texas algebra i: module 1, topic 2 pacing guide

1

Texas Algebra I: Module 1, Topic 2 Pacing Guide*1 Day Pacing = 45 min. Session

Lesson # Lesson Title/MATHia Unit Lesson Subtitle/MATHia Workspace Highlights TEKS Pacing*

Module 1Topic 2: SequencesELPS: 1.A, 1.C,1.E, 1.F, 1.G, 2.C, 2.E, 2.I, 3.D, 3.E, 4.B, 4.C, 5.B, 5.F, 5.G

1 Is There a Pattern Here? Recognizing Patterns and Sequences

Given ten contexts or geometric patterns, students write a numeric sequence to represent each problem. They represent each sequence as a table of values, state whether each sequence is increasing or decreasing, and describe the sequence using a starting value and operation. They determine that all sequences are functions and have a domain that includes only positive integers. Infinite sequence and finite sequence are defined.

A.9A 2

MATHiaRecognizing Patterns and Sequences

Describing Patterns in SequencesStudents determine the patterns in sequences and determine the next terms in sequences.

A.12C 1

2 The Password Is… Operations! Arithmetic and Geometric Sequences

Given 16 numeric sequences, students generate additional terms and describe the rule they used for each sequence. They sort the sequences into groups based upon common characteristics and explain their rationale. The terms arithmetic sequence, common difference, geometric sequence, and common ratio are defined with examples. They then categorize the given sequences based on the definitions and identify the common difference or common ratio where appropriate. Students then practice writing sequences with given characteristics.

A.12AA.12D

2

MATHiaRecognizing Patterns and Sequences

Graphs of Sequences

Students sort numeric sequences by whether they are arithmetic, geometric, or neither. They analyze the characteristics of graphs of arithmetic and geometric sequences. Students match graphs of sequences to their numeric representations.

A.12C 1

Mid Topic Assessment 1

2

Texas Algebra I: Module 1, Topic 2 Pacing Guide*1 Day Pacing = 45 min. Session

Lesson # Lesson Title/MATHia Unit Lesson Subtitle/MATHia Workspace Highlights TEKS Pacing*

3 Did You Mean: Recursion?Determining Recursive and Explicit Expressions from Contexts

Scenarios are presented that can be represented by arithmetic and geometric sequences. Students determine the value of different terms in each sequence. As the term number increases it becomes more time-consuming to generate the term value, which sets the stage for explicit formulas to be defined and used. Students practice using these formulas to determine the values of terms in both arithmetic and geometric sequences.

A.12D 2

MATHiaDetermining Recursive and Explicit Expressions

Writing Recursive FormulasStudents determine if sequences are arithmetic or geometric and determine recursive formulas for the sequences.

A.12C A.12D

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4 3 Pegs, N Discs Modeling Using Sequences

Students are introduced to the process of mathematical modeling, with each of the four activities representing a specific step in the process. Students are invited to play a puzzle game, observe patterns, and think about a mathematical question. Students then organize their information and pursue a given question by representing the patterns they noticed using mathematical notation. As a third step, students analyze their recursive and explicit formulas and use them to make predictions. Finally, students test their predictions and interpret their results.

A.9DA.12D

2

MATHiaDetermining Recursive and Explicit Expressions

Writing Explicit FormulasStudents determine if sequences are artihmetic or geometric and develop the explicit formulas for the sequences.

A.12C A.12D 1

End of Topic Assessment 1

TOPIC 2: Sequences • 1A

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.SequencesTopic 2 Overview

How is Sequences organized?

In Sequences, students progress from recognizing patterns of numbers, letters, and shapes to identifying arithmetic and geometric sequences. They explore sequences represented as lists of numbers, in tables of values, by equations, and as graphs on the coordinate plane. The intent of this topic is to move students from their intuitive understanding of patterns to a more formal approach of representing sequences as functions. In later modules, they will use the connection between arithmetic sequences and linear functions and between some geometric sequences and exponential functions to examine the structure of each function family.

As in Quantities and Relationships, students begin Sequences by analyzing sequences presented in scenarios. They infer a rule for each sequence, identify additional terms, and represent the sequence as a table. They explain why all sequences are functions and differentiate between a finite sequence and an infinite sequence. After articulating the differences between different sequences, students define arithmetic sequences as those with a common difference and geometric sequences as those with a common ratio. They then match sequences to corresponding graphs.

Once familiar with the structure of sequences, students write recursive and explicit formulas

for arithmetic and geometric sequences. Students return to the scenarios from the first lesson and write an arithmetic or geometric formula for each.

In the final lesson of the topic, students are introduced to the modeling process. Defined in four steps—Notice and Wonder, Organize and Mathematize, Predict and Analyze, and Test and Interpret—the modeling process gives students a structure for approaching real-world mathematical problems. Throughout the final lesson, students work through the process as they model situations using sequences.

What is the entry point for students?

Students have been analyzing and extending numeric patterns since elementary school. They have discovered and explained features of patterns. They have formed ordered pairs with terms of two sequences and compared the terms. In middle school, students have connected term numbers and term values as the inputs and outputs of a function.

After analyzing and describing patterns in various equations and graphs in Quantities and Relationships, they continue the process to recognize patterns in sequences. In addition to describing patterns of numbers as they did in grade 8, they will now write recursive and explicit formulas for relationships. They will use what they know about functions to recognize that every sequence is a function.

Click here to viewTEKS and ELPS

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http://cl.myemcp.com/a1tekselps

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1B • TOPIC 2: Topic Overview

How does a studentdemonstrate understanding?

How does a student demonstrate understanding?

Students will demonstrate understanding of the standards in Sequences if they can:• Understand that a sequence represents a

relationship between term numbers (inputs) and term values (outputs).

• State the appropriate domain for a sequence.• Distinguish between arithmetic and

geometric sequences.• Recognize that an arithmetic sequence has

a common difference between terms and a geometric sequence has a common ratio between terms.

• Determine the common difference between two terms in an arithmetic sequence and the common ratio between two terms in a geometric sequence represented in tables and graphs.

• Describe the graph of an arithmetic and geometric sequence.

• Explain that a recursive formula tells you how to determine the next value of a sequence from the previous value.

• Explain that an explicit formula tells you how to determine any value given the term number.

• Distinguish between explicit and recursive formulas.

• Write recursive and explicit formulas for any sequence, including those presented as real-world scenarios.

• Translate between explicit and recursive formulas.

• Decide when real-world problems model an arithmetic or geometric sequence.

• Utilize a modeling process to analyze and solve problems.

Why is Sequences important?

As students deepen their understanding of functions throughout this course and beyond, recognizing that all sequences are functions is an important building block. The study of functions is a major focus throughout high school mathematics. A rich understanding of arithmetic sequences, including their graphical and algebraic representations, is the foundation for linear functions. Likewise, it is important for students to recognize that some geometric sequences represent exponential functions, and that their graphs and equations have the defining characteristics of that function family. Examining patterns in numbers and the structure of their representations will help students to use functions to model real-world phenomena.

Finally, as students gain experience with more complex functions, the modeling process will help them to approach and solve problems that they encounter in the real world. By recognizing structure in sequences of numbers, they will be more aware of the possible functions that can model a scenario, which in turn allows them to solve more complicated problems.

How do the activities in XXX promote student expertise in the

mathematical practice standards?

How do the activities in Sequences promote student expertise in the mathematical process standards?

All Carnegie Learning topics are written with the goal of creating mathematical thinkers who are active participants in class discourse,

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TOPIC 2: Sequences • 1C

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so elements of habits of mind should be evident in all lessons. Students are expected to make sense of problems and work towards solutions, reason using concrete and abstract ideas, and communicate their thinking while providing a critical ear to the thinking of others.

Students use tools—tables, graphs, and equations—to model situations as arithmetic and

geometric sequences. They examine the structure of these sequences to recognize key and defining characteristics. Finally, students use the modeling process to solve a real-world sequence problem.

Materials NeededGlueQuarters, nickels, and dimes (optional)Scissors

New NotationsA recursive formula expresses each new term of a sequence based on the preceding term in the sequence. The recursive formulas to determine the nth term of an arithmetic sequence and a geometric sequence are shown.

Arithmetic Sequence Geometric Sequence

nth term

common difference

previous term

an 5 an 2 1 1 d

nth term

common ratio

previous term

gn 5 gn 2 1 ? r

The explicit formula represents the sequence as a function. The explicit formulas to determine the nth term of an algebraic sequence and a geometric sequence are shown.

Arithmetic Sequence Geometric Sequence

nth term

common difference

previous term number

an 5 a1 1 d(n 2 1)

nth term

1st term

common ratio

previous term number

gn 5 g1 ? r n 2 1

1st term

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1D • TOPIC 2: Topic Overview

The Modeling Process

Notice and WonderGather information, notice patterns, and formulate mathematical questions about what you notice.

Organize and MathematizeOrganize your information and represent it using mathematical notation.

Predict and AnalyzeExtend the patterns created, complete operations, make predictions, and analyze the mathematical results.

Test and InterpretInterpret your results and test your mathematical predictions in the real world. Make adjustments necessary.

Assessments

There are two assessments aligned to this topic: Mid-Topic Assessment and End of Topic Assessment.

NOTICE | WONDER

TEST | INTERPRET

REPORT

ORGANIZE | MATHEMATIZE

PREDICT | ANALYZE

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TOPIC 2: SUMMARY • M1-1

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Sequences Summary

A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term in a sequence is an individual number, figure, or letter in the sequence. Many different patterns can generate a sequence of numbers.

A sequence that continues on forever is called an infinite sequence. A sequence that terminates is called a finite sequence.

For example, consider the situation in which an album that can hold 275 baseball cards is filled with 15 baseball cards at the end of each week. A sequence to represent how many baseball cards can fit into the album after 6 weeks is 275 cards, 260 cards, 245 cards, 230 cards, 215 cards, and 200 cards. This sequence begins at 275 and decreases by 15 with each term. The pattern cannot continue forever since you cannot have a negative number of cards, so this is a finite sequence.

KEY TERMS• sequence• term of a sequence• infi nite sequence• fi nite sequence• arithmetic sequence• common diff erence

• geometric diff erence• common ratio• recursive formula• explicit formula• mathematical modeling

LESSON

1 Is There a Pattern Here?

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M1-2 • TOPIC 2: SEQUENCES

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is a constant. This constant is called the common difference and is typically represented by the variable d. The common difference of a sequence is positive if the same positive number is added to each term to produce the next term. The common difference of a sequence is negative if the same negative number is added to each term to produce the next term.

For example, consider the sequence 14, 16 1 __ 2 , 19, 21 1 __ 2 , … . The pattern of this sequence is to add 2 1__2

to each term to produce the next term. This is an arithmetic sequence, and the common difference d is 2 1 __ 2 .

A geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is a constant. The constant, which is either an integer or a fraction, is called the common ratio and is typically represented by the variable r.

For example, consider the sequence 27, 9, 3, 1, 1 __ 3 , 1 __ 9 . The pattern is to multiply each term by thesame number, 1 __ 3 , to determine the next term. Therefore, this sequence is geometric and thecommon ratio r is 1 __ 3 .

LESSON

2 The Password Is . . . Operations!

A recursive formula expresses each new term of a sequence based on a preceding term of the sequence. The recursive formula to determine the nth term of an arithmetic sequence is an 5 an21 1 d. The recursive formula to determine the nth term of a geometric sequence is gn 5 gn21 ? r. When using the recursive formula, it is not necessary to know the first term of the sequence.

For example, consider the geometric sequence 32, 8, 2, 1 __ 2 , . . .with a common ratio of 1 __ 4 . The 5th term of the sequence can bedetermined using the recursive formula.

The 5th term of the sequence is 1 __ 8 .

LESSON

3 Did You Mean: Recursion?

gn 5 gn21 ? rg5 5 g4 ? r

g5 5 1 __ 2 ? 1__4

g5 5 1__8


TOPIC 2: SUMMARY • M1-3

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An explicit formula for a sequence is a formula for calculating each term of the sequence using the index, which is a term’s position in the sequence. The explicit formula to determine the nth term of an arithmetic sequence is an 5 a1 1 d(n 2 1). The explicit formula to determine the nth term of a geometric sequence is gn 5 g1 ? r n21.

For example, consider the situation of a cactus that is currently 3 inches tall and will grow 1 __ 4 inch every month. The explicit formulafor arithmetic sequences can be used to determine how tall the cactus will be in 12 months.

