Lesson 2: Representing Pattern Relationships with Tables and Graphs

Property

Given two numerical patterns, $A$ and $B$, presented in a table, the relationship between their corresponding terms can often be described by an equation. If the difference between corresponding terms is constant, the relationship can be written as $B = A + c$, where $c$ is the constant difference.

Examples

• Consider the table:

The relationship is $B = A + 3$, since each term in Pattern B is 3 more than the corresponding term in Pattern A.

• Consider the table:

The relationship is $Y = X + 4$, since each term in Pattern Y is 4 more than the corresponding term in Pattern X.

Explanation

A table is a useful tool for organizing and comparing two related numerical patterns. By looking at the corresponding terms in each row or column, you can identify a consistent relationship between them. This relationship often involves adding or subtracting a constant value, which is equal to the difference between the starting values of the two patterns.

Property

A numerical pattern that relates two quantities can be represented by a set of ordered pairs, $(x, y)$. The first number in the pair, $x$, typically represents the position in the sequence (the term number), and the second number, $y$, represents the value of that term.

Examples

• Pattern: A pattern starts with 3 and follows the rule "add 3". The first three terms are 3, 6, 9. This can be represented by the ordered pairs $(1, 3)$, $(2, 6)$, and $(3, 9)$.
• Pattern: A pattern is generated by the rule $y = 4x$. For the first three terms, we substitute $x=1, 2, 3$:
• If $x=1$, $y = 4(1) = 4$. The ordered pair is $(1, 4)$.
• If $x=2$, $y = 4(2) = 8$. The ordered pair is $(2, 8)$.
• If $x=3$, $y = 4(3) = 12$. The ordered pair is $(3, 12)$.

Explanation

Ordered pairs provide a structured way to describe the relationship within a pattern. Each pair, $(x, y)$, connects a term''s position ($x$) to its specific value ($y$). This format helps organize the pattern''s data, making it easier to analyze and understand the rule governing the sequence. By converting a pattern into a set of ordered pairs, you are preparing the data to be graphed on a coordinate plane.

Property

An ordered pair $(x, y)$ represents a point on the coordinate grid. The first number, the x-coordinate, indicates the horizontal position from the origin. The second number, the y-coordinate, indicates the vertical position from the origin.

Examples

• The pattern "add 2" starting at 1 generates the ordered pairs $(1, 3)$, $(2, 5)$, and $(3, 7)$. When plotted on a coordinate grid, these points form a straight line.
• The pattern "multiply by 3" starting at 2 generates the ordered pairs $(1, 6)$, $(2, 18)$, and $(3, 54)$. When plotted, these points form a curve that increases rapidly.

Explanation

Plotting the ordered pairs of a pattern on a coordinate grid creates a graph. This graph provides a visual representation of the relationship between the numbers. By observing the plotted points, you can identify the nature of the pattern, such as whether it is linear (forms a straight line) or non-linear. This visual tool helps in analyzing and extending patterns.