Lesson 30 CCLS Analyze Patterns and...

©Curriculum Associates, LLC Copying is not permitted.L30: Analyze Patterns and Relationships310

Analyze Patterns and RelationshipsLesson 30 Part 1: Introduction

In previous lessons, you learned to identify and extend numerical patterns. Now you will describe the relationship between two patterns. Take a look at this problem.

Maria is working at the snack stand at a basketball game. Each frozen yogurt

costs $3, and each sandwich costs $6. Create a table to show the costs for buying

0, 1, 2, 3, 4, 5, or 6 frozen yogurts. Create another table to show the costs for the

same number of sandwiches. How do the costs of frozen yogurts compare to the

costs of an equal number of sandwiches?

Explore It

Use the math you already know to solve the problem.

What is the cost for buying 1 frozen yogurt?

What number do you add to the cost of 1 frozen yogurt to get the cost of 2 frozen yogurts?

Complete the table to show the cost for each number of frozen yogurts.

Number of Yogurts 0 1 2 3 4 5 6Cost ($)

What number do you add to the cost of 1 sandwich to get the cost of 2 sandwiches?

Complete the table to show the cost for each number of sandwiches.

Number of Sandwiches 0 1 2 3 4 5 6Cost ($)

How does the cost of a frozen yogurt compare to the cost of the same number of sandwiches?

CCLS5.OA.3

©Curriculum Associates, LLC Copying is not permitted.311L30: Analyze Patterns and Relationships

Lesson 30Part 1: Introduction

Find Out More

The costs of frozen yogurts and the costs of sandwiches form numerical patterns.

Cost of Frozen Yogurts

13 13 13 13 13 13

0 3 6 9 12 15 18

Cost of Sandwiches

16 16 16 16 16 16

0 6 12 18 24 30 36

You can make a table to help write ordered pairs to see relationships between corresponding terms in the two patterns. The first number of each pair is the cost of frozen yogurts, and the second number of the pair is the cost of the same number of sandwiches.

Cost of Yogurts (Add 3)

Cost of Sandwiches

(Add 6)Ordered

Pairs

0 0 (0, 0)

3 6 (3, 6)6 12 (6, 12)9 18 (9, 18)

12 24 (12, 24)15 30 (15, 30)18 36 (18, 36)

There is a relationship between the two numbers of the ordered pairs. The second number of each ordered pair is always twice the first number.

Reflect

1 Find the sum of the numbers of each ordered pair. What pattern do you notice?

Lesson 30

©Curriculum Associates, LLC Copying is not permitted.

L30: Analyze Patterns and Relationships312

Part 2: Modeled Instruction

Read the problem below. Then explore how to identify relationships between two numerical patterns.

In Level 1 of a game, you get 2 points for each correct answer. In Level 2, you get

6 points for each correct answer. Compare the number of points in Level 2 to the

number of points in Level 1 if you correctly answer 0, 1, 2, 3, 4, 5, or 6 questions.

Picture It

You can use a picture to help find each pattern.

Use the number of points for correct answers at each level to find the patterns.

Level 1 Level 2 Each correct answer earns 2 points. Each correct answer earns 6 points.

12 12 12 12 12 12 16 16 16 16 16 16

0, 2, 4, 6, 8, 10, 12 0, 6, 12, 18, 24, 30, 36

Model It

Use a table to show the number of points you get for correct answers in each level of the game.

Then write ordered pairs where the first number is a term from the first pattern and the second number is the corresponding term from the second pattern.

Points in Level 1

Points in Level 2

Ordered Pairs

0 0 (0, 0)2 6 (2, 6)4 126 188 24

10 3012 36

The total number of points in Level 1 goes up by 2 for each correct answer. The total number of points in Level 2 goes up by 6 for each correct answer.

Lesson 30


Part 2: Guided Instruction

Connect It

Now you will solve the problem from the previous page by looking at the ordered pairs.

2 Look at Picture It on the previous page. Describe the “rules” for finding the number of points for a correct answer in Level 1 and Level 2.

Level 1 rule:

Level 2 rule:

3 Complete the table on the previous page.

4 For each ordered pair, how does the first number compare to the second number?

5 Suppose the game has a third level. You get 9 points for each correct answer in that level. Explain how you could figure out how the terms in this pattern compare to the corresponding terms in the pattern for points in Level 2.

Try It

Use what you just learned about comparing two number patterns to solve this problem. Show your work on a separate sheet of paper.

6 School magnets cost $4, and shirts cost $24. Write a pattern for the costs of 0, 1, 2, 3, 4, and 5 magnets and a second pattern for the costs of 0, 1, 2, 3, 4, and 5 shirts. How do the corresponding terms of the two patterns compare?

Lesson 30



Read the problem below. Then explore what the graph looks like when you plot corresponding terms of number patterns on a graph.

