Analyze Patterns and Relationships

Key Concepts

• Proportionality
• Graph of the numerical sequences

Introduction

In this chapter, we will learn about proportionality, ordered pairs, graph of the numerical sequence.

Proportionality 

A proportional relationship is one in which two quantities vary directly with each other. 

The graph of a proportional relationship is a straight line passing through the origin (0, 0). 

If x and y have a proportional relationship, the constant of proportionality is the ratio of y to x. 

Example 1: 

The graph below shows the distance traveled and the time taken as proportional to each other. 

Notice that the graph is a straight line starting from the origin. 

Look at the values on both axes: 

• When the distance axis is 4, the time axis is 2. 
• When the distance axis shows 8, the time axis shows 4. 
• This means that when one of the variables doubles, the other variable also doubles. 
• This is the test for proportionality. 

Example 2: 

Nancy laundromat dry cleans Adele’s suits. 

The below graph shows that there is a proportional relationship between the number of suits Adele dry cleans, x, and the total cost (in dollars), y. 

What is the constant of proportionality? 

Solution: 

To find the constant of proportionality, first, identify the coordinates of one of the points on the line. 

Calculate the ratio of the y-coordinate to the x-coordinate. 

= 9 

The constant of proportionality is 9 dollars per suit. 

Graph of the numerical sequences 

Example 1: 

Magana runs 5 miles per day. Robin runs 10 miles per day. Both of them made a table using the rule 

‘Add 5’ to show Magana’s miles and the rule ‘Add 10’ to show Robin’s miles. Complete the table, compare their runs, and graph the ordered pair of the corresponding terms. 

Day	0	1	2	3	4
Meghana	0	5	10	15	20
Robin	0	10	20	30	40

Solution: 

Compare the numbers in Meghana and Robin sequence. 

Step 1: Each sequence begins with zero. 

Step 2: Then, each term in Robin’s pattern is 2 times greater than the corresponding terms in Meghana’s pattern. 

Step 3: Generate ordered pairs from the total miles Meghana and Robin have run after each day. 

(0, 0) (5, 10) (10, 20) (15, 30) (20, 40) 

Graph the ordered pairs. 

Example 2: 

Every month shank pays $200 for the car’s payment. He spends $50 for library membership. 

Write an ordered pair to represent how much Shank spends in 6 months for car payment and the library membership. 

Money	0	1	2	3	4	5	6
Library membership	0	50	100	150	200	250	300
Car payment	0	200	400	600	800	1000	1200

Compare the numbers in library membership and car payment sequence. 

Step1: Then, each term in car payment is 4 times greater than the corresponding terms in library membership. 

Step2: Generate ordered pairs from the total amount of library membership and car payment after each month. 

(0, 0) (50, 200) (100, 400) (150, 600) (200, 800) (250, 1000) (300, 1200) 

Step3: Graph the ordered pairs. 

Exercise

• Which graph shows a proportional relationship?

• What is the constant of proportionality for this table?

• Write the constant of proportionality for this table.

• Write ordered pairs from the graph.

• Draw the graph for the following table.

• Write rule for the following table.

Days	1	2	3	4
Miles	4	8	12	16

• Write rule for the following table.

Days	1	2	3	4
Pages	3	6	9	12

• Robin can read 15 pages in 5 days. How many pages does he read each day?
• Draw the graph for the following table.

•  Complete the missing pairs

Meghan’s distance	Robin’s distance
0	0
2	4
4
6	12
8
10	20

What have we learned

• Understand the proportionality in which two quantities vary directly with each other.
• Generating ordered pairs
• Generating a graph based on the ordered pairs

Concept Map