Patterns, Patterns, Patterns… - Carnegie Learning · Step 5 and 6: She continues this pattern two more times, alternating between black and green sets of beads. rite the first six

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orkshop resourCes • Course 1 • Course 2 • Course 3PATTERNS, PATTERNS, PATTERNS …Student Text

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2.1 Developing Sequences of Numbers from Diagrams and Contexts • 43

Legend tells us that when the inventor of the game of chess showed his work

to the emperor, the emperor was so pleased that he allowed the inventor to

choose any prize he wished. So the very wise inventor asked for the following:

1 gold coin for the first square on the chess board, 2 gold coins for the second

square, 4 coins for the third, and so on up to the 64th square. The emperor, not as

wise as the inventor, quickly agreed to such a cheap prize. Unfortunately, the

emperor could not afford to pay even the amount for just the 32nd square:

4,294,967,295 gold coins!

How many gold coins would the emperor have to pay for just the 10th square?

20th square? What pattern did you use to calculate your answers?

Patterns, Patterns, Patterns…Developing Sequences of Numbers from Diagrams and Contexts

Key Terms sequence

term

ellipsis

Learning GoalsIn this lesson, you will:

Write sequences of numbers generated from the creation

of diagrams and written contexts.

State varying growth patterns of sequences.

MS_Participant_Handbook.indd 173 4/15/14 3:10 PM

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PATTERNS, PATTERNS, PATTERNS …Student TextCONT’D

44 • Chapter 2 Linear Functions

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Often, only the first few terms of a sequence are listed, followed by an ellipsis. An ellipsis

is three periods, which stand for “and so on.”

1. What is the next term in Sequence A?

2. What is the third term in Sequence B?

3. What is the twenty-fifth term in Sequence C?

4. What is the twelfth term in Sequence D?

Problem 1 Sequences

The inventor from the story used his knowledge of sequences to his advantage to

gain riches.

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.

Here are some examples of sequences.

Sequence A: 2, 4, 6, 8, 10, 12,…

Sequence B:

, , , ,

Sequence C: A, B, C, D, E, F, G,…

Sequence D: , , , , ,


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Problem 2 Designing a Bead Necklace

Emily is designing a necklace by alternating black and green beads. To create her

necklace, she performs the following steps.

Step 1: She starts with one black bead.

Step 2: Next, she places one green bead on each side of the black bead.

Step 3: Then, she places two black beads on each side of the green beads.

Step 4: Then, she places three green beads on each side of the black beads.

Step 5 and 6: She continues this pattern two more times, alternating between black and

green sets of beads.

1. Write the first six terms in the sequence that represents this situation. Make sure each

term indicates the total number of beads on the necklace after Emily completes that

step. Finally, explain how you determined the sequence.

If you need help, draw the

sequence on the necklace.


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Copyright © 2014 Carnegie Learning, Inc.46 • Chapter 2 Linear Functions

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Problem 3 Crafting Toothpick Houses

Ross is crafting toothpick houses for the background of a diorama. He creates one house

and then adds additional houses by adjoining them as shown.

A diorama is a three-dimensional

natural scene in which models of people, animals,

or plants are seen against a background.

1. Write the first eight terms in the sequence that represents this

situation. The first term should indicate the number of

toothpicks used for one house. The second term should

indicate the total number of toothpicks needed for two houses, and so on.

Explain your reasoning.

2. How is the number of toothpicks needed to build each house represented in

the sequence?



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Problem 4 Taking Apart a Card Trick

Matthew is performing a card trick. It is important that he collect the cards shown in a

particular order. Each turn, he collects all of the cards in the right-most column, and all the

cards in the bottom row.

1. Write a sequence to show the number of cards removed during each of the first

five turns.

2. Write a sequence to show the number of cards remaining after each of the first

five turns.

3. What pattern is shown in each sequence?


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Problem 5 Arranging Pennies

Lenny is making arrangements with pennies. He has made three penny arrangements

and now he wants to make five more arrangements. Each time he adds another

arrangement, he needs to add one more row to the base than the previous row in the

previous arrangement.

1. Write the first eight terms in the sequence that represents this situation. Each term

should indicate the total number of pennies in each arrangement.