In 12 months, the cactus will be 5 3 __ 4 inches tall.

A process called mathematical modeling involves explaining patterns in the real world based on mathematical ideas. The four basic steps of the mathematical modeling process are Notice and Wonder, Organize and Mathematize, Predict and Analyze, and Test and Interpret.

For example, consider a theater that has 25 rows of seats. The first three rows have 16, 18, and 20 seats, respectively. The ushers working at this theater need to know how many seats their sections have when they are directing people.

The first step of the modeling process, Notice and Wonder, is to gather information, look for patterns, and formulate mathematical questions about what you notice. In the example, each row seems to have 2 more seats than the previous row.

The second step of the modeling process, Organize and Mathematize, is to organize the information and express any patterns you notice using mathematical notation. A table can be used to represent the given information about the first three rows in the theater. The recursive pattern shown in the table can be expressed as Sn 5 Sn21 1 2.

The third step of the modeling process, Predict and Analyze, is to analyze the mathematical notation and make predictions. The fourth row will have 22 seats and the fifth row will have 24 seats. The pattern can be expressed using the explicit formula Sn 5 16 1 2(n 2 1).

LESSON

4 3 Pegs, N Discs

an 5 a1 1 d(n 2 1)a12 5 3 1 1 __ 4 (12 2 1)

a12 5 3 1 1 __ 4 (11)

a12 5 5 3__4

Row Number of Seats1 162 183 20


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M1-4 • TOPIC 2: SEQUENCES

The fourth and final step of the modeling process, Test and Interpret, is to test and interpret the information. A graph can be constructed for the explicit formula. The graph is discrete because rows and seats are integer values.

This information can be used to determine that an usher working in rows 15 and 16 will have 44 and 46 seats, respectively.

43Rows

Num

ber

of S

eats

6

8

4

2

0 21 5

10

12

14

16

18

20

22

24

x

y


LESSON 1: Is There a Pattern Here? • 1

Is There a Pattern Here?Recognizing Patterns and Sequences

1

Lesson OverviewStudents begin by exploring various patterns in Pascal’s triangle. Sequences and term of a sequence are defined. Given ten geometric patterns or contexts, students write a numeric sequence to represent each problem. They are guided to represent each sequence as a table of values and conclude that all sequences are functions. Students then organize the sequences in a table, state whether each sequence is increasing or decreasing, and describe the sequence using a starting value and operation. They determine that all sequences have a domain that includes only positive integers. Infinite sequence and finite sequence are defined and included as another characteristic for students to consider as they write sequences.

Algebra 1 Exponential functions and equations (9) The student applies the mathematical process standards when using properties of exponential

functions and their related transformations to write, graph, and represent in multiple waysexponential equations and evaluate, with and without technology, the reasonableness of theirsolutions. The student formulates statistical relationships and evaluates their reasonablenessbased on real-world data. The student is expected to:

determine the domain and range of exponential functions of the form f(x) 5 ab x andrepresent the domain and range using inequalities;

Algebra 1 Number and algebraic methods (12) The student applies the mathematical process standards and algebraic methods to write, solve,

analyze, and evaluate equations, relations, and functions. The student is expected to: decide whether relations represented verbally, tabularly, graphically, and symbolically define afunction;write a formula for the n th term of arithmetic and geometric sequences, given the value ofseveral of their terms;

ELPS1.A, 1.C, 1.E, 1.F, 1.G, 2.C, 2.E, 2.I, 3.D, 3.E, 4.B, 4.C, 5.B, 5.F, 5.G

MATERIALSNone

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(A)

(A)

(D)

2 • TOPIC 2: Sequences

Lesson Structure and Pacing: 2 DaysDay 1Engage

Getting Started: A Pyramid of PatternsStudents explore various patterns in Pascal’s triangle. They first explore patterns on their own and then are guided to recognize and explain specific patterns.

DevelopActivity 1.1: Patterns to Sequences to TablesStudents are given the definitions of sequences and term of a sequence. Given ten geometric patterns or contexts, students describe each pattern, determine the next few figures or numbers in the patterns, write a numeric sequence for each pattern, and represent each sequence using a table of values.

Day 2Activity 1.2: Looking at Sequences More CloselyStudents organize sequences in a table, state whether each sequence is increasing or decreasing and describe the sequence using a starting value and operation. Infinite sequence and finite sequence are defined, and examples are provided.

DemonstrateTalk the Talk: Searching for a SequenceStudents are provided characteristics, including the newly defined terms infinite and finite, to build sequences.

Essential Ideas • A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters,

or other objects. • A term of a sequence is an individual number, figure, or letter in the sequence. • A sequence can be written as a function. The domain includes only positive integers.• An infinite sequence is a sequence that continues forever, or never ends. • A finite sequence is a sequence that terminates, or has an end term.



Getting Started: A Pyramid of Patterns

Facilitation NotesIn this activity, students identify patterns using the first 7 rows of Pascal’s triangle.

Have students work with a partner or in a group to complete Questions 1 through 5. Share responses as a class.

Questions to ask• How is each row of Pascal’s Triangle generated?• What pattern exists in the diagonals of the triangle?• Explain the symmetry in the triangle.• How is each row related to the power of 2?

Differentiation strategyTo extend the activity, have students research Pascal’s Triangle to discover other patterns such as the hockey stick pattern, parallelogram pattern, and Fibonacci numbers. Explain that Pascal’s triangle will be revisited because of its connections to higher mathematics.

SummaryPascal’s Triangle is a famous geometric and numeric figure that generates many patterns.

Activity 1.1Patterns to Sequences to Tables

Facilitation NotesIn this activity, students are given the definitions of sequences and term of a sequence. Given ten geometric patterns or contexts, students describe each pattern, determine the next few figures or numbers in the patterns, write a numeric sequence for each pattern, and represent each sequence using a table of values.


As students work, look for• Arithmetic errors that prevent may prevent them from

recognizing patterns.• Language that demonstrates a generalization of patterns.

ENGAGE

DEVELOP



Questions to ask for Positive Thinking• How do the number of dots in each figure compare to the number of

dots in the figure before it? • How does the sequence relate to the diagram?• Why can all sequences be represented by a table of values?

Questions to ask for Family Tree• How do the number of parents in one generation compare to the

number of parents in the generation that follows it? • How many parents do you think are in the 5th generation? Why? • How does this sequence relate to the family relationships?• How is this pattern different than the one in the Positive

Thinking problem?Questions to ask for A Collection of Squares

• How do the number of squares in figure 1 compare to the number of squares in figure 2? Figure 2 to figure 3? Figure 3 to figure 4?

• How does the sequence relate to the diagram?• Explain how the pattern in this sequence is different than the

patterns in the other problems.Questions to ask for Al’s Omelets

• How does the number of eggs left after making 1 omelet compare to the number of eggs left after making 2 omelets? 2 omelets compare to 3 omelets? 3 omelets to 4 omelets?

• How does the sequence relate to the scenario?• Out of the sequences you have written, what sequence has a pattern

most similar to this one? Explain why.Questions to ask for Donna’s Daisies

• How do the number of daisies in the 2nd column compare to the number of daisies in the 1st column? The 3rd column to the 2nd column? The 4th column to the 3rd column?

• What is the total number of daisies in the 8th column? 9th column? 10th column? Why?

• Explain the pattern in the sequence in words.• How does the pattern in this sequence compare to the others you

have written? Questions to ask for Troop of Triangles

• How do the number of shaded triangles in figure 2 compare to the number of shaded triangles in figure 1? Figure 3 to figure 2? Figure 4 to figure 3?

• Out of the sequences you have written, what sequence has a pattern most similar to this one? Explain why.

• What sequence would represent the number of white triangles?



Questions to ask for Gamer Guru• Why did it make sense that the first term of the sequence is 550

rather than 500?• How does the sequence relate to the scenario?• Out of the sequences you have written, what sequence has a pattern

most similar to this one? Explain why.Questions to ask for Polygon Party

• What makes a figure a polygon?• What is the name of each polygon based on its number of sides?• What is the term used for a polygon that has all sides the same

length and all angles the same measure?• How does the sequence relate to the diagram?• Out of the sequences you have written, what sequence has a pattern

most similar to this one? Explain why.Questions to ask for Pizza Contest

• Create a diagram and demonstrate how the sequence is generated.• What would the sequence be if the terms represented the number

of slices rather than the size of the slice? • Out of the sequences you have written, what sequence has a pattern

most similar to this one? Explain why.Questions to ask for Coin Collecting

• How does the sequence relate to the scenario?• Compare this sequence to the other increasing sequences. Which

ones have a growth pattern similar to this one? How would you describe the growth pattern?

Differentiation strategies• To support students who struggle, demonstrate how any sequence

can be converted into a table of values. Allow tables to be set up horizontally or vertically.

Term number 1 2 3 4

Term value 25, 21, 17, 13,

• To assist all students with Polygon Party, accept non-regular polygons with the correct number of sides or demonstrate how to sketch regular polygons by making a circle, placing points relatively equidistant around the circle, and then connecting consecutive points with line segments.

• To extend the activity• Have students revisit Troop of Triangles. Ask them to write a

sequence for the total number of the smallest triangles in each figure, explain the pattern, and connect the pattern to the pattern in A Collection of Squares.



• Have students revisit Troop of Triangles. Ask them to write a sequence for the total number of triangles of any size in each figure and explain the pattern.

SummaryAll numeric sequences can be represented as a function. The independent variable is the term number beginning with 1, and the dependent variable is the term of the sequence.

Activity 1.2Looking at Sequences More Closely

Facilitation NotesIn this activity, students organize sequences in a table, state whether each sequence is increasing or decreasing, and describe the sequence using a starting value and operation. Infinite sequence and finite sequence are defined, and examples are provided.


Questions to ask• How many sequences could be described as increasing? Decreasing?• How many sequence patterns involve multiplication or division? • How many sequence patterns involve addition or subtraction?• Can any of the sequences be described as an even or odd sequence?

Why or why not? • Do any of the sequences begin at zero? • Can you think of a sequence that would begin at zero? What is

an example? • What determines the first term of a sequence?• Is the last term of the sequence the end term of the sequence? Why

or why not? • Are all numbers divisible by 4? Explain why or why not.

Ask a student to read the definitions and example following Question 5 aloud, then complete Question 6 as a class.

Questions to ask• What is the difference between an infinite sequence and a

finite sequence?• How can you determine whether a sequence is infinite or finite?



• Are all sequences either infinite or finite? Why or why not? • What is another example of an infinite sequence? A finite sequence?• Is it possible to visually represent zero blocks?

Summary The domain of a sequence is the set of term numbers, and the range of a sequence is the set of term values. A sequence that continues on forever is called an infinite sequence, and a sequence that terminates is called a finite sequence.

Talk the Talk: Searching for a SequenceFacilitation NotesIn this activity, students build sequences to fit given criteria.

Have students work with a partner or in a group to complete Questions 1 and 2. Share responses as a class.

Questions to ask• What is the definition of a sequence?• Why do all sequences have the same domain?• Does every sequence contain term numbers and term values?

Explain.• Does each term number of a sequence correspond to a unique

term value?• Can you create a different sequence that meets these criteria?• How did you demonstrate a sequence that is decreasing

by multiplication?• What is another way to demonstrate decreasing by multiplication?

Differentiation strategyTo extend the activity, have students create their own sequences. Have the class categorize them by increasing or decreasing, type of operation used, and infinite or finite.

SummarySequences can be built from a list of characteristics. Characteristics may include a starting value, whether the sequence is increasing or decreasing, operations used between consecutive terms, and whether the sequence is finite or infinite.

DEMONSTRATE




Is There a Pattern Here?Recognizing Patterns and Sequences

1

Learning Goals• Recognize and describe patterns. • Represent patterns as sequences. • Predict the next term in a sequence.• Represent a sequence as a table of values.

Since early elementary school, you have been recognizing and writing patterns involving shapes, colors, letters, and numbers. How are patterns related to sequences and how can sequences be represented using a table of values?

Key Terms• sequence • term of a sequence • infinite sequence • finite sequence

Warm Up Write the next three terms in each pattern and explain how you generated each term.

1. J, F, M, A, M, J, J, A, S, . . .

2. S, M, T, W, . . .

3. 5, 10, 15, 20, . . .

4. 100, 81, 64, 49, . . .

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Warm Up Answers

1. O, N, D; They are the first letter of each month.

2. T, F, S; They are the first letter of each day of the week.

3. 25, 30, 35; They are all increasing by 5.

