Pattern Partners Worksheet
Unit: Sequence Showdown
PATTERN PARTNERS
Partner A Name
Partner B Name
Class/Date
Mission Brief: Fold or focus on your own side first! Count the blocks in each step of your sequences, write down the numbers, and decide if the growth is Arithmetic (+/-) or Geometric (x/÷). Work out the pattern rule and extend it to find the next three term counts.
FOLD OR CUT HERE
A
Partner A's Decoders
Sequence A1 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Sequence A2 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Sequence A3 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Sequence A4 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
B
Partner B's Decoders
Sequence B1 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Sequence B2 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Sequence B3 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Sequence B4 (Figure blocks) Visual Cipher
S1
S2
S3
S4
Counts: , , ,
Arithmetic Geometric
Constant Change (d or r):
Next 3 Term Counts: __, __, __
Unit 3: Linear & Exponential Growth Ready to combine forces? Turn over! Page 1 of 2
Phase 2: Combined Analysis
THE JOINT CODEBREAKER
COOPERATIVE CRITICAL THINKING
1
The Pattern Matchmaker
Compare your decoded sequences from Page 1. Match each of Partner A's sequences to the Partner B sequence that has a **similar type and rate of growth** (e.g., they add or multiply by the same value).
| Partner A Sequence | Partner B Match | The Shared Secret (What makes them match?) |
|---|
| A1 (2, 5, 8...) | Sequence: _________ | Both are and have a constant change of |
| A2 (1, 2, 4...) | Sequence: _________ | Both are and have a constant change of |
| A3 (9, 7, 5...) | Sequence: _________ | Both are and have a constant change of |
| A4 (16, 8, 4...) | Sequence: _________ | Both are and have a constant change of |
2
Conceptual Showdown
Challenge 1: The Race to 1,000 (Comparing Growth Rates)
Compare A2 (1, 2, 4, 8...) and B2 (3, 6, 12, 24...). Both sequences multiply by 2. Which sequence will cross 1,000 first, and why? Describe the mathematical relationship between their terms.
Challenge 2: The Infinite Half (Geometric Decay)
Look at A4 (16, 8, 4...) and B4 (24, 12, 6...). Both decay by multiplying by 1/2. If you continue this pattern forever, will either sequence ever reach 0 or a negative number? Why or why not?
Challenge 3: The Big Synthesis (Arithmetic vs. Geometric)
In your own words, what is the core visual and numerical difference in how Arithmetic sequences grow/shrink compared to how Geometric sequences grow/shrink? (Think addition vs. multiplication).
Unit 3: Linear & Exponential Growth Great collaboration! Show your teacher when complete. Page 2 of 2
Pattern Partners Teacher Guide
Sequence Showdown • Professional Resource
TEACHER GUIDE
Facilitation & Keys
Cooperative Math Structure
The Pattern Partners Worksheet uses a highly visual, physical block representation of sequences. By counting physical block groups rather than starting with abstract numbers, students develop a more tactile, intuitive understanding of arithmetic and geometric sequences.
Recommended Lesson Flow (45-50 Minutes)
01: Launch (10m) Review sequences on the board. Define Arithmetic (+ / -) vs Geometric (× / ÷) with block drawings.
02: Solo (15m) Students work independently on their side. Encourage physical counting and translation of shapes to numerical counts first.
03: Team (15m) Partners unfold pages, sit face-to-face, and match sequences based on similar block-growth behaviors on Page 2.
04: Debrief (10m) Debrief as a class. Highlight the explosive nature of multiplication and discuss the asymptote concept in Challenge 2.
Common Misconceptions
- Division in Figures: In Sequences A4 and B4, students might confuse decay (halving the blocks) with simple subtraction. Emphasize that in subtraction, the step-by-step change must be constant (like subtracting 2). Halving subtracting different amounts (e.g. -8, then -4, then -2) is Geometric because it uses a constant ratio of 1/2.
- Growth Comparison: Students often incorrectly assume that because A2 and B2 both double (×2), the one that starts smaller will catch up. Emphasize that a higher starting value maintains a proportional lead.
