##
* *Reflection: Staircase of Complexity
Arithmetic Sequences Day 3 - Section 3: Closure

Students showed a great deal of creativity with this ticket out. While some students had to "mimic" the problems that I put up as exemplars, others were able to come up with unique problems. To create a question like this, students really need to have the common difference in mind. No students came out with a non-integer common difference which shows that they understood the concept of having the number of jumps be a factor of the difference between the two known terms.

Determining the explicit formula was not as much of an issue at the end of class as it was at the start. This shows that the practice and instruction around writing functions is starting to pay-off.

*Staircase of Complexity: Closure-Reflection*

# Arithmetic Sequences Day 3

Lesson 4 of 19

## Objective: SWBAT determine an explicit rule for an arithmetic sequence.

#### Open

*15 min*

This opening activity (arith_sequence_day3_open.pdf) is all about students examining the structure of a sequence (MP7). A major goal of studying linear sequences and linear equations is learning to identity the pattern between the input and output variables. If a student can determine the rate at which a sequence is changing, then they know are well on their way to modeling the relationship.

The sequences in this Opening are scaffolded to develop this understanding. In the first sequence, students can simply list all of the missing terms to find the common difference, the rate of change. In the second, while it is still possible to list all of the missing terms, it becomes cumbersome (or tedious) form many students.

Put the first question up and let students work on it by themselves for 2-3 minutes. Then, let them share their ideas with a partner (MP3). Take some time to let students share their ideas with the class. There are a lot of ways to think about these types of problems. Because of this, students can learn a great deal from one another by hearing each other's ideas. As students are sharing, frequently check with the class to ensure they are following the student's (or pair of students') reasoning for finding the common difference. You can also ask students to find the explicit formula for each sequence as an extension.

Go through the same process with the second question, again emphasizing different ways of thinking about the same problem. Many students will start to look at the number of jumps from one term to the other. For example, in the second example this would be 8 jumps from the 2nd term to the 10th. They will then compare this to the distance they have to cover. In the second example the distance from 3 to 27 is 24. When students divide these values they can calculate the common difference as a ratio (i.e., 24/8=3). This ratio of rise to run is the rate of change or slope of a linear function.

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#### Independent Practice

*20 min*

This part of the lesson provides students an opportunity to practice working with arithmetic sequences. One of the difficulties that students face is becoming familiar with notation for describing sequences. This practice worksheet (arith_sequence_day3_practice.doc) provides a nice introduction to the notation. On this worksheet I let students get right to work. As questions arise we discuss them. In my experience, students can figure out a great deal of the notation on their own, even it is not completely intuitive.

#### Resources

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#### Closure

*5 min*

This Ticket Out (arith_sequence_day3_close.pdf) will assess whether or not students can create a sequence based on the work they had done in class. I post the questions from the beginning of class and tell students to make up a similar question of their own. I remind students that the question on the left is simpler than the question on the right. They should choose to model their question after the one they feel is most appropriate for them.

I instruct students to put their question on one side of their paper and the solution to their question on the reverse. If time permits, students can switch papers and answer each other's questions.

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- LESSON 1: Patio Problem: Sequences and Functions
- LESSON 2: Arithmetic Sequences
- LESSON 3: Arithmetic Sequences Day 2 and Quiz
- LESSON 4: Arithmetic Sequences Day 3
- LESSON 5: Arithmetic Sequences: Growing Dots
- LESSON 6: Graphing Linear Functions Using Tables
- LESSON 7: Linear Functions
- LESSON 8: Discrete and Continuous Functions
- LESSON 9: Rate of Change
- LESSON 10: Slope as a Rate of Change - Day 1 of 2
- LESSON 11: Slope as a Rate of Change - Day 2 of 2
- LESSON 12: Graphing Linear Functions Using Intercepts
- LESSON 13: Graphing Linear Functions Using Slope Intercept Form
- LESSON 14: Comparing Graphs of Linear Functions Using Dynamic Algebra
- LESSON 15: Using Linear Functions for Modeling
- LESSON 16: Writing the Equation for a Linear Function (Day 1 of 2)
- LESSON 17: Writing the Equation of A Linear Function (Day 2 of 2)
- LESSON 18: Graph Linear Equations Practice
- LESSON 19: Solving Two Variable Inequalities Part 2