Patterns and Relationships

🔢 Patterns and Relationships

Part 1 of 5 — Reading a Pattern and Finding Its Rule

Topics in This Part

🔑 Key Concept: A pattern is a list of numbers that follows a rule. Once you find the rule, you can keep the pattern going forever — and that one skill is the foundation for everything else in this lesson.

What Is a Number Pattern?

A number pattern (also called a sequence) is a list of numbers in order, where each number is found using the same rule.

Look at this pattern:

$3,\; 6,\; 9,\; 12,\; 15,\; \ldots$

To get from one number to the next, you add 3 every time. The number you add (or subtract, or multiply) the same way each step is the rule.

The dots ($\ldots$) mean "keep going." Because the rule is add 3, the next number is $15 + 3 = 18$.

💡 Tip: To find a rule, ask "What did I do to the number to get the next one?" Check it against every pair, not just the first two.

Concept Check 🎯

Extending a Pattern

Once you know the rule, you extend the pattern by applying the rule again and again.

Example: Continue $4,\ 7,\ 10,\ 13,\ \ldots$

The rule is add 3 (since $4+3=7$, $7+3=10$, $10+3=13$). So:

$13 + 3 = 16, \qquad 16 + 3 = 19, \qquad 19 + 3 = 22$

The pattern continues $16,\ 19,\ 22,\ \ldots$

Example (subtraction): Continue $30,\ 25,\ 20,\ 15,\ \ldots$

The rule is subtract 5, so the next terms are $15 - 5 = 10$ and $10 - 5 = 5$.

⚠️ Watch out: A rule must work for the whole pattern. If "add 4" works at the start but breaks later, it is not the rule — keep looking.

Extend the Pattern 🧮

Find the rule, then enter the next number in each pattern.

1) $6,\ 12,\ 18,\ 24,\ ?$ 2) $50,\ 45,\ 40,\ 35,\ ?$ 3) $1,\ 3,\ 9,\ 27,\ ?$

Find the Rule, Then Extend 🔽

Look at the pattern $2,\ 8,\ 32,\ 128,\ \ldots$

Where This Is Going

You can now read a pattern, find its rule, and extend it. That is the engine for the rest of the lesson.

In Part 2, you will run two patterns at the same time and discover that the two lists are connected — that connection is what we mean by a relationship.

🔢 Patterns and Relationships

Part 2 of 5 — Two Patterns at Once

🔑 The Idea: When you generate two patterns from two rules and start them at the same place, the terms line up in pairs. Comparing those pairs reveals a relationship between the two patterns.

Generating Two Patterns

Let's build two patterns side by side. Both start at $0$.

• Pattern A rule: add 2
• Pattern B rule: add 4

We apply each rule step by step and write the results in a table.

The numbers that sit in the same row are called corresponding terms. For example, $4$ in Pattern A corresponds to $8$ in Pattern B.

💡 Corresponding terms are partners — they were made at the same step. Always compare a number to its partner in the same row.

Concept Check 🎯

Use the table above (Pattern A = add 2, Pattern B = add 4).

Spotting the Relationship

Now compare each corresponding pair from the table:

Every term in Pattern B is twice the matching term in Pattern A. We say:

$\text{Pattern B} = 2 \times \text{Pattern A}$

This is the relationship between the patterns. It makes sense: Pattern B added $4$ each step while Pattern A added only $2$ — exactly double the step, so the terms stay double too.

🔑 Key Idea: To find the relationship, line up corresponding terms and ask "What do I do to the A-number to get the B-number?" — the same answer must work for every row.

Build the Table 🧮

Pattern C starts at $1$ with the rule add 3. Pattern D starts at $1$ with the rule add 6.

1) Step 3, Pattern D $= ?$ 2) Step 4, Pattern C $= ?$ 3) Step 4, Pattern D $= ?$

Name the Relationship 🔽

Two patterns both start at $0$. Pattern X uses add 3; Pattern Y uses add 9. Here are their corresponding terms:

🔢 Patterns and Relationships

Part 3 of 5 — Ordered Pairs

🔑 Why it matters: When two patterns line up in a table, each pair of corresponding terms can be written as an ordered pair $(x, y)$. Ordered pairs are the bridge from a table to a graph.

What Is an Ordered Pair?