The scouts have a choice of making a model plane or a model boat. The materials

for the plane cost $2, and the materials for the boat cost $4. Write and graph

ordered pairs to compare the cost of making one or more planes to the cost of

making the same number of boats.

Picture It

You can use a picture to find the costs of making various numbers of each model.

Each pattern of numbers below shows the cost of making 0, 1, 2, 3, and 4 models.

Planes Boats

12 12 12 12

14 14 14 14

0, 2, 4, 6, 8 0, 4, 8, 12, 16

Model It

You can use a table to help understand the problem.

List the cost for the planes and boats in a table. Then write the corresponding costs as ordered pairs.

Planes (x) Boats (y) Ordered Pairs (x, y)0 0 (0, 0)2 4 (2, 4)4 8 (4, 8)6 12 (6, 12)8 16 (8, 16)

Part 3: Modeled Instruction

Lesson 30


Connect It

Now you will solve the problem from the previous page by graphing ordered pairs.

7 Explain how to write the ordered pairs from the patterns.

8 Plot the ordered pairs on the graph to the right. The first number shows the location on the x-axis. The second number shows the location on the y-axis.

The point (2, 4) has been plotted for you.

9 How do the coordinates of corresponding terms of the patterns compare?

10 Suppose you connect the points. What would the graph look line?

11 What directions would you give to someone to get from one point to the next on the graph? How do your directions relate to the rules for the patterns?

Try It

Use what you just learned about comparing two number patterns on a graph to solve this problem. Show your work on a separate sheet of paper.

12 Use the rules “add 1” and “add 3” to write and graph ordered pairs made up of corresponding terms from the two patterns. Start each pattern with 0. Describe the graph.

Part 3: Guided Instruction

y

x

18161412108642

2 4 6 8 10 120

Student Model

Part 5: Guided Practice Lesson 30



Does it matter how many terms you write for each sequence?

Pair/Share

How are these patterns different from other patterns in this lesson?

Pair/Share

Study the model below. Then solve problems 13–15.

Look at the following two number patterns.

Pattern 1: 6, 5, 4, 3, 2, 1, 0

Pattern 2: 24, 20, 16, 12, 8, 4, 0

What is the relationship between corresponding terms in the two

patterns?

Look at how you could show your work using ordered pairs.

The first coordinate is a term from Pattern 1, the second coordinate is a corresponding term from Pattern 2. Ordered pairs: (6, 24), (5, 20), (4, 16), (3, 12), (2, 8), (1, 4), (0, 0)

Solution:

13 One sequence starts at 0 and has the rule “add 8.” Another sequence starts at 0 and has the rule “add 4.” Write each sequence of numbers. How do the corresponding terms in the sequences compare?

Show your work.

Solution:

Each term in Pattern 2 is four times the

corresponding term in Pattern 1.

The student wrote ordered pairs to identify a relationship between corresponding terms.

How do I generate the sequences?

Part 5: Guided Practice Lesson 30


Find the difference between the coordinates in each ordered pair. Do you see another pattern?

Pair/Share

Does Mike’s answer make sense?

Pair/Share

14 Identify the pattern in each column. Then complete the table below. Form ordered pairs for the corresponding terms. Describe the relationship between corresponding terms.

Show your work.

x y Ordered Pairs (x, y)4 18 2

12 3

Solution:

15 The ordered pairs (2, 12), (3, 18), and (4, 24) were formed by corresponding terms in two sequences. How do the values of the y-coordinates compare to the values of the x-coordinates? Circle the letter of the correct answer.

A 10 more

B 2 times as much

C 1 ·· 6 times as much

D 6 times as much

Mike chose C as the correct answer. How did he get that answer?

What are the rules for each pattern?

What rule works for all the points?

Part 6: Common Core Practice Lesson 30



Solve the problems. Mark your answers to problems 1–3 on the Answer Form to the right. Be sure to show your work.

1 What rule could be used to create the pattern 9, 18, 27, 36, 45, 54, . . . ?

A multiply each term by 2 to get the next term

B multiply each term by 9 to get the next term

C add 9 to get the next term

D add 3 to get the next term

2 Describe the relationship between corresponding terms in these two patterns.

Pattern 1: 3, 6, 9, 12, 15, 18, . . .

Pattern 2: 18, 36, 54, 72, 90, 108, . . .

A Each term in Pattern 1 is 6 times the corresponding term in Pattern 2.

B Each term in Pattern 2 is 6 times the corresponding term in Pattern 1.

C Each term in Pattern 1 is 15 times the corresponding term in Pattern 2.

D Each term in Pattern 2 is 15 times the corresponding term in Pattern 1.

3 Tickets at an outdoor play cost $2 for students and $8 for adults. Jason creates two patterns to compare the costs. He writes ordered pairs in the form (student cost, adult cost) for the corresponding number of tickets. Which ordered pair could be on Jason’s list of ordered pairs?

