Pattern Hunters Worksheet Pattern Hunters
Discovery Worksheet: Quadratic Sequences
Name: ____________________________________
Date: ____________________________________
Part 1: The Warm-Up
Calculate the differences between consecutive terms for the sequence below. Draw arrows between the numbers to show your work.
4
7
12
19
Show your difference calculations here:
Part 2: Video Analysis
Watch the second example in the video (\(a_n = n^2 + 3\)).
Terms Found:
4, 7, 12, 19
Differences:
3, 5, 7
Why is this NOT an arithmetic sequence? Use a keyword from the video.
Part 3: The Great Discovery
We know the "First Differences" (3, 5, 7) aren't the same. But what happens if we find the difference of the differences ?
Sequence A: \(3, 5, 7, \dots\) (The First Differences from above)
3 5 7
Find the "Second Difference"
Sequence B: \(1, 4, 9, 16\) (The formula is \(a_n = n^2\))
1 4 9 16
First Differences
Second Differences
Sequence C: \(2, 8, 18, 32\) (The formula is \(a_n = 2n^2\))
2 8 18 32
1st Diff
1st Diff
1st Diff
2nd Diff
2nd Diff
Pattern Summary
Complete the following statement based on your findings:
For all sequences involving an \(n^2\) term, the (first / second) difference is always .
Part 4: Predictions
1. If a sequence has a formula \(a_n = 3n^2 - 1\), what can you predict about its second differences?
2. If you find a sequence where the second difference is always 10, what kind of term must be in its formula?
Pattern Hunter Mission Complete
Difference Detectors Slides 10th Grade Mathematics
Pattern Hunters
Unlocking the Secrets of Quadratic Sequences
Differences
Patterns
Equations
Warm-Up Challenge
Analyze the following sequence. What do you notice about how it grows?
4
7
12
19
"Is there a common difference? Calculate it now!"
Expert Analysis
Embedded media
Sequence Breakdown
4, 7, 12, 19
The Problem
"Wait, there isn't a common difference? What's going on?"
The Second Level
What if we find the difference of the differences ?
4 7 12 19
3 5 7
2 2
It's CONSTANT!
The Math Behind the Magic
Key Connection
If the Second Difference is constant, the formula contains an:
\(n^2\)
Linear formulas (\(3n+4\)) have constant first differences.
Quadratic formulas (\(n^2+3\)) have constant second differences.
Final Prediction
A new sequence has first differences of 10, 15, 20, 25...
Q1
What is the constant second difference?
Q2
What power of n is in the equation?
"Submit your answers on the back of your worksheet."
Sequence Talk Cards Sequence Talk Cards
Deepening the Discovery: Quadratic Patterns
Card #1
The "Why" Question
"Why do we need a second difference to find a pattern in an \(n^2\) sequence, but only a first difference for a linear sequence?"
Hint: Think about the 'rate of change' or how the numbers accelerate.
Card #2
The Visualization
"If you graphed the first differences of \(a_n = n^2\), what shape would they make? (Look at 3, 5, 7...)"
Hint: Look at the values 3, 5, 7. How do they grow?
Card #3
The Constant Challenge
"If the second difference is 0 , what does that tell you about the sequence? Is it still quadratic?"
Think back: What kind of sequence has a second difference of 0?
Card #4
The Prediction Trick
"If the first three terms are 1, 4, 10 , can you find the second difference yet? What's the minimum number of terms we need?"
Try calculating: 1 -> 4 (diff 3), 4 -> 10 (diff 6). Now what?
Cut along the solid lines to create individual discussion prompts for small groups.
Pattern Discovery Teacher Guide Instructional Guide
Pattern Discovery Guide
A facilitation framework for identifying patterns in quadratic sequences for 10th Grade Math.
Duration
45 - 50 Minutes
Focus
Second Differences (\(n^2\))
Standards
HSF-IF.A.3
Lesson Flow
1
Warm-Up: The "Mystery" Growth (5 min)
Distribute the Pattern Hunters Worksheet . Students analyze 4, 7, 12, 19.
Teacher Prompt: "Last week we looked at sequences where we add the same number every time. Is that happening here? Prove it."
2
Video Viewing: Analysis (5 min)
Play the video starting at 1:34 . This shows the calculation of the sequence \(n^2+3\).
Key Moment (2:15): Pause here. Ask, "If these differences (3, 5, 7) aren't the same, what can we say for sure about the formula?" (Answer: It's not linear/arithmetic).
3
The Discovery: Second Level (20 min)
Direct students to "The Great Discovery" section of the worksheet. They find the differences of the differences.
Check for Understanding Ensure students realize that the 'second differences' for \(n^2\) sequences will always be 2 (or a multiple if there's a coefficient).
Common Misconception Students might subtract incorrectly (e.g., \(4-7=-3\) instead of \(7-4=3\)). Emphasize order: \(a_{n+1} - a_n\).
4
Guided Discussion (10 min)
Use the Sequence Talk Cards to facilitate small group or whole class debate.
Big Idea: A constant second difference means the rate of change is changing linearly, which only happens with quadratic functions.
Teacher Talk Cheat Sheet
Prompt A: The Constant 2
"We found that for \(n^2\) and \(n^2+3\), the second difference was 2. For \(2n^2\), it was 4. What do you think the second difference would be for \(3n^2\)? Why?"
Prompt B: Real World
"If gravity makes objects accelerate, and acceleration means a changing speed, do you think falling objects follow a linear or quadratic sequence?"
© 2026 Lenny Education • Pattern Discovery Unit
Pattern Hunters Answer Key Answer Key
Pattern Hunters Worksheet
Teacher Use Only
Part 1: The Warm-Up
4
+3
7
+5
12
+7
19
Part 2: Video Analysis
Keywords:
It is NOT arithmetic because it does not have a "common difference."
Part 3: The Discovery
Sequence A (Second Differences):
2
2
Sequence B (1, 4, 9, 16):
First Diffs:
357
Second Diffs:
22
Sequence C (2, 8, 18, 32):
First Diffs:
61014
Second Diffs:
44
For all sequences involving an \(n^2\) term, the SECOND difference is always CONSTANT (or the same).
Part 4: Predictions
1. Prediction for \(a_n = 3n^2 - 1\):
The second differences will be constant (specifically, they will be 6).
2. Constant second difference of 10:
The formula must include an \(n^2\) term (it is a quadratic sequence).