4. 36, 25, 16; They are all decreasing perfect square numbers beginning with 10.

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1. List at least 3 patterns that you notice.

2. Describe the pattern for the number of terms in each row.

3. Describe the pattern within each row.

4. Describe the pattern that results from determining the sum of each row.

5. Determine the next two rows in Pascal’s Triangle. Explain your reasoning.

A Pyramid of Patterns

Pascal’s Triangle is a famous pattern named after the French mathematician and philosopher Blaise Pascal. A portion of the pattern is shown.

1

233

111 1

11 4 46

11 6 615 1520

11155 10101

GETTING STARTED


Answers

1. Answers may vary.2. In each row, the number

of terms is 1 more than the row above it.

3. Each row is symmetrical.4. The sum of each row

generates the pattern 1, 2, 4, 8, 16, 32, 64.

5. 1 7 21 35 35 21 7 1

1 8 26 56 70 56 28 8 1

Each row begins and ends with one. All other terms are the sum of the numbers to the immediate left and right of it in the row above.

ELL TipCreate an anchor chart to identify and ensure students’ understanding of patterns. Display examples of mathematical and non-mathematical patterns on the chart. Non-mathematical examples may include the design of a tablecloth, patterns on a tile floor, and patterns in nature (waves, leaves). Discuss how Pascal’s Triangle contains patterns, and ask students to identify the patterns on the triangle. Ask students to create examples of patterns involving numbers as well as other items, such as colors or clothing.




Patterns to Sequences to Tables

AC TIVIT Y

1.1

A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term of a sequence is an individual number, figure, or letter in the sequence.

Ten examples of sequences are given in this activity. For each sequence, describe the pattern, draw or describe the next terms, and represent each sequence numerically.

1. Positive Thinking

a. Analyze the number of dots. Describe the pattern.

b. Draw the next three figures of the pattern.

c. Represent the number of dots in each of the seven figures as a numeric sequence.

d. Represent the number of dots in each of the first seven figures as a function using a table of values.

Term Number 1 2 3 4 5 6 7

Term Value

All numeric sequences can be represented as functions. The independent variable is the term number beginning with 1, and the dependent variable is the term of the sequence.


Answers

1a. Each figure has 4 fewer dots than the figure before it.

1b.

1c. 25, 21, 17, 13, 9, 5, 11d.

Term Number

Term Value

1 25

2 21

3 17

4 13

5 9

6 5

7 1

ELL TipReview the term sequence. Discuss the relationship between patterns and sequences. Before beginning the exercises in the activity, ask students to create their own pattern of dots, similar to the example given in Activity 1.1. Ask for volunteers to explain how their pattern can be represented as a numeric sequence.




2. Family TreeJessica is investigating her family tree by researching each generation, or set, of parents. She learns all she can about the first four generations, which include her two parents, her grandparents, her great-grandparents, and her great-great-grandparents.

a. Think about the number of parents. Describe the pattern.

b. Determine the number of parents in the fifth and sixth generations.

c. Represent the number of parents in each of the 6 generations as a numeric sequence. Then represent the sequence using a table of values.

3. A Collection of Squares

a. Analyze the number of small squares in each figure. Describe the pattern.

b. Draw the next three figures of the pattern.

c. Represent the number of small squares in each of the first seven figures as a numeric sequence. Then represent the sequence using a table of values.

Term Number

Term Value

Term Number

Term Value


Answers

2a. Each generation has 2 times the number of parents as the generation after it.

2b. The fifth generation has 25 5 32 parents, and the sixth generation has 26 5 64 parents.

2c. 2, 4, 8, 16, 32, 64.

Term Number

Term Value

1 2

2 4

3 8

4 16

5 32

6 64

3a. The number of small squares are decreasing perfect squares, beginning with the square of 7.

3b.

3c. 49, 36, 25, 16, 9, 4, 1

Term Number

Term Value

1 49

2 36

3 25

4 16

5 9

6 4

7 1

ELL TipSome non-mathematical terms that appear in this lesson are generation, omelet, decals, and auditorium. Create a vocabulary chart that shows each term followed by a picture and synonyms that describe each term in students’ native language.




4. Al’s OmeletsAl’s House of Eggs N’at makes omelets. Al begins each day with 150 eggs to make his famous Bestern Western Omelets. After making 1 omelet, he has 144 eggs left. After making 2 omelets, he has 138 eggs left. After making 3 omelets, he has 132 eggs left.

a. Think about the number of eggs Al has left after making each omelet. Describe the pattern.

b. Determine the number of eggs left after Al makes the next two omelets.

c. Represent the number of eggs left after Al makes each of the first 5 omelets as a numeric sequence. Then represent the sequence using a table of values.

5. Donna’s DaisiesDonna is decorating the top border of her bedroom walls with a daisy pattern. She is applying decals with each column having a specific number of daisies.

a. Think about the number of daisies in each column. Describe the pattern.

b. Determine the number of daisies in each of the next two columns.

c. Represent the number of daisies in each of the first 8 columns as a numeric sequence. Then represent the sequence using a table of values.

Term Number

Term Value

Term Number

Term Value


Answers

4a. Al has 6 fewer eggs after making each omelet.

4b. After making 4 omelets, Al has 126 eggs. After making 5 omelets, Al has 120 eggs.

4c. 150, 144, 138, 132, 126, 120

Term Number

Term Value

1 150

2 144

3 138

4 132

5 126

5a. The number of daisies repeat in the pattern 3, 4, 2.

5b. The seventh column has 3 daisies, and the eighth column has 4 daisies.

5c. 3, 4, 2, 3, 4, 2, 3, 4

Term Number

Term Value

1 3

2 4

3 2

4 3

5 4

6 2

7 3

8 4




6. Troop of Triangles

a. Analyze the number of dark triangles. Describe the pattern.

b. Draw the next two figures of the pattern.

c. Represent the number of dark triangles in each of the first 6 figures as a numeric sequence. Then represent the sequence using a table of values.

Term Number

Term Value


Answers

6a. The second figure has 2 more triangles than the first, the third figure has 3 more triangles than the second, and the fourth figure has 4 more triangles than the third.

6b.

6c. 1, 3, 6, 10, 15, 21

Term Number

Term Value

1 1

2 3

3 6

4 10

5 15

6 21




7. Gamer GuruMica is trying to beat his high score on his favorite video game. He unlocks some special mini-games where he earns points for each one he completes. Before he begins playing the mini-games, Mica has 500 points. After completing 1 mini-game he has a total of 550 points, after completing 2 mini-games he has 600 points, and after completing 3 mini-games he has 650 points.

a. Think about the total number of points Mica gains from mini-games. Describe the pattern.

b. Determine Mica’s total points after he plays the next two mini-games.

c. Represent Mica’s total points after completing each of the first 5 mini-games as a numeric sequence. Be sure to include the number of points he started with. Then represent the sequence using a table of values.

Term Number

Term Value


Answers

7a. Mika gains 50 points for each mini-game he plays.

7b. After playing 4 mini-games, Mica has 700 points. After playing 5 mini-games, Mica has 750 points.

7c. 500, 550, 600, 650, 700, 750

Term Number

Term Value

1 500

2 550

3 600

4 650

5 700

6 750




8. Polygon Party

a. Analyze the number of sides in each polygon. Describe the pattern.

b. Draw the next two figures of the pattern.

c. Represent the number of sides of each of the first 6 polygons as a numeric sequence. Then represent the sequence using a table of values.

Term Number

Term Value


Answers

8a. Each figure is a regular polygon that has one more side than the previous polygon.

8b.

8c. 3, 4, 5, 6, 7, 8

Term Number

Term Value

1 3

2 4

3 5

4 6

5 7

6 8




9. Pizza ContestJacob is participating in a pizza-making contest. Each contestant has to bake the largest and most delicious pizza they can. Jacob’s pizza has a 6-foot diameter! After the contest, he plans to cut the pizza so that he can pass the slices out to share. He begins with 1 whole pizza. Then, he cuts it in half. After that, he cuts each of those slices in half. Then he cuts each of those slices in half, and so on.

a. Think about the size of each slice in relation to the whole pizza. Describe the pattern.

b. Determine the size of each slice compared to the whole pizza after the next two cuts.

c. Represent the size of each slice compared to the whole pizza after each of the first 5 cuts as a numeric sequence. Include the whole pizza before any cuts. Then represent the sequence using a table of values.

Term Number

Term Value


Answers

9a. After every cut, each slice is 1 __ 2 the size of each slice in the previous cut.

9b. Each slice is 1 ___ 16 of the original after 4 cuts. Each slice is 1 ___ 32 of the original after 5 cuts.

9c. 1, 1 __ 2 , 1 __ 4 , 1 __ 8 , 1 ___ 16 , 1 ___ 32

Term Number

Term Value

1 1

2 1 __ 2

3 1 __ 4

4 1 __ 8

5 1 ___ 16

6 1 ___ 32




10. Coin CollectingMiranda’s uncle collects rare coins. He recently purchased a rare coin for $5. He claims that the value of the coin will triple each year. So even though the coin is currently worth $5, next year it will be worth $15. In 2 years it will be worth $45, and in 3 years it will be worth $135.

a. Think about how the value of the coin changes each year. Describe the pattern.

b. Determine the value of the coin after 4 years and after 5 years.

c. Represent the value of the coin after each of the first 5 years as a numeric sequence. Include the current value. Then represent the sequence using a table of values.

Term Number

Term Value


Answers

10a. Each year the coin value is 3 times greater than its value in the previous year.

10b. After four years the coin value will be equal to 135(3) 5 405, and after five years the coin value will be equal to 405(3) 5 1215.

10c. 5, 15, 45, 135, 405, 1215

Term Number

Term Value

1 5

2 15

3 45

4 135

5 405

6 1215




There are many different patterns that can generate a sequence of numbers. For example, you may have noticed that some of the sequences in the previous activity were generated by performing the same operation using a constant number. In other sequences, you may have noticed a different pattern.

The next term in a sequence is calculated by determining the pattern of the sequence, and then using that pattern on the last known term of the sequence.

1. For each sequence in the previous activity, write the numeric sequence, record whether the sequence increases or decreases, and describe the sequence by stating the first term and the operation(s) used to create the sequence. The first one has been completed for you.

Looking at Sequences More Closely

AC TIVIT Y

1.2

Problem Name Numeric Sequence

Increases or Decreases Sequence Description

Positive Thinking 25, 21, 17, 13, 9, 5, 1 Decreases Begin at 25. Subtract 4 from

each term.

Family Tree

A Collection of Squares

Al’s Omelets

Donna’s Daisies

Troop of Triangles

Gamer Guru

Polygon Party

Pizza Contest

Coin Collecting


Problem Sequence I/D DescriptionPositive Thinking 25, 21, 17, 13, 9, 5, 1 D Begin at 25. Subtract 4 from each term.Family Tree 2, 4, 8, 16, 32, 64 I Begin at 2. Multiply each term by 2.A Collection of Squares

49, 36, 25, 16, 9, 4, 1 D Begin at 72. Decrease each base by 1 while retaining the square.

Al’s Omelets 150, 144, 138, 132, 126, 120

D Begin at 150. Subtract 6 from each term.

Donna’s Daisies 3, 4, 2, 3, 4, 2, 3, 4 I/D Begin at 3. Continue with 4 and 2. Repeat this 3, 4, 2 pattern

Answers

1. See bottom of this page and bottom of the next page.




2. Which sequences are similar? Explain your reasoning.

3. What do all sequences have in common?

4. Consider a sequence in which the first term is 64 and each term after that is calculated by dividing the previous term by 4. Margaret says that this sequence ends at 1 because there are no whole numbers that come after 1. Jasmine disagrees and says that the sequence continues beyond 1. Who is correct? If Margaret is correct, explain why. If Jasmine is correct, predict the next two terms of the sequence.

5. What is the domain of a sequence? What is the range?


Answers

2. Sample answer. Some sequences increase/decrease by adding or subtracting: Positive Thinking, Al’s Omelets, Troop of Triangles, Gamer Guru, Polygon Party

Some sequences increase/decrease by multiplying or dividing: Family Tree. Pizza Contest, Coin Collecting

Some follow neither of those patterns: A Collection of Squares, Donna’s Daisies

3. All of the sequences are functions.

4. Jasmine is correct. Even though the sequence begins with whole numbers, this does not mean that it must contain only whole numbers. After 1, the next two terms of the sequence are 1 4 4 5 1 __ 4 and 1 __ 4 4 4 5 1 ___ 16 .

