Equations with Variables Scaffolded Packet
8th Grade Math • Unit 1: Equations Standard 8.EE.C.7b
The Codebreaker's Mission: Solving Multi-Step Equations
Name: ______________________
Date: ______________________
Period: ____________________
Standard 8.EE.C.7b Hook • The Codebreaker's Mystery CCSS 8.EE.7b
In computer programming, video game physics, and space navigation, secret unknown numbers are locked inside parentheses equations. Your mission today: use Distribution to unlock the parentheses doors, group your clues by Combining Like Terms, and use Inverse Operations to crack the code and isolate the mystery variable!
Color Code:
🟣 Coefficient (Number on letter) 🔵 Variable (Mystery letter: x, b) 🟠 Constant (Plain number: +5, -7)
🍕
Step-by-Step Story: What is Distribution?
Explanation
Think of distributing as a pizza delivery driver who MUST deliver to every friend inside the room:
Step 1: Driver Arrives
🚪 3
Outside driver is 3.
Step 2: Visits Friend 1
3 × x = 3x
Delivers to friend x.
Step 3: Visits Friend 2
3 × 4 = 12
Delivers to friend 4.
Step 4: Total Party
3x + 12
Nobody left out!
💡
Why Distribution IS Multiplication:
The outside number is simply the count of equal groups! In 3(x + 4), you have 3 groups of x (\(3 \times x = 3x\)) and 3 groups of 4 (\(3 \times 4 = 12\)). That's why distribution is pure multiplication!
a(b + c) = ab + ac
Step-by-Step Bridge: From Repeated Addition to Distribution
1. Repeated Add
(x + 4) + (x + 4) + (x + 4)
3 identical bags
2. Group Same Types
(x+x+x) + (4+4+4)
Combine like items
3. Multiply Groups
3(x) + 3(4)
Multiply each part
4. Distributed
3x + 12
Final expanded form!
Page 1 of 6 • Lesson Hook, Concept Guide & Mathematical Bridge
8th Grade Math • Lesson 7 Visual Scaffolds for Abstract Problems
Step-by-Step Visual Models: Scales & Tape Diagrams
Step-by-Step Balance Scale Model: Why Inverse Operations Work 3-Step Physics Model
Step 1: Balanced Equation
4x - 13 = 15
Left pan has 4x minus 13. Right pan has 15.
Step 2: Add +13 to BOTH Pans
4x = 28
Cancels the -13, scale remains perfectly level!
Step 3: Split into 4 Piles
x = 7
Divide both sides by 4 to find 1 pile (x).
Fraction Tape: \(\frac{x}{4} = -7\) 4 Equal Slices
• Step 1: Whole x is cut into 4 equal boxes.
• Step 2: Each box holds a value of -7.
-7
-7
-7
-7
Step 3: Multiply 4 × (-7): x = -28
Word Problem Tape: \(4p - 4 = 48\) Juwan's Pizza
• Step 1: 4 pizzas minus $4 coupon = $48.
• Step 2: Add back $4 coupon → 4 pizzas = $52.
p
p
p
p
Step 3: 52 ÷ 4: 1 pizza = $13
Step-by-Step Area Box Method: Outer Wall × Roof Step 1: Wall • Step 2: Roof • Step 3: Multiply
-3
-3(x) = -3x
-3(-5) = +15
\(\frac{1}{2}\)
\(\frac{1}{2}\)(2x) = 1x
\(\frac{1}{2}\)(-4) = -2
Page 2 of 6 • Step-by-Step Visuals: 3-Step Balance Scales & Tape Diagrams
8th Grade Math • Lesson 7 Part 2: Order of Operations & Solving Protocol
Order of Operations Model: PEMDAS vs. SADMEP
The Order of Operations Rule: Forward to Simplify • Reverse to Solve PEMDAS vs. Reverse SADMEP
⬇️ Forward PEMDAS (To Simplify) Building
P - Parentheses → Distribute to unlock ( )
E - Exponents
M / D - Multiplication & Division (L → R)
A / S - Addition & Subtraction → Combine Like Terms
⬆️ Reverse SADMEP (To Solve/Undo) Unwrapping
1. S / A - Undo + / - first (Inverse on constant)
2. D / M - Undo × / ÷ second (Divide coefficient)
3. E - Undo Exponents
4. P - Variable is isolated!
Guided Model 1 Solve: 7 + 4(x - 5) = 15
Order of Ops Step
Equation Line
What you do
1. PEMDAS: Distribute
7 + 4x - 20 = 15
4(x)=4x, 4(-5)=-20
2. PEMDAS: Combine
4x - 13 = 15
7 - 20 = -13
3. SADMEP: Undo -13
4x = 28
+ 13 to both sides
4. SADMEP: Divide 4
x = 7
÷ 4 on both sides
Guided Model 2 Simplify: 2x - 3(x - 5) + 3 = 10
1. Distribute -3
2x - 3x + 15 + 3 = 10
-3 × (-5) = +15
2. Combine Terms
-x + 18 = 10
Choice B Equivalent
Order of Ops Reminder: You cannot add x + 4 inside parentheses because they are not like terms. Distribution multiplies them to unlock the parentheses!
