Lesson 10

Situations and Sequence Types

10.1: Describing Growth (5 minutes)

Warm-up

In previous courses, students worked with percent change and representing percent change with an exponential function. They learned about the relationship between a growth rate \(r\) and growth factor \(1+r.\) The purpose of this warm-up is to prepare students to recognize 20% growth and use a growth factor of 1.2 to represent it when it appears later in this lesson.

Student Facing

  1. Here is a geometric sequence. What is the growth factor? 16, 24, 36, 54, 81
  2. One way to describe its growth is to say it’s growing by \(\underline{\hspace{.25in}}\)% each time. What number goes in the blank for the sequence 16, 24, 36, 54, 81? Be prepared to explain your reasoning.

Student Response

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

Activity Synthesis

The goal of this discussion is to check that students understand the difference between growth rate and growth factor when talking about a sequence. Begin by selecting students to share how they calculated the growth factor and why the number they chose for the percent makes sense.

Remind students that in a previous course, they learned that 1.5 is known as the growth factor for this function, and 0.5 is known as the growth rate. Growth rate is often expressed as a percentage, so 50%. The result of increasing a quantity by 50% can be calculated by multiplying it by 1.5. More generally, the result of increasing a quantity by \(r\%\) can be calculated by multiplying it by \(1+\frac{r}{100}\). It is not required that students memorize the vocabulary growth rate and growth factor at this time, but they should recognize that both are ways to describe how the value of a sequence changes from term to term.

Conclude the discussion by asking students:

  • “How can you define the \(n^{\text{th}}\) term of the sequence?” (\(f(n)=16(1.5)^n\) for \(n\ge0\).)
  • “Where in that function do you see the growth rate and growth factor?” (The growth factor is the value being repeatedly multiplied for different values of \(n\). The growth rate describes how much more (or less) the value of one term is when compared to the previous term using a percent, so in this case it is the growth factor minus 1, which is 0.5.)

If students need additional practice differentiating between growth rate \(r\) and growth factor \(r+1\), display the following questions for all to see and ask students to calculate the growth factor and growth rate for each.

  1. 64, 80, 100, 125 (1.25 and 25%)
  2. 64, 112, 196, 343 (1.75 and 75%)
  3. 64, 128, 256, 512 (2 and 100%)
  4. 125, 100, 80, 64 (0.8 and 20% (decrease))

After a brief work time, select students to share how they calculated their solutions.

10.2: Finding Population Patterns (15 minutes)

Activity

This is the first of two activities where students define sequences with equations and use their equations to answer questions about the context (MP2). The population values were purposefully chosen in order for students to focus on creating representations (like tables and graphs) and not on calculating “best fit.”

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

Launch

Arrange students in groups of 2. Display the table for all to see.

years since 1990 Population \(A\) Population \(B\)
0 23,000 3,125
1 29,000 3,750
2 35,000 4,500
3 41,000 5,400
4 47,000 6,480

Ask students, "What are some ways you could figure out if the sequences represented by the populations are arithmetic, geometric, or neither?" (I could use a spreadsheet to calculate if the terms have a common difference or a growth factor.) Give students quiet work time and then time to share their work with a partner. Select 2–3 groups to share their ideas with the class while recording them for all to see.

Student Facing

The table shows two animal populations growing over time.

years since 1990 Population \(A\) Population \(B\)
0 23,000 3,125
1 29,000 3,750
2 35,000 4,500
3 41,000 5,400
4 47,000 6,480
  1. Are the sequences represented by Population \(A\) and Population \(B\) arithmetic or geometric? Explain how you know.
  2. Write an equation to define Population \(A\).
  3. Write an equation to define Population \(B\).
  4. Does Population \(B\) ever overtake Population \(A\)? If so, when? Explain how you know.

Student Response

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

Anticipated Misconceptions

If students incorrectly identify the type of sequence from the table, encourage these students to graph the points or, if they have already made an equation, check that their equation generates the sequence correctly. 

Activity Synthesis

The goal of this discussion is for students to understand how different representations of functions are useful in different ways. For example, using technology to make a graph from the definition of the \(n^{\text{th}}\) term of each sequence can lead to the solution efficiently while testing values or trying to write an equation to solve may be more time consuming or challenging.

Select students to share what kind of sequence is represented by Population A along with their recursive or \(n^{\text{th}}\) term definitions. Then do the same for population \(B\). Record for all to see any representations made by students that they used to write their equations, such as extra rows in the table, graphs, or spreadsheets. If not brought up in student explanations, ask how each representation shows different features of the sequence, for example, the growth factor or rate of change.

Conclude the discussion by inviting students to share how they determined whether Population B would ever overtake Population A. Display any representations used by students, such as graphs, equations, or spreadsheets, for all to see during explanations.

