Lesson 2

Patterns of Growth

Let’s compare different patterns of growth.

2.1: Which One Doesn’t Belong: Tables of Values

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

 

2.2: Growing Stores

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?


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.)

2.3: Flow and Followers

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. \( 80 \boldcdot 81\)

 

Summary

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.