A (8, 2)

B (4, 10)

C (10, 16)

D (6, 24)

1 B C D

2 B C D

3 B C D

Answer Form

Number Correct 3

Part 6: Common Core Practice Lesson 30


4 Create two patterns starting with 0 using the rules below. Then describe the relationship between corresponding terms of the two patterns.

Pattern 1: add 3 Pattern 2: add 12

Show your work.

5 Begin at 0 and use the rule “add 2” and “add 5” to complete the table in Part A.

Part A

Add 2 Add 5 Ordered Pairs

Part B

Graph the ordered pairs. Describe your graph.

Part C

Describe the relationship between the corresponding terms of the two patterns.

y

x20

2

4

6

8

10

12

14

16

18

20

4 6 8 10 12 14 16 18 20

Analyze Patterns and RelationshipsLesson 30

L30: Analyze Patterns and Relationships 315©Curriculum Associates, LLC Copying is not permitted.

(Student Book pages 310–319)

Lesson objecTives

• Generate a numeric sequence given a rule.

• Identify apparent relationships between corresponding terms of two sequences.

• Graph ordered pairs on a coordinate plane.

PReRequisiTe skiLLs

• Use addition, subtraction, multiplication, and division.

• Recognize and extend patterns.

• Be familiar with the coordinate plane.

• Plot ordered pairs.

vocAbuLARy

corresponding terms: the numbers that are in the same place in two or more related patterns (e.g., the first term in one pattern and the first term in a related pattern are corresponding)

ordered pair: a pair of numbers that locate a point on a coordinate plane

Review the following terms.

coordinate plane: a two-dimensional space formed by two perpendicular number lines called axes

The LeARning PRogRession

In Grade 5, students perform several related tasks to demonstrate the relationship between two numerical patterns, including graphing ordered pairs made of the corresponding terms of the two patters.

Students build on this in Grade 6 as they continue on a path that leads to the study of the concept of functions.

ccLs Focus

5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

sTAnDARDs FoR MATheMATicAL PRAcTice: SMP 1, 2, 7, 8 (see page A9 for full text)

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Prerequisite Skills 5.OA.3

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Tools for Instruction

Interactive Tutorials ✓ ✓

L30: Analyze Patterns and Relationships316©Curriculum Associates, LLC Copying is not permitted.

Part 1: introduction Lesson 30

AT A gLAnce

Students read a word problem and answer a series of questions that lead them to create two numerical sequences, which they record in tables. Students use the tables to compare the cost of two items.

sTeP by sTeP

• Tell students that this page models using a table to generate a pattern for a given rule and comparing the costs of two items.

• Have students read the problem at the top of the page.

• Work through Explore It as a class.

• Ask students how they figured out how much to add to the cost of 1 sandwich to find the cost of 2 sandwiches. [Add the cost of one more sandwich, $6.]

• Guide students to see that the cost of 1 sandwich, $6, is two times the cost of 1 yogurt, $3.

• Ask student pairs or groups to explain their answers for the remaining questions.

sMP Tip: Students demonstrate their understanding of a problem (SMP 1) by organizing the given information in tables. Encourage students to continue to use tables to organize and make sense of information described in problems.

• What pattern do you see you in the table for the cost of yogurts?

Student responses should indicate an understanding that as the number of yogurts purchased increases by 1, the cost increases by $3.

• How would you describe how to create a table that shows the costs if someone buys 0, 1, 2, 3, or 4 items that cost $10 each?

Student responses should indicate an understanding that as the number of items someone buys increases by 1, the cost increases by $10.

Mathematical Discourse

©Curriculum Associates, LLC Copying is not permitted.L30: Analyze Patterns and Relationships310

Analyze Patterns and RelationshipsLesson 30 Part 1: introduction

in previous lessons, you learned to identify and extend numerical patterns. now you will describe the relationship between two patterns. Take a look at this problem.

Maria is working at the snack stand at a basketball game. Each frozen yogurt

costs $3, and each sandwich costs $6. Create a table to show the costs for buying

0, 1, 2, 3, 4, 5, or 6 frozen yogurts. Create another table to show the costs for the

same number of sandwiches. How do the costs of frozen yogurts compare to the

costs of an equal number of sandwiches?

explore it

use the math you already know to solve the problem.

What is the cost for buying 1 frozen yogurt?

What number do you add to the cost of 1 frozen yogurt to get the cost of 2 frozen yogurts?

Complete the table to show the cost for each number of frozen yogurts.

number of yogurts 0 1 2 3 4 5 6

cost ($)

What number do you add to the cost of 1 sandwich to get the cost of 2 sandwiches?

