Math Core

You already know how to make a number pattern from a rule, like "start at 0 and add 3." When you make two patterns side by side, you can pair up their terms and graph them on the coordinate plane. The picture helps you see how the two patterns are related.

Two patterns side by side

Here are two patterns. Both start at 0.

Pattern A: add 1 each time.
Pattern B: add 3 each time.

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

Now compare the terms in each column. Each term in Pattern B is 3 times the matching term in Pattern A: $1 \times 3 = 3$ , $2 \times 3 = 6$ , $3 \times 3 = 9$ , and so on. That makes sense, because Pattern B grows 3 times as fast.

Making ordered pairs

Put the two terms in each column together to make an ordered pair. The Pattern A term is the x-coordinate, and the Pattern B term is the y-coordinate:

(0, 0) \quad (1, 3) \quad (2, 6) \quad (3, 9) \quad (4, 12)

Pairing two patterns

Match the terms that are in the same position. The term from the first pattern is $x$ and the term from the second pattern is $y$ . Then plot each ordered pair.

The ordered pairs from Pattern A (add 1) and Pattern B (add 3).Open in grapher →

The points line up in a straight row going up to the right. Each time you move 1 unit right, you move 3 units up. The graph shows the "3 times" relationship too.

Worked example: Saving money

Leo saves $3 each week. Mia saves $6 each week. Both start with $0. Make both patterns for weeks 0 through 4 and describe how they are related.

Week	0	1	2	3	4
Leo (add 3)	0	3	6	9	12
Mia (add 6)	0	6	12	18	24

The ordered pairs (Leo, Mia) are $(0, 0)$ , $(3, 6)$ , $(6, 12)$ , $(9, 18)$ and $(12, 24)$ .

Mia always has twice as much as Leo, because she adds twice as much each week.

Worked example: Patterns with different starts

Pattern A starts at 1 and adds 1. Pattern B starts at 3 and adds 1. Write the first four ordered pairs and describe the relationship.

Pattern A: 1, 2, 3, 4. Pattern B: 3, 4, 5, 6.

The ordered pairs are $(1, 3)$ , $(2, 4)$ , $(3, 5)$ and $(4, 6)$ .

Both patterns add the same amount, so the gap between them never changes. Each term in Pattern B is 2 more than the matching term in Pattern A.

Each point is 1 right and 1 up from the one before it.Open in grapher →

Tip

To check a relationship, test it on every column, not just one. In the first table, $0 \times 3 = 0$ , $1 \times 3 = 3$ and $4 \times 3 = 12$ all work, so "3 times" is a good description.

Common mistake

Only pair terms in the same position. The 3rd term of Pattern A goes with the 3rd term of Pattern B. And keep the order: the first pattern gives $x$ , the second gives $y$ .

Practice

Practice 1

A pattern starts at 0 and adds 5 each time. What is the 5th term? (The 0 is the 1st term.)

Type a number.

Practice 2

Pattern A starts at 0 and adds 2. Pattern B starts at 0 and adds 6. What ordered pair do you make from the 4th terms of the two patterns?

Enter a point like (2, -3)

Practice 3

Use the patterns from the last problem. Each term in Pattern B is how many times the matching term in Pattern A?

Type a number.

Practice 4

Pattern A starts at 0 and adds 1. Pattern B starts at 0 and adds 4. The terms are paired as (Pattern A, Pattern B) and graphed. Which point is not on the graph?

Practice 5

These points come from two patterns. Pattern A starts at 0 and adds 1. Pattern B starts at 1 and adds 2.

The first four ordered pairs.Open in grapher →

What is the next ordered pair?

Enter a point like (2, -3)

Practice 6

A baker uses 1 cup of sugar and 2 cups of flour for each batch of muffins. The cups of sugar make the pattern 0, 1, 2, 3, … and the cups of flour make the pattern 0, 2, 4, 6, … How many cups of flour does the baker use with 5 cups of sugar?

Type a number.

Practice 7

Pattern A starts at 2 and adds 2: 2, 4, 6, 8, … Pattern B starts at 5 and adds 2: 5, 7, 9, 11, … When the Pattern A term is 20, what is the matching Pattern B term?

Type a number.