Up until this lesson, students have defined sequences using mostly informal language such as “each new term is 3 more than the previous term” or “multiply the current term by 2 to get the next term.” The purpose of this Warm-up is for students to use that type of language to describe a pattern of dots and make sense of the general structure. This will be useful when students continue working with the same pattern and use function notation to recursively define the sequence in a later activity. While students may notice and wonder many things about these images, the relationship between the number of dots and the step number is the important discussion point.

This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

This sequence is neither arithmetic nor geometric. This was a deliberate choice to promote student discussion. Since the pattern to generate the sequence is more complex, students will be less likely to use algebra and more likely to focus on the recursive relationship between terms.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the relationship between the number of dots and the step number does not come up during the conversation, ask students to discuss this idea.

Highlight how the number of dots in each new row matches the step number.

• If “questioning vs. telling,” a shared reason could focus on your belief that students are capable mathematical thinkers and your desire to understand how students are making meaning of the mathematics.
• If “listening,” a shared reason could be that sometimes you want to sit quietly with a group just to listen and hear student thinking and not because you think the group needs help or is off-track.

After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.

Continuing with the dot pattern from the Warm-up, the goal of this activity is for students to understand that, since sequences are functions, we can define sequences recursively and express them using function notation.

Students use a table to express regularity in repeated reasoning (MP8) as they write an expression for using . They also consider possible inputs for , reasoning that only integer values make sense. This leads to students expanding their understanding of sequences as functions whose inputs are restricted to the integers.

Continuing the work of using function notation to write recursive definitions for sequences, students use this activity to practice writing such definitions. Students have previously seen several of the sequences in this activity since the focus of this activity is on the notation and not on making sense of the sequences. The sequence 1, 3, 7, 15, 31, . . . is from the Tower of Hanoi puzzle and is the only sequence in the activity that is neither arithmetic nor geometric.

Monitor for groups who wrote their definitions for the first two sequences in different, but correct, ways. For example, for the sequence 80, 40, 20, 10, 5:

• , 
• ,
• ,

In each of these approaches, the growth factor (or rate of change) is displayed in a different way, which corresponds to students’ previous work with writing non-recursive functions in more than one form. Showing multiple correct recursive definitions will help students build their understanding of recursively defined functions and discourage students from memorizing definitions.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as "I notice . I wonder "). A typical discussion question is “What is the same and what is different?” prompting students to compare their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).