Lesson 6

The purpose of this warm-up is for students to recall some of the ways functions can be represented, such as tables, graphs, equations, and descriptions. The focus here is on exponential and linear functions, and identifying either the rate of change or growth factor. Students will continue to use this skill later in the unit when they review writing explicit equations for exponential and linear situations.

Display the questions for all to see throughout the discussion. For each problem, select students to share how they identified either the growth factor or rate of change. Highlight any comments linking these sequences to linear or exponential functions, such as that the rate of change would be the slope of the line that the terms of the sequence are on when graphed.

In this partner activity, students take turns matching a sequence to a definition. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

One sequence and one definition do not have matches. Students are tasked with writing the corresponding match.

Monitor for students using clear reasoning as they create a recursive definition for the sequence 18, 20, 22, 24 to share during the discussion.

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

Arrange students in groups of 2. Tell students that for each sequence, one partner finds its matching recursive definition and explains why they think it matches. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next sequence, the students swap roles. If necessary, demonstrate this protocol before students start working.

Ensure that students notice that one sequence and one definition do not have matches, and they are tasked with writing the corresponding match for each.

Some students may not be sure how to begin matching terms in a sequence to a definition. Encourage them to start by picking a definition and calculating the first few terms of the sequence it represents.

Once all groups have completed the matching, ask “How did you decide which definitions to match to sequence 3, 6, 12, 24 and sequence 18, 36, 72, 144 when they both involve doubling?” (They are both geometric with a growth factor of 2, but since they have different first terms you can use those to match the sequences to \(M\) and \(L\).)

Next, invite previously identified students to share the recursive definition they created for sequence 18, 20, 22, 24 and their strategy for writing it.

If time allows and students need extra practice graphing functions, assign students one function each from the task statement to sketch a graph for. After work time, select students to share their sketches, displaying them for all to see and compare.

The purpose of this task is for students to write a recursive definition for a sequence that represents a mathematical context and to create other representations of the sequence.

Monitor for groups who create their recursive definitions in different ways to share during the whole-class discussion. For example, some students may first create a table showing step number and the associated values while others may draw additional steps.

Allow students to use graph paper to sketch their graphs if needed. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If not brought up by students, ask “How would you describe the total number of small squares in Step 3 compared to the total number of small squares in Step 2?” (There are 9, or \(3^2\), more squares in Step 4 than in Step 3.)

Arrange students in groups of 2. Encourage them to check in with their partner frequently as they work through the task.

Students may not be sure where to begin with the graph since no axes are provided in the task statement. Encourage these students to first figure out what values they need to plot before drawing, scaling, and labeling their axes.

The goal of this discussion is for students to share how they reasoned about a recursive definition for \(S\) and how they created their graph for Steps 1 to 7. Invite previously identified groups to share how they created their definitions, making sure to display for all to see any additional representations they used to help their thinking. After these students have shared, ask "Did anyone use a different strategy for writing their definition?" and invite any new students to share their thinking.

Conclude the discussion by reviewing graphing strategies. Select 2–4 students to share how they created their sketch of \(S\). In particular, focus on how the scale on each axis was chosen.