5. The domain of a sequence is all integers beginning with 1; the range of a sequence varies depending upon the function.

Problem Sequence I/D DescriptionTroop of Triangles 1, 3, 6, 10, 15, 21 I Begin at 1. Add 2, then add 3, then add 4, …Gamer Guru 500, 550, 600, 650, 700, 750 I Begin at 500. Add 50 to each term.Polygon Party 3, 4, 5, 6, 7, 8 I Begin at 3. Add 1 to each term.

Pizza Contest 1, 1 __ 2 , 1 __ 4 , 1 __ 8 , 1 ___ 16 , 1 ___ 32 D Begin at 1. Multiply each term by 1 __ 2 (or divide each term by 2)

Coin Collecting 5, 15, 45, 135, 405, 1215 I Begin at 5. Multiply each term by 3.




6. Does the pattern shown represent an infinite or finite sequence? Explain your reasoning.

If a sequence continues on forever, it is called an infinite sequence. If a sequence terminates, it is called a finite sequence.

For example, consider an auditorium where the seats are arranged according to a specific pattern. There are 22 seats in the first row, 26 seats in the second row, 30 seats in the third row, and so on. Numerically, the sequence is 22, 26, 30, . . . , which continues infinitely. However, in the context of the problem, it does not make sense for the number of seats in each row to increase infinitely. Eventually, the auditorium would run out of space! Suppose that this auditorium can hold a total of 10 rows of seats. The correct sequence for this problem situation is:

22, 26, 30, 34, 38, 42, 46, 50, 54, 58.

Therefore, because of the problem situation, the sequence is a finite sequence.

An ellipsis is three periods, which means “and so on.” An infinite sequence can be represented using an ellipsis.


Answers

6. This is a finite sequence. Each stage results by taking away the bottom row of the pyramid until the pyramid no longer exists.

ELL TipsThe terms infinite and finite are cognates in many languages and may be easily identified by students. Discuss how the word finite is related to the words final or finish. Also discuss how infinite is an antonym of finite. Model the use and meanings of the terms with an example such as, “There are a finite number of positive factors for the number 100, but there are an infinite number of multiples of the number 100.” Ask students to create their own sentences showing the contrast between a finite amount and an infinite amount of something.




NOTES TALK the TALK

Searching for a Sequence

In this lesson you have seen that many different patterns can generate a sequence of numbers.

1. Explain why the definition of a function applies to all sequences.

2. Create a sequence to fit the given criteria. Describe your sequence using figures, words, or numbers. Provide the first four terms of the sequence. Explain how you know that it is a sequence.

a. Create a sequence that begins with a positive integer, is decreasing by multiplication, and is finite.

b. Create a sequence that begins with a negative rational number, is increasing by addition, and is infinite.


Answers

1. In a function, each independent value has one corresponding dependent value. In a sequence, each term number (independent value) has one term (dependent value) that corresponds to it.

2a. Sample answer. 27, 9, 3, 1, 1 __ 3 2b. Sample answer. 22.5, 21.5, 20.5, 0.5, ...


LESSON 2: The Password Is... Operations! • 1

The Password Is... Operations!Arithmetic and Geometric Sequences

2

Lesson OverviewGiven 16 numeric sequences, students generate several additional terms for each sequence and describe the rule they used for each sequence. They sort the sequences into groups based upon common characteristics of their choosing and explain their rationale. The terms arithmetic sequence, common difference, geometric sequence, and common ratio are then defined, examples are provided, and students respond to clarifying questions. They then categorize the sequences from the beginning of the lesson as arithmetic, geometric or neither and identify the common difference or common ratio where appropriate. Students begin to create graphic organizers, identifying four different representations for each arithmetic and geometric sequence. In the first activity, they glue each arithmetic and geometric sequence to a separate graphic organizer and label them, and in the second activity, the corresponding graph is added. The remaining representations are completed in the following lessons. This lesson concludes with students writing sequences given a first term and a common difference or common ratio and identifying whether the sequences are arithmetic or geometric.


analyze, and evaluate equations, relations, and functions. The student is expected to: decide whether relations represented verbally, tabularly, graphically, and symbolically define afunction;write a formula for the n th term of arithmetic and geometric sequences, given the value ofseveral of their terms;


Essential Ideas • An arithmetic sequence is a sequence of numbers in which the difference between any two

consecutive terms is a positive or negative constant. This constant is called the commondifference and is represented by the variable d.

MATERIALSScissorsGlue


(A)

(D)


• A geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is a constant. This constant is called the common ratio and is represented by the variable r.

• The graph of a sequence is a set of discrete points.• The points of an arithmetic sequence lie on a line. When the common difference is a positive, the

graph is increasing, and when the common difference is a negative, the graph is decreasing. • The points of a geometric sequence do not lie on a line. When the common ratio is greater than 1,

the graph is increasing; when the common ratio is between 0 and 1, the graph is decreasing; and when the common ratio is less than 0, the graph alternates between increasing and decreasing between consecutive points.


Getting Started: What Comes Next, and How Do You Know?Students generate several additional terms for 16 different numeric sequences and describe the rule they used for each sequence. They sort the sequences into groups based upon common characteristics of their choosing and explain their rationale.

DevelopActivity 2.1: Defining Arithmetic and Geometric SequencesStudents are provided the definitions of arithmetic sequence, common difference, geometric sequence, and common ratio. Examples are provided, and students respond to clarifying questions. They then categorize the sequences from the beginning of the lesson as arithmetic, geometric, or neither and identify the common difference or common ratio where appropriate. Students begin to create graphic organizers, identifying four different representations for each sequence. In this activity, students glue each arithmetic and geometric sequence to a separate graphic organizer.

Day 2Activity 2.2: Matching Graphs and SequencesStudents match graphs to their corresponding numeric sequence and then add the graphs to each graphic organizer.

DemonstrateTalk the Talk: Name That Sequence!Students are given a first term and a common difference or common ratio, and they must identify the unique sequence it describes and state whether the sequence is arithmetic or geometric.



Getting Started: What Comes Next, and How Do You Know?

Facilitation NotesIn this activity, students cut out sequence cards, generate additional terms for 16 different numeric sequences, and then describe the rule they used for each sequence. A sort activity is used to categorize the sequences based upon common characteristics.

Have students work with a partner or in a group to complete Questions 1 through 3. Make sure that students understand that they are just describing a pattern; they do not have to write a rule. Share responses as a class.

Differentiation strategyFor students who struggle, reduce the number of sequences while maintaining variety.As students work, look forStrategies and phrases they use to determine the next terms of the sequences.Questions to ask

• How did you determine the next term in the sequence? • Is there another rule that can be used to determine that

same sequence?• Is the sequence increasing or decreasing? How do you know? • How many sequences involve addition or subtraction? • Which sequences involve addition by the same number each time?• Which sequences involve addition by numbers in a pattern

each time?• How many sequences involve multiplication or division?• What other operations are used to generate the sequences?

SummaryDifferent operations can be used to generate sequences.

Activity 2.1Defining Arithmetic and Geometric Sequences

Facilitation NotesIn this activity, students are provided the definitions of arithmetic sequence, common difference, geometric sequence, and common ratio. Examples are provided, and students respond to clarifying questions. They then

ENGAGE

DEVELOP



categorize the sequences from the beginning of the lesson as arithmetic, geometric, or neither and identify the common difference or common ratio where appropriate. Students begin to create graphic organizers, identifying four different representations for each sequence. In this activity, students glue each arithmetic and geometric sequence to a separate graphic organizer.

Ask a student to read the introduction and definitions aloud. Review the worked example as a class. Have students work individually or with a partner to complete Question 1 and discuss as a class. Then have students work with a partner or in a group to complete Question 2. Share responses as a class

MisconceptionStudents may confuse the term arithmetic (noun) with the term arithmetic (adjective). Emphasize how to pronounce arithmetic when it is an adjective rather than a noun.Questions to ask for Question 1

• Think about a sequence such as 1, 2, 3, 4 . . . where x is any real number. Is there a difference between adding a negative x to each term of the sequence and subtracting a positive x from each term of the sequence?

• Is there a difference between adding 2 to each term of the sequence and subtracting 2 from each term in the sequence?

• If the common difference of the sequence is 4, how would you describe the rule used to generate the next terms using addition?

• If the common difference of the sequence is 4, how would you describe the rule used to generate the next terms using subtraction?

Questions to ask for Question 2• How many of the sixteen sequences used a rule that is described by

the use of addition or subtraction?• How is the common difference evident in the description of

each pattern?

Ask a student read the definitions following Question 2 aloud. Review the worked example as a class.




MisconceptionStudents already have an understanding of the terms arithmetic and geometry. Address how previous use of these terms is the same and different as how they are used with sequences.Questions to ask

• Explain the difference between a common ratio and a common difference.

• If the common ratio is changed from 2 to 3, will the terms increase more rapidly or more slowly? Why? Is the new sequence increasing or decreasing?

• If the common ratio is changed from 2 to 1, what will be the first 5 terms? Is the new sequence increasing or decreasing?

• If the common ratio is changed from 2 to 1 __ 3 , will the terms increase more rapidly or more slowly? Why? Is the new sequence increasing or decreasing?

• If the common ratio is changed from 2 to 22, will the terms increase more rapidly or more slowly? Why? Is the new sequence increasing or decreasing?

• Is the common ratio of a sequence the number which each term is divided by or multiplied by?


Questions to ask • Is each term of this sequence multiplied by 3 or multiplied by 1 __ 3 ?• How many of the sixteen sequences used a rule that is described by

the use of multiplication? • Is the common ratio stated in the description of each pattern?

Where?• Can you think of a sequence that is different than Dante’s and Kira’s? • Describe a third sequence that would also begin with these first two

terms. How would you describe the pattern? Does it have a common ratio or a common difference?

• Is there a different arithmetic sequence that satisfies these first two terms?



• Is there a different geometric sequence that satisfies these first two terms?

• Are all sequences considered either geometric or arithmetic sequences? Why or why not?

• If every term in a sequence is the same number, what is the common difference?

SummaryAn arithmetic sequence is a sequence of numbers in which a positive or negative constant, called the constant difference, is added to each term to produce the next term. A geometric sequence is a sequence of numbers in which you multiply each term by a constant, called the common ratio, to determine the next term.

Activity 2.2Matching Graphs and Sequences

Facilitation NotesIn this activity, students cut out and match several graphs to the appropriate numeric sequence and then attach the graphs to each graphic organizer.


Questions to ask• Which graphs appear to be linear? What information does this give

you about the sequence? • Which graphs appear to be exponential? What information does this

give you about the sequence? • Which graphs appear to be increasing? What information does this

give you about the sequence? • Which graphs appear to be decreasing? What information does this

give you about the sequence?• How can determining the bounds of the y-axis be helpful in matching

the graphs to the appropriate sequence? • How can determining the y-intercept be helpful in matching the

graphs to the appropriate sequence? • How can the coordinates of the first term be helpful in matching the

graphs to the appropriate sequence?



Summary All sequences are functions. The graph of a sequence is a set of discrete points. The points of an arithmetic sequence lie on a line. When the common difference is a positive, the graph is increasing, and when the common difference is a negative, the graph is decreasing. The points of a geometric sequence do not lie on a line. When the common ratio is greater than 1, the graph is increasing; when the common ratio is between 0 and 1, the graph is decreasing; and when the common ratio is less than 0, the graph alternates between increasing and decreasing between consecutive points.

Talk the Talk: Name That Sequence!Facilitation NotesIn this activity, students are given a first term and a common difference or common ratio. Using those criteria, they write the first five terms of a unique sequence and state whether the sequence is arithmetic or geometric.


Differentiation strategyTo extend the activity, have students design their own problems.

• Ask students to write a first term and either common difference or common ratio. Give the information to their partner and ask them to generate the first few terms in the sequence.

• Ask students to create a sequence using their own rule, then ask their partner to identify the rule.

Questions to ask• What two pieces of information are needed to generate a sequence?• Explain why this information always provides a unique sequence.• How can you determine whether a sequence is arithmetic or

geometric from the sequence of numbers? From its graph?

SummaryA unique sequence can be described by a first term and common difference or common ratio.

