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### Activity

# Algebra Sleuth: Proof that 1 = 2?

Challenge your high school student to find the flaw in this short mathematical proof that one is equal to two. This activity provides a good review of basic math principals and the structure of mathematical proofs. It’s also a good reminder that knowing math principals is good protection against getting tricked.

### What You Need:

- Proof that 1 = 2 (see below)

### What You Do:

- Show your teen the proof.
- Ask her to tell you which step is invalid. She should determine both which number is wrong, and why.
- Help her keep going until she understands the answer.

### The Proof that 2 = 1

1) a = b 1) Given

2) a^{2} = ab 2) Multiply both sides by a

3) a^{2}-b^{2} = ab-b^{2} 3) Subtract b^{2} from both sides

4) (a+b)(a-b) = b(a-b) 4) Factor both sides

5) (a+b) = b 5) Divide both sides by (a-b)

6) a+a = a 6) Substitute a for b

7) 2a = a 7) Addition

8) 2 = 1 8) Divide both sides by a

### Solution:

Part one: Step five is wrong. The rules of mathematics do not allow us to divide by zero.

Since a and b are equal, (a-b) = 0. Therefore, we cannot divide by (a-b)!

Note: To explain why you can't divide something by zero, ask your student how she would divide a pizza into 0 pieces. Impossible! The fewest number of pieces she could make would be one piece—the whole pizza!