Patterns as Pairs

Start at 0 and add 3 gives 0, 3, 6, 9. Start at 0 and add 6 gives 0, 6, 12, 18.

How the terms relate

Each term of the second is exactly double the matching term of the first.

Making pairs

Pair them up: (0, 0), (3, 6), (6, 12), (9, 18).

On the grid

Plotted, those points sit in a straight line. The relationship becomes a picture.

Two patterns, one graph

Generate one sequence with "add 3" and another with "add 6", both starting at 0. Pair the terms: (0,0), (3,6), (6,12), (9,18). Plotting these pairs turns two separate patterns into a single picture of how they relate.

Spotting the relationship

In that example, every y-value is twice its x-value. That relationship was never stated in either rule — it emerged from the pairing. Finding such relationships is the beginning of algebra, and it starts as a genuinely visual activity.

Why the points line up

Plotted points from two constant-step rules always fall on a straight line, because each step across is matched by the same step up every time. The steadiness of the rules is what produces the straightness of the line.

Using the graph to predict

Once you see the line, you can read off values you never calculated. If x is 15, y is 30. A graph is not just a record of the numbers you have — it is a tool for finding the ones you have not.