Decoder Slides Pattern Decoders
Can you solve the mystery of the hidden rule?
Mission: CRITICAL
The Cold Case
Evidence #001
What's the Secret?
"It looks like subtraction... but wait, look at the second row! The rule changed! Or did it?"
To solve this case, we need to meet our two prime suspects...
The Two Suspects
Suspect: ADDITIVE
Changes by adding or subtracting a constant value.
y = x + a
"Wait for me! Let's stay the same distance apart."
Suspect: MULTIPLICATIVE
Changes by multiplying by a constant value (the rate).
y = ax
"Catch me if you can! I'm growing fast!"
Clue 1: The Table
Case Study
x (Input) y (Output) 1 4 2 8 3 12 4 16
Ask Yourself:
1 Can I add the same number to every x to get y?
2 Can I multiply every x by the same number to get y?
"Detective, what is the rule for this table?"
Expert Level: Hidden Multipliers
Warning!
Sometimes the multiplier is a decimal . Does it still count as multiplicative?
YES!
The Decimal Trap:
"Wait! y is smaller than x. How can that be multiplication?"
y = 0.5 \(\cdot\) x
Multiplying by 0.5 is the same as taking half .
"Multiplicative rules don't always make things bigger!"
Clue 2: The Graph
Origin Check
0 x y
How to Spot Them:
Multiplicative: Always passes through (0,0).Additive: Starts somewhere else on the y-axis.
"The origin is the smoking gun!"
Time to Decode!
Guided Notes
Fill in your field manual as we walk through the first suspects together.
Problem Set
Apply your detective skills to hunt down the rules in new data.
Sort It Out
Categorize tables and graphs by their secret relationships.
Guided Discovery Notes Field Notes: Pattern Decoder
TEKS 5.4D: Identifying Additive vs. Multiplicative Relationships
Agent Name:
Date:
The Mission Goal
How can I tell the difference between an additive pattern and a multiplicative pattern just by looking at a table or a graph?
Additive Relationship
THE RULE:
y = x + ____
TABLE CLUE:
You ________ the same number to every \(x\) to get \(y\).
GRAPH CLUE:
Does not pass through the _________________ (0,0).
Multiplicative Relationship
THE RULE:
y = ____ \(\cdot\) x
TABLE CLUE:
You ________ the same number by every \(x\) to get \(y\).
GRAPH CLUE:
Always passes through the _________________ (0,0).
Investigation #1: The Table Suspect
x (Input) y (Output) 2 10 4 20 6 30
Step 1: Test for Addition
2 + ____ = 10 4 + ____ = 20
Is it the same number? ________
Step 2: Test for Multiplication
2 \(\cdot\) ____ = 10 4 \(\cdot\) ____ = 20
Is it the same number? ________
VERDICT: This is a _________________________ pattern.
Investigation #2: The Graph Suspect
0 x y
Exhibit B: Line Analysis
Look closely at the graph. Locate the Origin (0,0) .
Check the Origin:
Does the line pass through (0,0)?
YES
NO
THE DECODE:
Because it does not start at zero, this is an _________________________ pattern.
Expert Investigation: The Decimal Secret
Wait! \(y\) is getting smaller!
Is it additive? \(4 - 2 = 2\), but \(10 - 2 = 8\). NO.
Test for decimal multiplication by dividing: y \(\div\) x
Rule Hunt Worksheet Rule Hunt
Special Agent Training Level 1
Agent:
Briefing: Decode the Patterns
Each table and graph below contains a secret relationship. Your job is to decode the rule and identify if it is additive or multiplicative . Use the scaffold boxes to test your theories!
1
The Number Cruncher
Test Your Clues
Add Test:
3 + ____ = 9
5 + ____ = 15
Multiply Test:
3 \(\cdot\) ____ = 9
5 \(\cdot\) ____ = 15
Rule Equation: y = ____________
Verdict
Additive
Multiplicative
2
The Visual Clue
0 x y
Analyze the Origin
Does the line pass through the origin (0,0)?
YES
NO
Final Verdict
This pattern type is:
REASON: __________________________________________________
3
The Decimal Ghost
Detective Tip
If the numbers are getting smaller , but it's not subtraction, the multiplier is a decimal between 0 and 1!
