Skill: Write a complete mathematical justification

A complete mathematical justification does more than give an answer. It includes:

1. A clear claim—what you think is true.
2. Representations—numbers, equations, a table, or a diagram that show the important information.
3. Logical steps—an explanation of why each step is valid.
4. A conclusion—a sentence that connects the work back to the question.

Example

Problem: Which fraction is greater, $\frac{3}{4}$ or $\frac{5}{8}$? Justify your answer.

Step 1: State a claim.

I claim that $\frac{3}{4}$ is greater than $\frac{5}{8}$.

Step 2: Represent the fractions with the same denominator.

To compare fractions, it helps to use equal-sized parts. The denominators $4$ and $8$ can both be represented using eighths.

Since $4 \times 2=8$, multiply the numerator and denominator of $\frac{3}{4}$ by $2$:

$$\frac{3}{4}=\frac{3\times 2}{4\times 2}=\frac{6}{8}.$$

This does not change the value of the fraction because the numerator and denominator were multiplied by the same number.

Step 3: Compare the equivalent fractions.

Now compare $\frac{6}{8}$ and $\frac{5}{8}$. They have the same denominator, so compare their numerators:

$$6>5,$$

which means

$$\frac{6}{8}>\frac{5}{8}.$$

Because $\frac{3}{4}=\frac{6}{8}$, it follows that

$$\boxed{\frac{3}{4}>\frac{5}{8}}.$$

Therefore, $\frac{3}{4}$ is greater than $\frac{5}{8}$ because it represents $6$ eighths, while $\frac{5}{8}$ represents only $5$ eighths.