2. Explain why the pattern does not increase by the same amount each time.



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Problem 6 Building Stairs

Dawson is stacking cubes in configurations that look like stairs. Each new configuration

has one additional step.

1. Write the first five terms in the sequence that represents this situation. Each term

should indicate the number of faces shown from the cubes shown. The bottom faces

are not shown. The first cube has 5 shown faces. Explain your reasoning.

2. Predict the number of shown faces in a stair configuration that is 7 cubes high.

Show your work.

A configuration is another way

of saying an arrangement

of things.


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Problem 7 Arranging Classroom Tables

Some schools purchase classroom tables that have trapezoid-shaped tops rather than

rectangular tops. The tables fit together nicely to arrange the classroom in a variety of

ways. The number of students that can fit around a table is shown in the first diagram.

The second diagram shows how the tables can be joined at the sides to make one

longer table.

2 5

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43

1. Write the first 5 terms in the sequence that represents this situation. Each term should

indicate the total number of students that can sit around one, two, three, four, and five

tables. Explain your reasoning.

2. The first trapezoid table seats five students. Explain why each additional table does

not have seats for five students.



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There are all kinds

of sequences!


Problem 8 Drawing Flower Petals

Draw a flower in a series of stages. The figure shows a pair of flower petals as the starting

point, Stage 0. In each stage, draw new petal pairs in the middle of every petal pair

already drawn.

● In Stage 1, you will draw petals.




indicate the number of new petals drawn in that stage. Explain your reasoning.


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Problem 9 Babysitting

Every Friday, Sarah earns $14 for babysitting. Every Saturday, Sarah spends $10 going

out with her friends.

1. Write a sequence to show the amounts of money Sarah has every Friday after

babysitting and every Saturday after going out with her friends for five consecutive

weeks. The sequence should have 10 terms. Explain your reasoning.

Problem 10 Recycling

The first week of school, Ms. Sinopoli asked her class to participate in collecting cans for

recycling. The students started bringing in cans the second week of school. They

collected 120 cans per week.

1. Write a sequence to show the running total of cans collected through the first nine

weeks of school. Explain your reasoning.



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Problem 11 Selling Tickets

Sam is working at the ticket booth during a basketball game. His cash box has

two $10 bills, five $5 bills, and twenty $1 bills. Tickets cost $3.

1. How much money does Sam have at the beginning of the basketball game?

2. Write a sequence to show the amount of cash Sam has available to start selling

tickets, and the amounts available after selling one ticket, two tickets, three tickets,

four tickets, and five tickets. Explain your reasoning.

Talk the Talk

There are many different patterns that can generate a sequence.

Some possible patterns are:

● adding or subtracting by the same number each time,

● multiplying or dividing by the same number each time,

● adding by a different number each time, with the numbers being part of a pattern,

● alternating between adding and subtracting.

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.


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Look back at Problems 2 through 11.

1. Describe the pattern of each sequence by completing the table shown.

Sequence NameIncreases or Decreases

Describe the Pattern

Designing a Bead Necklace

Crafting a Toothpick House

Taking Apart a Card Trick (1)

Arranging Pennies

Building Stairs

Arranging Classroom Tables

Drawing Flower Petals

Babysitting

Recycling

Selling Tickets

2. Which sequences are similar? Explain your reasoning.

Be prepared to share your solutions and methods.



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2.1 Developing Sequences of Numbers from Diagrams and Contexts • 43A

Essential Ideas

•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.

•A diagram can be used to show how each term changes as the sequence progresses.

• There are many different patterns that can generate a sequence of numbers. Some possible patterns are:

• adding or subtracting by the same number each time.

•multiplying or dividing by the same number each time.

• adding by a different number each time, with the numbers being part of a pattern.

• alternating between adding and subtracting.

Common Core State Standards for Mathematics8.FFunctions

Define,evaluate,andcomparefunctions.

1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Key Terms sequence

term

ellipsis


Write sequences of numbers generated

from the creation of diagrams and

written contexts.

State varying growth patterns of

sequences.



