Multiplying & Dividing Rational Expressions Maze - Engaging math activity for students to practice simplifying rational expressions.

Let's solve the "Multiplying & Dividing Rational Expressions Maze" step by step. The goal is to begin at the "Start Here!" square, simplify each rational expression, and follow the correct path (based on simplified answers) until we reach the "Finished!" square.

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🔹 Step 1: Start at the "START HERE!" square

Expression:
$$
\frac{x - 2}{x + 5} \cdot \frac{x + 5}{x - 1}
$$

#### Simplify:
- Cancel $ x + 5 $ in numerator and denominator.
$$
= \frac{x - 2}{\cancel{x + 5}} \cdot \frac{\cancel{x + 5}}{x - 1} = \frac{x - 2}{x - 1}
$$

✔ Answer: $ \frac{x - 2}{x - 1} $

Now look for the next square with this expression as an input. There’s a box directly to the right:

> Next square: $ \frac{x - 2}{x - 1} $

So move to that square.

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🔹 Step 2: Next Expression

$$
\frac{x + 6}{x - 6} \div \frac{x^2 + 4x - 12}{x - 6}
$$

Recall: To divide, multiply by the reciprocal.

First, factor $ x^2 + 4x - 12 $:
$$
x^2 + 4x - 12 = (x + 6)(x - 2)
$$

So:
$$
\frac{x + 6}{x - 6} \div \frac{(x + 6)(x - 2)}{x - 6}
= \frac{x + 6}{x - 6} \cdot \frac{x - 6}{(x + 6)(x - 2)}
$$

Cancel common terms:
- $ x + 6 $ cancels
- $ x - 6 $ cancels

Leaves:
$$
\frac{1}{x - 2}
$$

✔ Answer: $ \frac{1}{x - 2} $

Now find the next square with this answer.

Look at the options from this square — there’s a box labeled:
> $ \frac{1}{x - 2} $

Yes! So go to that box.

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🔹 Step 3: Next Expression

$$
\frac{x^2 - 4}{x^2 - 3x - 40} \cdot \frac{x + 5}{x + 2}
$$

Factor all expressions:

- $ x^2 - 4 = (x - 2)(x + 2) $
- $ x^2 - 3x - 40 = (x - 8)(x + 5) $

So:
$$
\frac{(x - 2)(x + 2)}{(x - 8)(x + 5)} \cdot \frac{x + 5}{x + 2}
$$

Cancel:
- $ x + 2 $ cancels
- $ x + 5 $ cancels

Left with:
$$
\frac{x - 2}{x - 8}
$$

✔ Answer: $ \frac{x - 2}{x - 8} $

Now look for a box with this expression. There's one below it:

> $ \frac{x - 2}{x - 8} $

So go to that square.

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🔹 Step 4: Next Expression

$$
\frac{x + 5}{2x + 12} \cdot (x^2 + 6x)
$$

First, factor:
- $ 2x + 12 = 2(x + 6) $
- $ x^2 + 6x = x(x + 6) $

So:
$$
\frac{x + 5}{2(x + 6)} \cdot x(x + 6)
$$

Cancel $ x + 6 $:
$$
= \frac{x + 5}{2} \cdot x = \frac{x(x + 5)}{2}
$$

But wait — check if this matches any of the outputs.

Wait — let's see what the output paths are from this square.

The possible paths are:
- $ x + 1 $
- $ \frac{6(x + 5)}{x} $
- $ \frac{1}{3(x - 3)} $

Our result $ \frac{x(x + 5)}{2} $ doesn't match any of these. But we must have made a mistake in logic.

Wait — perhaps I need to trace which path leads to a matching output, but we already followed the correct path so far?

Wait — actually, the maze has multiple branches. Let's double-check.

Wait — after $ \frac{x - 2}{x - 8} $, the next square should be connected via a path labeled $ \frac{x - 2}{x - 8} $. Looking at the diagram:

From the $ \frac{x^2 - 4}{x^2 - 3x - 40} \cdot \frac{x + 5}{x + 2} $ box, the path down says $ \frac{x - 2}{x - 8} $, which connects to a box below.

