications, 1988. Software.

3 periods

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