DEMONSTRATE




2The Password Is... Operations!Arithmetic and Geometric Sequences

Learning Goals• Determine the next term in a sequence. • Recognize arithmetic sequences and geometric sequences.• Determine the common difference or common ratio for

a sequence.• Graph arithmetic and geometric sequences. • Recognize graphical behavior of sequences. • Sort sequences that are represented graphically.

You have represented patterns as sequences of numbers—a relationship between term numbers and term values. What patterns appear when sequences are represented as graphs?

Key Terms• arithmetic sequence • common difference • geometric sequence • common ratio

Warm UpWrite the next three terms in each sequence and explain how you generated each term.

1. 22, 4, 28, 16, . . .

2. 60, 53, 46, 39, 32, . . .

3. 1, 5, 17, 53, 161, 485, . . .

4. 4, 10, 16, 22, . . .


Warm Up Answers

1. 232, 64, 2128; multiply the previous term by 22

2. 25, 18, 11; subtract 7 from the previous term

3. 1457, 4373, 13,121; multiply the previous term by 3, then add 2

4. 28, 34, 40; add 6 to the previous term




GETTING STARTED

What Comes Next, and How Do You Know?

Cut out Sequences A through P located at the end of the lesson.

1. Determine the unknown terms of each sequence. Then describe the pattern under each sequence.

2. Sort the sequences into groups based on common characteristics. In the space provided, record the following information for each of your groups.

• List the letters of the sequences in each group.

• Provide a rationale as to why you created each group.

3. What mathematical operation(s) did you perform in order to determine the next terms of each sequence?


Answers

1. A: 720, 1440, 2880; multiply by 2

B: 4, 6, 8; add 2 C: 2162, 2486, 21458;

multiply by 3 D: 26, 37, 50; square the

term and then add 1 E: 25, 2

29 ___ 4 , 2 19 ___ 2 ;

subtract 9 __ 4 F: 0.1234, 0.01234,

0.001234; multiply by 0.1 G: 26, 7, 28, 9;

consecutive numbers, every other number negative

H: 0, 4, 8; add 4 I: 23, 34; add consecutive

odd numbers J: 2

5 ___ 16 , 2 5 ___ 32 ; multiply by 1 __ 2

K: 0.5, 21, 22.5; subtract 1.5

L: 71, 65: subtract 1, then 2, then 3,...

M: 2 1 ___ 16 , 1 ___ 64 ; divide by 24

N: 1391.2, 1370.7, 1350.2; subtract 20.5

O: √ ____

22 , √ ____

23 ; square roots of decreasing consecutive integers

P: 2324, 972; multiply by 23

2. Sample answer. A, C, F, J, M, and P;

sequences that change by multiplying or dividing by the same number each time

B, E, H, K, and N; sequences that change by adding or subtracting by the same number each time

D, G, I, L, and O: sequences that change in some other way

3. Some sequences required addition,

subtraction, multiplication or division by the same number each time. Some sequences involved operations such as squaring or taking the square root, operations with consecutive numbers, or switching signs each time.

ELL TipReview the term rationale and create a list of synonyms for the term. Ask students for examples of when rationale is used in different contexts.




For some sequences, you can describe the pattern as adding a constant to each term to determine the next term. For other sequences, you can describe the pattern as multiplying each term by a constant to determine the next term. Still other sequences cannot be described either way.

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is a constant. In other words, it is a sequence of numbers in which a constant is added to each term to produce the next term. This constant is called the common difference. The common difference is typically represented by the variable d.

The common difference of a sequence is positive if the same positive number is added to each term to produce the next term. The common difference of a sequence is negative if the same negative number is added to each term to produce the next term.

Defining Arithmetic and Geometric Sequences

AC TIVIT Y

2.1

When you add a negative number, it is the same as subtracting a positive number.

Consider the sequence shown.

11, 9, 7, 5, . . .

The pattern is to add the same negative number, 22, to each term to determine the next term.

Sequence: 11 , 9 , 7 , 5 , . . .

This sequence is arithmetic and the common difference d is 22.

add ]2 add ]2 add ]2

Worked Example

Remember:

A1_SE_M01_T02_L02_TEA.indd 3 9/9/20 7:12 PMELL TipAsk students what is meant by consecutive terms. Discuss how the word consecutive is used in mathematical and non-mathematical situations. Have students provide mathematical examples of consecutive and non-consecutive terms relating to sequences.




1. Suppose a sequence has the same starting number as the sequence in the worked example, but its common difference is 4.

a. How would the pattern change?

b. Is the sequence still arithmetic? Why or why not?

c. If possible, write the first 5 terms of the new sequence.

2. Analyze the sequences you cut out in the Getting Started.

a. List the sequences that are arithmetic.

b. Write the common difference of each arithmetic sequence you identified.

NOTES


Answers

1a. The sequence would increase by 4 instead of decreasing by 2.

1b. Yes. The sequence is still arithmetic because the difference between each consecutive term is constant.

1c. 11, 15, 19, 23, 272a. Sequences B, E, H, K,

and N2b. Sequence B: d 5 2 Sequence E: d 5 2 9 __ 4 Sequence H: d 5 4 Sequence K: d 5 21.5 Sequence N: d 5 220.5




A geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is a constant. In other words, it is a sequence of numbers in which you multiply each term by a constant to determine the next term. This integer or fraction constant is called the common ratio. The common ratio is represented by the variable r.


1, 2, 4, 8, . . .

The pattern is to multiply each term by the same number, 2, to determine the next term.

Sequence: 1 , 2 , 4 , 8 , . . .

This sequence is geometric and the common ratio r is 2.

multiply by 2

multiply by 2

multiply by 2

3. Suppose a sequence has the same starting number as the sequence in the worked example, but its common ratio is 3.


b. Is the sequence still geometric? Explain your reasoning.

c. Write the first 5 terms of the new sequence.

Worked Example


Answers

3a. The sequence would still increase, but the terms would be different. The sequence would increase more rapidly.

3b. Yes. The sequence is still geometric because the ratio between any two consecutive terms is constant.

3c. 1, 3, 9, 27, 81




4. Suppose a sequence has the same starting number as the sequence in the worked example, but its common ratio is 1 __ 3 .


b. Is the sequence still geometric? Why or why not?


5. Suppose a sequence has the same starting number as the sequence in the worked example, but its common ratio is 22.


b. Is the sequence still geometric? Explain your reasoning.


NOTES


Answers

4a. The sequence would decrease.


4c. 1, 1 __ 3 , 1 __ 9 , 1 ___ 27 , 1 ___ 81 , 1 ____ 243 5a. The sequence would

decrease and increase and contain alternating positive and negative integers.


5c. 1, 22, 4, 28, 16, 232




6. Consider the sequence shown.270, 90, 30, 10, . . .

Devon says that he can determine each term of this sequence by multiplying each term by 1 __ 3 , so the common ratio is 1 __ 3 . Chase says that he can determine each term of this sequence by dividing each term by 3, so the common ratio is 3. Who is correct? Explain your reasoning.

7. Consider the sequences you cut out in the Getting Started. List the sequences that are geometric. Then write the common ratio on each Sequence Card.

8. Consider the sequences that are neither arithmetic nor geometric. List these sequences. Explain why these sequences are neither arithmetic nor geometric.


Answers

6. Devon is correct. The next term in the sequence can be determined by multiplying the previous term by 1 __ 3 . Chase is correct in that he can determine the sequence by dividing each term by 3, but the common ratio represents the number by which each term is multiplied. Each term in this sequence is not multiplied by 3, it is multiplied by 1 __ 3 .

7. Sequence A: r 5 2 Sequence C: r 5 3 Sequence F: r 5 0.1 Sequence J: r = 1 __ 2 Sequence M: r = 2 1 __ 4 Sequence P: r 5 238. Sequences D, G, L, and

O neither arithmetic nor geometric because there is no common difference or common ratio for any of these sequences.




10. Using the terms given in Question 9, write a sequence that is neither arithmetic nor geometric. Then, have your partner tell you what the pattern is in your sequence.

11. How many terms did your partner need before the pattern was recognized?

12. Consider the sequence 2, 2, 2, 2, 2. . . Identify the type of sequence it is and describe the pattern.

13. Begin to complete the graphic organizers located at the end of the lesson to identify arithmetic and geometric sequences. Glue each arithmetic sequence and each geometric sequence to a separate graphic organizer according to its type. Discard all other sequences.

9. Consider the first two terms of the sequence 3, 6, . . .

Dante says, “This is how I wrote the sequence for the given terms.”

3, 6, 9, 12, . . .

Kira says, “This is the sequence I wrote.”

3, 6, 12, 24, . . .

Who is correct? Explain your reasoning.


Answers

9. Both are correct. From the first two terms, Dante or Kira did not know whether the sequence was arithmetic or geometric. Dante assumed it was arithmetic with a common difference of 3. Kira assumed it was geometric with a common ratio of 2.

10. Sample answer. Sequence 3, 6, 9, 15,

24, . . . ; each term is the sum of the two previous terms.

11. Answers will vary.12. Sample answers. This sequence could

be arithmetic in that you could add 0 to each term.

This sequence could be geometric in that you could multiply each term by 1.

This sequence could be neither arithmetic nor geometric in that the term 2 could just be repeating.

13. Sequence A: geometric Sequence B: arithmetic Sequence C: geometric Sequence E: arithmetic Sequence F: geometric Sequence H: arithmetic Sequence J: geometric Sequence K: arithmetic Sequence M:

geometric Sequence N: arithmetic Sequence P: geometric




As you have already discovered when studying functions, graphs can help you see trends of a sequence—and at times can help you predict the next term in a sequence.

1. The graphs representing the arithmetic and geometric sequences from the previous activity are located at the end of this lesson. Cut out these graphs. Match each graph to its appropriate sequence and glue it into the Graph section of its graphic organizer.

2. What strategies did you use to match the graphs to their corresponding sequences?

3. How can you use the graphs to verify that all sequences are functions?

Matching Graphs and Sequences

AC TIVIT Y

2.2


Answers

1. Sequence A, Graph 1 Sequence B, Graph 4 Sequence C, Graph 2 Sequence E, Graph 5 Sequence F, Graph 3 Sequence H, Graph 6 Sequence J, Graph 9 Sequence K, Graph 7 Sequence M, Graph 10 Sequence N, Graph 8 Sequence P, Graph 112. Answers may vary.3. Sample answer. The graphs all pass the

vertical line test.




NOTES TALK the TALK

Name That Sequence!

Write the first five terms of each sequence described and identify the sequence as arithmetic or geometric.

1. The first term of the sequence is 8 and the common difference is 12.

2. The first term of the sequence is 29 and the common ratio is 22.

3. The first term of the sequence is 0 and the common difference is 26.

4. The first term of the sequence is 23 and the common ratio is 2 1 __ 4 .


Answers

1. 8, 20, 32, 44, 56; arithmetic

2. 29, 18, 236, 72, 2144; geometric

3. 0, 26, 212, 218, 224; arithmetic

4. 23, 3 __ 4 , 2 3 ___ 16 , 3 ___ 64 , 2 3 ____ 256 ; geometric




A

45, 90, 180, 360, —–— , —–— ,

—–— , . . .

B

]4, ]2, 0, 2, —–— , —–— ,

—–— , . . .

C

]2, ]6, ]18, ]54, —–— , —–— ,

—–— , . . .

D

2, 5, 10, 17, —–— , —–— ,

—–— , . . .

E

4, 7 __ 4 , ] 1 __ 2 , ]

11 ___ 4 —–— , —–— ,

—–— , . . .

F

1234, 123.4, 12.34, 1.234, —–— ,

—–— , —–— , . . .

G

1, ]2, 3, ]4, 5 —–— , —–— ,

—–— , —–— , . . .

H

]20, ]16, ]12, ]8, ]4, —–— ,

—–— , —–— , . . .

Sequence Cards✂





I

]1, 2, 7, 14, —–— , —–— , . . .

J

]5, ] 5 __ 2 , ]

5 __ 4 , ] 5 __ 8 , —–— , —–— ,

—–— , . . .

K

6.5, 5, 3.5, 2, —–— , —–— ,

—–— , . . .

L

86, 85, 83, 80, 76, —–— , —–— , . . .

M

] 16, 4, ] 1, 1 __ 4 , —–— , —–— , . . .

N

1473.2, 1452.7, 1432.2, 1411.7,

—–— , —–— , —–— , . . .

O

√ __

5 , 2, √ __

3 , √ __

2, 1, 0, √ ___

21 , —–— , —–— , . . .

P

24, 12, 236, 108, —–— , —–— , . . .

