Lesson 2

Patterns of Growth

2.1: Which One Doesn’t Belong: Tables of Values (5 minutes)

Warm-up

This warm-up prompts students to compare four tables. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. To allow all students to access the activity, each item has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

Student Facing

Which one doesn't belong?

Table A

\(x\) \(y\)
1 8
2 16
3 24
4 32
8 64

Table B

\(x\) \(y\)
0 0
2 16
4 32
6 48
8 64

Table C

\(x\) \(y\)
0 1
1 4
2 16
3 64
4 256

Table D

\(x\) \(y\)
0 4
1 8
2 12
3 16
4 20

 

Student Response

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

Activity Synthesis

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. To encourage more participation, after each person shares, ask if anyone chose the same table for the same reason, or if they have alternative reasons for claiming a particular table as the one that does not belong. It may be useful to introduce the language of common differences and common factors at this time. For example, in Table B, for every increase of 2 in \(x\), the difference between \(y\)-values is 16. However, in Table C, for every increase of 1 in \(x\), the \(y\)-values are related by a factor of 4. 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.

2.2: Growing Stores (20 minutes)

Activity

In the previous lesson, students compared two patterns presented via a description and a graph. In this activity, they make the same kind of comparisons using a table. A table is particularly helpful for illustrating that:

  • linear functions grow by equal differences over equal intervals
  • exponential functions grow by equal factors over equal intervals

At this stage, students simply observe these properties for a few particular functions and in the context of concrete problems. Toward the end of the unit, they will prove these properties are true for any linear or exponential function.

Launch

Display the following table for all to see. It shows the number of coffee shops worldwide that a company had in its first 10 years. Ask, what do you notice? What do you wonder?

Invite students to share what they noticed and wondered about.

Students may notice that:

  • the number of stores nearly doubled between 1987 and 1988
  • the number continued to grow by more than 1.5 times per year (except from 1990 to 1991 and from 1991 to 1992)
  • they added close to 1,000 stores in 10 years

They may wonder:

  • How many stores do they have now? (22,519 in 2015, more than 24,000 in 2017)
  • Have the stores increased by roughly the same number since 1996?
  • How was the company able to add so many stores each year?
year number of stores
1987 17
1988 33
1989 55
1990 84
1991 116
1992 165
1993 272
1994 425
1995 677
1996 1,015
...  
2015  

Tell students that they will now look at a couple of possible ways a company can expand its business. Give students access to a spreadsheet tool or a calculator. Arrange students in groups of 2. If time is limited, ask one partner to complete the questions for Part A and the other to do so for Part B.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

A food company currently has 5 convenience stores. It is considering 2 plans for expanding its chain of stores.

Plan A: Open 20 new stores each year.

  1. Use technology to complete a table for the number of stores for the next 10 years, as shown here.
    year number of stores difference from previous year
    0 5  
    1 25  
    2    
    3    
    4    
    5    
    6    
    7    
    8    
    9    
    10    
    1. What do you notice about the difference from year to year?
    2. If there are \(n\) stores one year, how many stores will there be a year later?
    1. What do you notice about the difference every 3 years?
    2. If there are \(n\) stores one year, how many stores will there be 3 years later?

Plan B: Double the number of stores each year.

  1. Use a technology to complete a table for the number of stores for the next 10 years under each plan, as shown here.
    year number of stores difference from
    previous year
    factor from
    previous year
    0 5    
    1      
    2      
    3      
    4      
    5      
    6      
    7      
    8      
    9      
    10      
    1. What do you notice about the difference from year to year?
    2. What do you notice about the factor from year to year?
    3. If there are \(n\) stores one year, how many stores will there be a year later?
    1. What do you notice about the difference every 3 years?
    2. What do you notice about the factor every 3 years?
    3. If there are \(n\) stores one year, how many stores will there be 3 years later?

Student Response

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

Student Facing

Are you ready for more?

Suppose the food company decides it would like to grow from the 5 stores it has now so that it will have at least 600 stores, but no more than 800 stores 5 years from now.

  1. Come up with a plan for the company to achieve this where it adds the same number of stores each year.
  2. Come up with a plan for the company to achieve this where the number of stores multiplies by the same factor each year. (Note that you might need to round the outcome to the nearest whole store.)

Student Response

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

Anticipated Misconceptions

Students may struggle to write an expression using the variable \(n\). Ask them to tell you any patterns they notice in the tables and then encourage them to use those patterns to write an expression for \(n\).

