Discrete and Continuous Functions
Objective
Big Idea
Opening
Today's lesson allows students to compare discrete and continuous functions. They will also examine and compare linear and exponential functions. We begin today's class by reading through the four different scenarios in questions #1 through #4 of Connecting the Dots: Piggies and Pools. I remind students that "create a mathematical model" means to explore what the tables, graphs, and equations will look like for these situations. I like for them to represent the scenario in all of these ways. I let students get to work in pairs or small groups.
Investigation
As students work together, I circulate and look for students who are struggling. Some common issues might include:
- Using function notation: I'll try to get students to write explicit equations using function notation. The problems have specified what the letter for the input in each problem should be, so students should be able to write f(n) instead of y =something with n.
- The first two problems are linear and may be fairly straightforward for students. I remind them to use multiple representations. The second two problems might be harder for students to work through. For Question #3, at this point in the unit, I'll let a student go on for a while taking 3% of what is in the bank account and adding it to the previous amount. This allows the student to see the recursive pattern quite clearly. When s/he struggles to write an explicit function, then I'll ask about how much of the value is retained, along with the additional 3%. Here, I'm trying to get students to see that they are multiplying by the full value (100%) and adding an additional 3%, thereby multiplying by 1.03. I think this is more valuable work than just showing them an exponential growth equation (this comes up later in the unit).
- Students also struggle with Question #4 where they have to subtract 3% of the water from the pool. The same reasoning for Question #3 applies here. I prompt students to think about how much of the water remains in the pool, rather than how much is drained. I do let students do this problem with two steps for a while though so they can understand it both ways.
- My students are not great at comparing and contrasting beyond the obvious. I may have to prompt them to think about the differences in how the functions are growing (some continuously and some at set intervals).
Discussion
We start today's discussion by comparing the first two problems. I often take pictures of examples of student work (tables, equations, graphs) and project them on the smartboard. Any way of sharing where students can see other student's work would be fine. I ask students what's similar about these two situations. I want to elicit that both are linear and both have positive rates of change. Next I ask students if both graphs should be graphed as lines or if either/both of them should be dots. We talk about whether or not it makes sense to look at intervals in between whole numbers. Students should be able to see that Savannah's bank account only grows once per day, it is not changing in between the days. The pool in Question #2 however, is always filling. It does make sense to take a look at what's happening at intervals smaller than minutes because the water is still filling inside of the pool. We talk here about discrete versus continuous graphs, a topic that we briefly discussed in the Multiple Representations unit.
This is also a good time to remind students about what the domain of a function means. I ask them what the domain (inputs) can be for both Question #1 and Question #2. I find that contrasting these situations helps students to understand what makes sense as values that can be used as inputs for a function.
Next, we compare Problems 3 and 4. Students will recognize these two problems as being similar to the geometric sequences they have worked on previously. We can talk about the differences in the 3% change, one that is happening at the end of each month and one that is happening continuously as the pool drains. We again look at the differences in the domain for both of these functions. I let students know that functions that are not discrete are not sequences even though they may look similar in terms of growth.
Lastly, we compare Questions 1 and 3. This comparison is more about students highlighting the difference between the kind of growth we are looking at. I want them to see the clear distinction between a pattern that is growing linearly and one that is growing exponentially.
Closing
I want to students to take away keys ideas about the differences between discrete and continuous functions. I end today's class with an exit ticket reflection question that uses the following prompt:
How can you determine if a function is discrete or continuous? Can you give an example of each?
Citations
Connecting the Dots: Piggies and Pools is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.
http://www.mathematicsvisionproject.org/secondary-1-mathematics.html
