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Learn to create two number patterns, spot relationships, and plot points on a coordinate plane.
People have been fascinated by number patterns for thousands of years. A number pattern is a list of numbers that follows a rule. When you count by 2s (2, 4, 6, 8…), that's a pattern! People noticed these patterns in nature, in buildings, and in the stars.
Today you will learn to create two patterns at the same time, compare them, and draw them on a grid called a coordinate plane. This skill helps you see how two sets of numbers are connected. It's like a superpower for math!
Before we dive in, let's learn some important words and ideas. These building blocks will help you understand everything in this lesson.
Let's look at two patterns side by side. Pattern A uses the rule "Add 3" starting at 0. Pattern B uses the rule "Add 6" starting at 0. The diagram below shows both patterns and the points they make on a coordinate plane.
Look at the dots on the graph. They form a perfectly straight line going up! This happens because the two patterns have a special relationship. Every number in Pattern B is exactly 2 times the matching number in Pattern A. When 6 is always double 3, and 12 is always double 6, the dots will always line up.
Let's look at why the two patterns are related. When both patterns start at the same number and each uses a simple "add" rule, we can figure out the relationship by comparing the rules.
Here's the big idea: since 6 = 2 × 3, every time you add 3 to Pattern A, you add 6 to Pattern B. That means Pattern B is always growing twice as fast as Pattern A. That's why every term in Pattern B is double the matching term in Pattern A.
Let's try a different example with new rules. This time, Pattern A uses the rule "Add 2" starting at 0, and Pattern B uses the rule "Add 8" starting at 0. Let's build the table and graph together.
| Position | Pattern A (Add 2) | Pattern B (Add 8) | Ordered Pair (A, B) |
|---|---|---|---|
| Start | 0 | 0 | (0, 0) |
| 1st | 2 | 8 | (2, 8) |
| 2nd | 4 | 16 | (4, 16) |
| 3rd | 6 | 24 | (6, 24) |
| 4th | 8 | 32 | (8, 32) |
See the pattern? In this example, both patterns start at 0 and use "add" rules. When that is the case, you can divide the bigger rule by the smaller rule to find how many times larger Pattern B is than Pattern A. Here, 8 ÷ 2 = 4, so Pattern B is always 4 times Pattern A. Keep in mind: this shortcut only works when both patterns start at 0. If the starting numbers are different, you need to compare the actual terms directly to find the relationship. The dots always make a straight line!
Let's work through a complete example from start to finish. Both patterns start at 0. Pattern A follows the rule "Add 4" and Pattern B follows the rule "Add 12."
Here are some things to watch out for when you work with two number patterns and graph them.
| Helpful Tip ✓ | Common Mistake ✗ |
|---|---|
| Always write both patterns in a table so you can line up matching terms. | Comparing terms that are NOT in the same position (like the 2nd term of A with the 3rd term of B). |
| In an ordered pair, the Pattern A number always comes first (x) and Pattern B comes second (y). | Switching the x and y values. Putting Pattern B first will flip your graph! |
| Label your axes so you remember which pattern is which. | Forgetting labels and then getting confused about which axis is which. |
| Use equal spacing on each axis. Count by the same amount for every line on the grid. | Making uneven spaces on the number lines, which makes dots land in the wrong spots. |
| When both patterns start at 0 and use "add" rules, you can divide the bigger rule by the smaller rule to quickly find the relationship. Always check by comparing the actual terms too. | Using the divide-the-rules shortcut when the patterns do NOT start at 0. The shortcut only works when both starting numbers are 0. |
The skills you are learning right now are building blocks for bigger math ideas. In middle school, you'll use these same ideas but with fancier names. Let's peek at how today's lesson connects to what comes next.
| What You Learn Now | What It Becomes Later |
|---|---|
| Following an "add" rule to make a pattern | Writing equations like y = 3x (algebra) |
| Making ordered pairs from two patterns | Using input-output tables for functions |
| Plotting dots on a coordinate plane | Graphing lines and curves in algebra |
| Noticing dots make a straight line | Understanding slope and linear relationships |
| Explaining why one pattern is 2× or 3× the other | Writing and proving mathematical relationships |
So when you work with number patterns and coordinate planes now, you're actually getting a head start on algebra! Every dot you plot is practice for graphing equations. Keep it up!
In this lesson, you learned how to generate two number patterns using two different rules, such as "Add 3" and "Add 6." You practiced identifying the relationship between corresponding terms — for example, when both patterns start at 0 and one rule is double the other, each term in the second pattern is twice the matching term in the first pattern. When both patterns start at 0, you can find this by dividing the bigger rule by the smaller rule. Always verify the relationship by checking the actual terms, especially if the patterns do not start at 0.
You also learned how to form ordered pairs by pairing up matching terms, then graph those pairs on a coordinate plane. The first pattern's terms go on the x-axis and the second pattern's terms go on the y-axis. When both patterns start at 0 and use "add" rules, the dots always form a straight line. These skills are your first steps toward algebra and graphing equations!