“Write and solve the equation for each model” means translating a visual equality into algebra. Each side of the model represents a sum of tiles; equality means the two sums are the same number. Once the equation is written, the solution is the value of the variable that makes both sides equal.
Assumption used throughout: the model uses three tile types—one variable tile and two integer tiles.
| Tile in the model | Algebraic meaning | Numeric value |
|---|---|---|
| Variable tile labeled x | One unknown quantity | \(x\) |
| Positive unit tile labeled +1 | Adds one to the sum | \(+1\) |
| Negative unit tile labeled −1 | Subtracts one from the sum | \(-1\) |
Each side of the model becomes an expression formed by adding its tiles. The equality sign is the balance point: \[ \text{(sum of left tiles)} = \text{(sum of right tiles)}. \]
Model-to-equation structure
A variable tile contributes \(x\) each time it appears. Positive and negative unit tiles contribute \(+1\) and \(-1\). Multiple tiles of the same type add up as repeated addition: \[ x + x + x = 3x, \qquad (+1)+(+1)+(+1)=3, \qquad (-1)+(-1)=-2. \]
Equations and solutions for the models
| Model | Equation written from the tiles | Solution for \(x\) | Check (substitution) |
|---|---|---|---|
| Model A | \(x + 3 = 7\) | \(x = 4\) | \(4 + 3 = 7\) |
| Model B | \(2x - 2 = 4\) | \(x = 3\) | \(2\cdot 3 - 2 = 4\) |
| Model C | \(x - 5 = -1\) | \(x = 4\) | \(4 - 5 = -1\) |
Algebraic justification (rigorous equalities)
Model A. \[ x+3=7 \quad\Rightarrow\quad x=7-3 \quad\Rightarrow\quad x=4. \]
Model B. \[ 2x-2=4 \quad\Rightarrow\quad 2x=4+2 \quad\Rightarrow\quad 2x=6 \quad\Rightarrow\quad x=\frac{6}{2}=3. \]
Model C. \[ x-5=-1 \quad\Rightarrow\quad x=-1+5 \quad\Rightarrow\quad x=4. \]
Sign conventions and common errors
A negative unit tile contributes \(-1\) each time it appears, so five negative tiles add to \(-5\), not \(+5\). Multiple variable tiles add as \(x+x=2x\), not \(x^2\). Equality remains true only when identical operations are applied to both sides, which is the algebraic counterpart of maintaining balance in the model.