Representing, Conversing, Listening: MLR 7 Compare and Connect. Create a need for students to use language to make sense of representations of sequences, and use their representations to find a term that meets a condition (the year when Population B overtakes Population A). As students prepare their work for display, look for approaches that extend the sequences in a recursive fashion, and approaches that make use of the definition for the \(n^{\text{th}}\) term. When students are comparing the approaches and representations used, prompt for and amplify comments about what is easier to see and understand in each, and comments about what might make each more complete or clear. Call students’ attention to the different ways students reasoned about when Population B overtakes Population A, and to the different ways these methods are made visible in each representation. Design Principles: Maximize Meta-awareness, Support Sense-making

10.3: Finding Square Patterns (15 minutes)

Activity

This is the second of two activities where students define sequences with equations and use their equations to answer questions about the context. Similar to the previous activity, students are asked to "write an equation" without being told the type, so the choice is theirs to define the sequences recursively or for the \(n^{\text{th}}\) term. Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Monitor for students who use different strategies to identify the type of sequences represented by the black and white squares. Here are some examples of strategies students may use:

  • Graphing the number of white squares for each image number to see that it is arithmetic.
  • Graphing the number of black squares for each image number to see that it is not arithmetic.
  • Making a table of total squares, black squares, and white squares for each image number and looking at the difference between successive terms to determine that \(W\) is arithmetic while \(B\) is neither.
  • Visualizing ways that the white squares grow as \(W(1) = 10, W(n) = W(n-1) + 4\) for \(n \ge2\) or \(W(n) = 4n + 6\) for \(n\ge1\) (or equivalent) in order to tell that \(W\) is arithmetic.
  • Visualizing ways that the black squares grow as \(B(n) = n(n+1)\) for \(n\ge1\) or \(B(1) = 2, B(n) = B(n-1) + 2n\) for \(n\ge2\) (or equivalent) in order to tell that \(B\) is neither arithmetic nor geometric.

If a strategy is not used, you do not have to introduce it.

Launch

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use color to highlight connections between the ways that the black and white squares grow and the corresponding equations. 
Supports accessibility for: Visual-spatial processing

Student Facing

Define the sequence \(W\) so that \(W(n)\) is the number of white squares in Step \(n\), and define the sequence \(B\) so that \(B(n)\) is the number of black squares in Step \(n\).

A pattern of white and black squares. Step 1. Step 2. Step 3.
  1. Are the sequences \(W\) and \(B\) arithmetic, geometric, or neither? Explain how you know.
  2. Write an equation for sequence \(W\).
  3. Write an equation for sequence \(B\).
  4. Is the number of black squares ever larger than the number of white squares? Explain how you know.

Student Response

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

Student Facing

Are you ready for more?

A definition for the \(n^{\text{th}}\) term of the Fibonacci sequence is surprisingly complicated. Humans have been interested in this sequence for a long time—it is named after an Italian mathematician who lived from around 1175 to 1250. The first person known to have stated the \(n^{\text{th}}\) term definition, though, was Abraham de Moivre, a French mathematician who lived from 1667 to 1754. So, this definition was unknown for hundreds of years! Here it is:

\(F(n) = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)\)

  1. Which form (recursive or \(n^{\text{th}}\) term) is more convenient to use for finding \(F(5)\)? What about \(F(10)\)? \(F(100)\)?

  2. What are some advantages and disadvantages of each form?

Student Response

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

Anticipated Misconceptions

Some students may be unsure how to start identifying what type of sequences are represented by the white and black squares. Encourage these students to think back to strategies and representations used previously.

Activity Synthesis

Select previously identified students to share their strategy for identifying the type of sequence represented by sequences \(W\) and \(B\) in the order shown in the Activity Narrative. Display any representations used for all to see during the explanations. In particular, if any student created a definition for the \(n^{\text{th}}\) term of \(B\), invite them to share their strategy since it is neither linear nor exponential, but rather quadratic.

Some important connections to make during the discussion are:

  • Arithmetic sequences always have a rate of change that can be interpreted in representations of the sequence.
  • Graphs of arithmetic sequences appear linear where the slope is the rate of change.
  • The definition for the \(n^{\text{th}}\) term of an arithmetic sequence can be written like a linear equation.
  • Geometric sequences always have a growth factor that can be interpreted in representations of the sequence.
  • Graphs of geometric sequences appear exponential and you can calculate the growth factor and growth rate given the coordinates of points on the graph.
  • The definition for the \(n^{\text{th}}\) term of a geometric sequence can be written like an exponential equation.

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is to highlight different representations used by students during the lesson for the populations and number of squares and reflect on their use. Ask students "Is one representation easier to use than another?" Students might note that that depends on how the information is given and what question is being asked. For example, graphing the definition for the \(n^{\text{th}}\) term might be easier for determining whether and when Population \(B\) ever overtakes Population \(A\). But the table might be easier for determining whether the sequences are arithmetic, geometric, or neither. Using a spreadsheet might be most efficient for both tasks.

10.4: Cool-down - Two Bacteria Populations (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Some situations can be accurately modeled with geometric sequences, arithmetic sequences, or sequences that are neither geometric nor arithmetic.

For example, here is a pattern of black squares surrounded by white squares, growing in steps.

A pattern of black squares surrounded by white squares. Step 0. Step 1. Step 2.

The number of white squares in each step grows (8, 13, 18. . .), with 5 more white squares each time. Since the same number of squares is added each time, the number of white squares forms an arithmetic sequence. The definition for the \(n^{\text{th}}\) term of \(W\), where \(W(n)\) is the number of white squares in Step \(n\), is \(W(n) = 5n + 8\) for \(n\ge0\).

Geometric sequences are involved in situations such as population growth and scaling. For example, the sequence of areas we got when we imagined cutting a piece of paper in half at each Step \(n\) in an earlier lesson.

Many situations lead to sequences that are neither geometric nor arithmetic. For example, consider these patterns of dots where a new row of \(n\) dots introduced in each step:

A dot pattern.

The number of dots in each step grows (1, 3, 6, 10, . . .), but there is no constant being multiplied or added to get from term to term. If we create a graph of this sequence showing the number of dots as a function of the step number, the dots would form neither a linear nor an exponential shape. This sequence is neither geometric nor arithmetic, but it does have a pattern that we can define with an equation.