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PATTERNS, PATTERNS, PATTERNS …Teacher Implementation GuideCONT’D

43B • Chapter 2 Linear Functions

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OverviewSequences and terms in a sequence are introduced. Sequences that involve numbers, figures, and

letters are provided, and students determine the next term in each sequence. Different contexts and

diagrams are provided for students to develop an understanding of sequences. It is important that all

students discuss all problems, as each problem demonstrates a different type of pattern.

Next, students will write sequences from written contexts only. In the last problem, students

summarize the ten sequences generated in this lesson, first by documenting whether each sequence

was increasing or decreasing, and then by defining the growth pattern of each sequence.


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Warm Up

1. List six consecutive numbers.

Six consecutive numbers are 3, 4, 5, 6, 7 and 8.

2. List six consecutive even numbers.

Six consecutive even numbers are 2, 4, 6, 8, 10 and 12.

3. List six consecutive multiples of seven.

Six consecutive multiples of seven are 7, 14, 21, 28, 35 and 42.

4. List six consecutive multiples of five that are decreasing.

Six consecutive multiples of five that are decreasing are 35, 30, 25, 20, 15 and 10.

5. List six consecutive prime numbers.

Six consecutive prime numbers are 2, 3, 5, 7, 11 and 13.

2.1 Developing Sequences of Numbers from Diagrams and Contexts • 43C


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Legend tells us that when the inventor of the game of chess showed his work

to the emperor, the emperor was so pleased that he allowed the inventor to

choose any prize he wished. So the very wise inventor asked for the following:

1 gold coin for the first square on the chess board, 2 gold coins for the second

square, 4 coins for the third, and so on up to the 64th square. The emperor, not as

wise as the inventor, quickly agreed to such a cheap prize. Unfortunately, the

emperor could not afford to pay even the amount for just the 32nd square:

4,294,967,295 gold coins!

How many gold coins would the emperor have to pay for just the 10th square?

20th square? What pattern did you use to calculate your answers?


Key Terms sequence

term

ellipsis


Write sequences of numbers generated from the creation

of diagrams and written contexts.

State varying growth patterns of sequences.



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•How could you determine the next term in Sequence C?

•How could you determine the next term in Sequence D?

•How did you determine the third term in Sequence B?

•How did you determine the twenty-fifth term in Sequence C?

•How did you determine the twelfth term in Sequence D?

•Was it necessary to list all the terms leading up to the term you were trying to determine?

Problem 1Sequence, term in a sequence, and ellipsis are defined. Examples of sequences that involve numbers, figures, and letters are provided. Students determine the next term in each sequence.

Grouping

•Ask a student to read the introduction to Problem 1 aloud. Discuss the definitions and worked example as a class.

•Have students complete Questions 1 through 4 with a partner. Then share the responses as a class.

Discuss Phase, Problem 1

• Explain what a sequence is in your own words.

•Do all sequences have terms? How many terms are in a sequence?

• Is the term in a sequence a number, a figure, or a letter?

•How have you heard the word sequence or term used outside of the math class? How is the usage of these words outside of the math class related to their meaning in math class?

Share Phase, Questions 1 through 4

•How did you determine the next term in Sequence A?

•How could you determine the next term in Sequence B?


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Often, only the first few terms of a sequence are listed, followed by an ellipsis. An ellipsis

is three periods, which stand for “and so on.”

1. What is the next term in Sequence A?

The next term is 14.

2. What is the third term in Sequence B?

The third term is a pentagon.

3. What is the twenty-fifth term in Sequence C?

The twenty-fifth term is the letter Y.

4. What is the twelfth term in Sequence D?

The twelfth term is an arrow pointing upward.

Problem 1 Sequences

The inventor from the story used his knowledge of sequences to his advantage to

gain riches.

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.

Here are some examples of sequences.

Sequence A: 2, 4, 6, 8, 10, 12,…

Sequence B:

, , , ,

Sequence C: A, B, C, D, E, F, G,…

Sequence D: , , , , ,



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NoteIn mathematical usage, the terms sequence and series have different meanings. This chapter addresses sequences, not series; it would be inaccurate to use sequence and series interchangeably in the mathematical sense of the terms.

