6.1: Reading Representations (5 minutes)
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.
For each sequence shown, find either the growth factor or rate of change. Be prepared to explain your reasoning.
- 5, 15, 25, 35, 45, . . .
- Starting at 10, each new term is \(\frac52\) less than the previous term.
- \(g(1)=\text-5, g(n)=g(n-1)\boldcdot \text-2\) for \(n\ge2\)
\(n\) \(f(n)\) 1 0 2 0.1 3 0.2 4 0.3 5 0.4
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.
6.2: Matching Recursive Definitions (15 minutes)
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.
Design Principle(s): Support sense-making; Maximize meta-awareness
Supports accessibility for: Organization; Attention; Social-emotional skills
Take turns with your partner to match a sequence with a recursive definition. It may help to first figure out if the sequence is arithmetic or geometric.
- For each match that you find, explain to your partner how you know it’s a match.
- For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
There is one sequence and one definition that do not have matches. Create their corresponding match.
- 3, 6, 12, 24
- 18, 36, 72, 144
- 3, 8, 13, 18
- 18, 13, 8, 3
- 18, 9, 4.5, 2.25
18, 20, 22, 24
- \(G(1)=18, G(n)=\frac12 \boldcdot G(n-1), n\ge2\)
- \(H(1)=3, H(n)=5 \boldcdot H(n-1), n\ge2\)
- \(J(1)=3, J(n)=J(n-1)+5, n\ge2\)
- \(K(1)=18, K(n)=K(n-1)-5, n\ge2\)
- \(L(1)=18, L(n)=2 \boldcdot L(n-1), n\ge2\)
- \(M(1)=3, M(n)=2 \boldcdot M(n-1), n\ge2\)
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.
6.3: Squares of Squares (15 minutes)
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.
Supports accessibility for: Language; Organization
Here is a pattern where the number of small squares increases with each new step.
- Write a recursive definition for the total number of small squares \(S(n)\) in Step \(n\).
- Sketch a graph of \(S\) that shows Steps 1 to 7.
- Is this sequence geometric, arithmetic, or neither? Be prepared to explain how you know.
Are you ready for more?
Start with a circle. If you make 1 cut, you have 2 pieces. If you make 2 cuts, you can have a maximum of 4 pieces. If you make 3 cuts, you can have a maximum of 7 pieces.
- Draw a picture to show how 3 cuts can give 7 pieces.
- Find the maximum number of pieces you can get from 4 cuts.
- From 5 cuts.
- Can you find a function that gives the maximum number of pieces from \(n\) cuts?
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.
Display the poster created earlier for all to see. Arrange students in groups of 3–4 and assign each group one of the example sequences. Tell groups to create a graph and visual pattern showing the first five values of their sequence. After work time, invite groups to share their representations. Add these to the poster for future reference.
6.4: Cool-down - Represent this Sequence (5 minutes)
Cool-downs for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
Here are some ways to represent a sequence. Each representation gives a different view of the same sequence.
- A list of terms. Here’s a list of terms for an arithmetic sequence \(D\): 4, 7, 10, 13, 16, . . . We can show this sequence is arithmetic by noting that the difference between consecutive terms is always 3, so we can say this sequence has a rate of change of 3.
- A table. A table lists the term number \(n\) and value for each term \(D(n)\). It can sometimes be easier to detect or analyze patterns when using a table.
- A graph. The graph of a sequence is a set of points, because a sequence is a function whose domain is a part of the integers. For an arithmetic sequence, these points lie on a line since arithmetic sequences are a type of linear function.
An equation. We can define sequences recursively using function notation to make an equation. For the sequence 4, 7, 10, 13, 16, . . ., the starting term is 4 and the rate of change is 3, so \(D(1) = 4, D(n) = D(n-1) + 3 \) for \(n\ge2\). This type of definition tells us how to find any term if we know the previous term. It is not as helpful in calculating terms that are far away like \(D(100)\). Some sequences do not have recursive definitions, but geometric and arithmetic sequences always do.