Explaining Reasoning - Pre Algebra

1. What property justifies going from $2x + 3 = 13$ to $2x = 10$?

   The subtraction property of equality: subtracting 3 from both sides keeps the equation balanced.

2. An area model for $3(x + 4)$ is a rectangle of width 3 split into lengths $x$ and 4. What are the areas of the two parts?

   $3x$ and 12. Together they give $3x + 12$, which shows the distributive property.

3. Which number is a counterexample to "squaring a positive number always makes it larger": 2 or $\frac{1}{2}$?

   $\frac{1}{2}$, since $(\frac{1}{2})^2 = \frac{1}{4}$, which is smaller than $\frac{1}{2}$.

4. Which pair is a counterexample to "the sum of two primes is always even": 3 and 5, or 2 and 3?

   2 and 3. Their sum, 5, is odd; $3 + 5 = 8$ is even and fits the claim.

5. Why is $6 \div 0$ undefined?

   Division asks what number times 0 gives 6. Every number times 0 is 0, so no such number exists.

6. Why does $(-3)^4 = 81$ but $-3^4 = -81$?

   In $-3^4$ the exponent applies only to 3, and the negative is applied after. In $(-3)^4$ the parentheses make $-3$ the base.

7. What is a counterexample?

   A single example that shows a general claim is false.

8. Why do several examples not prove a general statement?

   Examples check only some cases. A general statement must hold for every case, and one untested case could fail.

9. A student writes $\frac{2}{3} + \frac{1}{4} = \frac{3}{7}$. What is the error, and what is the correct sum?

   The student added numerators and denominators. With a common denominator, $\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$.

10. What factored expression shows that the sum of the odd numbers $2m + 1$ and $2n + 1$ is even?

    $2(m + n + 1)$. The sum is 2 times a whole number, so it is even.

11. A student writes $5 + 3 \times 2 = 16$. What is the error, and what is the correct value?

    The student added before multiplying. Multiplication comes first: $5 + 6 = 11$.

12. What property justifies going from $2x = 10$ to $x = 5$?

    The division property of equality: dividing both sides by 2 keeps the equation balanced.

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