Number Patterns and Relationships

In math, a number pattern (or sequence) is a list of numbers that follow a specific rule. Sometimes, we need to look at two different patterns at the same time to see how they relate to each other.

Generating Number Patterns

Let's say we have two different patterns that both start at $0$.

• Pattern A Rule: Add $2$
• Pattern B Rule: Add $6$

Let's generate the first few terms for each pattern:

• Pattern A: $0, 2, 4, 6, 8, \dots$
• Pattern B: $0, 6, 12, 18, 24, \dots$

Finding the Relationship

To find the relationship between the two patterns, we compare their corresponding terms (the numbers in the same position in each list).

• 1st terms: $0$ and $0$
• 2nd terms: $2$ and $6$
• 3rd terms: $4$ and $12$
• 4th terms: $6$ and $18$

Look closely at the pairs: $2 \times 3 = 6$, $4 \times 3 = 12$, and $6 \times 3 = 18$. The relationship is that each term in Pattern B is exactly $3$ times the corresponding term in Pattern A.

Forming and Plotting Ordered Pairs

We can take the corresponding terms from our two patterns and write them as ordered pairs $(x, y)$. Let's use Pattern A for the $x$-coordinate and Pattern B for the $y$-coordinate.

Our ordered pairs are: $(0, 0), (2, 6), (4, 12), (6, 18)$

You can plot these points on a coordinate plane:

1. Start at the origin $(0,0)$.
2. For the pair $(2, 6)$, move $2$ units to the right along the $x$-axis, and then move $6$ units up along the $y$-axis. Plot the point.
3. Repeat this for the rest of the ordered pairs.

Because of the steady relationship between the two rules, if you connect the plotted points, they will form a straight line!