Question

1. use the pattern to answer the questions.

step 1 step 2 step 3
images of pentagons in each step
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: box
b. how many pentagons are in step 0? box
c. rate of change (slope): box

Explanation:

Step 1: Analyze the number of pentagons in each step

• Step 1: Count the pentagons. There are 3 pentagons.
• Step 2: Count the pentagons. There are 4 pentagons.
• Step 3: Count the pentagons. There are 5 pentagons.

Step 2: Describe the pattern (part a)

Looking at the number of pentagons in each step: Step 1 has 3, Step 2 has 4, Step 3 has 5. So the pattern is that the number of pentagons increases by 1 as the step number increases by 1.

Step 3: Find the number of pentagons in Step 0 (part b)

We know the pattern is linear (since the number of pentagons increases by 1 each step). Let's denote the number of pentagons as \( y \) and the step number as \( x \). We can use the linear equation \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y - intercept (number of pentagons in Step 0, \( x = 0 \)).

From Step 1 (\( x = 1 \), \( y = 3 \)) and the slope (which we will find in part c, but we can also use the pattern). If at Step 1 (\( x = 1 \)) we have 3 pentagons, and the number of pentagons increases by 1 for each increase in step number by 1, then for Step 0 (\( x = 0 \)), we subtract 1 from the number of pentagons in Step 1. So \( 3- 1=2 \).

Step 4: Find the rate of change (slope, part c)

The rate of change (slope) \( m \) for a linear relationship \( y = mx + b \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points: \((x_1,y_1)=(1,3)\) and \((x_2,y_2)=(2,4)\). Then \( m=\frac{4 - 3}{2 - 1}=\frac{1}{1}=1 \). We can also check with \((2,4)\) and \((3,5)\): \( m=\frac{5 - 4}{3 - 2}=\frac{1}{1}=1 \).

Answer:

s:
a. Describe the pattern: The number of pentagons increases by 1 as the step number increases by 1. (Step 1: 3, Step 2: 4, Step 3: 5)
b. How many pentagons are in Step 0? \(\boldsymbol{2}\)
c. Rate of change (slope): \(\boldsymbol{1}\)