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Unit 6, Lesson 2

Patterns of Growth

Algebra 1

2.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Which Three Go Together?

Materials

None

Activity Narrative

This Warm-up prompts students to compare four tables. 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.

Launch

Arrange students in groups of 2–4. Display the tables for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three tables that go together and can explain why. Next, tell students to share their response

Activity

None

Which three go together? Why do they go together?

Table A


1	8
2	16
3	24
4	32
8	64

Table B


0	0
2	16
4	32
6	48
8	64

Table C


0	1
1	4
2	16
3	64
4	256

Table D


0	4
1	8
2	12
3	16
4	20

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Invite each group to share one reason why a particular set of three go together. 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 three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as “common differences” or “common factors.”

2.2

Activity

Instructional Routines

Notice and Wonder

Materials

To Gather

Spreadsheet technology

Activity Narrative

In this activity, students use tables to make comparisons between 2 patterns. 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 that these properties are true for any linear or exponential function.

Launch

Display the 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?"

Ask students to share the things that they noticed and wondered about.

Students may notice:

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.

Students may wonder:

How many stores do they have now?
Have the stores increased by roughly the same number since 1996?
How do you find the differences and factors from previous years?

year	number of stores	difference from previous year	factor from previous year
1987	17	-	-
1988	33
1989	55
1990	84
1991	116
1992	165
1993	272
1994	425
1995	677
1996	1,015
...

Ask students how they might calculate values for the last 2 columns. (For the difference, subtract the number of stores in the row from the number of stores in the row above it. For example, in the 1988 row, the difference is 16 because . For the factor, divide the number of stores in the row by the number of stores in the row above it. For example, in the 1988 row, the factor is about 1.94 because .)

Give students access to a spreadsheet tool or a calculator. It is ideal if each student has their own device. Arrange students in groups of 2. If time is limited, ask one partner to complete the questions for Plan A and the other to do so for Plan B.

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a calculator.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization

2.3

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

Students study two stories characterized by different growth patterns, and they 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.

Launch

None

Activity

None

Lesson Synthesis

Tables of values and expressions that represent two different patterns of growth can be used to notice relationships between variables. 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 the 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?” (The water is growing at a constant rate of 3 centimeters each minute. The number of outcomes doubles with each additional coin.)
“Which table shows common differences? Which one shows common factors?” (The water height shows common differences, and the number of outcomes 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?” ()

to access Cool-down.

Student Lesson 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 week	factor from previous week
0	10	-	-
1	15	5	1.5
2	20	5	1.33
3	25	5	1.25
4	30	5	1.2

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	difference from previous day	factor from previous day
0	1	-	-
1	5	4	5
2	25	20	5
3	125	100	5
4	625	500	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.

Notice that in the situation with the student running errands, the difference is constant from week to week, while the factor changes. In the situation about a rumor spreading, the difference changes from day to day, but the factor is constant. This can give us clues to how we might write out the pattern in each situation.

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.

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	factor 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. What do you notice about the factor from year to year?
3. If there are 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 stores one year, how many stores will there be 3 years later?

Plan B: Double the number of stores each year.

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	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 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 stores one year, how many stores will there be 3 years later?

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 a 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.

0 1 2 3 4

80 240 720 2,160 6,480
0 1 2 3 4

80 83 86 89 92

	0	1	2	3	4
	80	240	720	2,160	6,480

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Lesson 2

Patterns of Growth

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSF-BF.A

HSF-LE.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

HSF-IF.B.4

Building Toward

HSF-LE.A.1

HSF-LE.A.2

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

HSF-IF.C.9

Building Toward

HSF-LE.A.1

HSF-LE.A.2