Discrete and Continuous Functions

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Objective

SWBAT determine if a function is discrete or continuous based on the situation that it models.

Big Idea

Discrete situations can be modeled by functions that are continuous. The domain and range help to determine how the graph of a function will appear.

Launch

20 minutes

Have students work on discrete_continuous_functions_launch.pdf on their own.  The time-frame will depend on the needs of your individual students, but I would recommend about 8 minutes for the first question and about 12 minutes for the second question.  As always, if there is good student thinking and discourse going on, let the segment run long.  You can always take time from the next lesson to revisit the ideas in this one.

That said:

Give students time to read and understand the problem (MP1).  Students should try to develop a plan individually first before sharing and refining their plan with a partner.  While students are working on each of these problems make notes to yourself regarding how they are attempting to solve.  For example, are students using an iterative process of repeated addition?  Are they using multiplication and division to determine the amount of time it will take to get to the desired value?  Both of these are examples of students reasoning quantitatively (MP2).  Other students may have been able to determine a function that will help them get to the answer f(x) = 3x + 5 and are using this to solve the problem.  This would show that they are reasoning more abstractly (MP2).

These ideas will also come into play when students begin to construct their mathematical models (MP4).  They may use the function to find values of ordered pairs that they can plot.  They may also use the values from their iterative process.  In either case, listen for students who are starting to understand that the rate of change is constant (3 cents each day or 2 gallons every 5 minutes).  This is a major mathematical understanding which will need to be developed over time.

Don't be too helpful when students are working.  Encourage them to ask questions and learn from their peers (MP3) and try to make meaning for themselves around the representations.  The main objective here is that students begin to realize the difference between the functions in the two problems.

While both functions look very similar in their structure, one question often arises: Are both functions continuous?  Listen for students who are starting to understand this difference (in the first problem the number of days is discrete.  3 pennies each day implies that he won't be given 1.5 pennies each half day, etc.  In the second problem the time is continuous.  2 gallons every five minutes can be broken down to 1 gallon every 2.5 minutes, .5 gallons every 1.25 minutes, etc.)

Discussion

15 minutes

When students are sharing their work during the discussion of discrete_continuous_functions_discussion, sequence the order that work is shared.  Try to have students who approached the problem from a numerical standpoint go first. Then, have students present more abstract, algebraic approaches (MP3).

Teaching Point: It is going to be difficult for students to grasp the concept that the function f(x) = 5 + 3x, which is continuous, models the first problem. A graph of this function would need to be discrete based on the actual domain and range of the function.  Try to use domain and range, based on the context of the problem, as the tools to help students determine whether or not the graph is continuous or not.  You can compare/contrast this with the second problem in which the domain and range would indicate that the graph would be continuous.

 The second slide in this section can be used along with a non-verbal cue to see if students are grasping the idea of continuous and discrete graphs.  Let students use a hand signal to show whether they are thinking the situation would be discrete or continuous and then have one or two students from each side give an argument to support their conjecture (MP3).  

Possible Explanations for the 4 Function prompts:

(1) This function is discrete because you cannot have a fraction of a telephone call.

(2) An argument can be made either way for this situation.  If the balance is just recorded at the end of the day that would be discrete.  However, if the balance is recorded moment by moment throughout the day the graph could be continuous.  Let students argue their point on this one.  The mathematical discourse will be great!

(3) Discrete because you can't have a fraction of a pet.

(4) This function would be continuous because the ball always has a height based on the time since it has been thrown.

 

Closure

5 minutes

Discrete_continuous_functions_closure is a nice Ticket out the Door which will require the students to do some writing to support their ideas (MP3).  You will be able to tell from their explanation if they are beginning to understand the idea of continuous and discrete.  

Rather than simply giving them a situation, they are being given an "incomplete" situation.  The students will need to determine what further information would need to be know to say whether the situation is discrete or continuous.  For example, if Mark earns $7.15 per hour is that just for each full hour that he works?  What if he works for half of the hour, does he get half of the $7.15?  

Students' answers will give you some insight into their thinking about the situation and how they are grasping the concept.