Activity Synthesis

To help students compare and contrast the two tables, ask:

  • “How are the tables alike? Or how was the process of completing the two tables the same?” (The tables both follow a simple pattern: each table row is calculated by applying the same single arithmetic operation to the previous row. This pattern also holds for every other row or every third row, and so on. The entries in both tables increase as we go down the table.)
  • “How are the tables different? Or how was the process of completing the two tables different?” (In Plan A, 20 new stores are added each year; the repeated arithmetic operation is addition. In Plan B, the number of stores is doubled each year; the repeated arithmetic operation is multiplication. Plan B grows much more quickly than Plan A.)

Students may notice in the table that the difference from the number of stores the previous year is the same as the number of stores the previous year. This is because of the doubling pattern: to find the number of stores in the next year, the number from the previous year is added to itself (or doubled).

Representing, Conversing: MLR7 Compare and Connect. As students share their expressions for Plan A and Plan B, call students’ attention to the related mathematical operations within the context of each situation. Consider asking, “Why does this expression include multiplication and this one does not?” Use this opportunity to emphasize the meanings of “difference from previous year” and “factor from previous year” while highlighting connections to the related expressions. This will help students make connections between a specific feature of one table with another using precise mathematical language.
Design Principle(s): Maximize meta-awareness; Optimize output

2.3: Flow and Followers (10 minutes)

Activity

Students study two stories characterized by different growth patterns, and match expressions and tables with the two situations. The numbers in the two scenarios are identical so students need to focus on the mathematical relationships and the operations in the expressions to successfully complete the activity. If students are evaluating the expressions, encourage them to reason about the operations without calculating anything.

Student Facing

Here are verbal descriptions of 2 situations, followed by tables and expressions that could help to answer one of the questions in the situations.

  • Situation 1: A person has 80 followers on social media. The number of followers triples each year. How many followers will she have after 4 years?
  • Situation 2: A tank contains 80 gallons of water and is getting filled at rate of 3 gallons per minute. How many gallons of water will be in the tank after 4 minutes?

Match each representation (a table or an expression) with one situation. Be prepared to explain how the table or expression answers the question.

A. \(80 \boldcdot 3\boldcdot 3 \boldcdot 3\boldcdot 3\)

B.

\(x\) 0 1 2 3 4
\(y\) 80 240 720 2,160 6,480

C. \(80 + 3+3+3+3\)

D. \(80 + 4 \boldcdot 3\)

E.

\(x\) 0 1 2 3 4
\(y\) 80 83 86 89 92

F. \( 80 \boldcdot 81\)

 

Student Response

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

Activity Synthesis

Focus the discussion on how students went about matching the cards and the situations. Record students' reasoning for all to see. In particular, highlight observations about common differences and common factors. Ask questions such as:

  • “Besides evaluating the expression, how could you tell (a certain expression) matched one of the situations?”
  • “How could you tell (a certain table) matched one of the situations?”
  • “How is the number of followers growing every 2 years?”
  • “How is the amount of water in the tank increasing every 2 minutes?”
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension. Use color coding and annotations to highlight important connections between the text of each situation and the corresponding tables and expressions. Begin by asking students what they noticed to help identify how common differences and common factors appear in each representation.
Supports accessibility for: Visual-Spatial Processing; Conceptual processing

Lesson Synthesis

Lesson Synthesis

We looked at tables of values and expressions that represent two different patterns of growth. Ask students to summarize the key features of the two patterns using tables and situations from a classroom activity, or new tables and situations such as these.

This table shows the height, in centimeters, of water in a swimming pool as it is being filled.

time in minutes height in centimeters
0 50
1 53
2 56
3 59
4 62

This table shows total number of possible outcomes (heads or tails) when flipping a certain number of coins.

number of coins number of outcomes
1 2
2 4
3 8
4 16
5 32

Ask questions such as:

  • How are the values in each table growing?
  • Which table shows common differences? Which one shows common factors?
  • What expression can we write to find the height of the water after 10 minutes?
  • What expression can we write to find the number of possible outcomes for 7 coins?

2.4: Cool-down - Meow Island and Purr Island (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Here are two tables representing two different situations.

  • A student runs errands for a neighbor every week. The table shows the pay he may receive, in dollars, in any given week.
    number of errands pay in dollars difference from previous pay
    0 10  
    1 15 5
    2 20 5
    3 25 5
    4 30 5
  • A student at a high school heard a rumor that a celebrity will be speaking at graduation. The table shows how the rumor is spreading over time, in days.
    day people who have
    heard the rumor
    factor from previous
    number of people
    0 1  
    1 5 5
    2 25 5
    3 125 5
    4 625 5

Once we recognize how these patterns change, we can describe them mathematically. This allows us to understand their behavior, extend the patterns, and make predictions.

In upcoming lessons, we will continue to describe and represent these patterns and use them to solve problems.