Equation Rescue Worksheet
Resource Math Reteach Grade 7 • Two-Step Equations
Equation Rescue
Name:
Date:
Period:
Goal: Isolate the variable!
Key Math Vocabulary
Reference these words as you solve!
Variable x
The mystery letter we want to get completely alone on one side of the equals sign.
Coefficient 3x
The number stuck to the variable by multiplication (e.g., in \(3x\), the coefficient is 3).
Inverse Operation + ↔ -
The opposite operation that "undoes" a step. We use them to peel away numbers!
Inverse Pairs: Addition ( + ) undoes Subtraction ( - ) • Multiplication ( • ) undoes Division ( ÷ )
The 2-Step Solving Rule (Reverse PEMDAS)
Work from the outside in! Always undo the constant first, then undo the coefficient.
Step 1: Undo + or - Step 2: Undo × or ÷
I Do
Teacher Model: Watch How We Undo Each Step
Solve: \(2x + 4 = 12\)
1 Undo the Addition
The constant is + 4. The inverse of \(+ 4\) is \(- 4\) on both sides.
\(2x + 4 = 12\)
\(\quad - 4 \quad - 4\)
\(2x = 8\)
2 Undo the Multiplication
\(2x\) means \(2 \cdot x\). The inverse of multiplying by 2 is divide by 2.
\(\frac{2x}{2} = \frac{8}{2}\)
\(x = 4\)
Quick Check
Plug \(x = 4\) back into the original equation to verify!
\(2(4) + 4 = 12\)
\(8 + 4 = 12\)
\(12 = 12 \checkmark\)
We Do
Guided Practice: Work Together With Your Teacher
Draw your line down the equals sign!
Problem A: \(3x + 6 = 15\)
Step 1: Undo \(+ 6\) with inverse: - 6 on both sides
Step 2: Undo multiply by 3 with inverse: ÷ 3 on both sides
Final Solution:
x =
Problem B: \(4x - 5 = 11\)
Step 1: Undo \(- 5\) with inverse: + 5 on both sides
Step 2: Undo multiply by 4 with inverse: ÷ 4 on both sides
Final Solution:
x =
Two-Step Equations Reteach • Resource Room Math Turn page for "You Do" practice ➔
You Do
Independent Practice: Your Turn!
Checklist: 1. Undo constant (+ / -) • 2. Undo coefficient (× / ÷)
1. Solve for x: \(5x + 3 = 23\)
Show both inverse steps below: Balance line: |
Solution:
x =
2. Solve for x: \(2x - 7 = 9\)
Show both inverse steps below: Balance line: |
Solution:
x =
3. Solve for x: \(3x + 8 = 20\)
Show both inverse steps below: Balance line: |
Solution:
x =
4. Solve for x: \(6x - 4 = 26\)
Show both inverse steps below: Balance line: |
Solution:
x =
Show What You Know • Error Detective & Exit Question
5. Error Detective:
Leo tried to solve \(4x + 6 = 18\). His first step was dividing both sides by 4.
What inverse operation should he have done first?
6. Quick Exit Solve: \(7x - 5 = 16\)
Show Step 1 and Step 2 cleanly:
Final:
x =
Self-Assessment: How confident do you feel? ⚪ Need Help ⚪ Getting There ⚪ Got It! Page 2 of 2
Equation Rescue Answer Key
Teacher Guide & Key Resource Room Math • Grade 7
Equation Rescue Answer Key
Instructional Focus: Inverse Operations
Resource Room Verbal Script & Metaphor: "The Present & The Wrapping Paper"
Tell students: "The variable \(x\) is a gift. The coefficient is the gift box taped tight to the gift. The constant (+ or -) is the wrapping paper on the outside. To get to the gift, what do we unwrap first? The wrapping paper! (Step 1: Undo addition/subtraction). Then open the box! (Step 2: Undo multiplication/division)."
Vocabulary Mastery Check Ensure students can verbalize these definitions:
Variable The unknown letter (\(x\)). Goal: isolate it on one side (\(x = \text{number}\)).
Coefficient The number touching the letter (\(3x \rightarrow 3\)). It multiplies the variable.
Inverse Operation The opposite action that cancels or "undoes" a number: \(+ \leftrightarrow -\) and \(\cdot \leftrightarrow \div\).
I Do Model Equation: \(2x + 4 = 12\)
Target: \(x = 4\)
Step 1 Think-Aloud Prompt:
"I see \(+ 4\) as my constant. The inverse of adding 4 is subtracting 4. I must do it to both sides of my equals line to keep the scale balanced. \(12 - 4 = 8\). Now I have \(2x = 8\)."