Complete the table to show the cost for each number of sandwiches.

number of sandwiches 0 1 2 3 4 5 6

cost ($)

How does the cost of a frozen yogurt compare to the cost of the same number of sandwiches?

ccLs5.oA.3

$3

3

6

Possible answer: The cost of a sandwich is always two times the cost of the

same number of yogurts.

0 3 6 9 12 15 18

0 6 12 18 24 30 36


Part 1: introduction Lesson 30

AT A gLAnce

Students write ordered pairs consisting of corresponding terms from two numerical patterns. Students use the ordered pairs to identify the relationship between the two patterns.

sTeP by sTeP

• Read Find Out More as a class.

• Point out that it is important to list all the numbers from one pattern in one column of the table and all the numbers from the other pattern in the other column of the table.

• Remind students to write the x-value of the ordered pair first and the y-value second.

sMP Tip: Comparing two numerical patterns builds on analyzing patterns (SMP 8). Occasionally, remind students that a pattern can be identified by determining what number is repeatedly added or multiplied to the previous number in the pattern.

Point out that the phrase ordered pair contains the word order. Explain that the order of the coordinates in an ordered pair is important because all the x-coordinates represent the terms of one pattern and all the y-coordinates represent the terms of the other pattern.

eLL supportEncourage students to think about everyday places or situations where numerical patterns may occur and how comparing them is useful. Have volunteers share their ideas.

Examples: cost of a number of sunglasses compared to the cost of a number of watches, weight of a number of apples compared to the weight of a number of watermelons, cost of a number of boxes of pencils compared to the cost of a number of boxes of markers

Real-World connection


Lesson 30Part 1: introduction

Find out More

The costs of frozen yogurts and the costs of sandwiches form numerical patterns.

cost of Frozen yogurts

13 13 13 13 13 13

0 3 6 9 12 15 18

cost of sandwiches

16 16 16 16 16 16

0 6 12 18 24 30 36

You can make a table to help write ordered pairs to see relationships between corresponding terms in the two patterns. The first number of each pair is the cost of frozen yogurts, and the second number of the pair is the cost of the same number of sandwiches.

cost of yogurts(Add 3)

cost of sandwiches

(Add 6)ordered

Pairs

0 0 (0, 0)

3 6 (3, 6)

6 12 (6, 12)

9 18 (9, 18)

12 24 (12, 24)

15 30 (15, 30)

18 36 (18, 36)

There is a relationship between the two numbers of the ordered pairs. The second number of each ordered pair is always twice the first number.

Reflect

1 Find the sum of the numbers of each ordered pair. What pattern do you notice?

each sum is 9 more than the previous one.

318 L30: Analyze Patterns and Relationships


Lesson 30Part 2: Modeled instruction

AT A gLAnce

Students use a picture and a table to identity the relationship between two numerical patterns.

sTeP by sTeP

• Read the problem at the top of the page as a class.

• Read Picture It. Have students use their finger to trace the arrows above the numbers, saying the pattern aloud as they do so. [0 plus 2 is 2, 2 plus 2 is 4, 4 plus 2 is 6, etc.]

• Read Model It. Point out that the numbers in the columns for the points in Levels 1 and 2 are in the same order as the picture above. Emphasize that writing the numbers in the same order is important.

help students relate pictures of numerical sequences to tables.

• Draw the picture for the number of points scored in Level 1 on a strip of paper. Be sure to include the arrows and labels above them.

• Rotate the strip of paper so the numbers are listed vertically and place it next to the Points in Level 1 column of the table.

• Point out that the numbers align and the arrows indicating how much each number is greater than the one before it are still accurate.

• Ask students how they can use this to identify a pattern from a table. [Determine how each number in a table is related to the one before it. You could draw arrows from one number to the next as you compare them.]

• Have students repeat this process for the number of points scored in Level 2.

visual Model• What is another way you could model each sequence

of numbers of points?

Responses may indicate using counters. You could display counters for each number in the sequence.

• How could you find the number of points scored for 10 correct answers in Level 1?

Responses may include continuing the pattern or multiplying 10 by 2.


Lesson 30



Part 2: Modeled instruction

Read the problem below. Then explore how to identify relationships between two numerical patterns.

In Level 1 of a game, you get 2 points for each correct answer. In Level 2, you get

6 points for each correct answer. Compare the number of points in Level 2 to the

number of points in Level 1 if you correctly answer 0, 1, 2, 3, 4, 5, or 6 questions.

Picture it

you can use a picture to help find each pattern.

Use the number of points for correct answers at each level to find the patterns.

Level 1 Level 2

Each correct answer earns 2 points. Each correct answer earns 6 points.

12 12 12 12 12 12

16 16 16 16 16 16

0, 2, 4, 6, 8, 10, 12 0, 6, 12, 18, 24, 30, 36

Model it

use a table to show the number of points you get for correct answers in each level of the game.