Problem 2Students write a sequence to represent designing a bead necklace. The terms of the sequence follow the pattern of even numbers. The context helps students make sense of this pattern.

GroupingHave students complete Problems 2 through 5 with a partner. Then share the responses as a class.

Share Phase, Problem 2

• Explain how many black beads and green beads are added to the necklace during each of the first 6 steps.

• Explain how you can determine how many beads would be added to the necklace in the next step without drawing the beads.

•Will there be more green or black beads on the necklace when you are done? Explain?

•How did you determine the values in your sequence?

•Did you draw all the beads on the necklace to complete the sequence? If so, did you count all the beads or just the newly-added beads to determine the next term in the sequence? If not, explain your thinking process.

•How much did the numbers in your sequence grow by each time? What do those numbers represent in the context of the problem? Why are all those numbers even?

•Why are all the numbers in your sequence odd?

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Problem 2 Designing a Bead Necklace

Emily is designing a necklace by alternating black and green beads. To create her

necklace, she performs the following steps.

Step 1: She starts with one black bead.

Step 2: Next, she places one green bead on each side of the black bead.

Step 3: Then, she places two black beads on each side of the green beads.

Step 4: Then, she places three green beads on each side of the black beads.

Step 5 and 6: She continues this pattern two more times, alternating between black and

green sets of beads.

1. Write the first six terms in the sequence that represents this situation. Make sure each

term indicates the total number of beads on the necklace after Emily completes that

step. Finally, explain how you determined the sequence.

The sequence is 1, 3, 7, 13, 21, 31. The number of beads increases by 2, then 4,

then 6, then 8, and then 10.

If you need help, draw the

sequence on the necklace.



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Problem 3Students write a sequence to represent crafting toothpick houses. The terms of the sequence increase by a constant value; however, the initial term is one more than the amount of increase between consecutive terms. The context helps students make sense of this discrepancy between the initial value and the amount of increase between consecutive terms.


•How many houses require the use of 6 new toothpicks?

•How many houses require the use of 5 new toothpicks?

•What operation was used to determine the terms in your sequence?

•What do you notice about every other term in your sequence? Is there a pattern?

•Why is every other term in your sequence even?

•Why is every other term in your sequence odd?

•Did you draw all the houses to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process.

• If a house is comprised of six toothpicks, then why doesn’t your sequence grow by six each time?


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Problem 3 Crafting Toothpick Houses

Ross is crafting toothpick houses for the background of a diorama. He creates one house

and then adds additional houses by adjoining them as shown.

A diorama is a three-dimensional

natural scene in which models of people, animals,

or plants are seen against a background.

1. Write the first eight terms in the sequence that represents this

situation. The first term should indicate the number of

toothpicks used for one house. The second term should

indicate the total number of toothpicks needed for two houses, and so on.


The sequence is 6, 11, 16, 21, 26, 31, 36, 41. The first house uses six toothpicks.

The remaining houses use only an additional five toothpicks each because they

each share a side with the previous house.

2. How is the number of toothpicks needed to build each house represented in

the sequence?

The sequence starts with a 6. That is the number of toothpicks needed to build the

first house. The sequence increases by 5 each time because that is the number of

toothpicks needed for each house built after the first house.



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Problem 4Students write a sequence to represent a card trick. This is the only problem in this lesson where the sequences are decreasing. The first sequence decreases by a constant value of two. The second sequence decreases by a pattern of odd numbers, with the odd numbers being the terms in the first sequence. The context helps students make sense of why the sequences are decreasing and their patterns of decrease.


•How did you determine the terms in the first sequence?

•How did you determine the terms in the second sequence?

•Did you use the diagram of cards to complete the sequence? If so, explain how you wrote the sequence from the diagram. If not, explain.

• If the original diagram has eight columns and six rows, why are only 13 cards removed instead of 14 cards to determine the first term in the first sequence?

•How is the decrease by 2 in the first sequence represented in the diagram?

•How is the decrease in the second sequence represented in the diagram?

•Add the first terms of the two sequences together, the second terms, etc. and use the sum in each instance to create a new sequence. What familiar pattern do you notice in the new sequence?

