7.1: Which One Doesn’t Belong: Recursive Definitions (5 minutes)
This is the first Which One Doesn't Belong routine in the course. In this routine, students are presented with four figures, diagrams, graphs, or expressions with the prompt “Which one doesn’t belong?” Typically, each of the four options “doesn’t belong” for a different reason, and the differences are mathematically significant. Students are prompted to explain their rationale for deciding that one option doesn’t belong and given opportunities to make their rationale more precise.
This warm-up prompts students to compare four definitions of sequences. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the definitions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.
Which one doesn’t belong?
\(f(n) = f(n-1) - 5\) for \(n\ge2\)
\(f(n) = \frac 1 2 \boldcdot f(n-1)\) for \(n\ge2\)
\(f(n) = 10 \boldcdot f(n-1)\) for \(n \ge1\)
\(f(n) = f(n-1) + n^2\) for \(n\ge2\)
For groups who have trouble getting started, suggest that they simply look for ways that the four definitions are alike and different.
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. In particular, highlight students who created other representations such as tables, graphs, or a list of terms as they reasoned about the activity.
7.2: Info Gap: Ways To Represent A Sequence (30 minutes)
This is the first Info Gap activity in the course. See the launch for extended instructions for facilitating this activity successfully.
This Info Gap activity gives students an opportunity to determine and request the information needed to represent sequences in different ways.
The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Three pairs of cards are provided for this Info Gap with the expectation that one pair will be used for a class demonstration of how the Info Gap routine works.
Here is the text of the cards for reference and planning:
Tell students they will represent sequences in different ways.
This is the first time students do the Information Gap cards instructional routine, so it is important to demonstrate the routine in a whole-class discussion before they do the routine with each other. For this Info Gap, three sets of cards are provided so that you can demonstrate with one set, leaving two remaining sets so that each student has a chance to work with both the problem card and the data card.
Explain the Info Gap routine: students work with a partner. One partner gets a problem card with a math question that doesn’t have enough given information, and the other partner gets a data card with information relevant to the problem card. Students ask each other questions like “What information do you need?” and are expected to explain what they will do with the information. Once the partner with the problem card has enough information to solve the problem, both partners can look at the problem card and solve the problem independently. This graphic illustrates a framework for the routine:
Display Problem Card 1 for all to see. Tell students that, as a class, they are playing the role of the person with the problem card while you play the role of the person with the data card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question.
Explain that you will be playing the role of the person with the data card. Ask students, “What specific information do you need to represent the first five terms of the sequence?” Select students to ask their questions. Respond to each question with, “Why do you need that information?” Once students provide a mathematically-valid reason, provide only data that appears on the data card. Once the class has created the representations asked for on the problem card, also display the data card so that students can see what it looks like. Reiterate that the person with the data card should only provide information that appears on their card, after the person with the problem card explains why they need the information.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles. Provide access to graph paper.
Design Principle(s): Cultivate Conversation
Supports accessibility for: Memory; Organization
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
- Silently read the information on your card.
- Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
- Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
- Read the problem card, and solve the problem independently.
- Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
- Silently read your card and think about what information you need to answer the question.
- Ask your partner for the specific information that you need.
- Explain to your partner how you are using the information to solve the problem.
- When you have enough information, share the problem card with your partner, and solve the problem independently.
- Read the data card, and discuss your reasoning.
Are you ready for more?
Make a visual pattern (for example, using dots or boxes), starting with Step 0, so the pattern for Step \(n\) contains \(n^2 + 3n + 3\) dots.
After students have completed their work, share the correct answers and ask students to discuss the process of representing the sequences in different ways. Display the cards for all to see throughout the discussion. Some students may be surprised to see the different structures of the cards even though they were about the same topic. Allow time for students to share what questions they found helpful for solving their problem card and how they came up with them.
Highlight connections between the representations. For example, a recursive definition might include \(f(1)=16\) which corresponds to an entry in a table and the point \((1,16)\) on a graph.
Display this sequence \(J\) for all to see: 2, 8, 32, 128, . . . and give students a few minutes to represent it in as many ways as they can think of. Then, ask them to check in with their partner and see if their partner came up with any different ways. Time permitting, ask one student to demonstrate or explain each type of representation:
- Using function notation: \(J(1)=2,J(2)=8,J(3)=32,J(4)=128\) where the input is the term number and the output is the value of the term
- Using a table of inputs and outputs
- Writing a recursive definition like \(J(1)=2, J(n)=4 \boldcdot J(n-1), n\ge2\)
- Sketching a graph that includes the points \((1,2),(2,8),(3,32),(4,128)\)
If the optional lesson prior to this lesson was not used, take the time now to display the poster created earlier for all to see and add on visual patterns and graphs of each of the sequences for the first five terms. If time allows, arrange students in groups of 3–4 and assign each group one of the example sequences to create these representations and add their work to the poster.
7.3: Cool-down - It’s Geometric (5 minutes)
Student Lesson Summary
Sometimes we only need a little bit of information to say a lot about a function. Let's say we know the function \(H\) is a geometric sequence with a growth factor of \(\frac23\) and a starting term of 20.25. From here, we can calculate that the terms in the sequence after 20.25 are 13.5, 9, 6, 4 and so on because in a geometric sequence we multiply the current term by the growth factor to get to the next term.
We can also make a table of values showing how the terms are calculated. Or we can make a graph, which would help us see that \(H\) isn't linear if we didn't already know it is a geometric sequence.
|2||\(20.25 \boldcdot \frac23=13.5\)|
|3||\(20.25 \boldcdot \frac23\boldcdot\frac23=9\)|
|4||\(20.25 \boldcdot \frac23\boldcdot\frac23\boldcdot\frac23=6\)|
|5||\(20.25 \boldcdot \frac23 \boldcdot\frac23\boldcdot\frac23\boldcdot\frac23=4\)|
Using function notation, we can say that \(H(1)=20.25, H(n)=H(n-1)\boldcdot\frac23\) for \(n\ge2.\)