An ordered pair is two numbers written inside parentheses, separated by a comma:

$(x,\; y) \quad=\quad (\text{first number},\; \text{second number})$

It is called ordered because the order matters — $(3, 5)$ is not the same as $(5, 3)$.

When we make ordered pairs from two patterns, we use:

• the first pattern for the first number (we call it $x$)
• the second pattern for the second number (we call it $y$)

Example: Pattern A and Pattern B from Part 2 had corresponding terms $4$ and $8$. As an ordered pair, that is $(4,\ 8)$.

⚠️ Order matters! The $x$-value (from the first pattern) always comes first, and the $y$-value (from the second pattern) comes second.

Concept Check 🎯

Making Ordered Pairs from Two Rules

You can build ordered pairs straight from two rules. Both patterns below start at $0$.

• First pattern ($x$) rule: add 1
• Second pattern ($y$) rule: add 5

Notice the relationship: each $y$ is 5 times its $x$. So the ordered pairs all follow $y = 5 \times x$.

💡 Shortcut: If you know the relationship (here, $y = 5x$), you can write the ordered pair for any $x$ without finishing the whole table.

Make the Pairs 🧮

Pattern $x$ starts at $0$ and adds $2$. Pattern $y$ starts at $0$ and adds $10$.

Fill in the missing numbers.

1) Step 3, the $y$-value $= ?$ 2) Step 4, the $x$-value $= ?$ 3) The relationship is $y = ? \times x$. Enter the missing number.

From Table to Pairs 🔽

A table has corresponding terms $x = 3$ and $y = 12$ in one row.

🔢 Patterns and Relationships

Part 4 of 5 — Graphing on the Coordinate Plane

🔑 The Big Payoff: When you plot the ordered pairs from two patterns on a coordinate plane, the points form a straight line. Seeing the line makes the relationship visible.

The Coordinate Plane

The coordinate plane is made of two number lines that cross at right angles:

• the $x$-axis runs left–right (horizontal)
• the $y$-axis runs up–down (vertical)
• they meet at the origin, the point $(0, 0)$

To plot an ordered pair $(x, y)$:

1. Start at the origin $(0, 0)$.
2. Move right along the $x$-axis by the $x$-value.
3. Move up along the $y$-axis by the $y$-value.
4. Make a dot.

Example: To plot $(3, 5)$, go right 3, then up 5, and mark the point.

⚠️ Right, then up. Always do the $x$-value (right) first and the $y$-value (up) second — just like the order in the pair.

Concept Check 🎯

Patterns Make Straight Lines

Let's graph the pairs from a two-pattern table. Both patterns start at $0$; $x$ adds $1$ and $y$ adds $2$, so the relationship is $y = 2 \times x$.

When you plot $(0,0), (1,2), (2,4), (3,6), (4,8)$, the dots line up perfectly — they make a straight line climbing to the right.

This always happens when both patterns grow by a steady amount each step:

🔑 Key Idea: A constant "add this much each step" rule for both patterns gives ordered pairs that fall on a straight line. The faster $y$ grows compared to $x$, the steeper the line.

Read the Graph Pairs 🧮

A pattern table gives the ordered pairs $(0,0),\ (1,3),\ (2,6),\ (3,9)$.

1) To plot $(3, 9)$, how many units do you move right? $?$ 2) To plot $(3, 9)$, how many units do you move up? $?$ 3) These pairs follow $y = ? \times x$. Enter the missing number.

Plot It Correctly 🔽

You are plotting the ordered pair $(5, 2)$ on the coordinate plane.

🔢 Patterns and Relationships

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) find a rule and extend a pattern, (2) generate two patterns and compare corresponding terms, (3) write ordered pairs, and (4) plot them to reveal a straight line. Let's put it all together.

Quick Reference

💡 Big picture: A rule builds a pattern → two patterns make ordered pairs → ordered pairs make a graph. Each idea is built from the one before it.

⚠️ Two reminders: order matters in $(x, y)$, and a relationship must hold for every corresponding pair, not just one.

Mixed Practice 🎯

Put It Together 🧮

Pattern $x$ starts at $0$ and adds $3$. Pattern $y$ starts at $0$ and adds $9$.

1) When $x = 6$, what is the corresponding $y$? $?$ 2) Written as an ordered pair, the row is $(6,\ ?)$. Enter the missing $y$-coordinate. 3) The relationship is $y = ? \times x$. Enter the missing number.

Exit Quiz ✅

Answer all three to finish the lesson.