Radical Chain Link Strips
Paper Chain Activity
RADICAL CHAIN RELAY • STRIPS
Groups of 3–4: Cut strips, thread each new link through the previous ring, and tape/glue to build a 20-link loop!
Part 1 • Strips A–J
GLUE / TAPE
I HAVE \(2\sqrt{3}\)
STRIP A
SEEKING \(\sqrt{72}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(6\sqrt{3}\)
STRIP B
SEEKING \(\sqrt{80}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(3\sqrt{6}\)
STRIP C
SEEKING \(\sqrt{180}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(3\sqrt{5}\)
STRIP D
SEEKING \(\sqrt{128}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(12\sqrt{2}\)
STRIP E
SEEKING \(\sqrt{175}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(5\sqrt{3}\)
STRIP F
SEEKING \(\sqrt{200}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(6\sqrt{5}\)
STRIP G
SEEKING \(\sqrt{63}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(8\sqrt{2}\)
STRIP H
SEEKING \(\sqrt{150}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(5\sqrt{7}\)
STRIP I
SEEKING \(\sqrt{40}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(4\sqrt{3}\)
STRIP J
SEEKING \(\sqrt{90}\)
ALIGN & TAPE
Cut along dashed border lines. Each strip forms one link of your physical paper chain!
Page 1 of 3
Paper Chain Activity
RADICAL CHAIN RELAY • STRIPS
Thread each new strip through the previous loop before taping. Look for the largest perfect square factor!
Part 2 • Strips K–T
GLUE / TAPE
I HAVE \(4\sqrt{5}\)
STRIP K
SEEKING \(\sqrt{98}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(4\sqrt{7}\)
STRIP L
SEEKING \(\sqrt{54}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(2\sqrt{6}\)
STRIP M
SEEKING \(\sqrt{288}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(6\sqrt{2}\)
STRIP N
SEEKING \(\sqrt{75}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(3\sqrt{10}\)
STRIP O
SEEKING \(\sqrt{12}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(5\sqrt{6}\)
STRIP P
SEEKING \(\sqrt{112}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(7\sqrt{2}\)
STRIP Q
SEEKING \(\sqrt{45}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(2\sqrt{10}\)
STRIP R
SEEKING \(\sqrt{48}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(10\sqrt{2}\)
STRIP S
SEEKING \(\sqrt{108}\)
ALIGN & TAPE
GLUE / TAPE
I HAVE \(3\sqrt{7}\)
STRIP T
SEEKING \(\sqrt{24}\)
ALIGN & TAPE
Loop Check: Strip 20 loops back through Strip 1 to close the circular chain garland!
Page 2 of 3
Student Recording Sheet
PAPER CHAIN ASSEMBLY & PROOF TRACKER
Physical Loop Verification
Group / Team Name
Class Period
Team Members (3–4 Students)
1. Cut Strips
Cut out all 20 link strips. Divide them equally among teammates.
2. Make Ring 1
Pick any starting strip. Tape its ends together into your first ring.
3. Thread & Match
Simplify the radical, slide matching strip through ring, and tape.
4. Close Loop!
Slide Link 20 through Link 1 and tape to form one giant paper circle!
Part 1: Your 20-Link Paper Chain Sequence
Record the strip letters in the order connected
LINK 1 (START) Strip __
Link 2 Strip __
Link 3 Strip __
Link 4 Strip __
Link 5 Strip __
Link 6 Strip __
Link 7 Strip __
Link 8 Strip __
Link 9 Strip __
Link 10 Strip __
Link 11 Strip __
Link 12 Strip __
Link 13 Strip __
Link 14 Strip __
Link 15 Strip __
Link 16 Strip __
Link 17 Strip __
Link 18 Strip __
Link 19 Strip __
LOOPS TO LINK 1! Strip __
Part 2: Show Your Work (Prove Any 4 Radical Simplifications)
\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)
Proof 1: Strip: [ ]
Radical: \(\sqrt{\hspace{20px}}\) Factorization: \(\sqrt{\hspace{12px}\cdot\hspace{12px}} =\)
Simplest Form: ____________
Proof 2: Strip: [ ]
Radical: \(\sqrt{\hspace{20px}}\) Factorization: \(\sqrt{\hspace{12px}\cdot\hspace{12px}} =\)
Simplest Form: ____________
Proof 3: Strip: [ ]
Radical: \(\sqrt{\hspace{20px}}\) Factorization: \(\sqrt{\hspace{12px}\cdot\hspace{12px}} =\)
Simplest Form: ____________
Proof 4: Strip: [ ]
Radical: \(\sqrt{\hspace{20px}}\) Factorization: \(\sqrt{\hspace{12px}\cdot\hspace{12px}} =\)
Simplest Form: ____________
Part 3: Team Reflection & Error Analysis
1. Incomplete Factorization Alert:
A teammate simplified \(\sqrt{48}\) as \(2\sqrt{12}\). Why does this stall your paper chain, and what is the true simplest form?
