Recursive vs. Explicit

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SWBAT use their knowledge of recursive formulas to write an explicit formula for an input output table by sketching models of animal pens.

Big Idea

Students Compare the Direct Explicit formula of y= mx + b of an Arithmetic Sequence to the Recursive Formula that is the rule or pattern of the sequence.

Warm up

5 minutes

I begin the lesson with a Warm Up that I expect to take about 10 minutes for the students to complete and for me to review with the class.  The instructions ask my students to write a recursive formula for the sequence given in an input-output table. I will walk around the room during the Warm Up looking for different methods that students use, even if the method is incorrect. Based on my observations I will choose three students to write their responses on the board.  The video below demonstrates the types of strategies I see from my students on this task. 

Even though the instructions for the Warm Up were to write a recursive formula, comparing these different methods improves student understanding of the difference between explicit and recursive formulas. Students will revisit this topic on the Exit Ticket.



20 minutes

Today's exploration involves solving a problem related to building animal pens. My students are to sketch (or use toothpicks to form) the panels of animal pens that are joined together. Most of my students prefer to sketch, but I like to give them a concrete option. The task is for the students to determine how many panels (toothpicks) are needed to build 20 square animal pens that are connected.  

Below are examples of student work that show different methods that I expect my students to use:

  • Student 1 found a numeric pattern, and wrote the explicit formula.
  • Student 2 actually drew out 20 pens to find how many panels. Unfortunately, he ended up with an incorrect answer.

This activity encourages students to consider the idea of restricting the domain for the number of pens to the counting numbers. This topic often comes up in my class, so I want students to consider the idea again today. For example, you cannot build the first animal pen with less than 4 panels, so the minimum value for the domain is four. The maximum value of the domain is how many panels it takes to build 20 connected pens, although this is not a limit.

At the end of this activity, we compare 2 methods: sketching 20 animal pens vs. using a formula. I try to steer this conversation in ways that give students the opportunity to consider both Mathematical Practice 4 and Mathematical Practice 6. I will ask questions like:

  • What did x and y represent in the table? 
  • Why is it important to label a graph accurately?
  • Why is it important to define variables clearly?

I want my students to be consistent with the variable that they choose to use.  Either use x or n for the input variable, but do not use them interchangeably in the same problem.

After this discussion, we will also talk about the difference between using an Explicit and a Recursive formula for the problem.

Independent Practice

15 minutes

If time permits, I will assign the Writing Arithmetic Formulas worksheet for students to work on in class. If we students do not complete it in class, I will ask them to complete it for homework.I instruct students to first identify the tables as representing linear or non-linear functions.  Then, write an explicit and a recursive formula for each linear function.  

Students do not have to write formulas for non-linear functions at this time.  After students complete writing both types of formulas for each Arithmetic Sequence, then students are to graph all of the functions on the graphs on the next page including the non-linear functions. Students should observe that the linear functions form a line on the graph.  The functions that the students select as non-linear should not form a line on the graph. 

By reviewing how to write an explicit formula and a recursive formula for a linear function in the Warm Up, and the Exit slip, students should be able to complete the Independent Practice with little assistance.


Exit Slip

5 minutes

With about five minutes left to go in the lesson I will hand students today's Exit Slip.  On the Exit Slip students are to summarize the difference between an explicit and recursive formula for an Arithmetic Sequence.  I use this to assess the students quickly on the focus of explicit formulas using the input and output values, and recursive only using the output values.  I collect the Exit Slips to provide immediate feedback to the students that is that might benefit them before starting the Independent Practice.  I write a few examples from the Exit Slips without providing student names, to have a brief discussion before the Independent Practice.