That box is:

> $ \frac{x^2 + 2x - 3}{x^2 + 4x + 3} \div (x - 1) $

So now compute that.

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🔹 Step 5: New Expression

$$
\frac{x^2 + 2x - 3}{x^2 + 4x + 3} \div (x - 1)
$$

Divide by $ x - 1 $ means multiply by $ \frac{1}{x - 1} $

Factor both quadratics:

- $ x^2 + 2x - 3 = (x + 3)(x - 1) $
- $ x^2 + 4x + 3 = (x + 3)(x + 1) $

So:
$$
\frac{(x + 3)(x - 1)}{(x + 3)(x + 1)} \cdot \frac{1}{x - 1}
$$

Cancel:
- $ x + 3 $
- $ x - 1 $

Leaves:
$$
\frac{1}{x + 1}
$$

✔ Answer: $ \frac{1}{x + 1} $

Now look for a path labeled $ \frac{1}{x + 1} $

From this square, the path going down says $ \frac{1}{x + 1} $, leading to:

> $ \frac{x^2 - 4}{x + 3} \div \frac{x + 2}{x^2 - 9} $

So go to that square.

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🔹 Step 6: Next Expression

$$
\frac{x^2 - 4}{x + 3} \div \frac{x + 2}{x^2 - 9}
$$

Factor:
- $ x^2 - 4 = (x - 2)(x + 2) $
- $ x^2 - 9 = (x - 3)(x + 3) $

So:
$$
\frac{(x - 2)(x + 2)}{x + 3} \div \frac{x + 2}{(x - 3)(x + 3)}
= \frac{(x - 2)(x + 2)}{x + 3} \cdot \frac{(x - 3)(x + 3)}{x + 2}
$$

Cancel:
- $ x + 2 $
- $ x + 3 $

Left with:
$$
(x - 2)(x - 3)
$$

✔ Answer: $ (x - 2)(x - 3) $

Now look for a path labeled $ (x - 2)(x - 3) $

From this square, there's a path labeled $ (x - 2)(x - 3) $ leading to:

> $ \frac{x - 2}{x - 3} $

So go to that square.

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🔹 Step 7: Final Square

$$
\frac{x - 2}{x - 3}
$$

This is the last box before "FINISHED!"

And indeed, the path from here goes to "FINISHED!"

✔ We’ve reached the end!

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✔ Final Answer Path:

1. Start: $ \frac{x - 2}{x + 5} \cdot \frac{x + 5}{x - 1} = \frac{x - 2}{x - 1} $
2. → $ \frac{x + 6}{x - 6} \div \frac{x^2 + 4x - 12}{x - 6} = \frac{1}{x - 2} $
3. → $ \frac{x^2 - 4}{x^2 - 3x - 40} \cdot \frac{x + 5}{x + 2} = \frac{x - 2}{x - 8} $
4. → $ \frac{x^2 + 2x - 3}{x^2 + 4x + 3} \div (x - 1) = \frac{1}{x + 1} $
5. → $ \frac{x^2 - 4}{x + 3} \div \frac{x + 2}{x^2 - 9} = (x - 2)(x - 3) $
6. → $ \frac{x - 2}{x - 3} $
7. → FINISHED!

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🟩 Summary of Correct Path:

- Start
- → $ \frac{x - 2}{x - 1} $
- → $ \frac{1}{x - 2} $
- → $ \frac{x - 2}{x - 8} $
- → $ \frac{1}{x + 1} $
- → $ (x - 2)(x - 3) $
- → $ \frac{x - 2}{x - 3} $
- → FINISHED!

✔ This is the correct solution path through the maze.

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💡 Tips for Solving These Mazes:

- Always factor completely first.
- Remember: Division → Multiply by Reciprocal
- Cancel common factors in numerator and denominator.
- Match your simplified answer to the label on the path.

Let me know if you'd like a visual version or printable guide!