Lesson 2

Introducing Geometric Sequences

2.1: Notice and Wonder: A Pattern in Lists (5 minutes)

Warm-up

This is the first Notice and Wonder activity in the course. Students are shown four sequences. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to write down things they notice and things they wonder. After students have had a chance to write down their responses, ask several students to share things they noticed and things they wondered. Record these for all to see. Often, the goal is to steer the conversation to wondering about something mathematical that the class is about to focus on. The purpose is to make a mathematical task accessible to all students with these two approachable questions. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued.

The purpose of this task is to re-introduce growth factor. (Students likely encountered it in an earlier course when they studied exponential functions.) Students notice and describe that each sequence is characterized by the same type of relationship between consecutive terms.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or precise language, and then restate their observation with more precise language in order to communicate more clearly.

The last two sequences may present a challenge since the growth factor is less than 1. The purpose of including these sequences is to encourage students to notice and make use of structure (MP7). If they notice that in the first two sequences, each pair of consecutive terms has the same quotient, they could inspect the quotients in the last sequence. These two sequences also give opportunity to point out that we still use "growth factor" even when the terms are decreasing.

Launch

Display the four sequences 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 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to record what they notice and wonder.
Supports accessibility for: Language; Organization

Student Facing

What do you notice? What do you wonder?

  • 40, 120, 360, 1080, 3240
  • 2, 8, 32, 128, 512
  • 1000, 500, 250, 125, 62.5
  • 256, 192, 144, 108, 81

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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 sequence. 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 the idea of each consecutive term in a sequence growing by the same factor or having a common ratio does not come up during the conversation, ask students to discuss this idea. Students may describe how the “same thing” is happening with consecutive terms. Encourage students to use the word term, and to be specific when they describe what is happening, for example

  • “In the first sequence, you always multiply a term by 3 to get the next term.”
  • “In the last sequence, if you divide any term by the previous term, you always get \(\frac34\).”

Tell students that this “same thing” is called the growth factor or common ratio, and that we will use the first of these. For example, in the second list, the growth factor is 4 because \(8=4\boldcdot2\), \(32=4\boldcdot8\)\(128=4\boldcdot32\), and \(512=4\boldcdot128\).

Emphasize that the growth factor is defined to be the multiplier from one term to the next; said another way, the quotient of a term and the previous term. For example, students may want to say that the pattern of the third sequence is “divide by 2 each time.” This is true, but the growth factor is \(\frac12\), because it is the number you multiply by to get the next term.

In earlier courses, students may have learned that a ratio has two or more parts. In more advanced courses (like this one), ratio is sometimes used as a synonym for quotient.

2.2: Paper Slicing (20 minutes)

Activity

In this activity, students generate two geometric sequences from a mathematical situation. The purpose is to create representations of geometric sequences using tables and graphs. At this time, students do not need to write equations for the situation since that work will be the focus of a future lesson.

Monitor for students sketching neat and accurate graphs to highlight during the whole-class discussion. Note that future lessons will focus on a reasonable domain for sequences when regarded as functions. In this activity, students may “connect the dots” in the graphs with lines, but it’s not crucial to address at this time whether or not that’s a sensible thing to do.

Launch

Arrange students in groups of 2. Invite them to read the introduction to the task. If time allows, distribute scissors and blank paper (or copies of the blackline master, if using) and let students work with their partner to carry out the paper cutting while completing the first few rows of the table. If time is limited, conduct a demonstration of the paper cutting.

Tell students to pause after completing the first few rows of the table and then ask,

  • “What happens to the number of pieces after each cut?”
  • “What happens to the area of each piece after each cut?”

Ask students to share their responses with a partner, and then invite a few groups to share their response with the class. Ensure students can articulate that as a result of a cut, the number of pieces doubles, and the area of each piece is halved. Students then proceed with the remainder of the activity.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students consider the context of this problem. Without revealing the questions that follow, display and read aloud only the task statement that describes Clare's actions. Ask students to write down possible mathematical questions that could be asked about the situation. Invite 3–4 students to share their questions with the class. Listen for and amplify any questions that include language related to looking for patterns. 
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Provide Access by Recruiting Interest. Begin with a whole-class demonstration. If time does not allow for students to carry out paper cutting themselves, demonstrate the initial cuts so that students can complete the first few rows of the table.
Supports accessibility for: Memory; Conceptual processing

Student Facing

Clare takes a piece of paper, cuts it in half, then stacks the pieces. She takes the stack of two pieces, then cuts in half again to form four pieces, stacking them. She keeps repeating the process. 

number
of cuts
number
of pieces
area in square inches
of each piece
0    
1    
2    
3    
4    
5    
  1. The original piece of paper has length 8 inches and width 10 inches. Complete the table.
  2. Describe in words how you can use the results after 5 cuts to find the results after 6 cuts.
  3. On the given axes, sketch a graph of the number of pieces as a function of the number of cuts. How can you see on the graph how the number of pieces is changing with each cut?
  4. On the given axes, sketch a graph of the area of each piece as a function of the number of cuts. How can you see how the area of each piece is changing with each cut?
Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “number of cuts”. Vertical axis from 0 to 50 by 5’s, labeled “number of pieces”.