Test for the Decimal Relationship:
Row 1: 1 \(\div\) 4 =
Row 2: 2 \(\div\) 8 =
Is the multiplier consistent? ________
Case Verdict
Equation Rule: y = ________ \(\cdot\) x
Additive
Multiplicative
Final Challenge: Complete the Set
Look at the rule below. Generate the missing numbers in the table and identify the pattern type.
Secret Rule: y = x + 7
Fill in the missing coordinates:
Rule Sorter Activity Rule Sort: The Big Debate
Case Classification Mat
Additive relationship
\(y = x + a\)
Multiplicative relationship
\(y = ax\)
Sorting Evidence Cards
Cut along the dotted lines
Evidence Card A
<table class="w-full border-collapse border border-slate-300 text-sm font-mono mb-4"><tbody><tr class="bg-slate-50"><th class="p-2 border">x</th><td class="p-2 border text-center">2</td><td class="p-2 border text-center">4</td><td class="p-2 border text-center">6</td></tr><tr><th class="p-2 border">y</th><td class="p-2 border text-center font-bold">5</td><td class="p-2 border text-center font-bold">7</td><td class="p-2 border text-center font-bold">9</td></tr></tbody></table>
Evidence Card B
<table class="w-full border-collapse border border-slate-300 text-sm font-mono mb-4"><tbody><tr class="bg-slate-50"><th class="p-2 border">x</th><td class="p-2 border text-center">2</td><td class="p-2 border text-center">4</td><td class="p-2 border text-center">6</td></tr><tr><th class="p-2 border">y</th><td class="p-2 border text-center font-bold">4</td><td class="p-2 border text-center font-bold">8</td><td class="p-2 border text-center font-bold">12</td></tr></tbody></table>
Evidence Card C
0
Passes through origin
Evidence Card D
0
Does not pass through origin
Evidence Card E
y = x + 12
Evidence Card F
y = 0.25x
Mystery Multiplier
Evidence Card G
<table class="w-full border-collapse border border-slate-300 text-sm font-mono mb-4"><tbody><tr class="bg-slate-50"><th class="p-2 border">x</th><td class="p-2 border text-center">10</td><td class="p-2 border text-center">20</td><td class="p-2 border text-center">30</td></tr><tr><th class="p-2 border">y</th><td class="p-2 border text-center font-bold">15</td><td class="p-2 border text-center font-bold">30</td><td class="p-2 border text-center font-bold">45</td></tr></tbody></table>
Multiply by 1.5?
Evidence Card H
y = 0.5x Pattern Teacher Guide Teacher Guide: The Great Pattern Debate
Pacing, Answers, & Mission Briefing
Small Group (25 Min)
Pacing Breakdown
0-5 MIN
Slide Hook & Suspect Intro (Include Decimal Slide)
5-15 MIN
Guided Discovery Notes (Investigations 1-3)
15-22 MIN
Rule Hunt Worksheet (Page 2 Focus: Decimals)
22-25 MIN
Rule Sort Activity (Exit Ticket Check)
High-Level Questions
"In Investigation #3, why did some of you think it was subtraction at first?" (Address the 'smaller number' misconception).
"If I have a multiplicative relationship with a multiplier like 0.5, where does the graph start?" (Reinforce (0,0)).
"How can we use division (\(y \div x\)) to prove a pattern is multiplicative when the whole number test fails?"
Master Answer Keys
Rule Hunt Worksheet
Problem 1
\(y = 3x\) (Multiplicative)
Problem 2
Origin: NO. \(y = x + 3\) (Additive)
Problem 3 (Decimals)
y = 0.25x (Multiplicative)
Multiplier: 0.25 or 1/4
Final Challenge
(2,9), (4,11), (6,13). Type: Additive
Rule Sorter Cards
Additive
Card A (Table, \(+3\))
Card D (Graph, non-origin)
Card E (\(y = x + 12\))
Multiplicative
Card B (\(\cdot 2\))
Card C (Origin check)
Card F (\(y = 0.25x\))
Card G (\(y = 1.5x\))
Card H (\(y = 0.5x\))
Critical Mission Intel
The "Larger" Fallacy
Students assume multiplication must result in a larger product. They will often misidentify \(y = 0.5x\) as subtraction (\(x - a\)) because they only look at the first row. Force the multi-row check.
Division as a Tool
Teach students to use In-to-Out Division (\(y \div x\)). If they get the same decimal quotient for every pair, they have found a multiplicative relationship.