Page 3 of 6 • Order of Operations Model (PEMDAS vs. Reverse SADMEP)
8th Grade Math • Lesson 7 Part 3: Independent Practice
Independent Practice & Exit Ticket
Problem 1 3(2x - 1) = 8
1. Distribute: 6x - 3 = 8
2. Add 3: 6x = 11
3. Divide by 6: x = \(\frac{11}{6}\)
Answer: x = \(\frac{11}{6}\) (Choice A)
Problem 2 x + 2(x + 7) = 20
1. Distribute: x + 2x + 14 = 20
2. Combine: 3x + 14 = 20
3. Solve: 3x = 6 → x = 2
Answer: x = 2 (Choice C)
Problem 3 4b - 3(6 + 2b) = 12
1. Distribute: 4b - 18 - 6b = 12
2. Combine: -2b - 18 = 12
3. Solve: -2b = 30 → b = -15
Answer: b = -15
Problem 4 2(x - 1) + 3x = 3
1. Distribute: 2x - 2 + 3x = 3
2. Combine: 5x - 2 = 3
3. Solve: 5x = 5 → x = 1
Answer: x = 1 (Choice A)
Exit Ticket Assessment Demonstrate mastery on Lesson 7:
Target: 100%
Equation: 3 - 2(x + 5) = 10 Line-by-line solution
1. Distribute -2:
3 - 2x - 10 = 10
2. Combine 3 - 10:
-2x - 7 = 10
3 & 4. Inverse & Solve:
-2x = 17 → x = -8.5
Multiple Choice Check:
A. -11.5 B. 11.5 ✓ C. -8.5 D. 8.5
Page 4 of 6 • Independent Practice & Exit Ticket Assessment
Skill Builder • Foundation Workshop Number Line & Fluency Sprint
Number Line Practice & Multiplication Fluency
Number Line Practice Tracks: Draw or Follow the Hops Hop Right (+) • Hop Left (-)
Track 1: 4 × (+3) = 12 4 hops of +3 to the right
0 +3 +6 +9 +12
Track 2: 3 × (-5) = -15 3 hops of -5 to the left
0 -5 -10 -15
1-Minute Fluency Sprint Multiply rapidly to build automaticity:
Goal: 12/12 Correct
#1 3 × 6 = 18
#2 4 × (-5) = -20
#3 -2 × 7 = -14
#4 -3 × (-4) = +12
#5 5 × (-8) = -40
#6 -6 × (-3) = +18
#7 7 × 4 = 28
#8 -9 × 2 = -18
#9 -5 × (-5) = +25
#10 \(\frac{1}{2}\) × 16 = 8
#11 -4 × (-6) = +24
#12 8 × (-3) = -24
Quick Sign Reference: (+) × (+) = (+) (+) × (-) = (-) (-) × (+) = (-) (-) × (-) = (+)
Page 5 of 6 • Number Line Practice & Multiplication Fluency Sprint
8th Grade Math • Lesson 7 Extension Tiered Practice & Mastery
Tiered Mastery & Mixed Review Workspaces
A Tier 1: Multi-Step Equations with Negative Distribution Solve line-by-line
Problem 5 4 - 2(1 + x) = 12
Trap: Do NOT do 4 - 2 = 2! Distribute -2 first.
1. Distribute: 4 - 2 - 2x = 12
2. Combine: 2 - 2x = 12
3. Subtract 2: -2x = 10 → x = -5
Solution: x = -5
Problem 6 -3x - 2(x + 1) = 13
Distribute -2 to both x and 1.
1. Distribute: -3x - 2x - 2 = 13
2. Combine: -5x - 2 = 13
3. Add 2: -5x = 15 → x = -3
Solution: x = -3
B Tier 2: Mixed Review • Double Distribution & Expressions Simplify completely
Simplify: 8r - 5(2r + 3) + 6r - 12
1. Distribute -5: 8r - 10r - 15 + 6r - 12
2. Group r's: (8r - 10r + 6r) = 4r
3. Group #'s: (-15 - 12) = -27
Simplified = 4r - 27
Simplify: -3r - 4(2r - 2) + 3 - r
1. Distribute -4: -3r - 8r + 8 + 3 - r
2. Group r's: (-3r - 8r - r) = -12r
3. Group #'s: (+8 + 3) = +11
Simplified = -12r + 11
Tier 3: Olympiad Challenge Problem Deep Extension
What is the value of x in the equation: -(ax - 7) - 5(3 - ax) = -16 when a = -2?
1. Substitute a = -2: -(-2x - 7) - 5(3 - (-2x)) = -16 → -(-2x - 7) - 5(3 + 2x) = -16
2. Distribute both: 2x + 7 - 15 - 10x = -16
3. Combine like terms: -8x - 8 = -16 → -8x = -8 → x = 1
Page 6 of 6 • Tiered Practice & Mixed Review Workspaces