 

Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “number of cuts”. Vertical axis from 0 to 100 by 5’s, labeled “area of each piece”.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Clare has a piece of paper that is 8 inches by 10 inches. How many pieces of paper will Clare have if she cuts the paper in half \(n\) times? What will the area of each piece be?

  2. Why is the product of the number of pieces and the area of each piece always the same? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students struggle to remember the meaning of area, ask them what they remember about the idea from their previous courses. A workable definition is “the number of one-inch squares covered by the piece of paper.” Using the blackline master may assist in this understanding. (Length times width is how one computes the area of the region bounded by a rectangle, but it doesn’t explain what area means.)

Activity Synthesis

The goal of this discussion is for students to identify the growth factor for each sequence and learn that sequences with a growth factor are called geometric sequences.

Invite previously identified students to share their graphs with the class. Display the two sequences in this activity (1, 2, 4, 8, 16, 32 and 80, 40, 20, 10, 5, 2.5) for all to see.

Here are some possible questions for discussion:

  • “When did you stop cutting the paper and complete the table using a pattern?”
  • “What was the pattern you noticed?” (To find the number of pieces, multiply the previous number by 2. To find the area, multiply the previous number by \(\frac12.\))
  • “How did you find the results after six cuts?” (Multiply 32 by 2 to get 64, and divide 2.5 by 2 to get 1.25.)
  • “What is the growth factor for each sequence?” (It’s 2 for the number of pieces and \(\frac12\) for the area.)
  • “How can you see the growth factor in each graph?” (For the number of pieces, the height of each plotted point is twice the height of the previous plotted point. For the area, the height of each plotted point is half the height of the previous plotted point.)

Tell students that the sequences they have seen today (in this activity and in the warm-up) have a special name: geometric sequences. Geometric sequences are characterized by a growth factor. In a geometric sequence if you divide any term by the previous term, you always get the same value: the growth factor for the sequence. Reiterate that the growth factor for the area sequence is \(\frac12\) because it’s what you multiply by to get the next term.

Some students may notice the similarity between a geometric sequence and an exponential function. Invite these students to share their observations, such as how both are defined by a growth factor. Tell students that geometric sequences are a type of exponential function and that their knowledge of exponential functions will help them describe geometric sequences during this unit. If students do not bring up the connection to exponential functions, ask "What do you remember about exponential functions?" Record student responses for all to see and invite comparisons between exponential functions and geometric sequences.

2.3: Complete the Sequence (10 minutes)

Activity

The purpose of this task is to provide students with practice working with geometric sequences and identifying the growth factor of a sequence.

Launch

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a calculator, or graphing software. Some students may benefit from a checklist or list of steps to use the calculator or software. 
Supports accessibility for: Organization; Conceptual processing; Attention

Student Facing

  1. Complete each geometric sequence.
    1. 1.5, 3, 6, ___, 24, ___
    2. 40, 120, 360, ___, ___
    3. 200, 20, 2, ___, 0.02, ___
    4. \(\frac 1 7\), ___, \(\frac 9 7\), \(\frac {27} 7\), ___
    5. 24, 12, 6, ___, ___
  2. For each sequence, find its growth factor.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

In the lead up to writing recursive definitions for sequences, it is important for students to understand that for geometric sequences the growth factor is defined to be the multiplier from one term to the next. Said another way, the growth factor is the quotient of a term and the previous term. For example, many students will want to say that the pattern of the third sequence is “divide by 10 each time.” This is true, but the growth factor is \(\frac{1}{10}\).

For each sequence, invite a student to share how they completed the sequence and determined the growth factor. Highlight the method of dividing any term by the previous term to find the growth factor. Emphasize that the presence of a growth factor is what makes a sequence a geometric sequence.

Lesson Synthesis

Lesson Synthesis

Arrange students in groups of 2. Ask each group to come up with a new geometric sequence and be prepared to explain why it is a geometric sequence. After a brief work time, select 3–4 groups to share their sequence and why it is a geometric sequence. Encourage students to use precise language as they share with the class, such as term, multiplier, and growth factor.

If time allows, ask students if they think the sequence 2, 2, 2, 2, 2 is a geometric sequence. (Yes, it has a growth factor of 1.)

2.4: Cool-down - A Possible Geometric Sequence (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Consider the sequence 2, 6, 18, . . . How would you describe how to calculate the next term from the previous?

In this case, each term in this sequence is 3 times the term before it.

A sequence. The number 2, followed by 6, 18, and two blanks. Starting underneath 2, and repeating to the right 4 times, an arrow going to the right labeled "times 3."

A way to describe this sequence would be: the starting term is 2, and the \(\text{current term} = 3 \boldcdot \text{previous term}\).

This is an example of a geometric sequence. A geometric sequence is one where the value of each term is the value of the previous term multiplied by a constant. If you know the constant to multiply by, you can use it to find the value of other terms.

This constant multiplier (the “3” in the example) is often called the sequence’s growth factor or common ratio. To find it, you can divide consecutive terms. This can also help you decide whether a sequence is geometric.

The sequence 1, 3, 5, 7, 9 is not a geometric sequence because \(\frac31 \neq \frac53 \neq \frac75\). The sequence 100, 20, 4, 0.8, however, is because if you divide each term by the previous term you get 0.2 each time: \(\frac{20}{100} = \frac{4}{20} = \frac{0.8}{4} = 0.2\).