Analyze and Graph Number Patterns - 5th Grade Math

SECTION 1

Why Do We Study Number Patterns?

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.

~3000 BC • Ancient Egypt

Egyptian builders used number patterns to design pyramids. They followed rules to make each layer the right size.

~500 BC • Ancient Greece

A mathematician named Pythagoras studied patterns in numbers and music. He showed that patterns can explain how things work.

1200s • Fibonacci's Famous Pattern

An Italian mathematician named Fibonacci discovered a special pattern (1, 1, 2, 3, 5, 8…) that appears in flowers, pinecones, and seashells.

1600s • Coordinate Plane Invented

René Descartes invented a way to show number patterns as pictures on a grid. We still use his idea today!

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!

SECTION 2

Core Ideas You Need to Know

Before we dive in, let's learn some important words and ideas. These building blocks will help you understand everything in this lesson.

Rule

A rule tells you what to do to get the next number. For example, "Add 3" or "Multiply by 2."

Sequence (Pattern)

A sequence is the list of numbers you get when you follow a rule. Each number in the list is called a term.

Corresponding Terms

Corresponding terms are numbers in the same position in two different patterns. The 1st term matches the 1st term, the 2nd matches the 2nd, and so on.

Ordered Pair

An ordered pair is two numbers written like (x, y). The first number tells you how far to go right. The second tells you how far to go up.

Coordinate Plane

A coordinate plane is a grid with two number lines that cross. One goes across (x-axis) and one goes up (y-axis).

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KEY TAKEAWAY

Think of two number patterns like two friends walking together. They both start at the same time and take steps, but one friend might take bigger steps than the other. An ordered pair is like a snapshot of where both friends are at the same moment. The coordinate plane is a map that shows all their snapshots as dots!

SECTION 3

See It on a Coordinate Plane

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.

The coordinate plane shows ordered pairs made from two patterns. Pattern A (Add 3) values go along the bottom (x-axis), and Pattern B (Add 6) values go up the side (y-axis). Notice all the dots line up in a straight line!

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.

SECTION 4

The Math Behind the Patterns

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.

PATTERN A RULE

Start at 0, then add 3 each time → 0, 3, 6, 9, 12, …

Start at 0 and keep adding 3. The 1st term is 0, the 2nd term is 3, the 3rd term is 6, the 4th term is 9, and so on.

PATTERN B RULE

Start at 0, then add 6 each time → 0, 6, 12, 18, 24, …

Start at 0 and keep adding 6. The 1st term is 0, the 2nd term is 6, the 3rd term is 12, the 4th term is 18, and so on.

THE RELATIONSHIP

Pattern B term = 2 × Pattern A term

Since 6 is 2 × 3, every number in Pattern B will always be twice the matching number in Pattern A. This is true for any pair of matching terms!

ORDERED PAIR FORMAT

(Pattern A term, Pattern B term) → (x, y)

We write the Pattern A number first (the x-value), then the Pattern B number second (the y-value). These pairs become dots on the coordinate plane.

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.

SECTION 5

How to Build Two Patterns Step by Step

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)

Two patterns built with rules Add 2 and Add 8

With rules Add 2 and Add 8 (both starting at 0), each Pattern B term is 4 times the matching Pattern A term (because 8 ÷ 2 = 4). The dots make a straight line that goes up steeply.

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!

SECTION 6

Worked Example: Add 4 and Add 12

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."

Full Example: Add 4 vs. Add 12

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Step 1 — Write Out Pattern A

Start at 0 and keep adding 4. Write the first 5 terms: 0, 4, 8, 12, 16.

Pattern A: 0, 4, 8, 12, 16
2
Step 2 — Write Out Pattern B

Start at 0 and keep adding 12. Write the first 5 terms: 0, 12, 24, 36, 48.

Pattern B: 0, 12, 24, 36, 48
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Step 3 — Find the Relationship

Compare matching terms. 12 ÷ 4 = 3. Check: 24 ÷ 8 = 3. Check: 36 ÷ 12 = 3. Every Pattern B term is 3 times the matching Pattern A term. This makes sense because the rule for B (add 12) is 3 times the rule for A (add 4). Note: this times relationship works here because both patterns start at 0.

Pattern B = 3 × Pattern A
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Step 4 — Form Ordered Pairs

Pair up matching terms as (Pattern A, Pattern B). The first number goes on the x-axis, and the second number goes on the y-axis.

(0, 0), (4, 12), (8, 24), (12, 36), (16, 48)
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Step 5 — Plot on the Coordinate Plane

For each ordered pair, go right to the x-value and up to the y-value. Place a dot there. For (4, 12), go right 4 and up 12. For (8, 24), go right 8 and up 24. Connect the dots to see the straight line.

