Days%2'3:% The mathematics behind what I intend to teach: In a nutshell the concepts behind the mathematics that will be introduced to the students will be the formal definition of a function having a limit as x approaches infinity, namely, the statement: f(x) has a limit, L, as x approaches infinity if and only if for every epsilon greater than zero there exist a real number M such that if x is greater than M, then −< . The key notion from this statement is the understanding that after a certain point on the graph, if the function does indeed have a limit, the subsequent points should be restricted (or bounded) between a certain error or tolerance. Indirect to these concepts, students will be (very briefly) introduced to reading math symbols and translating mathematical expressions (e.g. −<) into written sentences, although this is only a minor stepping stone to the overall concept of a function having a limit as x approaches infinity. Ideas I want my students to learn and the ways of thinking I want them to have: I want my students to be thinking about limits in the way the statement is written. There’s a certain order to reading the formal definition written above (i.e. first one must consider any epsilon greater than zero, next they have to see if such an M exist, et cetera). I also want students to think about the behavior of the functions at certain values of x’s (which in this case for limits a x approaches infinity, I want my students to consider the function only after certain M values). What they should be tuning into is the fact that the function remains “bounded,” for a lack of a better word, around the limit value after a certain M value. Moreover, I want my students to understand that this M value isn’t anything inherently special. Certainly, at times for many functions, M may be defined as an odd, messy function of epsilon, but the important thing I wantmy students is understand is that we only need just one M value for every epsilon to work – by work, I mean that the function stays bounded between ± for x > M. Materials and teaching: Much of this day will be in the hands of the students themselves. While there is a worksheet for them to work with and fill out, it merely serves as a guide for their own self discovery. They will use GSP to create graphs of various functions and make observations and conclusions from those graphs. The teacher will, like always, start off the class day by leading the class in a discussion about last night’s homework. Also before the main activity, the teacher will give specific instructions about the functions the students will be looking at, and in particular note that the students should only concern themselves for nonnegative values of x. While the students are engaged in the main activity, the teacher will be cycling around from student to student. There are particular portions of their activity that might need teacher input. For example, there is a place where the students themselves come up with their own error bounds and limit values to test. There is a chance that students will choose such numbers that will not produce the contradiction this discovery yearns for, so the teacher should be ready to give his/her own error bounds/limit values for the students to test. Otherwise, the students are in complete control of the activity. This activity in and of itself can be split up into two portions. The first half of the activity has students looking at functions where the students themselves determined had limits at infinity whereas the second half has students looking at functions that they determined did not have limits at infinity. Since this activity is very graph-centered and needs time for self-reflection on the students’ parts, this would be an ideal place to separate the activity into two days. Materials• Geometer’s Sketch Pad (or some other graphing software that is user-friendly and allows students to manipulate their graphs) • Calculator • Activity worksheet The graphing utility will allow students to examine various functions through “zooming” in and out. From this, students will discover that they can find an M value (or many M values as a matter of fact) that satisfies the condition that they were looking for – which is one of the major component to the formal definition of a limit of a function at infinity. The other major component can be acquired through the use of the graphic utility as well. Visually, students can make a determination about a function’s limit (if it has one) by seeing that the function does or does not stay within the bounds of their epsilon strips. The activity worksheet has students calculating function values for arbitrarily large numbers, so here are the two ways students can view limits: graphically and