Using Innovative Approaches To Explore Functions
Use student-directed resource sheets, multiple approaches, and group work instead of drill and kill.
By Donna Iadipaolo
When I learned about mathematical functions in high school, I was required to work toward proficiency by struggling through problem after grueling problem. Usually, we used the same method over and over again. It was essentially a drill-and-kill experience. As an educator, I've come to believe that this method isn't best when teaching students about mathematical functions. I think we should promote the use of “yellow pages,” a method in which students try multiple problem solving approaches, collaborate with one another, and make group presentations.
Yellow Pages as an Individualized Resource Sheet
“Yellow pages” are resource sheets that are created by students and describe their individual problem solving process. On these sheets, students note common pitfalls or their individual misunderstandings. Yellow pages can help students develop study skills, practice independent thinking skills, and can be used as a way to monitor progress. In addition, yellow pages can be used to support skill-development and conceptual understanding of a particular topic. For instance, some students might not remember exactly how to apply logarithmic rules when approaching a problem. If a student is trying to solve 2^x=25 algebraically, s/he might note how to correctly take the log of each side of the equation in the following manner: log 2^x=log 25; x log 2= log 25; x=(log 25)/(log 2).
Common Core Standard F-LE.4 specifically deals with this Algebra 2 concept of using logarithms to solve exponential equations. To solve algebraically in the above instance, the teacher needs to review the “change of base formula”: Logb x = loga x/loga b. Teachers can also remind students that logs without a subscript 10 as the base are actually base 10. Furthermore, if anxiety exists in association with logs, the teacher can also use their exponential form to aid in understanding.
Multiple Approaches to Problem Solving
Encouraging students to use different problem solving methods with a particular problem helps them develop confidence because they are able to cross check their answers. For instance, students could use two different graphical approaches to solve 2^x=x+3. Students could convert the equation to a function in the following manner: 0=x+3-2^x, and then graph y=x+3-2^x. Another way to represent the solution graphically would be to graph each side of the equation 2^x=x+3. In other words, they could write each side of the equation as a function and then graph each function on graph paper or a graphing calculator to determine where the two graphs intersect. Such a method allows students to understand how parts of a function make up the graphical representation of the function as a whole. The important mathematical topics and tasks that are supported when using multiple methods to solve problems include the NCTM standards of “apply and adapt a variety of appropriate strategies to solve problems” as well as “select, apply, and translate among mathematical representations to solve problems.”
Collaboration Not Competition
Students should work in groups, explore the different methods, and then create a poster board that explains the multiple methods their group used. They can then present their methods to the class as a whole. This helps students connect mathematics with communication and collaboration. When students discuss which information to place on their poster board, they are practicing the NCTM standard that requires them to “organize and consolidate their mathematical thinking through communication.” Presenting their methods to the entire class utilizes the NCTM standard to “communicate their mathematical thinking coherently and clearly to peers, teachers, and others.” If students are given a problem that has an exponential and linear component, students could approach the problem using graphical representations of the problem, as well as other problem-solving techniques, such as guess and check, create a table, and solve algebraically. Using a variety of representations and problem-solving techniques allows students to make connections within mathematics as well. What follows are more innovative lessons on functions.
Mathematical Functions Lessons:
Students explore how temperature and another variable relate with three sets of data. Students graph and explore ways to identify the functions created by the data. Students compare and contrast the nature of the data and graphs.
Investigating the Sine and Cosine Functions
Students explore how sine and cosine functions are created. Students collect data using a unit circle to find the sine and cosine at various angles. Students then graph their data on the coordinate plane and discover the original attributes of the sine and cosine function.
Students explore how quadratic functions model shooting a basketball. Students impose a coordinate grid on the path of a basketball being thrown and determine corresponding data points. Students then use various methods to determine the equation of the function.
Students collect data in a paper folding and M&M investigation and then analyze the exponential functions created. Students use graphing calculators to create scatter plots that form exponential functions. Students use exponential functions to predict future rhino populations.