Invisible 1: Have students physically write a small '1' in front of isolated letters (e.g., \( -w \rightarrow -1w \)).
Guided Problem Solution: \( 7y + 9 - 4y + 2 \)
Boxes: \( 7y - 4y = \mathbf{\color{#2563eb}{3y}} \)
Circles: \( +9 + 2 = \mathbf{\color{#d97706}{+11}} \)
Final: \( 3y + 11 \)
Problem #1: \( 6a + 5 + 3a + 4 \) \( 9a + 9 \)
• Group \( a \)-terms: \( 6a + 3a = \mathbf{9a} \)
• Group constants: \( +5 + 4 = \mathbf{+9} \)
Combine: \( 9a + 9 \)
Ensure students do not write \( 18a \) (adding letters and numbers).
Problem #2: \( 8m - 3 + 2m + 9 \) \( 10m + 6 \)
• Group \( m \)-terms: \( 8m + 2m = \mathbf{10m} \)
• Group constants: \( -3 + 9 = \mathbf{+6} \)
Combine: \( 10m + 6 \)
Watch the negative: \( -3 + 9 \) is positive 6.
Problem #3: \( 9k + 12 - 2k - 5 \) \( 7k + 7 \)
• Group \( k \)-terms: \( 9k - 2k = \mathbf{7k} \)
• Group constants: \( +12 - 5 = \mathbf{+7} \)
Combine: \( 7k + 7 \)
Check that students box \( -2k \) with the subtraction symbol.
Problem #4: \( 4w + 10 - w - 6 \) \( 3w + 4 \)
• Group \( w \)-terms: \( 4w - 1w = \mathbf{3w} \)
• Group constants: \( +10 - 6 = \mathbf{+4} \)
Combine: \( 3w + 4 \)
Reinforce the invisible 1: \( -w \) means \( -1w \).
Equation Essentials • Teacher Answer Key Key Page 2 of 4
Teacher Guide & Answer Key
Balance & River Method
Drawing the vertical line straight through the equals sign helps students with spatial orientation difficulties isolate left side vs. right side.
Key check question: "Did you write the exact same inverse operation on both sides of the line?"
#1: \( m + 9 = 21 \) \( m = 12 \)
Operation shown: Adding 9
Inverse operation: Subtract 9
\( m + 9 - 9 = 21 - 9 \)
\( m = 12 \)
Check: \( 12 + 9 = 21 \) ✓ (Scale balanced!)
#2: \( k - 6 = 14 \) \( k = 20 \)
Operation shown: Subtracting 6
Inverse operation: Add 6
\( k - 6 + 6 = 14 + 6 \)
\( k = 20 \)
Check: \( 20 - 6 = 14 \) ✓ (Scale balanced!)
#3: \( 5p = 35 \) \( p = 7 \)
Operation shown: Multiplying by 5
Inverse operation: Divide by 5
\( \frac{5p}{5} = \frac{35}{5} \)
\( p = 7 \)
Check: \( 5(7) = 35 \) ✓ (Scale balanced!)
#4: \( \frac{w}{3} = 8 \) \( w = 24 \)
Operation shown: Dividing by 3
Inverse operation: Multiply by 3
\( \frac{w}{3} \times 3 = 8 \times 3 \)
\( w = 24 \)
Check: \( 24 \div 3 = 8 \) ✓ (Scale balanced!)
Equation Essentials • Teacher Answer Key Key Page 3 of 4
Teacher Guide & Answer Key
Progress Monitoring
Guided Problem Solution: \( 2y - 4 = 12 \)
Step 1: Add 4 → \( 2y = \mathbf{\color{#e11d48}{16}} \)
Step 2: Divide by 2 → \( y = \mathbf{\color{#e11d48}{8}} \)
Answer: \( y = 8 \)
Order of Ops
\( 20 - 4 \times (2 + 1) \)
1. \( 2 + 1 = 3 \)
2. \( 4 \times 3 = 12 \)
3. \( 20 - 12 = 8 \)
Answer = 8
Like Terms
\( 5x + 8 + 3x - 2 \)
1. \( 5x + 3x = 8x \)
2. \( +8 - 2 = +6 \)
3. Combine: \( 8x + 6 \)
Answer = \( 8x + 6 \)
Two-Step Equation
\( 4k - 2 = 14 \)
1. Add 2: \( 4k = 16 \)
2. Divide 4: \( k = 4 \)
3. Check: \( 4(4) - 2 = 14 \) ✓
Answer: \( k = 4 \)
Sample IEP Objective: "Given visual organizers and graphic supports, student will apply order of operations, combine like terms, and solve one- and two-step linear equations with 80% accuracy across 3 consecutive trials."
Level 1: Emerging (0 - 50%) Requires verbal prompting for each operation step; confuses signs.
Level 2: Developing (51 - 79%) Identifies correct inverse operation; occasional arithmetic or negative sign slip.
Level 3: Mastered (80 - 100%) Independently applies funnels, shape groupings, and inverse balance steps.
Equation Essentials • Teacher Answer Key Key Page 4 of 4
Mastery • Putting It Together
SADMEP Order
The Order of Operations in Reverse:
Step 1 Undo Addition or Subtraction first!
Clear away the constant number on the variable's side.
Step 2 Undo Multiplication or Division second!
Divide or multiply to leave the variable completely alone.
Worked Example
\( 3x + 5 = 23 \)
\( - 5 \) \( - 5 \)
\( 3x = 18 \)
\( \div 3 \) \( \div 3 \)
\( x = 6 \)
Always check your work: \( 3(6) + 5 = 18 + 5 = 23 \) ✓ True!
Algebra Toolkit Complete
One operation per line. PEMDAS from top to bottom.
Box terms with their variables and carry the sign in front.
Draw your river, do the inverse to both sides equally.
"Math is not about being fast. It's about being step-by-step!"