✂





Graph Cards✂

y3000

x10

Graph 1

x10

Graph 2y

100

–900

1350y

x10

Graph 3

–150

y

–10

10

x10

Graph 4

y10

–10

Graph 5

x10

y20

–20

x10

Graph 6





y10

–10

x10

Graph 7

y9

–21

x10

Graph 10

y

0 x

1500

10

Graph 8

y

0 x

–10

10

Graph 9

y1050

Graph 11

–450

x10

✂





Arithmetic Sequence

GraphSequence

Explicit FormulaRecursive Formula





Arithmetic Sequence

GraphSequence






Geometric Sequence

GraphSequence






GraphSequence




LESSON 3: Did You Mean: Recursion? • 1

Did You Mean: Recursion?Determining Recursive and Explicit Expressions from Contexts

3

Lesson OverviewScenarios are presented that can be represented by arithmetic and geometric sequences. Students determine the value of terms in each sequence. The term recursive formula is defined and used to generate term values. As the term number increases, it becomes more time consuming to generate the term value. This sets the stage for explicit formulas to be defined and used. Students practice using these formulas to determine the values of terms in both arithmetic and geometric sequences.

Algebra 1 Number and algebraic methods(12) The student applies the mathematical process standards and algebraic methods to write, solve,

analyze, and evaluate equations, relations, and functions. The student is expected to:identify terms of arithmetic and geometric sequences when the sequences are given infunction form using recursive processes;write a formula for the n th term of arithmetic and geometric sequences, given the value ofseveral of their terms;


Essential Ideas• A recursive formula expresses each new term of a sequence based on a preceding term of

the sequence.• An explicit formula for a sequence is a formula for calculating each term of the sequence using

the term’s position in the sequence.• The explicit formula for determining the nth term of an arithmetic sequence is

an 5 a1 1 d(n 2 1), where n is the term number, a1 is the first term in the sequence,an is the nth term in the sequence, and d is the common difference.

• The explicit formula for determining the nth term of a geometric sequence is gn 5 g1 ? r (n21),where n is the term number, g1 is the first term in the sequence, gn is the nth term in thesequence, and r is the common ratio.

MATERIALSGraphic organizers from The Password Is . . . Operations!


(C)

(D)


Lesson Structure and Pacing: 1 DayEngage

Getting Started: Can I Get a Formula?A scenario is given that can be represented by an arithmetic sequence. Students complete a table of values listing each term number and the value of the first ten terms. This is an introduction to the problem situation presented in Activity 3.1.

DevelopActivity 3.1: Writing Formulas for Arithmetic SequencesStudents use two worked examples to understand recursive and explicit formulas for arithmetic sequences. They use this understanding to write recursive and explicit formulas for the sequence described by the problem situation from the Getting Started. The problem situation is then changed, and students answer questions about the new problem situation by rewriting the explicit formula.

Activity 3.2: Writing Formulas for Geometric SequencesStudents are given a new problem situation and determine that the situation can be represented by a geometric sequence. They analyze two worked examples to understand recursive and explicit formulas for geometric sequences. Students then use this understanding to write recursive and explicit formulas for the sequence described by the problem situation. The problem situation is then changed, and they answer questions about the new problem situation by rewriting the explicit formula.

Activity 3.3: Writing Recursive and Explicit FormulasStudents use what they now know about recursive and explicit formulas for arithmetic and geometric sequences to write both types of formula for each of the sequences they studied in the previous lesson.

DemonstrateTalk the Talk: Pros and ConsStudents write paragraphs to describe the advantages and disadvantages of using recursive and explicit formulas to determine term values of arithmetic and geometric sequences.



Getting Started: Can I Get a Formula?

Facilitation NotesIn this activity, a scenario is given that can be represented by an arithmetic sequence. Students complete a table of values listing each term number and the value of the first ten terms. This is an introduction to the problem situation presented in Activity 3.1.

Have students work with a partner or in a group to complete the table of values and answer Questions 1 through 4. Share responses as class.

Questions to ask• What is the difference between an arithmetic sequence and a

geometric sequence? Can you spot the difference when you first read the scenario, or do you need to observe entries listed in an organizational table?

• Does this scenario describe an arithmetic sequence or a geometric sequence? How did you determine the sequence type?

• Why is the number of home runs not the same as the term number? Does this affect how you solve for each donation amount?

• How did you determine the donation amount if the team hits 2 home runs? 9 home runs?

• What strategy can you use to calculate the nth term?• What elements of a scenario are absolutely necessary to represent

a situation as an arithmetic sequence?

SummaryAn arithmetic sequence can be used to model a situation by creating additional term values using the common difference. The term numbers and term values can be organized in a table.

Activity 3.1Writing Formulas for Arithmetic Sequences

Facilitation NotesIn this activity, students analyze two worked examples to understand recursive and explicit formulas for arithmetic sequences. They use this understanding to write recursive and explicit formulas for the sequence described by Rico’s donations to the baseball team. The problem situation is then changed, and students answer questions about the new problem situation by rewriting the explicit formula.

DEVELOP

ENGAGE



Ask a student to read the definition aloud. Review the worked example as a class.

Have students work with a partner or in a group to complete the table of values and answer Questions 1 and 2. Share responses as class.

Questions to ask• What is the purpose of writing a recursive formula?• What information is needed to write a recursive formula?• What do the subscripts in the formula represent?• Why is n 2 1 rather than n added to d in the recursive formula?• When n 5 1, what is the result? Explain why this makes sense.• If n 5 5, what does each term in the recursive formula represent?• Can you determine the 11th term in this sequence without using a

recursive formula?• Why would you rather use a recursive formula in this situation?• Would you want to use a recursive formula to identify the 200th or

1000th term value in this sequence? Why not?

Ask a student to read the information and definition following Question 2 aloud. Review the worked example as a class.

Have students work with a partner or in a group to complete Questions 3 through 5. Share responses as class.

Differentiation strategyTo assist all students, help them connect the new terminology to words they already know. Recursive has the prefix re- and means repeating something, in this case, repeating the same operation to get the next term. Explicit means clearly, such as giving explicit directions; in this case, the explicit formula is more clear or direct.Questions to ask

• What elements are needed to write an explicit formula?• Why does the explicit formula use multiplication when a common

difference means addition is used in the sequence?• Why is (n 2 1) rather than n multiplied by d?• When n 5 1, what is the result? Explain why this makes sense.• How do the terms of the recursive formula relate to the terms in the

explicit formula?• What is the purpose of writing an explicit formula?• Can you determine the 50th term in this sequence without using an

explicit formula?• Why would you rather use an explicit formula in this situation?• Would you want to use an explicit formula or a recursive formula

to identify the 202th or 935th term value in this sequence? Why not?



SummaryAn arithmetic sequence can be represented using a recursive formula or an explicit formula. The explicit formula is more efficient in determining any term value without having to calculate all the terms before it.

Activity 3.2Writing Formulas for Geometric Sequences

Facilitation NotesIn this activity, students are given a problem situation that can be represented by a geometric sequence. They analyze two worked examples to understand recursive and explicit formulas for geometric sequences. They then write and use recursive and explicit formulas for the sequence described by the problem situation.

Ask a student to read the introduction aloud.

Have students work with a partner or in a group to complete Question 1. Share responses as class.

Questions to ask• Does this sequence have a common difference or a common ratio?

How do you know?• How did you determine the common ratio?• Why is the number of cell divisions not the same as the term

number? Does this affect how you determine the total number of cells?

• How did you determine the total number of cells after 2 cell divisions?

• Do you need to know the 4th term value to determine the 5th term value?


• Do you need to know the 9th term value to determine the 10th term value?


Ask a student to read the description of the recursive formula associated with a geometric sequence aloud. Review the worked example as a class.

Have students work with a partner or in a group to complete Question 2. Share responses as class.



Questions to ask• Is there more than one strategy to calculate the 20th term value?• What is the first term? • What is the common ratio?• How did you determine the term number?

Ask a student to read the description of the explicit formula associated with a geometric sequence aloud. Review the worked example as a class.

Have students work with a partner or in a group to complete Questions 3 through 5. Share responses as class.

Questions to ask• Why does the explicit formula use exponents when a common ratio

means multiplication was used in the sequence?• How do the terms of the recursive formula relate to the terms in the

explicit formula?• How do you know when it is better to use the recursive formula and

when it is better to use the explicit formula?• How do you know what term number to use when solving the

formula? Is that always the case?• How are the term values of a geometric series affected when r is a

negative value?• How is the scenario in Question 4 different from Question 3? How

do these changes affect your formula? MisconceptionStudents sometimes misunderstand the meaning of the first term value of a sequence. Sequences always start with term number 1. Based upon the phrasing of the scenario, the first term number usually represents a starting value, and the 2nd term represents the first time the operation is performed. For example, the first term is the number of cells after 0 divisions, not after one division, so the 100th term represents the number of cells after 99 divisions, not after 100 divisions. Sometimes students get this concept, but go in the reverse direction. As students solve these problems, have them explain the value they substitute in the formula and the meaning of the result. The clarification now will help later when students connect sequences and functions, and realize that the first term of a sequence is not the same as the y-intercept.

Summary A geometric sequence can be represented using a recursive formula or an explicit formula. The explicit formula is more efficient to determine any term value without having to calculate all the terms before it.



Activity 3.3Writing Recursive and Explicit Formulas

Facilitation NotesIn this activity, students use what they know about recursive and explicit formulas for arithmetic and geometric sequences to write both types of formulas for each of the sequences they studied in the previous lesson.

Have students work with a partner or in a group to complete this activity. Share responses with the class.

As students work, look for• Arithmetic sequences written two different ways when d is a

negative value. • Proper use of parentheses when r is a negative value.

Differentiation strategyTo extend the activity, show students how to use graphing calculators to identify a specified term. These steps show how to determine the 20th term in the sequence 3, 10, 17, 24, 31 . . . using a graphing calculator.

Step 1: Enter the first value of the sequence, 3. Then press ENTER to register the first term. The calculator can now recall that first term.

Step 2: From that term, add the common difference, 7. Press ENTER. The next term should be calculated. The calculator can now recall the formula as well.

Step 3: Press ENTER and the next term should be calculated.Step 4: Continue pressing ENTER until you determine the nth term

of the sequence you want to determine. Keep track of how many times you press ENTER so you know when you have the 20th term.

These steps show how to use a graphing calculator to generate two sequences at the same time to determine a certain term in a sequence.

Step 1: Within a set of brackets, enter the first term number followed by a comma and then the first term value of the sequence, {1,3}. Press ENTER.

Step 2: Provide direction to the calculator to increase the term number by 1 and the term value by the common difference. Type: {Ans(1)+1, Ans(2)+7}. Press ENTER.

Step 3: Continue pressing ENTER until you reach the nth term number and value you want to determine.



Questions to ask• Where did you get the information from the sequence to create the

recursive formula?• Did you go back to the original sequence or use the recursive

formula to write the explicit formula? Why?• What information is contained in the explicit formula that is not in

the recursive formula?• How are the recursive and explicit formulas related for an arithmetic

sequence? A geometric sequence?

Summary Recursive and explicit formulas can be used to generate arithmetic and geometric sequences.

Talk the Talk: Pros and ConsFacilitation NotesIn this activity, students write paragraphs describing the advantages and disadvantages of using recursive and explicit formulas to determine term values of arithmetic and geometric sequences.

Have students work with a partner or in a group to complete Questions 1 and 2. Share responses as a class.

Questions to ask• What information is needed to create an explicit formula for an

arithmetic sequence?• What information is needed to create a recursive formula for a

geometric sequence?• Which formula requires knowledge of the previous term and the

common difference?• Which formula requires knowledge of the term’s position in

the sequence?• Which formula is used to generate the next term and depends on

knowledge of the previous term?• Which formula is used to generate any term and depends on

knowledge of the term number?

SummaryThere are advantages and disadvantages to using either an explicit or recursive formula to represent an arithmetic or geometric sequence.

DEMONSTRATE




Learning Goals• Write recursive formulas for arithmetic and geometric

sequences from contexts. • Write explicit expressions for arithmetic and

geometric sequences from contexts. • Use formulas to determine unknown terms of

a sequence.

You have learned that arithmetic and geometric sequences always describe functions. How can you write equations to represent these functions?

Key Terms• recursive formula• explicit formula

Warm Up The local bank has agreed to donate $250 to the annual turkey fund to help feed families in need. In addition, for every bank customer that donates $50, the bank will donate $25.