Then write ordered pairs where the first number is a term from the first pattern and the second number is the corresponding term from the second pattern.

Points in Level 1

Points in Level 2

orderedPairs

0 0 (0, 0)

2 6 (2, 6)

4 12

6 18

8 24

10 30

12 36

The total number of points in Level 1 goes up by 2 for each correct answer. The total number of points in Level 2 goes up by 6 for each correct answer.

(4, 12)(6, 18)(8, 24)

(10, 30)(12, 36)


Lesson 30Part 2: guided instruction

AT A gLAnce

Students revisit the problem on page 312 to use ordered pairs to compare the two number patterns. Then, students use ordered pairs to answer a word problem.

sTeP by sTeP

• Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 312.

• Ask, If you score 3 points for each correct answer in Level 1, what would be the “rule” for finding the number of points for a correct answer in Level 1? [add 3]

• Review how to write an ordered pair from a table. [write the x-coordinate first and the y-coordinate second]

• Emphasize that the number of points in Level 2 is always 3 times the number of points in Level 1 no matter how many questions are answered correctly.

• Be sure students understand that they can compare two sequences of numbers by plotting the corresponding ordered pairs on a coordinate plane and comparing how the points increase.

TRy iT soLuTions

6 Solution: $0, $4, $8, $12, $16, $20 and $0, $24, $48, $72, $96, $120. The cost of a shirt is 6 times the cost of a magnet; Students may create a table or write ordered pairs where the x-coordinates are the cost of magnets and the y-coordinates are the cost of shirts.

sMP Tip: Students identify how patterns are constructed (SMP 7) by comparing each term to the previous term. They also identify the relationship between two sequences by comparing corresponding terms of the sequences.

solve the problem using an alternate method.

Materials: counters

• Organize student pairs.

• Tell students that 1 counter represents 1 point in the game described on page 312.

• Have one student use the counters to model the number of points earned in Level 1.

• Have the other student use counters to model the number of points earned in Level 2.

• Ask students to explain the pattern in each model to their partner.

• Ask students to compare the two patterns.

hands-on Activity

ERROR ALERT: Students who wrote 20 subtracted the cost of one magnet from the cost of one shirt.

Lesson 30


Part 2: guided instruction

connect it

now you will solve the problem from the previous page by looking at the ordered pairs.

2 Look at Picture It on the previous page. Describe the “rules” for fi nding the number of points for a correct answer in Level 1 and Level 2.

Level 1 rule:

Level 2 rule:

3 Complete the table on the previous page.

4 For each ordered pair, how does the fi rst number compare to the second number?

5 Suppose the game has a third level. You get 9 points for each correct answer in that level. Explain how you could fi gure out how the terms in this pattern compare to the corresponding terms in the pattern for points in Level 2.

Try it

use what you just learned about comparing two number patterns to solve this problem. show your work on a separate sheet of paper.

6 School magnets cost $4, and shirts cost $24. Write a pattern for the costs of 0, 1, 2, 3, 4, and 5 magnets and a second pattern for the costs of 0, 1, 2, 3, 4, and 5 shirts. How do the corresponding terms of the two patterns compare?

see table.

each number increases by 2 (add 2).

each number increases by 6 (add 6).

Possible answer: each first number is one third of the second number.

Possible answer: you could make a third pattern that shows 9 points for each

number of correct answers. Then you could write ordered pairs with Level 2

points as the first number and Level 3 points as the second number. Look for

relationships in the ordered pairs.

The terms in the second pattern are always six times the corresponding terms

in the first pattern.



Lesson 30Part 3: Modeled instruction

AT A gLAnce

Students use a picture and a table to understand a word problem. Then, students write ordered pairs from the table.

sTeP by sTeP

• Read the problem at the top of the page as a class.

• Read Picture It. Point out that the difference between each number in the sequences and the one before it is always the same.

• Ask students how the “plus 2” written above each arrow for the cost of making planes relates to the context of the problem. [It is equal to the cost of materials for 1 plane.]

• Read Model It. Emphasize that the ordered pairs are created by writing the cost for the number of planes in the x-coordinate position and the cost for the number of boats in the y-coordinate position.

You may wish to explore with students how patterns looks using arrays. Write the numbers 2, 4, 6, 8 on the board. Ask students how you could show these numbers with arrays of dots. Show 1 row of 2 dots, 2 rows of 2 dots, 3 rows of 2 dots, and 4 rows of 2 dots. Below those arrays, show 1 row of 4 dots, 2 rows of 4 dots, 3 rows of 4 dots, and 4 rows of 4 dots. Ask students to describe what they see within each pattern as well as between the two patterns.

visual Model• Is it easier to compare the costs of the boats and

planes using a picture, a table, or ordered pairs? Explain your reasoning.

