Study the dot patterns in parts (a) and (b) below. Assume that each pattern continues to increase by the same number of dots and in the same locations for each figure. For each pattern, sketch the $4^{\text{th}}$ and $5^{\text{th}}$ figures. Then predict how many dots will be in the $100^{\text{th}}$ figure. 

1. ${\large\begin{array}{c c} \;\;\bullet\\ \;\;\bullet &\bullet\\ \;\;\bullet\\ \end{array}\\ \text{Figure 1}} \qquad \qquad$

   ${\large\begin{array}{c c c} \bullet\\ \bullet\\ \bullet &\bullet & \bullet\\ \bullet\\ \bullet \end{array}\\ \\ \;\text{Figure 2}} \qquad \qquad$

   ${\large\begin{array}{c c c c} \bullet\\ \bullet\\ \bullet\\ \bullet &\bullet & \bullet & \bullet\\ \bullet\\ \bullet\\ \bullet\\ \end{array}\\ \quad \; \text{Figure 3}}$

   Hint 1 (a):

   Look at the vertical dots. How are they growing? What about the horizontal ones?

   Hint 2 (a):

   With each new figure, the vertical line gets one dot on the top and one dot on the bottom. The horizontal line gets one dot on the right side each time.

   How can you find each figure number in the number of dots? Can you find groups of two dots in the second figure? Can you find groups of three dots in the third figure? Use this to help you with the $100^{\text{th}}$ figure.

   Hint 3 (a):

   Partial Answer (a):

   ${\large\begin{array}{c c c c c} \bullet\\ \bullet\\ \bullet\\ \bullet\\ \bullet &\bullet &\bullet &\bullet &\bullet\\ \bullet\\ \bullet\\ \bullet\\ \bullet\\ \end{array}\\ \qquad\text{Figure 4}}$

   ${\large\begin{array}{c c c c c} \bullet\\ \bullet\\ \bullet\\ \bullet\\ \bullet\\ \bullet &\bullet &\bullet &\bullet &\bullet &\bullet\\ \bullet\\ \bullet\\ \bullet\\ \bullet\\ \bullet \end{array}\\ \qquad \;\; \text{Figure 5}}$

   The $100^{\text{th}}$ figure will have $3$ groups of hundred $+ 1$.  

2. ${\large\begin{array}{c} \quad \; \bullet\\ \quad \; \bullet\\ \end{array}\\ \text{Figure 1}} \qquad \qquad$

   ${\large\begin{array}{c c} \; \; \bullet &\bullet\\ \; \; \bullet &\bullet\\ \end{array}\\ \text{Figure 2}}$

   ${\large\begin{array}{c c c} \bullet &\bullet &\bullet\\ \bullet &\bullet &\bullet\\ \end{array}\\ \;\; \text{Figure 3}}$

   Hint (b):

   How many columns of two are in each figure? How many would you expect in Figures $4$ and $5$? What about Figure $100$?

3. For each pattern, explain how you made your prediction for the $100^{\text{th}}$ figure.

   Hint (c):

   Did you use numbers or diagrams? How do you know that your answer makes sense?