Numerical Patterns and Graphing Ordered Pairs

A rule makes a pattern

Start with a number and apply a rule over and over.

Rule: add $3$. Start at $0$.

$$0, \; 3, \; 6, \; 9, \; 12, \; 15, \; \ldots$$

Rule: add $6$. Start at $0$.

$$0, \; 6, \; 12, \; 18, \; 24, \; 30, \; \ldots$$

Each term comes from the one before it. The rule and the starting number together fix the whole sequence — change either and you get a different pattern.

Comparing two patterns

Line them up term by term.

Term	Pattern A (add $3$)	Pattern B (add $6$)
1st	$0$	$0$
2nd	$3$	$6$
3rd	$6$	$12$
4th	$9$	$18$
5th	$12$	$24$

Every term of B is twice the matching term of A.

That is no coincidence. B adds twice as much each step and starts in the same place, so it stays exactly twice as far along the whole way.

Ordered pairs from the patterns

Take the matching terms as a pair, first pattern first:

$$(0, 0), \; (3, 6), \; (6, 12), \; (9, 18), \; (12, 24)$$

Each pair is one point on the coordinate plane: the $x$ from Pattern A, the $y$ from Pattern B.

Graphing them

The points fall in a straight line through the origin. That picture is the relationship “B is twice A”, drawn.

Every point sits twice as high as it is far right, which is exactly what doubling looks like.

Why graphing the pairs is worth doing

The table and the graph carry the same information, and the graph makes one thing plain that the table does not: whether the relationship is steady.

Points in a straight line mean the multiple never changes. Points that curve or scatter mean it does. Spotting that from a column of numbers takes checking every row; spotting it from a graph takes a glance.

The straight-line-through-the-origin shape is worth recognizing now, because it turns up again with a name. It is exactly what a proportional relationship looks like, and the “twice” here is what later gets called the constant of proportionality.

When the patterns start differently

If the starting numbers differ, the relationship is usually not a simple multiple.

A: start at $2$, add $3$ → $2, 5, 8, 11, 14$ B: start at $0$, add $6$ → $0, 6, 12, 18, 24$

A	$2$	$5$	$8$	$11$
B	$0$	$6$	$12$	$18$
B ÷ A	$0$	$1.2$	$1.5$	$1.6$

The quotients keep changing, so B is not a fixed multiple of A. Plotting these pairs would still give a straight line, but one that misses the origin — because the patterns did not start together.

Worked examples

Common mistakes

Practice problems