Student responses should indicate the method that is easier for them and include a valid reason. Sample response: A table is easier because the information is organized in columns and the two sequences are side-by-side.

• If a problem does not say how many numbers to list for each sequence, how many should you list? Why?

Student responses should indicate an understanding that you need to list at least 4 or 5 numbers in order to determine the pattern.


Lesson 30



Read the problem below. Then explore what the graph looks like when you plot corresponding terms of number patterns on a graph.

The scouts have a choice of making a model plane or a model boat. The materials

for the plane cost $2, and the materials for the boat cost $4. Write and graph

ordered pairs to compare the cost of making one or more planes to the cost of

making the same number of boats.

Picture it

you can use a picture to find the costs of making various numbers of each model.

Each pattern of numbers below shows the cost of making 0, 1, 2, 3, and 4 models.

Planes boats

12 12 12 12

14 14 14 14

0, 2, 4, 6, 8 0, 4, 8, 12, 16

Model it

you can use a table to help understand the problem.

List the cost for the planes and boats in a table. Then write the corresponding costs as ordered pairs.

Planes (x) boats (y) ordered Pairs (x, y)

0 0 (0, 0)

2 4 (2, 4)

4 8 (4, 8)

6 12 (6, 12)

8 16 (8, 16)

Part 3: Modeled instruction


Lesson 30Part 3: guided instruction

AT A gLAnce

Students revisit the problem on page 314 to graph ordered pairs. Then, students use the graph to compare the two patterns.

sTeP by sTeP

• Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 314.

• Review plotting ordered pairs. [The x-coordinate indicates how many places to move to the right, and the y-coordinate indicates how many places to move up.]

• Ensure students recognize that both of the axes are scaled by 2s, so moving 1 unit right or up represents an increase of 2, and moving 2 units up represents an increase of 4.

• Ask a volunteer to explain what the point (2, 4) represents in this situation. [It compares the cost of 1 plane ($2) and 1 boat ($4).]

TRy iT soLuTions

12 Solution: (0, 0), (1, 3), (2, 6), (3, 9), (4, 12), (5, 15); Students plot a point for each ordered pair on a coordinate plane. The y-coordinates are 3 times the x-coordinates; Students may create a table to write the ordered pairs and then graph the ordered pairs. Students may compare the points using a table or a coordinate plane.

Make and test a conjecture.

• Tell students they are going to explore how the graph changes if they list the cost of boats as the x-coordinate and the cost of planes as the y-coordinate.

• Have students make a conjecture about how the graph will look. [Possible answer: The points will not form a line.]

• Show the original table from page 314. Make a new table with the cost of boats in the x-column and the cost of planes in the y-column.

• List and graph the corresponding ordered pairs.

• Have a volunteer describe how the two graphs compare. [Possible answer: The points do not go up as steeply.]

• Ask students if the relationship between the cost of boats and the cost of planes changed. Why? [Possible answer: The relationship did not change because the costs did not change.]

visual Model

ERROR ALERT: Students who wrote that the x-coordinates are 3 times the y-coordinates graphed the y-values on the x-axis and the x-values on the y-axis.

Lesson 30


connect it

now you will solve the problem from the previous page by graphing ordered pairs.

7 Explain how to write the ordered pairs from the patterns.

8 Plot the ordered pairs on the graph to the right. The fi rst number shows the location on the x-axis. The second number shows the location on the y-axis.

The point (2, 4) has been plotted for you.

9 How do the coordinates of corresponding terms of the patterns compare?

10 Suppose you connect the points. What would the graph look line?

11 What directions would you give to someone to get from one point to the next on the graph? How do your directions relate to the rules for the patterns?

Try it

use what you just learned about comparing two number patterns on a graph to solve this problem. show your work on a separate sheet of paper.

12 Use the rules “add 1” and “add 3” to write and graph ordered pairs made up of corresponding terms from the two patterns. Start each pattern with 0. Describe the graph.

Part 3: guided instruction

y

x

18161412108642

2 4 6 8 10 120

The first number is a term from the first pattern, and the second number is

the corresponding term from the second pattern.

The second number is always twice the first

number.

To get from one point to the next, move 2 to the right and 4 up. The rule for

the first pattern was add 2, the rule for the second pattern was add 4.

if you connect the points, they make a straight line.

(0, 0), (1, 3), (2, 6), (3, 9), (4, 12), (5, 15); The second numbers are 3 times the

first numbers. The points are all on the same line on the graph.

18161412108642

y

x 2 4 6 8 10 120



Lesson 30Part 4: guided Practice

AT A gLAnce

Students use lists, tables, and ordered pairs to solve word problems involving sequences of numbers.

sTeP by sTeP

• Ask students to solve the problems individually and label each column in their tables.

• When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.

soLuTions

Ex Writing ordered pairs is shown as one way to find the relationship between terms. Students could also make lists, a table, or a coordinate graph.