Step 2 Think-Aloud Prompt:
"\(2x\) means 2 times \(x\). The inverse of multiplying by 2 is dividing by 2. I divide both sides by 2. \(8 \div 2 = 4\). So, \(x = 4\)!"
We Do
Guided Practice Solutions & Common Errors
Problem A: \(3x + 6 = 15\) Solution: \(x = 3\)
\(3x + 6 = 15\)
\(\quad - 6 \quad - 6\) (Inverse of +6 is -6)
\(3x = 9\)
\(\div 3 \quad \div 3\) (Inverse of •3 is ÷3)
\(x = 3\)
Watch out: Students often subtract 6 from 15 but forget to write the minus 6 on the left.
Problem B: \(4x - 5 = 11\) Solution: \(x = 4\)
\(4x - 5 = 11\)
\(\quad + 5 \quad + 5\) (Inverse of -5 is +5)
\(4x = 16\)
\(\div 4 \quad \div 4\) (Inverse of •4 is ÷4)
\(x = 4\)
Watch out: Students see \(- 5\) and accidentally subtract 5 again (\(11 - 5 = 6\)) instead of adding!
Diagnostic Fixes for Resource Room Obstacles
Dividing First: If a student divides first, ask: "Can you easily divide 5 by 2?" Show them how waiting to divide makes the math much cleaner.
One-Sided Operations: If they subtract from only one side, use hand scales or physical weights to demonstrate how the equation tips over.
Equation Rescue Teacher Guide • Page 1 of 2 Turn page for "You Do" solutions & accommodations ➔
You Do Key
Independent Practice Worked Solutions
Problems 1 - 4 & Exit Checks
1. \(5x + 3 = 23\) x = 4
\(5x + 3 = 23\)
\(\quad - 3 \quad - 3\) (Undo +3)
\(5x = 20\)
\(\div 5 \quad \div 5\) (Undo •5)
\(x = 4\)
Check: \(5(4) + 3 = 20 + 3 = 23 \checkmark\)
2. \(2x - 7 = 9\) x = 8
\(2x - 7 = 9\)
\(\quad + 7 \quad + 7\) (Undo -7)
\(2x = 16\)
\(\div 2 \quad \div 2\) (Undo •2)
\(x = 8\)
Check: \(2(8) - 7 = 16 - 7 = 9 \checkmark\)
3. \(3x + 8 = 20\) x = 4
\(3x + 8 = 20\)
\(\quad - 8 \quad - 8\) (Undo +8)
\(3x = 12\)
\(\div 3 \quad \div 3\) (Undo •3)
\(x = 4\)
Check: \(3(4) + 8 = 12 + 8 = 20 \checkmark\)
4. \(6x - 4 = 26\) x = 5
\(6x - 4 = 26\)
\(\quad + 4 \quad + 4\) (Undo -4)
\(6x = 30\)
\(\div 6 \quad \div 6\) (Undo •6)
\(x = 5\)
Check: \(6(5) - 4 = 30 - 4 = 26 \checkmark\)
Show What You Know: Solutions & Exemplar Explanations
5. Error Detective Exemplar Answer:
Answer: Leo should have subtracted 6 from both sides first.
Teacher note: In reverse order of operations (reverse PEMDAS), we always undo addition and subtraction (the constant) before undoing multiplication (the coefficient).
6. Exit Solve Exemplar (\(7x - 5 = 16\)):
\(7x - 5 = 16 \rightarrow 7x = 16 + 5 = 21\)
\(7x = 21 \rightarrow x = 21 \div 7 = \mathbf{3}\)
Final: x = 3
Resource Room Accommodations & Scaffolding Strategies
Highlighter Strategy Have students highlight the constant in yellow and the coefficient in blue. Yellow step always happens first!
Physical Balance Line Draw a bold vertical line through the equals sign. Whatever is written on the left must be mirrored on the right.
Fact Families Reference Provide a clean times table so working memory is preserved for the algebraic steps rather than computation.
IEP Goal Progress Tracker (Resource Math) Target: 4 out of 5 correct with independent steps
Level 1 (Emerging) Needs full prompt for Step 1 & 2
Level 2 (Developing) Identifies inverse but misses balance
Level 3 (Proficient) Solves with reverse order correctly
Level 4 (Mastery) Solves & self-checks independently
Equation Rescue Teacher Guide & Key • Resource Room Math Page 2 of 2
