##
* *Reflection: Problem-based Approaches
Tower Task: Exploring Explicit Formulas - Section 4: Closure

I always enjoy teaching this lesson because students can use something concrete (wooden cubes) to actually build the towers and examine the pattern. Students did a great job with finding the pattern. Looking at page 2 of the student work, you can see that I provided additional scaffolds for some of my struggling learners. Rather than leaving the question in paragraph form, I broke it down into a step-by-step approach that they could more easily follow.

This was, however, the first year that I did not have any groups of students come up with the explicit function by looking at the pattern and constructing the towers. Students were very uncomfortable with the idea of generalizing the pattern for any height tower.

I was able to question students and guide them to determining how to write a generalized formula. There are a few ways to think about the generalization. The group that got the closest and helped to lead the class discussion noticed that the "legs" of the tower were always one block shorter than the tower height. This eventually led to a student from another group coming up with x-1 for the length of each leg. Once we had that algebraic representation most groups were able to come up with the explicit formula and check that it worked.

In the closing activity, I wanted to see if students were grasping the difference between recursive and explicit formulas. From the attached student work you can see that students were more comfortable with describing the recursive formula than the explicit. In either case, this ticket out showed me that we still had some work to do with understanding recursive and explicit formulas.

# Tower Task: Exploring Explicit Formulas

Lesson 3 of 18

## Objective: SWBAT define a situation both recursively and explicitly and explain the relationship between the formulas.

#### Do Now

*3 min*

**Slide 1: **As students are entering the room they can begin working on this error analysis question. I like to use error analysis as a way to have students attend to **MP3** on critiquing another student’s work. Finding errors or proving that a solution is correct forces students to a deeper level of understanding than simply carrying out a process or procedure.

#### Resources

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#### Opening

*5 min*

Pass out the tower task worksheet or display it on the projector for students to read and begin going through the problem solving process. Because this is at the beginning of the year, I like to take some time to model the habit of taking time to understand the problem. The first thing I do is ask students to read the problem a second time. Students often rush through reading a math problem and basically skim for key words, numbers, etc. Having students slow down and process is a crucial step in solving any problem. Next, I ask students to talk with their partners to decide how they will define a 5-block tall tower or a 10-block tall tower. During this time I pass out 25, 1-inch cubes to each pair of students. The number 25 is chosen strategically so that students can build towers 1-5 but then will have to abstractly determine how to find the remaining towers. Once students are finished with their discussions about how to define the height of the tower, I do not have groups share out and I do not discuss the definition. It is important to the problem solving process that students understand the problem and work through it (**MP1**).

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#### Closure

*5 min*

**Slide 3: **Before students leave I want to make sure they are starting to develop an understanding of the difference between recursive and explicit formulas. This understanding will be crucial as we move forward in the functions unit. Students can certainly cite specific examples from the problem that they just solved but I want to make sure that their definition shows some generalized understanding of the difference between the two types of equations.

#### Resources

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- LESSON 1: PRE-ALGEBRA: Evaluating Expressions
- LESSON 2: Defining Functions Recursively
- LESSON 3: Tower Task: Exploring Explicit Formulas
- LESSON 4: Function Notation
- LESSON 5: Understanding Domain and Range
- LESSON 6: Multiple Representation of Functions
- LESSON 7: Piecewise and Step Functions
- LESSON 8: Mirror Task: Understanding Equivalent Functions
- LESSON 9: Modeling with Functions
- LESSON 10: Functions Practice and Assessment
- LESSON 11: Introduction to Piecewise Functions: Dance-a-Thon Question
- LESSON 12: More with Piecewise Functions
- LESSON 13: Evaluating Functions Day 2
- LESSON 14: Transformation of Functions Day 1
- LESSON 15: Transformation of Functions Day 2
- LESSON 16: Transformations "How To" Guide
- LESSON 17: Functions Review Assignment
- LESSON 18: Functions Unit Assessment