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Problem 4 Taking Apart a Card Trick

Matthew is performing a card trick. It is important that he collect the cards shown in a

particular order. Each turn, he collects all of the cards in the right-most column, and all the

cards in the bottom row.

1. Write a sequence to show the number of cards removed during each of the first

five turns.

The sequence is 13, 11, 9, 7, 5.

2. Write a sequence to show the number of cards remaining after each of the first

five turns.

The sequence is 35, 24, 15, 8, 3.

3. What pattern is shown in each sequence?

In the first sequence, the number of cards decreases by 2 each time. In the

second sequence, the number of cards decreases by the values shown in the

first sequence starting with 11.



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Problem 5Students write a sequence to represent triangular arrangements of pennies. The terms of the sequence increase by adding consecutive integers. The context helps students make sense of this pattern.


•How did you determine the first eight terms in your sequences?

•How many new rows of pennies are added in each term?

•Did you draw all of the pennies to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process.

•How is the increase by consecutive numbers represented in the diagram?


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Problem 5 Arranging Pennies

Lenny is making arrangements with pennies. He has made three penny arrangements

and now he wants to make five more arrangements. Each time he adds another

arrangement, he needs to add one more row to the base than the previous row in the

previous arrangement.

1. Write the first eight terms in the sequence that represents this situation. Each term

should indicate the total number of pennies in each arrangement.


The sequence is 1, 3, 6, 10, 15, 21, 28, 36.

The number of pennies increases by 2, 3, 4, 5, 6, 7, and 8. Every increase is by one

more penny than the number contained in the previous arrangement’s bottom row.

2. Explain why the pattern does not increase by the same amount each time.

Every time a new row is added on to the base, the triangle gets wider and takes

one more penny than the previous row.



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Problem 6Students write a sequence to represent building stairs. This is the only problem in this lesson which requires students to interpret three-dimensional drawings. The terms of the sequence increase by adding consecutive odd integers. The context helps students make sense of this pattern.

GroupingHave students complete Problems 6 through 11 with a partner. Then share the responses as a class.

MisconceptionStudents may have difficulty drawing the stacked cubes and counting the number of exposed faces. Some students may benefit from using actual cubes to build the models.


•How did you determine the first five terms in your sequence?

•How many additional exposed faces appear between the first and second diagram?

•How many additional exposed faces appear between the second and the third diagram?

•Did you draw or construct the stacked cubes to complete the sequence? If so, explain how you wrote the sequence from your diagram or model. If not, explain your thinking process.

•How were you able to organize your counting process when counting the number of exposed faces of the cubes? What patterns did you notice?

•How much does the increase between the terms grow each time?

• The increase between the terms grows by two every time. How is this two represented in the diagram?

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Problem 6 Building Stairs

Dawson is stacking cubes in configurations that look like stairs. Each new configuration

has one additional step.

1. Write the first five terms in the sequence that represents this situation. Each term

should indicate the number of faces shown from the cubes shown. The bottom faces

are not shown. The first cube has 5 shown faces. Explain your reasoning.

The sequence is 5, 12, 21, 32, 45.

The number of exposed faces increases by 7, then 9, then 11, and then 13.

Every increase is by two more than in the previous configuration.

2. Predict the number of shown faces in a stair configuration that is 7 cubes high.

Show your work.

The number of exposed faces would be 77. Continuing the pattern: 45 1 15 5 60,

60 1 17 5 77.

A configuration is another way

of saying an arrangement

of things.



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•How is this problem similar to Problem 3 when you crafted toothpick houses?

•How is this problem different from Problem 3 when you crafted toothpick houses?

Problem 7Students write a sequence to represent arranging classroom tables. The terms of the sequence increase by a constant value; however, the initial term is two more than the amount of increase between consecutive terms. The context helps students make sense of this discrepancy between the initial value and the amount of increase between consecutive terms. This problem is similar to Problem 3 because adjacent figures are sharing sides thus reducing the total count for the term number. This problem is different from Problem 3 because the sequence represents the perimeter rather than the total number of sides in the diagram.


•How did you determine first five terms in your sequence?

•Did you draw all the tables to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process.

