Two Numerical Patterns: Generate, Compare, and Graph

In this lesson:

• Generate two sequences side by side in a table
• Discover the relationship between corresponding terms
• Graph ordered pairs and describe the pattern

What You Will Learn in This Lesson

By the end of this lesson, you should be able to:

1. Generate two sequences from two rules in a table
2. Identify the relationship between corresponding terms
3. Form ordered pairs and plot them on a coordinate plane
4. Predict later terms using the discovered relationship

What Happens When Two Rules Run Together?

You know how to follow one rule:

• Start at 0, add 3 → 0, 3, 6, 9, 12, ...

Today: two rules at the same time

• Rule A: Start at 0, add 3
• Rule B: Start at 0, add 6

Organizing Two Sequences in One Table

Each row pairs terms in the same position — these are corresponding terms.

Building the Add Three and Add Six Table

Start at 0 for both. Apply each rule step by step:

Completing the Add Three and Six Table

Six rows completed. Look across — how do the columns relate?

Your Turn: Build the Add Five Table

Add 5 from 0 / Add 10 from 0.

Fill in six rows, then advance.

How Do Corresponding Terms Compare Here?

Look across each row. How does Seq B relate to Seq A?

Compare Across Rows, Not Down Columns

• Down a column: "Goes up by 3" — that is the rule
• Across a row: "6 is double 3" — that is the relationship

The relationship connects Sequence A to Sequence B, position by position.

Testing Additive Versus Multiplicative Relationships

• Additive: 6 − 3 = 3, 12 − 6 = 6, 18 − 9 = 9 — differences change
• Multiplicative: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 18 ÷ 9 = 2 — ratio always 2

Check every row, not just the first.

The Multiplicative Relationship Stays Constant

• Differences: 3, 6, 9, 12, 15 — changing
• Ratios: 2, 2, 2, 2, 2 — constant

Seq B is always 2 times Seq A.

Your Turn: Find the Add Two Relationship

Check: 8 ÷ 2 = ?, 16 ÷ 4 = ?, 24 ÷ 6 = ?

Verify every row, then advance.

Checking the Multiplier Across Rule Pairs

Add 5 / Add 10: 10 ÷ 5 = 2, 20 ÷ 10 = 2 — always 2

Add 2 / Add 8: 8 ÷ 2 = 4, 16 ÷ 4 = 4 — always 4

The multiplier depends on the rules, not individual terms.

Turning Table Rows into Ordered Pairs

= right, = up

Plotting Add Three and Add Six Points

All six points form a straight line through the origin.

The Straight Line Confirms the Pattern

The line confirms: B is always 2 times A.

• Point off the line? Recheck your table or plotting
• Use a ruler — all points should touch the edge
• The line is a visual picture of the relationship

Predicting the Next Point Using the Relationship

The relationship is "always double." If Seq A has a new term of 18:

• = 18, so = 18 × 2 = 36
• The next point is (18, 36)

Predict without extending both sequences term by term.

Comparing Steepness of Two Different Multipliers

• Times 2 (Add 3 / Add 6): less steep line
• Times 4 (Add 2 / Add 8): steeper line

A bigger multiplier means a steeper line.

Quick Check: Which Multiplier Gives a Steeper Graph?

Which graph is steeper: a "times 3" or a "times 2" relationship?

Why?

Think about how far up each line goes for every step right.

How the Multiplier Controls the Line Steepness

• Times 2: go up 2 for every 1 right — gradual
• Times 3: go up 3 for every 1 right — steeper
• Times 4: go up 4 for every 1 right — steepest

The multiplier tells you how fast Seq B grows compared to Seq A.

Now Explore New Rule Pairs and Their Relationships

9 ÷ 3 = 3, 18 ÷ 6 = 3 — B is always 3 times A

The Multiplier Equals the Ratio of Rules

Multiplier = Rule B ÷ Rule A

Prediction Challenge: Add Seven and Add Twenty-One

Can you predict the multiplier without a table?

• Rule A: Add 7
• Rule B: Add 21
• Both start at 0

Predicted multiplier: 21 ÷ 7 = ?

Predict, then verify with two rows.

Practice: Tables, Relationships, and Predictions

Problem 1: Add 4 / Add 12, start at 0.
Build a five-row table. State the relationship.

Problem 2: Add 6 / Add 18, start at 0.
Predict the multiplier, then verify.

Problem 3: Ordered pairs: (0, 0), (5, 15), (10, 30).
What is the multiplier? What rules work?

Answers to the Practice Problems Above

Problem 1: (0,0), (4,12), (8,24), (12,36), (16,48)
B is always 3 times A

Problem 2: Predicted: 18 ÷ 6 = 3. Verified.

Problem 3: Multiplier = 15 ÷ 5 = 3
Possible rules: Add 5 / Add 15

Summary: What You Learned About Two Patterns

• Two-column table: organize sequences side by side
• Corresponding terms: compare across rows, not down
• Multiplicative relationship: ratio stays constant
• Seq A = , Seq B = — do not swap
• Points form a straight line through the origin

Where These Patterns Lead You Next

In upcoming lessons:

• Graph real-world data on coordinate planes (5.G.A.2)
• Explore ratios and proportional relationships (Grade 6)
• Write equations for linear relationships (Grade 8)

Today's patterns are the foundation for all of these!