Guiding Questions
- "Look at A1 and B1. How many blocks are we adding at each step? Why do they have different block totals if their rate of growth is identical?"
- "In Challenge 2, if we keep taking half of a pile of blocks, will we ever have 0 blocks? What happens mathematically when we divide a positive decimal by 2?"
- "How does looking at the physical shapes make it easier to see the difference between straight-line adding and curved multiplying?"
Conceptual Connections (CCSS Alignment)
F-IF.3 (Recognize Sequences) Establishes sequences as functions, defined recursively or explicitly, whose domain is a subset of the integers. This lesson focuses on translating visual patterns into recursive values.
F-LE.1 (Linear vs. Exponential Growth) Builds the fundamental distinction between linear growth (equal differences over equal intervals) and exponential growth (equal ratios over equal intervals).
Unit 3: Sequences Facilitation Solutions Key on Page 2 Page 1 of 2
Sequence Showdown • Reference Key
ANSWER KEY
MASTER KEY
Page 1 Solutions: Solo Decoder Sequences
Partner A Solutions
A1: Counts: 2, 5, 8, 11 → Arithmetic • Rule: +3 • Next: 14, 17, 20
A2: Counts: 1, 2, 4, 8 → Geometric • Rule: ×2 • Next: 16, 32, 64
A3: Counts: 9, 7, 5, 3 → Arithmetic • Rule: -2 • Next: 1, -1, -3
A4: Counts: 16, 8, 4, 2 → Geometric • Rule: ×1/2 • Next: 1, 0.5, 0.25
Partner B Solutions
B1: Counts: 4, 7, 10, 13 → Arithmetic • Rule: +3 • Next: 16, 19, 22
B2: Counts: 3, 6, 12, 24 → Geometric • Rule: ×2 • Next: 48, 96, 192
B3: Counts: 10, 8, 6, 4 → Arithmetic • Rule: -2 • Next: 2, 0, -2
B4: Counts: 24, 12, 6, 3 → Geometric • Rule: ×1/2 • Next: 1.5, 0.75, 0.375
Page 2: Matchmaker Table Key
| Partner A Sequence | Partner B Match | The Shared Secret (Reason) |
|---|
| A1 (2, 5, 8...) | Sequence B1 | Both are Arithmetic with a constant change of +3 (adding 3 blocks). |
| A2 (1, 2, 4...) | Sequence B2 | Both are Geometric with a constant change of ×2 (doubling the blocks). |
| A3 (9, 7, 5...) | Sequence B3 | Both are Arithmetic with a constant change of -2 (subtracting 2 blocks). |
| A4 (16, 8, 4...) | Sequence B4 | Both are Geometric with a constant change of ×1/2 (halving the blocks). |
Page 2: Conceptual Showdown Keys
Challenge 1: The Race to 1,000 (Growth Comparison)
Correct: Sequence B2 will cross 1,000 first. Why: B2 starts at 3, while A2 starts at 1. Since both double (×2) each term, every term of B2 is exactly three times larger than the corresponding term of A2. B2 crosses 1,000 on Term 10 (which has $3 \times 2^9 = 1,536$ blocks), whereas A2 crosses 1,000 on Term 11 ($1 \times 2^{10} = 1,024$ blocks).
Challenge 2: The Infinite Half (Geometric Decay)
Correct: Neither sequence will ever reach 0 or become negative. Why: Halving a positive number (multiplying by 1/2) always results in a positive number, no matter how microscopically small it gets (e.g., halving 0.000001 yields 0.0000005, which is still strictly greater than 0). This conceptually introduces the exponential horizontal asymptote.
Challenge 3: The Big Synthesis (Arithmetic vs. Geometric Growth)
Key Points: Arithmetic growth/decay is additive, adding or subtracting a constant value, yielding a steady, linear straight line. Geometric growth/decay is multiplicative, multiplying or dividing by a constant ratio, yielding rapid curved exponential changes (runaway expansion or rapid flattening).
Unit 3: Sequences Key Classroom-ready solutions Page 2 of 2
