Question

1. use the pattern to answer the questions.

step 1 step 2 step 3
(three green pentagons) (four green pentagons) (five green pentagons)
a. describe the pattern
b. how many pentagons are in step 0? ____
c. rate of change (slope): ____
d. write an equation to represent the
number of pentagons (y) in a given step (x).
a describe the pattern

Explanation:

Part a: Describe the pattern

Step 1: Count pentagons per step

Step 1: 3 pentagons; Step 2: 4 pentagons; Step 3: 5 pentagons.

Step 2: Identify the pattern

The number of pentagons increases by 1 as the step number increases by 1. The base (Step 0) and the rate of change can be inferred from this.

Step 1: Analyze the pattern

From Step 1 (x=1, y=3), Step 2 (x=2, y=4), Step 3 (x=3, y=5), the pattern is linear. The formula for a linear pattern is \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y-intercept (Step 0 value).

Step 2: Find the y-intercept

Using Step 1: \( 3 = m(1) + b \). From the pattern, \( m = 1 \) (since each step increases by 1). So \( 3 = 1 + b \), solving for \( b \) gives \( b = 2 \). Thus, Step 0 (x=0) has 2 pentagons.

Step 1: Recall slope formula

The slope \( m \) is calculated as \( \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the number of pentagons and \( \Delta x \) is the change in step number.

Step 2: Calculate slope

From Step 1 to Step 2: \( \Delta y = 4 - 3 = 1 \), \( \Delta x = 2 - 1 = 1 \). So \( m = \frac{1}{1} = 1 \).

Answer:

The number of pentagons increases by 1 for each subsequent step. Step 1 has 3, Step 2 has 4, Step 3 has 5.

Part b: Pentagons in Step 0