1. A sequence describes the relationship between the number of $50 donations and the amount of the bank’s donation. Is the sequence arithmetic or geometric?

2. How can you calculate the 10th term based on the 9th term?

3. What is the 20th term?

3Did You Mean: Recursion?Determining Recursive and Explicit Expressions from Contexts


Warm Up Answers

1. The sequence is arithmetic because the common difference is 25.

2. Add 25 to the ninth term.

3. The 20th term is $725.

ELL TipAssess students’ prior knowledge of the word donate. Create a list of synonyms for the word and discuss the distinction between donating money and giving money to a friend, for example. Ask for volunteers to share examples of scenarios of money donations.




GETTING STARTED

Can I Get a Formula?

While a common ratio or a common difference can help you determine the next term in a sequence, how can they help you determine the thousandth term of a sequence? The ten-thousandth term of a sequence?

Consider the sequence represented in this situation.

Rico owns a sporting goods store. He has agreed to donate $125 to the Centipede Valley High School baseball team for their equipment fund. In addition, he will donate $18 for every home run the Centipedes hit during the season. The sequence shown represents the possible dollar amounts that Rico could donate for the season.

125, 143, 161, 179, . . .

Number of HomeRuns

Term Number

(n)

Donation Amount(dollars)

0 1

1

2

3

4

5

6

7

8

9

1. Identify the sequence type. Describe how you know.

2. Determine the common difference or common ratio for the sequence.

3. Complete the table.

4. Explain how you can calculate the tenth term based on the ninth term.

Thinkabout:

Notice that the 1st term in this sequence is the amount Rico donates if the team hits 0 home runs.


Answers

1. The sequence is arithmetic. It is arithmetic because a constant is added to each term to produce the next term.

2. The common difference is 18.

3.

Number of Home

Runs

Term Number

(n)

Donation Amount (dollars)

0 1 125

1 2 143

2 3 161

3 4 179

4 5 197

5 6 215

6 7 233

7 8 251

8 9 269

9 10 287

4. To calculate the tenth term, add 18 to the ninth term.

ELL TipReview the terms common difference and common ratio. Create an anchor chart with two columns using the terms as the headers for each column. Discuss the similarities and differences between the terms and fill in the anchor chart with key ideas about each term. Ask students to give examples of sequences that have a common difference, as well as sequences that have a common ratio. Ensure students’ understanding of which term applies to an arithmetic sequence and which term applies to a geometric sequence.




Writing Formulas for Arithmetic Sequences

AC TIVIT Y

3.1

A recursive formula expresses each new term of a sequence based on the preceding term in the sequence. The recursive formula to determine the nth term of an arithmetic sequence is:

Consider the sequence showing Rico’s contribution to the Centipedes baseball team in terms of the number of home runs hit.

1. Use a recursive formula to determine the 11th term in the sequence. Explain what this value means in terms of this problem situation.

2. Is there a way to calculate the 20th term without first calculating the 19th term? If so, describe the strategy.

You only need to know the previous term and the common difference to use the recursive formula.

previousterm

commondifferencenth term an 5 an21 1 d

Consider the sequence 22, 29, 216, 223, . . .You can use the recursive formula to determine the 5th term. an 5 an 2 1 1 d a5 5 a5 2 1 1 (27)

The expression a5 represents the 5th term. The previous term is 223, and the common difference is 27. a5 5 a4 1 (27) a5 5 223 1 (27) a5 5 230

The 5th term of the sequence is 230.

Worked Example


Answers

1. a11 5 a10 1 18; a11 5 287 1 18; a11 5 305; Rico will donate a total of $305 if 10 home runs are hit.

2. Answers will vary.

ELL TipAsk students to identify what the prefix pre- means in the word preceding. Follow up with additional examples of words with the prefix pre-, including pretest, preview, and precooked. Define these words and then ask students to explain why preceding means “the term before” in the context of “the preceding term in the sequence”. Create a list of words beginning with the prefix pre- and have students add to it as they encounter additional words with this prefix in the lesson.




You can determine the 93rd term of the sequence by calculating each term before it, and then adding 18 to the 92nd term, but this will probably take a while! A more efficient way to calculate any term of a sequence is to use an explicit formula.

An explicit formula of a sequence is a formula to calculate the nth term of a sequence using the term’s position in the sequence. The explicit formula for determining the nth term of an arithmetic sequence is:

nthterm

commondifference

1st termprevious term number

an 5 a1 1 d(n21)

You can use the explicit formula to determine the 93rd term in this problem situation. an 5 a1 1 d(n 2 1)

a93 5 125 1 18(93 2 1)

The expression a93 represents the 93rd term. The first term is 125, and the common difference is 18. a93 5 125 1 18(92)

a93 5 125 1 1656

a93 5 1781

The 93rd term of the sequence is 1781.

This means Rico will contribute a total of $1781 if the Centipedes hit 92 home runs.

The 1st term in this sequence is the amount Rico donates if the team hits 0 home runs. So, the 93rd term represents the amount Rico donates if the team hits 92 home runs.

Worked Example

Remember:





3. Use the explicit formula to determine the amount of money Rico will contribute for each number of home runs hit.

a. 35 home runs b. 48 home runs

c. 86 home runs d. 214 home runs

Rico decides to increase his initial contribution and amount donated per home run hit. He decides to contribute $500 and will donate $75 for every home run the Centipedes hit.

4. Write the first 5 terms of the sequence representing the new contribution Rico will donate to the Centipedes.

5. Determine Rico’s contribution for each number of home runs hit.

a. 39 home runs b. 50 home runs

NOTES


Answers

3a. $7553b. $9893c. $16733d. $39774. 500, 575, 650,

725, 8005a. $34255b. $4250




Notice that the 1st term in this sequence is the total number of cells after 0 divisions (that is, the mother cell).

Writing Formulasfor Geometric Sequences

AC TIVIT Y

3.2

When it comes to bugs, bats, spiders, and—ugh, any other creepy crawlers—finding one in your house is finding one too many! Then again, when it comes to cells, the more the better. Animals, plants, fungi, slime, molds, and other living creatures are composed of eukaryotic cells. During growth, generally there is a cell called a “mother cell” that divides itself into two “daughter cells.” Each of those daughter cells then divides into two more daughter cells, and so on.

1. The sequence shown represents the growth of eukaryotic cells.

1, 2, 4, 8, 16, . . .

a. Describe why this sequence is geometric and identify the common ratio.

b. Complete the table of values. Use the number of cell divisions to identify the term number and the total number of cells after each division.

c. Explain how you can calculate the tenth term based on the ninth term.

Number of Cell Divisions

Term Number(n)

Total Number of Cells

0 1

1

2

3

4

5

6

7

8

9


Answers

1a. This sequence is geometric because each term is multiplied by a constant to produce the next term. The common ratio is 2.

1b.

Number of Cell

Divisions

Term Number

(n)

Total Number of Cells

0 1 1

1 2 2

2 3 4

3 4 8

4 5 16

5 6 32

6 7 64

7 8 128

8 9 256

9 10 512

1c. Multiply the ninth term by 2.

ELL TipReview the scientific terms given in the example for the activity. Ask students to make a list of terms such as cells, mother cells, daughter cells, petri dish, and hypothesis. Discuss how the terms are used in the activity and ask students to create a sentence using each term to demonstrate their understanding. Also ask students to create a list of synonyms for hypothesis.




The recursive formula to determine the nth term of a geometric sequence is:

Consider the sequence of cell divisions and the total number of resulting cells.

2. Write a recursive formula for the sequence and use the formula to determine the 12th term in the sequence. Explain what your result means in terms of this problem situation.

nthterm

commonratio

previousterm

gn 5 gn21 ? r


4, 12, 36, 108, . . .

You can use the recursive formula to determine the 5th term.

gn 5 gn21 • r

g5 5 g521 • (3)

The expression g5 represents the 5th term. The previous term is 108, and the common ratio is 3.

g5 5 g4 • (3)

g5 5 108 • (3)

g5 5 324

The 5th term of the sequence is 324.

Worked Example


Answers

2. g12 5 1024 ? 2; g12 5 2048; There are a total of 2048 cells after 11 divisions.




The explicit formula to determine the nth term of a geometric sequence is:

nthterm previous

term number

1stterm

gn 5 g1 ? r n21

You can use the explicit formula to determine the 20th term in this problem situation.

gn 5 g1 • r n21

g20 5 1 • 22021

The expression g20 represents the 20th term. The first term is 1, and the common ratio is 2.

g20 5 1 • 219

g20 5 1 • 524,288

g20 5 524,288

The 20th term of the sequence is 524,288.

This means that after 19 cell divisions, there are a total of 524,288 cells.

commonratio

Worked Example

The 1st term in this sequence is the total number of cells after 0 divisions. So, the 20th term represents the total number of cells after 19 divisions.

Remember:





3. Use the explicit formula to determine the total number of cells for each number of divisions.

a. 11 divisions b. 14 divisions

c. 18 divisions d. 22 divisions

Suppose that a scientist has 5 eukaryotic cells in a petri dish. She wonders how the growth pattern would change if each mother cell divided into 3 daughter cells.

4. Write the first 5 terms of the sequence for the scientist’s hypothesis.

5. Determine the total number of cells in the petri dish for each number of divisions.

a. 13 divisions b. 16 divisions

In the previous lesson you identified sequences as either arithmetic or geometric and then matched a corresponding graph.

1. Go back to the graphic organizers from the previous lesson. Write the recursive and explicit formulas for each sequence.

Writing Recursive and Explicit Formulas

AC TIVIT Y

3.3


Answers

3a. 20483b. 16,3843c. 262,1443d. 4,194,3044. 5, 15, 45, 135, 4055a. 7,971,6155b. 215,233,605

1. Sequence A:gn 5 gn 2 1 ? 2gn 5 45 ? 2n 2 1

Sequence B:an 5 an 2 1 1 2an 5 24 1 2(n 2 1)

Sequence C:gn 5 gn 2 1 ? 3gn 5 22 ? 3n 2 1

Sequence E:an 5 an 2 1 2 9 __ 4 an 5 4 2 9 __ 4 (n 2 1)

Sequence F:gn 5 gn 2 1 ?

1 ___ 10

gn 5 1234 ? ( 1 ___ 10 ) n 2 1

Sequence H:an 5 an 2 1 1 4an 5 220 1 4(n 2 1)

Sequence J:gn 5 gn 2 1 ?

1 __ 2

gn 5 25 ? ( 1 __ 2 ) n 2 1

Sequence K:an 5 an 2 1 2 1.5an 5 6.5 2 1.5(n 2 1)

Sequence M:gn 5 gn 2 1 ? (2 1 __ 4 )

gn 5 216 ? (2 1 __ 4 ) n 2 1

Sequence N:an 5 an 2 1 2 20.5an 5 1473.2 2 20.5(n 2 1)Sequence P:gn 5 gn 2 1 ? (23)gn 5 24 ? (23)n 2 1



NOTES


TALK the TALK

Pros and Cons

1. Explain the advantages and disadvantages of using a recursive formula.

2. Explain the advantages and disadvantages of using an explicit formula.


Answers

1. Sample answer. Advantage: It enables

you to make sense of the growth pattern of the sequence.

Disadvantage: It is not an efficient method when determining the term value for a large term number.

2. Sample answer. Advantage: It is an

efficient method when determining the term value for a large term number.

Disadvantage: It takes a little more effort to determine an explicit formula than it does to determine a recursive formula.


LESSON 4: 3 Pegs, N Discs • 1

43 Pegs, N DiscsModeling a Situation Using Sequences

Lesson OverviewStudents are introduced to the process of mathematical modeling in this lesson, with each of the four activities representing a specific step in the process. Students are first presented with the Towers of Hanoi puzzle game and invited to play the game, observe patterns, and think about a mathematical question. Students then organize their information and pursue a given question by representing the patterns they notice using mathematical notation. The third step of the modeling process involves analyzing recursive and explicit formulas the students have generated and using these formulas to make predictions. Finally, students test their predictions and interpret their results. They then reflect on the modeling process and summarize what is involved in each phase.