13 Solution: The numbers in the sequence with the rule “add 8” are twice as much as the numbers in the sequence with the rule “add 4.”

14 Solution: See completed student table above; The

y-coordinate is 1 ··

 4 the corresponding x-coordinate or

the x-coordinate is 4 times the y-coordinate.

15 Solution: D; Each y-coordinate is 6 times the corresponding x-coordinate.

Explain to students why the other two answer choices are not correct:

A is not correct because 10 more is not true for (3, 18) or (4, 24).

B is not correct because y is not 2 times as much as x in any of the 3 points.

Part 5: guided Practice Lesson 30


Find the difference between the coordinates in each ordered pair. Do you see another pattern?

Pair/share

Does Mike’s answer make sense?

Pair/share

14 Identify the pattern in each column. Then complete the table below. Form ordered pairs for the corresponding terms. Describe the relationship between corresponding terms.

Show your work.

x y ordered Pairs (x, y)

4 1

8 2

12 3

Solution:

15 The ordered pairs (2, 12), (3, 18), and (4, 24) were formed by corresponding terms in two sequences. How do the values of the y-coordinates compare to the values of the x-coordinates? Circle the letter of the correct answer.

A 10 more

b 2 times as much

c 1 ·· 6 times as much

D 6 times as much

Mike chose c as the correct answer. How did he get that answer?

What are the rules for each pattern?

What rule works for all the points?

Possible student work:

Possible answer: The y-coordinate is 1 ·· 4 times the

x-coordinate.

Possible answer: Mike compared the x-coordinates to the

y-coordinates instead of comparing the y-coordinates to the

x-coordinates.

(4, 1)(8, 2)

(12, 3)(16, 4)(20, 5)(24, 6)

162024

456

Student Model

Part 5: guided Practice Lesson 30



Does it matter how many terms you write for each sequence?

Pair/share

How are these patterns different from other patterns in this lesson?

Pair/share

study the model below. Then solve problems 13–15.

Look at the following two number patterns.

Pattern 1: 6, 5, 4, 3, 2, 1, 0

Pattern 2: 24, 20, 16, 12, 8, 4, 0

What is the relationship between corresponding terms in the two

patterns?

Look at how you could show your work using ordered pairs.

The first coordinate is a term from Pattern 1, the second coordinate is a corresponding term from Pattern 2. ordered pairs: (6, 24), (5, 20), (4, 16), (3, 12), (2, 8), (1, 4), (0, 0)

Solution:

13 One sequence starts at 0 and has the rule “add 8.” Another sequence starts at 0 and has the rule “add 4.” Write each sequence of numbers. How do the corresponding terms in the sequences compare?

Show your work.

Solution:

each term in Pattern 2 is four times the

corresponding term in Pattern 1.

The student wrote ordered pairs to identify a relationship between corresponding terms.

How do I generate the sequences?

Add 8: 0, 8, 16, 24, 32, 40, . . .

Add 4: 0, 4, 8, 12, 16, 20, . . .

Possible answer: The numbers in the sequence with

the rule “add 8” are twice the corresponding numbers in the

sequence with the rule “add 4.”


Lesson 30Part 5: common core Practice

AT A gLAnce

Students write and compare patterns of numbers to solve problems that might appear on a mathematics test.

sTeP by sTeP

• First, tell students that they will write and compare sequences of numbers. Then have students read the directions and answer the questions independently. Remind students to fill in the correct answer choices on the Answer Form.

• After students have completed the Common Core Practice problems, review and discuss correct answers. Have students record the number of correct answers in the box provided.

soLuTions

1 Solution: C; Compare each term of the pattern to the term before it.

2 Solution: B; Compare each term in Pattern 2 to the corresponding term in Pattern 1.

3 Solution: D; Determine for which ordered pair the x-coordinate divided by the student cost, $2, is equal to the y-coordinate divided by the adult cost, $8.

4 Solution: Each term in Pattern 2 is 4 times as much as the corresponding term in Pattern 1; List the first several terms for each sequence and compare corresponding terms.

5 Part A Solution: See table above.

Part B Solution: Students plot the ordered pairs from the table on a coordinate plane. See graph above.

Part C Solution: Each term in the “add 5” pattern is

2 1 ··

 2 times the corresponding term in the “add 2”

pattern.

Part 6: common core Practice Lesson 30



Solve the problems. Mark your answers to problems 1–3 on the Answer Form to the right. Be sure to show your work.

1 What rule could be used to create the pattern 9, 18, 27, 36, 45, 54, . . . ?

A multiply each term by 2 to get the next term

B multiply each term by 9 to get the next term

C add 9 to get the next term

D add 3 to get the next term

2 Describe the relationship between corresponding terms in these two patterns.

Pattern 1: 3, 6, 9, 12, 15, 18, . . .