2. Largest Factor Discovery:
Which strip in your chain had the largest perfect square factor? What square factor unlocked it?
Teacher Inspection: [ ] Closed 20-Link Ring Garland Verified [ ] Proof Work Checked
Teacher Signature: _______________________
Radical Chain Answer Key
Teacher Reference
RADICAL CHAIN RELAY • ANSWER KEY
20-Link Paper Chain
Quick-check strip link sequence, complete step-by-step factorization breakdown, reflection answers, and paper chain assembly tips.
Master 20-Link Paper Chain Sequence (Continuous Cycle)
5-Second Physical Check
Because this is a continuous interlocking loop, student groups can start their paper chain with any strip. Check that their physical chain rings follow this cyclical order:
Strip A → N → F → S → B → K → Q → D → H → P → L → C → G → T → M → E → I → R → J → O → (loops to Strip A)
Full Solution Sequence & Factorization Table
| Strip |
I Have |
Seeking → Simplification |
Next |
| A |
\(2\sqrt{3}\) |
\(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\) |
N |
| N |
\(6\sqrt{2}\) |
\(\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}\) |
F |
| F |
\(5\sqrt{3}\) |
\(\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}\) |
S |
| S |
\(10\sqrt{2}\) |
\(\sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3}\) |
B |
| B |
\(6\sqrt{3}\) |
\(\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}\) |
K |
| K |
\(4\sqrt{5}\) |
\(\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}\) |
Q |
| Q |
\(7\sqrt{2}\) |
\(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\) |
D |
| D |
\(3\sqrt{5}\) |
\(\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt{2}\) |
H |
| H |
\(8\sqrt{2}\) |
\(\sqrt{150} = \sqrt{25 \cdot 6} = 5\sqrt{6}\) |
P |
| P |
\(5\sqrt{6}\) |
\(\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}\) |
L |
| Strip |
I Have |
Seeking → Simplification |
Next |
| L |
\(4\sqrt{7}\) |
\(\sqrt{54} = \sqrt{9 \cdot 6} = 3\sqrt{6}\) |
C |
| C |
\(3\sqrt{6}\) |
\(\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}\) |
G |
| G |
\(6\sqrt{5}\) |
\(\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}\) |
T |
| T |
\(3\sqrt{7}\) |
\(\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}\) |
M |
| M |
\(2\sqrt{6}\) |
\(\sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2}\) |
E |
| E |
\(12\sqrt{2}\) |
\(\sqrt{175} = \sqrt{25 \cdot 7} = 5\sqrt{7}\) |
I |
| I |
\(5\sqrt{7}\) |
\(\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}\) |
R |
| R |
\(2\sqrt{10}\) |
\(\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\) |
J |
| J |
\(4\sqrt{3}\) |
\(\sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}\) |
O |
| O |
\(3\sqrt{10}\) |
\(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\) |
A |
Part 3: Student Debrief Key & Misconception Analysis
1. Largest Factor Alert (\(\sqrt{48}\)):
Why \(2\sqrt{12}\) stalls the chain: 12 is not square-free (\(12 = 4 \cdot 3\)). No strip has \(2\sqrt{12}\).
True Simplest Form: \(\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\) (or \(2\sqrt{4 \cdot 3} = 2(2\sqrt{3}) = 4\sqrt{3}\)). Links to Strip J!
2. Largest Factor in Chain Strategy:
Largest Factor: \(144\) in \(\sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2}\) (Strip M → E). (Also \(100\) in \(\sqrt{200}\), \(64\) in \(\sqrt{128}\)).
Recommended Strategy: Test prime quotients (\(\div 2, \div 3, \div 5\)) to identify if the other factor is a perfect square.
Assembly & Tape Tip
Clear tape works faster than liquid glue. Have groups pre-cut 1-inch pieces of tape. Strips form ~2.5-inch diameter rings that easily interlock.
Classroom Display
Completed 20-link circular garlands can be hung from the ceiling or whiteboard hooks to visually celebrate team mastery of radical simplification!
Fast-Finisher Extension
Challenge early finishers to construct a bonus "golden link" with a radical expression (e.g. \(\sqrt{300} = 10\sqrt{3}\)) that can splice into the loop.
Algebra 1 • Unit: Radicals & Irrational Numbers • Continuous Paper Chain Key Page 1 of 1