• If a table seats five students, why doesn’t your sequence grow by five each time?

•How much does each term grow when a table is added?


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Problem 7 Arranging Classroom Tables

Some schools purchase classroom tables that have trapezoid-shaped tops rather than

rectangular tops. The tables fit together nicely to arrange the classroom in a variety of

ways. The number of students that can fit around a table is shown in the first diagram.

The second diagram shows how the tables can be joined at the sides to make one

longer table.

2 5

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43


indicate the total number of students that can sit around one, two, three, four, and five

tables. Explain your reasoning.

The sequence is 5, 8, 11, 14, 17.

As the number of tables increases by 1, the number of students increases by 3.

2. The first trapezoid table seats five students. Explain why each additional table does

not have seats for five students.

There are seats for only three more students each time a table is added. That is

because two seats are lost where the tables connect, one from the first table and

one from the second table.



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Problem 8Students write a sequence to represent the numbers of petals in various stages when drawing a flower. Students follow the directions to draw the flower petals and generate the sequence of numbers. This is the only problem in this lesson where the sequence is increasing by a factor of 2. The context helps students make sense of this pattern.

MisconceptionStudents may have difficulty keeping track of what petals were drawn in which stage. If this is the case, have students use a different color of pencil for the petals generated in each stage. Also, encourage students to write the value for each term of the sequence as they complete each stage of the flower drawing, rather than writing all the values after the entire flower has been drawn.


•Without drawing, how could you determine the next term in the sequence?

•What is the growth pattern of this sequence?

•What other mathematical operation could you use to represent the growth of this sequence?

• In what way is the growth of the terms in this sequence different from all other problems?

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There are all kinds

of sequences!


Problem 8 Drawing Flower Petals

Draw a flower in a series of stages. The figure shows a pair of flower petals as the starting

point, Stage 0. In each stage, draw new petal pairs in the middle of every petal pair

already drawn.

● In Stage 1, you will draw 2 petals.



32

3

1

3

2

3

3 2

3

1

3

2

3

0

0


indicate the number of new petals drawn in that stage. Explain your reasoning.

The sequence is 2, 4, 8, 16, 32. For each new stage, the new number of petals

increases by multiplying the previous number of new petals by 2.



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•Problems 3 and 7 also increase by a constant value. Why is it that the sequence in this problem is a list of multiples, but that is not the case with the sequences in Problems 3 and 7?

Problem 9Students write a sequence to represent babysitting earnings. This is the only problem in this lesson where the sequence alternates between decreasing by one values and increasing by a different value. The context helps students make sense of this pattern.



•Why does this growth pattern make sense in context of the problem?

Problem 10Students write a sequence to represent recycling. This is the only problem in this lesson where the sequence begins with zero. Because the sequence begins with zero and the terms of the sequence increase by a constant value, the sequence represents a list of multiples.


•What is the first term of this sequence?

•Why is the first term of the sequence zero?

• Each term is a multiple of what number?


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Problem 9 Babysitting

Every Friday, Sarah earns $14 for babysitting. Every Saturday, Sarah spends $10 going

out with her friends.

1. Write a sequence to show the amounts of money Sarah has every Friday after

babysitting and every Saturday after going out with her friends for five consecutive

weeks. The sequence should have 10 terms. Explain your reasoning.

The sequence is 14, 4, 18, 8, 22, 12, 26, 16, 30, 20. This pattern is generated by

beginning with 14, subtracting 10, adding 14, subtracting 10, and continuing.

Problem 10 Recycling

The first week of school, Ms. Sinopoli asked her class to participate in collecting cans for

recycling. The students started bringing in cans the second week of school. They

collected 120 cans per week.

1. Write a sequence to show the running total of cans collected through the first nine

weeks of school. Explain your reasoning.

The sequence is 0, 120, 240, 360, 480, 600, 720, 840, 960. Every term is calculated

by adding 120 to the previous term. Since this sequence begins with 0, the terms

are multiples of 120.



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Problem 11Students write a sequence to represent selling tickets. In this problem, students must do some arithmetic to determine the initial value of the sequence. The sequence increases by a constant value.


•How did you determine the starting value of the sequence?