All five dots form a straight line!
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Step 6 — Explain the Relationship

Since 12 = 3 × 4, every time Pattern A grows by 4, Pattern B grows by 12 (which is 3 times as much). So Pattern B is always 3 times Pattern A.

The "Add 12" pattern grows 3 times faster than the "Add 4" pattern.

SECTION 7

Helpful Tips and Common Mistakes

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.

Tips vs. Common Mistakes

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KEY TAKEAWAY

Imagine you and a friend are saving money. You save $3 every week, and your friend saves $9 every week. Both of you start with $0. After 4 weeks, you have $12 and your friend has $36. Your friend always has 3 times as much as you because $9 is 3 × $3. This "times" relationship works because you both start at $0. If your friend had started with some money already saved, the relationship would be different!

SECTION 8

Connecting to Future Math

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

From 5th Grade Patterns to Middle School Algebra

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!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Pattern A uses the rule "Add 5" starting at 0, and Pattern B uses the rule "Add 10" starting at 0. Without writing out the patterns, what do you think the relationship is between Pattern B and Pattern A? Explain your thinking.

Show answer

Pattern B is always 2 times Pattern A. You can figure this out by comparing the two rules: 10 ÷ 5 = 2. Since both patterns start at 0 and Pattern B adds twice as much each time, every term in Pattern B will be double the matching term in Pattern A. For example, when Pattern A is at 5, Pattern B is at 10, and 10 = 2 × 5.

Question 2: BASIC CALCULATION

Pattern A: Add 4, start at 0. Pattern B: Add 12, start at 0. Write the first 5 terms of each pattern. Then write the 5 ordered pairs you would use to make a graph.

Show answer

Pattern A: 0, 4, 8, 12, 16. Pattern B: 0, 12, 24, 36, 48. The ordered pairs are (0, 0), (4, 12), (8, 24), (12, 36), and (16, 48). You match the terms in the same position: the 1st term of A pairs with the 1st term of B, the 2nd with the 2nd, and so on. The relationship is Pattern B = 3 × Pattern A because 12 ÷ 4 = 3.

Question 3: INTERMEDIATE

Pattern A: Add 3, start at 0. Pattern B: Add 9, start at 0. Generate the first 6 terms of each pattern. Find the relationship. If Pattern A's 10th term is 30, what is Pattern B's 10th term?

Show answer

Pattern A: 0, 3, 6, 9, 12, 15. Pattern B: 0, 9, 18, 27, 36, 45. The relationship is Pattern B = 3 × Pattern A, because 9 ÷ 3 = 3. You can check: 9 = 3 × 3, 18 = 3 × 6, 27 = 3 × 9, and so on. Since Pattern A's 10th term is 30, Pattern B's 10th term is 3 × 30 = 90. You don't need to list all the terms — just use the relationship!

Question 4: APPLIED

Maya earns $5 each week for chores. Her older sister Leah earns $15 each week. Both start with $0. Make a table showing their savings for weeks 0 through 5. Write the ordered pairs (Maya's savings, Leah's savings) and describe the relationship. If Maya has saved $40, how much has Leah saved?

Show answer

Maya's savings: $0, $5, $10, $15, $20, $25. Leah's savings: $0, $15, $30, $45, $60, $75. The ordered pairs are (0, 0), (5, 15), (10, 30), (15, 45), (20, 60), (25, 75). Leah's savings is always 3 times Maya's savings because $15 = 3 × $5. If Maya has saved $40, then Leah has saved 3 × $40 = $120. The relationship stays the same no matter how far you go in the pattern.

Question 5: CRITICAL THINKING

Jayden says, "If Pattern A uses 'Add 6' and Pattern B uses 'Add 6,' then Pattern B will always be equal to Pattern A." Is he correct? Now what if Pattern A starts at 0 but Pattern B starts at 5, and both use the rule 'Add 6'? Write the first 4 terms of each and explain what happens to the relationship.

Show answer

Jayden is correct for the first part. If both start at 0 and both add 6, the patterns are identical: 0, 6, 12, 18… and Pattern B = Pattern A. For the second part, Pattern A: 0, 6, 12, 18 and Pattern B: 5, 11, 17, 23. Now Pattern B is always 5 more than Pattern A (not a times relationship, but a plus relationship). The ordered pairs (0, 5), (6, 11), (12, 17), (18, 23) would still form a straight line on the graph, but the line would not go through (0, 0). This shows that the starting number matters and can change the relationship between the patterns.

SUMMARY

Lesson Summary

Summary

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!