Pattern 2: 18, 36, 54, 72, 90, 108, . . .

A Each term in Pattern 1 is 6 times the corresponding term in Pattern 2.

B Each term in Pattern 2 is 6 times the corresponding term in Pattern 1.

C Each term in Pattern 1 is 15 times the corresponding term in Pattern 2.

D Each term in Pattern 2 is 15 times the corresponding term in Pattern 1.

3 Tickets at an outdoor play cost $2 for students and $8 for adults. Jason creates two patterns to compare the costs. He writes ordered pairs in the form (student cost, adult cost) for the corresponding number of tickets. Which ordered pair could be on Jason’s list of ordered pairs?

A (8, 2)

B (4, 10)

C (10, 16)

D (6, 24)

1 A B C D

2 A B C D

3 A B C D

Answer Form

numbercorrect 3

Part 6: common core Practice Lesson 30


4 Create two patterns starting with 0 using the rules below. Then describe the relationship between corresponding terms of the two patterns.

Pattern 1: add 3 Pattern 2: add 12

Show your work.

5 Begin at 0 and use the rule “add 2” and “add 5” to complete the table in Part A.

Part A

Add 2 Add 5 Ordered Pairs

Part B

Graph the ordered pairs. Describe your graph.

Part C

Describe the relationship between the corresponding terms of the two patterns.

y

x20

2

4

6

8

10

12

14

16

18

20

4 6 8 10 12 14 16 18 20

Pattern 1: 0, 3, 6, 9, 12, 15, . . .

Pattern 2: 0, 12, 24, 36, 48, 60, . . .

All of the points are on the same line.

Possible answer: each term in Pattern 2 is four times the corresponding term in

Pattern 1.

each term in the second pattern is 2 1 ·· 2 times the corresponding term in the

first pattern.

y

x2

2

4

6

8

10

12

14

16

18

20

4 6 8 10 12 14 16 18 20

(0, 0)(2, 5)

(4, 10)(6, 15)(8, 20)

420

68

1050

1520

Differentiated instruction Lesson 30

L30: Analyze Patterns and Relationships324©Curriculum Associates, LLC Copying is not permitted.

Assessment and Remediation

hands-on Activity

challenge Activitycreate two numerical sequences with a given relationship.

List the following numbers on the board: 2, 4, 6, 8, 10, 12, 18, 24, 30. Challenge students to create a table so that each column is a numerical sequence and the numbers in one column are always the same number of times greater than the corresponding numbers in the other column.

use counters to compare two numerical sequences.

Materials: colored counters

• Have students work in pairs or groups. Distribute counters so that each pair or group has five different colors of counters to work with.

• Write the following problem on the board:

Describe the relationship between the corresponding terms of these two patterns:

Pattern 1: 1, 3, 5, 7, 9; Pattern 2: 2, 6, 10, 14, 18

• Tell students to use one color counter to model the first term of each pattern, another color to model the second term of each pattern, and so on.

• Ask students to describe the relationship between the corresponding terms of these two patterns:

Pattern 1: 1, 3, 5, 7, 9, . . . Pattern 2: 5, 15, 25, 35, 45, . . .

[Each term in Pattern 2 is five times as much as the corresponding term in Pattern 1.]

• For students who are struggling, use the chart below to guide remediation.

• After providing remediation, check students’ understanding. Ask students to describe the relationship between the corresponding terms of these two patterns:

Pattern 1: 1, 4, 7, 10, 13, . . . Pattern 2: 3, 12, 21, 30, 39, . . .

[Each term in Pattern 2 is three times as much as the corresponding term in Pattern 1.]

• If a student is still having difficulty, use Ready Instruction, Level 4, Lesson 8.

if the error is . . . students may . . . To remediate . . .

each term in Pattern 2 is 4 more than the corresponding term in Pattern 1.

have subtracted the first term of Pattern 1 from the first term of Pattern 2.

Have students read the first term of each pattern aloud, and then tell how many times greater the first term of Pattern 2 is than the first term of Pattern 1. Continue this process for the remaining terms, having students check that each term of Pattern 2 is always the same number of times more than the corresponding term of Pattern 1.

each term in Pattern 2 is ten more than the corresponding term in Pattern 1.

have identified the pattern in Pattern 2.

Have students check their answer by multiplying each term in Pattern 1 by 10 and comparing the result to the corresponding term in Pattern 2. When they get an answer that doesn’t check, ask them how they can correct their answer so it does check.

• Ask students how using colored counters could help them compare the two patterns. [Possible answer: The same color counters represent corresponding terms of the two patterns, so you should compare the same colored counters to each other.]

• Ask students to describe the relationship between the two patterns. [Each term of Pattern 2 is twice as much as the corresponding term in Pattern 1.]

Lesson 30 CCLS Analyze Patterns and...

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