• Explain how the growth pattern of this sequence connects to the context of the problem.

Talk the TalkStudents summarize the patterns of each sequence in Problems 2 through 11, and describe the similarities.

GroupingHave students complete the Questions 1 and 2 with a partner. Then share the responses as a class.

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Problem 11 Selling Tickets

Sam is working at the ticket booth during a basketball game. His cash box has

two $10 bills, five $5 bills, and twenty $1 bills. Tickets cost $3.

1. How much money does Sam have at the beginning of the basketball game?

Sam has $65 at the start of the game.

2. Write a sequence to show the amount of cash Sam has available to start selling

tickets, and the amounts available after selling one ticket, two tickets, three tickets,

four tickets, and five tickets. Explain your reasoning.

The sequence is 65, 68, 71, 74, 77, 80. Every term is calculated by adding 3 to the

previous term.

Talk the Talk

There are many different patterns that can generate a sequence.

Some possible patterns are:

● adding or subtracting by the same number each time,

● multiplying or dividing by the same number each time,

● adding by a different number each time, with the numbers being part of a pattern,

● alternating between adding and subtracting.

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.



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Share Phase, Questions 1 and 2

•Which sequences increase?

•Which sequences decrease?

•Which sequences have growth patterns that generate new terms by adding or subtracting by the same number?

•Which sequences have growth patterns that generate new terms by multiplying or dividing by the same number?

•Which sequences have growth patterns that generate new terms by adding a different number each time?

•Which sequences have growth patterns that generate new terms by alternating between adding and subtracting?

•How are the sequences from Problems 3, 7, 10 and 11 alike?

•How are the sequences from Problems 2, 5 and 6 alike?


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Look back at Problems 2 through 11.

1. Describe the pattern of each sequence by completing the table shown.

Sequence NameIncreases or Decreases

Describe the Pattern

Designing a Bead Necklace Increases 1 2 1 4 1 6 . . .

Crafting a Toothpick House Increases Starts at 6, adding by 5

Taking Apart a Card Trick (1) Decreases Subtracting 2

Arranging Pennies Increases 1 1 1 2 1 3 . . .

Building Stairs Increases 1 7 1 9 1 11 . . .

Arranging Classroom Tables Increases Starts at 5, adding 3

Drawing Flower Petals Increases Multiplying by 2

BabysittingIncreases and

Decreases 1 14 2 10 1 14 2 10 . . .

Recycling Increases Adding 120, starts at 0

Selling Tickets Increases Adding 3, starts at 65

2. Which sequences are similar? Explain your reasoning.

Answers will vary.

Arranging classroom tables, recycling, and selling tickets are similar because each

sequence is generated by adding the same number

each time.

Be prepared to share your solutions and methods.



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2.1 Developing Sequences of Numbers from Diagrams and Contexts • 54A

Follow Up

AssignmentUse the Assignment for Lesson 2.1 in the Student Assignments book. See the Teacher’s Resources

and Assessments book for answers.

Skills PracticeRefer to the Skills Practice worksheet for Lesson 2.1 in the Student Assignments book for additional

resources. See the Teacher’s Resources and Assessments book for answers.

AssessmentSee the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.

Check for Students’ UnderstandingDetermine the next three terms in each sequence of numbers. Show the pattern used to determine the next terms.

1. 17, 38, 33, 54, 49, _____, _____, _____,...

The next terms are 70, 65 and 86.

The pattern was 121, 25, 1 21, 25, ...

2. 109, 103, 97, 91, _____, _____, _____,...


The pattern was decreasing by a constant value of 6.

3. 34, 37, 42, 49, _____, _____, _____,...


The pattern was 13, 15, 17, … or increasing by consecutive odd numbers.

4. 1, 3, 9, 27, _____, _____, _____, …


The pattern was increasing by multiplying by 3 each time.

5. 68, 60, 51, 41, _____, _____, _____,…


The pattern was 28, 29, 210,… or decreasing by consecutive decreasing numbers.


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Patterns, Patterns, Patterns… - Carnegie Learning · Step 5 and 6: She continues this pattern two more times, alternating between black and green sets of beads. rite the first six