Boxes and Circles Handout
Name: Date:
Period:
Boxes & Circles Method
Solving Multi-Step Linear Equations
Visual Strategy Guide
1 Parentheses? ALWAYS Distribute! ONLY time to multiply!
2 Box & Circle
X-Box #
Keep sign inside!
3 ONE of Each! Total into 1 Box & 1 Circle per side 1 Box + 1 Circle
4 Cross the Line! Line through each '=' CHANGE THE SIGN!
5 Final Divide Divide by the box number Solve for \( x \)!
The Equal Sign Rule: Draw the Line & Change the Sign!
Draw a continuous vertical line straight down through each '=' sign. When any shape crosses that line, you MUST CHANGE ITS SIGN!
Memory Rhyme: "Cross the line & change the sign!"
Master Example: Line Straight Down Through Each '='
Equation: \( 2(x + 4) + 3x = x + 24 \)
1. Distribute \( 2x + 8 + 3x \)
=
\( x + 24 \)
2. Box & Circle
+2x +8 +3x
=
+1x +24
3. Condense Sides
5x +8
=
1x +24
4. Cross Line!
5x - 1x → 4x
=
24 - 8 → 16
5. Divide by 4 \( \frac{4x}{4} \)
=
\( \frac{16}{4} \) → \( x = 4 \)
Guided Walkthrough
Together: Solve \( 4(x + 2) + 2x = x + 23 \)
Draw your vertical line through each '=' sign!
1. Distribute
4 × x and 4 × 2
2 & 3. Condense
1 Box & 1 Circle per side
[ ] ( ) = [ ] ( )
4. Cross Line!
Change the sign!
5. Final Divide
Divide by box number
Remember: "Draw a line straight down through each '=' sign — cross the line & change the sign!"
Page 1 of 2
Boxes & Circles Practice Name:
Independent Practice Zone
1) Distribute → 2) Box & Circle → 3) Condense → 4) Line through '=' & change sign → 5) Divide
X-Box = Variable Circle = Number
Problem 1 Combine Same Side
\( 5x + 7 + 2x \)
=
\( 35 \)
Condense left side → line goes down through '='!
Left Side ← cross line & change sign → Right Side
=
=
=
Final Solution:
x =
Problem 2 Variables Both Sides
\( 6x + 5 \)
=
\( 2x + 29 \)
Line through '=': Cross line & change sign!
Left Side ← cross line & change sign → Right Side
=
=
=
Final Solution:
x =
Problem 3 Distribute First!
\( 3(x + 4) + 2x \)
=
\( 32 \)
Distribute → Condense → Line down through '='!
Left Side ← cross line & change sign → Right Side
=
=
=
Final Solution:
x =
Problem 4 Master Challenge
\( 2(2x + 1) + 3x \)
=
\( x + 26 \)
Distribute → Condense → Line down through '='!
Left Side ← cross line & change sign → Right Side
=
=
=
Final Solution:
x =
My Equation Self-Check Checklist
Tick each box as you complete your work:
1. Distributed parentheses first
2. Kept signs inside shapes
3. Condensed to 1 box & 1 circle per side
4. Drew line down through each '=' sign
Boxes and Circles Practice Worksheet
Name: Date:
Period:
Boxes & Circles Practice Worksheet
Rule: Follow the vertical line straight down through each '=' sign — Cross the line & change the sign!
X-Box Circle
Problem 1 Solve: \( 4x + 6 + 3x = 27 \)
Cross the line & change sign
\( 4x + 6 + 3x \)
=
\( 27 \)
Shapes:
=
Condense:
=
Cross Line:
=
Divide to Solve:
x =
Problem 2 Solve: \( 5x + 8 = 2x + 23 \)
Line through each '='!
\( 5x + 8 \)
=
\( 2x + 23 \)
Shapes:
=
ONE of each:
=
Cross Line:
=
Divide to Solve:
x =
Problem 3 Solve: \( 3(x + 4) + 2x = 32 \)
Multiply ONLY with Parentheses!
Distribute:
+ 2x
=
\( 32 \)
Condense:
=
Cross Line:
=
Divide to Solve:
x =
Notice the vertical line running straight down through each '=' sign! Page 1 of 2
Boxes & Circles Practice (Continued) Name:
Level Up: Challenge Problems
Follow the vertical line that runs down through each '=' sign!
Cross the line & change sign
Problem 4 Solve: \( 2(3x - 2) + 4x = 36 \)
Keep minus sign inside!
Distribute:
-
+ 4x
=
\( 36 \)
Condense:
=
Cross Line:
=
Divide to Solve:
x =
Problem 5 Solve: \( 4(x + 3) + x = 2x + 30 \)
Cross the line & change sign!
Distribute:
+ x
=
\( 2x + 30 \)
Condense:
=
Boxes and Circles Slides
Visual Math Strategy
Multi-Step Equations
X
&
The Boxes & Circles Method
Sort terms into shapes, condense each side, and cross the line through '=' to change the sign!
Algebra Strategy Workshop Step 1 of 5 → Solving Mastery
Foundations
Rule 1: Sort Every Term into Shapes
📦
The "X-Box"
For Every Variable Term
Put any term with an \( x \) inside a square box. Always keep the sign (+ or -) inside!
Examples: +3x -5x
⭕
Constant Circle
For Every Regular Number
Put any plain number inside a circle. Always keep the sign (+ or -) inside!
Examples: +8 -12
Golden Rule: The sign (+ or -) in front of each term travels INSIDE the shape!
Rule Zero
Parentheses? ALWAYS Distribute First!
THIS IS THE ONLY TIME WE MULTIPLY!
Crucial Rule
Outside number multiplies both inside terms:
\( 2(x + 5) \)
→ Multiply to break the brackets:
\( 2 \cdot x = 2x \) | \( 2 \cdot 5 = +10 \)
\( 2x + 10 \)
Once parentheses are multiplied out, you are ready to draw your Boxes and Circles!
Remember: Never draw shapes inside parentheses—distribute first!
Step 3 Strategy
Condense to ONE Box & ONE Circle per Side
Before crossing the line, total the shapes on the same side:
LEFT VARIABLE ONE BOX
LEFT NUMBER ONE ⭕
=
RIGHT VARIABLE ONE BOX
RIGHT NUMBER ONE ⭕
Add all boxes together on the left side
Add all circles together on the right side
Check: Is there at most ONE box and ONE circle on each side? If yes, draw your line!
Step 4 Strategy
Draw the Line Straight Down Through the '=' Sign!
"Cross the line & change the sign!" A continuous line starts at the top and goes straight down through each '=' sign!
Left Side 5x
=
Right Side 24
Crossing from left to right: \( +8 \) crosses line → becomes \( -8 \)
Crossing from right to left: \( +1x \) crosses line → becomes \( -1x \)
