Math Core

You already know how to follow a pattern rule like "start at 0, add 3." Now you'll make two patterns side by side and discover how they are connected. Pairing up the terms and graphing them shows the connection as a picture.

Two patterns side by side

Let's make two patterns, both starting at 0.

Pattern A: start at 0, add 3.
Pattern B: start at 0, add 6.

Line them up in a table so each term sits next to its partner:

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

Compare the partners: 3 and 6, 6 and 12, 9 and 18, 12 and 24. Each term in Pattern B is 2 times its partner in Pattern A. That makes sense: B adds 6 each step, and 6 is 2 times 3, so B grows twice as fast.

You can also say it the other way: each term in Pattern A is half of its partner in Pattern B.

Comparing two patterns

When two patterns both start at 0, compare what they add each time. If one adds 4 times as much as the other, then every term is 4 times its partner.

Ordered pairs

Write each pair of partners as an ordered pair: the Pattern A term first, then the Pattern B term.

(0, 0),\quad (3, 6),\quad (6, 12),\quad (9, 18),\quad (12, 24)

The first number tells you how far to go across (the $x$ -axis). The second number tells you how far to go up (the $y$ -axis). Now you can plot the pairs on a coordinate grid.

Pattern A across, Pattern B up. The points line up, and each one is twice as high as it is far across.Open in grapher →

The points make a straight line going up from $(0, 0)$ . Every point is twice as high as it is far across, because each $y$ is 2 times its $x$ .

Common mistake

Keep the order the same for every pair. If Pattern A is first in one pair, it must be first in all of them. $(3, 6)$ and $(6, 3)$ are different points.

Worked example: Build the pairs

Pattern A: start at 0, add 2. Pattern B: start at 0, add 8. Write the first 5 ordered pairs and describe how the terms are related.

Pattern A: 0, 2, 4, 6, 8

Pattern B: 0, 8, 16, 24, 32

Ordered pairs: $(0, 0), (2, 8), (4, 16), (6, 24), (8, 32)$ .

Pattern B adds 8 each time and Pattern A adds 2. Since $8 = 4 \times 2$ , each term in Pattern B is 4 times its partner in Pattern A.

Worked example: Read a graph

The graph shows ordered pairs made from two patterns. How is the second number related to the first?

Ordered pairs (0, 0), (2, 1), (4, 2), (6, 3), (8, 4).Open in grapher →

The pairs are $(0, 0), (2, 1), (4, 2), (6, 3), (8, 4)$ . The first numbers go up by 2 and the second numbers go up by 1. Each second number is half of the first number: half of 6 is 3, half of 8 is 4.

Worked example: A saving story

Omar saves $4 each week. His sister saves $12 each week. They both start with $0. When Omar has $20 saved, how much has his sister saved?

Omar: 0, 4, 8, 12, 16, 20. That is 5 weeks.

Sister: 0, 12, 24, 36, 48, 60.

She saves 3 times as much each week ( $12 = 3 \times 4$ ), so she always has 3 times as much: $3 \times 20 = 60$ . His sister has saved $60.

Tip

To check a relationship like "B is 3 times A," test it on at least three pairs, not just one.

Practice

Practice 1

The rule is start at 0, add 5. What is the 5th term?

Type a number.

Practice 2

Pattern A: start at 0, add 2. Pattern B: start at 0, add 6. Each term in Pattern B is how many times its partner in Pattern A?

Type a number.

Practice 3

Using the same two patterns (A: start at 0, add 2; B: start at 0, add 6), write the ordered pair made from the 4th terms (Pattern A first).

Enter a point like (2, -3)

Practice 4

Pattern A: start at 0, add 4. Pattern B: start at 0, add 12. What is the 6th term of Pattern B?

Type a number.

Practice 5

The graph shows ordered pairs made from two patterns. Which statement is true?

Ordered pairs (0, 0), (1, 4), (2, 8), (3, 12).Open in grapher →

Practice 6

Pattern A: start at 0, add 10. Pattern B: start at 0, add 5. Which is true about every pair of partner terms?

Practice 7

Jen saves $3 each week and her brother saves $9 each week. Both start with $0. When Jen has saved $15, how many dollars has her brother saved?

Type a number.

Practice 8

Pattern A: start at 0, add 3. Pattern B: start at 0, add 9. One term of Pattern A is 27. What is its partner in Pattern B?

Type a number.