Algebra 1 Exponential functions and equations (9) The student applies the mathematical process standards when using properties of exponential

functions and their related transformations to write, graph, and represent in multiple waysexponential equations and evaluate, with and without technology, the reasonableness of theirsolutions. The student formulates statistical relationships and evaluates their reasonablenessbased on real-world data. The student is expected to:

(D) graph exponential functions that model growth and decay and identify key features, includingy-intercept and asymptote, in mathematical and real-world problems;


analyze, and evaluate equations, relations, and functions. The student is expected to:

(9) write a formula for the n th term of arithmetic and geometric sequences, given the value ofseveral of their terms;


MATERIALSScissorsQuarters, nickels and dimes (optional)




Getting Started: Notice and WonderStudents are presented with the Towers of Hanoi game and the rules for playing. They play the game with a partner and record their observations, including any patterns they notice between the number of discs used and the minimum number of moves required to complete the game.

DevelopActivity 4.1: Organize and MathematizeStudents play the game again and record the minimum number of moves for 1, 2, and 3 discs in a table. They then record the numeric patterns they observe in the table and use mathematical notation to express this pattern as a sequence.

Day 2Activity 4.2: Predict and AnalyzeUsing results from the previous activity, students write a recursive and explicit formula to represent the pattern observed and then use these formulas to predict the minimum number of moves for 4 and 5 discs. Students also describe how to translate between the recursive and explicit formulas.Activity 4.3: Test and InterpretIn this final stage of the modeling process, students use their formulas to predict the minimum number of moves required for 4, 5, or n discs. They create and interpret graphs representing the recursive and explicit formulas for the sequence, and then they test their predictions on the game one last time.

DemonstrateTalk the Talk: A Modeling ProcessStudents reflect on the modeling process and summarize what is involved in each phase.

Essential Ideas • Mathematical modeling involves noticing patterns and formulating mathematical questions,

organizing information and representing this information using appropriate mathematical notation, analyzing mathematical representations and using them to make predictions, and then testing these predictions and interpreting the results.

• Both recursive and explicit formulas can be used for sequences that model situations.• Sequence formulas can be used to make predictions for real-world situations.



Getting Started: Notice and Wonder

Facilitation NotesIn this activity, students are presented with the game and the rules for playing. Students are then asked to play the game with a partner and record their observations, including any patterns they notice between the number of discs used and the minimum number of moves required to complete the game.

Have students work with a partner to complete this activity. Share responses as a class.

Differentiation strategiesTo assist all students,

• Allow students the freedom to any conjectures they want, but guide students to start thinking about the relationship between the number of discs and the minimum number of moves.

• Make the activity completely open-ended, then have students reflect on their actions and how they coincided with the modeling process.

• Allow students to use an online version of this game.Questions to ask

• Demonstrate how you solved the game.• Is there another way to solve the game?• Is there another way to solve the game with less moves?• What could be a mathematical question related to solving this game?• Could you predict the minimum number of moves if there were

more discs?

SummaryThe first step of the mathematical modeling process is to notice and wonder. Make observations, recognize patterns, and formulate mathematical questions.

ENGAGE



Activity 4.1Organize and Mathematize

Facilitation NotesIn this activity, students play the game again and record the minimum number of moves for 1, 2, and 3 discs in a table. Students then record the numeric patterns they observe in the table and use mathematical notation to express this pattern as a sequence.


As students work, look for• Clever ways students document moves so that they can be repeated.• Incorrect responses for the number of moves.• Different patterns, such as 12, 14, etc. or 3 2 1 1 to move from one

term value to the next.• Students who think they need more data to be confident in

their pattern.Questions to ask

• Can anyone demonstrate a solution with less moves?• How did you document your steps so that you could

remember them?• Why are you convinced that the game could not be solved with

less moves?• How does your pattern connect to solving the game?• Consider the sequence 1, 3, 7, … what operation(s) can be used on

the 1st term to generate the 2nd term? Can it also be used on the 2nd term to generate the 3rd term?

• Does anyone recognize another pattern in the data? If so, explain the pattern.

• How is this sequence different from others that you have written recursive and explicit formulas for?

• How can you modify what you know about writing formulas to express this sequence using mathematical notation?

SummaryThe second step of the modeling process is to organize your information and express the information and patterns in appropriate mathematical notation.

DEVELOP



Activity 4.2Predict and Analyze

Facilitation NotesIn this activity, using results from the previous activity, students write a recursive and explicit formula to represent the pattern observed and then use these formulas to predict the minimum number of moves for 4 and 5 discs. Students also describe how to translate between the recursive and explicit formulas.


MisconceptionThere is more than one pattern that fits this limited data. Allow students to continue with whatever pattern they recognize. Any errors will be addressed later in the modeling process.

Differentiation strategyFor students who struggle with how to represent their recursive formula as an explicit formula, discuss the fact that the common ratio of a geometric sequence is used as a multiplier in a recursive formula and as a base for an exponent in an explicit formula. Suggest they work with this basic premise and make modifications to the formula to generate the output values in the table.Questions to ask

• Do your predictions for 4 discs and 5 discs seem reasonable? Why or why not?

• How is this sequence different from others that you have written a recursive and explicit formulas for?

• What modifications did you make to your formula in order for it to generate the output values in the table?

• Explain how your recursive and explicit formulas are related to each other.

• How are your recursive and explicit formulas related to the solution of the game?

Summary The third step of the modeling process is to analyze your mathematical work and make predictions.



Activity 4.3Test and Interpret

Facilitation NotesIn this activity, the final stage of the modeling process, students use their formulas to predict the minimum number of moves required for 4, 5, or n discs. They create and interpret a graph representing the explicit formula for the sequence. They also use the explicit formula to compute the time it would take to complete the game using 25 discs.


As students work, look for• Incorrect game solutions for 4 and 5 discs.• Predictions that do not match the number of minimum number

of steps.• Patterns that students notice in solving the game that can be linked

to the formulas.Questions to ask

• Did the results of the this time game agree with your predictions? If not, how did you regroup to determine a new pattern and formula?

• How is multiplying by 2 connected to solving the game?• What are the independent and dependent variables?• Does the graph of the explicit formula appear to be linear? How do

you know?• Does the graph of the explicit formula appear to be exponential?

How do you know?• Can the graph be used to answer the question posed about the

length of time it would take to play the game using 25 discs? How so, or why not?

• How did you determine the time to complete the game with 25 discs?

• How did you determine the units used to report your solution?• 33,554,431 seconds is how many hours? How many days?

Summary The final step of the modeling process is to test your predictions and interpret your results. If your predictions aren't accurate, revisit your mathematical work and assumptions.



Talk the Talk: A Modeling ProcessFacilitation NotesIn this activity, students reflect on the modeling process; Notice and Wonder, Organize and Mathematize, Predict and Analyze, and Test and Interpret.

Have students work with a partner or in a group to complete this activity. Share responses as a class.

Questions to ask• Which part of the process involves gathering information?• Which part of the process involves recognizing patterns?• Which part of the process involves using mathematical notations?• Which part of the process involves performing operations?• Which part of the process involves working with results?• Which part of the process involves using the real world?• What is an example of something you might do in this part of

the process?• Should this modeling process be applied to solve all problems?• For what type of situations would this process be most useful?

SummaryThe mathematical modeling process includes the basic steps: (1) Notice and Wonder, (2) Organize and Mathematize, (3) Predict and Analyze, (4) Test and Interpret, and then report a solution.

DEMONSTRATE



Warm Up Answers

1. 9 2 1 (n 2 1)2. 20 1 20 (n 2 1)3. 1 1 3 __ 2 (n 2 1)

ELL TipReview the terms recursive formula and explicit formula. Ask students to create a chart displaying each formula along with a brief explanation of what the variables in each formula represent. Ensure students’ understanding of the distinction between the formulas as well as the application of each in various sequences.


Learning Goals• Model situations using recursive and

explicit formulas.• Translate between recursive and explicit expressions

of a mathematical model.• Explore the process of mathematical modeling.

You have written recursive and explicit formulas for arithmetic and geometric sequences. How can you model a real-world situation using both recursive and explicit formulas for sequences?

Key Term• mathematical modeling

Warm Up Write an explicit formula for each arithmetic sequence.

1. {9, 8, 7, 6, 5, . . .}

2. {20, 40, 60, 80 . . .}

3. {1, 5 __ 2 , 4, 5.5, 7, 17 ___ 2 . . .}

43 Pegs, N DiscsModeling Using Sequences




Answers

1. Answers will vary.


GETTING STARTED

Notice and Wonder

In this lesson, you will explore the process of mathematical modeling. The first step in modeling a situation mathematically is to gather information, notice patterns, and formulate mathematical questions about what you notice.

Let’s play a game.

The object of the game is to move an entire stack of discs or coins from the start circle to any of the other circles.

The rules of the game are simple:• You can only move one disc at a time.• You cannot put a larger disc on top of a smaller disc.

Let’s first play with 3 discs. To begin, place a quarter, nickel, and dime on top of each other in that order in a stack in the Start circle. Or, use the cutout discs at the end of the lesson, stacked from largest to smallest inside the Start circle.

1. Play this game several times with a partner. Record any patterns you notice.

Mathematical modeling is explaining patterns in the real world based on mathematical ideas.

Thinkabout:

Is there a relationship between the number of discs and the number of moves it takes to complete the game?

Start




Answers

1.

Number of Discs

Minimum Number of

Moves

1 1

2 3

3 7

2. Sample answer.

The minimum number of moves for three discs, D3, is 1 plus 2 times the minimum number of moves for D2, etc.

3. Sample answer.

The recursive pattern can be expressed as Dn 5 2Dn21 1 1 for n $ 1, assuming n is a positive integer.


Organize and MathematizeAC TIVIT Y

4.1

The second step in the modeling process is to organize your information and represent it using mathematical notation.

Consider the question from the previous activity. Is there a relationship between the number of discs and the minimum number of moves?

1. Play the game again and record your results in the table.

2. What pattern do you notice in your results?

3. Use mathematical notation to represent the pattern you have identified in your results. Explain your reasoning.

Number of Discs

Minimum Number of

Moves

1

2

3

Askyourself:

How do you know you did it in the least number of moves?




Answers

1.

Number of Discs

Minimum Number of

Moves

1 1

2 3

3 7

4 15

5 31

2. The recursive pattern can be expressed as Mn 5 2Mn21 1 1 for n $ 1, assuming n is a positive integer.

3. The pattern can be expressed using the explicit formula Mn 5 2n

2 1 for n $ 1, assuming n is a positive integer.


Step 3 of the modeling process is to extend the patterns you created, complete operations, make predictions, and analyze the mathematical results.

1. Use your results to extend the pattern in the table in the previous activity.

2. Write a recursive formula to represent the pattern shown in your table. What predictions does this formula make for the minimum number of moves required for 4 and 5 discs?

3. Write an explicit formula to represent the pattern shown in your table. What predictions does this formula make for the minimum number of moves required for 4 and 5 discs?

Predict and AnalyzeAC TIVIT Y

4.2




Answers

1. Answers will vary.2.

y

x0

5Number of Discs

Min

imum

Num

ber

of M

oves

25

50

10

Answers may vary.3. It would take

approximately 9320.6753 hours, or about 388 days, to complete.


The final step in the modeling process is to interpret your results and test your mathematical predictions in the real world. If your predictions are incorrect, you can revisit your mathematical work and make adjustments—or start all over!

1. Play the game again to demonstrate that your prediction for 4 discs and 5 discs is accurate. Record your observations.

2. Construct a graph to represent your explicit formula. Describe the characteristics of the graph in terms of

the situation.

Test and InterpretAC TIVIT Y

4.3

y

x

3. Suppose you could make 1 move every second. How long would it take to complete a game with 25 discs? Show your work.

Askyourself:

What is the level of accuracy appropriate for this situation?




AnswersSample answers.

Notice and WonderGather information, notice patterns, and formulate mathematical questions about what you notice.

Organize and MathematizeOrganize your information and represent it using mathematical notation.

Predict and AnalyzeExtend the patterns created, complete operations, make predictions, and analyze the mathematical results.

Test and InterpretInterpret your results and test your mathematical predictions in the real world. Make adjustments if necessary.


NOTES

© C

arne

gie

Lear

ning

, Inc

.

TALK the TALK

A Modeling Process

In this lesson, you used a modeling process to figure out whether the number of moves in the disc game is related to the number of discs. The basic steps of the mathematical modeling process are summarized in the diagram.

Summarize what is involved in each phase of this modeling process.

Notice and Wonder

Organize and Mathematize

Predict and Analyze

Test and Interpret





The Modeling Process

NOTICE | WONDER

TEST | INTERPRET

REPORT

ORGANIZE | MATHEMATIZE

PREDICT | ANALYZE





Discs

✂



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