Question

How can benchmarks be used to compare and order fractions

Answer 1

Benchmarks can be useful tools for comparing and ordering fractions by providing reference points that help visualize where a fraction falls on a number line. Here are some ways to use benchmarks for this purpose:

### 1. Use Common Benchmarks
Common benchmarks for comparing fractions include \(0\), \(\frac{1}{2}\), and \(1\).

- **0**: Any fraction that is greater than \(0\) is positive, and any fraction less than \(0\) is negative.
- **\(\frac{1}{2}\)**: This is a critical point; fractions can be compared to \(\frac{1}{2}\) to see if they are less than, equal to, or greater than half.
- **1**: This shows whether the fraction is less than or equal to 1.

### 2. Visualize on a Number Line
Using a number line can help:

- Draw a number line and mark `0`, `1/2`, and `1`.
- Place the fractions on the number line according to their values.
- This visual representation will help you see which fractions are larger or smaller than others.

### 3. Convert to Benchmarks
For more complex fractions, you can convert them into benchmark fractions:

- For instance, convert fractions to decimals or percentages (for example, \(\frac{1}{4} = 0.25\), \(\frac{3}{4} = 0.75\)).
- Compare these decimal representations to the benchmark values (like \(0.5\) for \(\frac{1}{2}\)).

### 4. Compare to \( \frac{1}{2} \) Specifically
If you're comparing several fractions, determine whether each fraction is greater than, less than, or equal to \(\frac{1}{2}\):

- Fractions greater than \(\frac{1}{2}\) are greater than many fractions like \(\frac{1}{3}\) or \(\frac{2}{5}\).
- Those less than \(\frac{1}{2}\) will rank comparatively lower.

### 5. Common Denominator
If you're working with fractions that have different denominators, convert them to a common denominator to compare them directly. This might not be a benchmark method per se, but finding a common denominator can streamline comparisons.

### Example
Let’s compare the fractions \(\frac{1}{4}\), \(\frac{3}{8}\), and \(\frac{5}{8}\):

1. Identify benchmarks: \(0\), \(\frac{1}{2}\), and \(1\).
2. Convert to decimal:
- \(\frac{1}{4} = 0.25\)
- \(\frac{3}{8} = 0.375\)
- \(\frac{5}{8} = 0.625\)
3. Compare to \(0.5\):
- \(0.25 < 0.5\)
- \(0.375 < 0.5\)
- \(0.625 > 0.5\)
4. Order from least to greatest: \(\frac{1}{4} < \frac{3}{8} < \frac{5}{8}\).

By using benchmarks, you can quickly and effectively compare and order fractions with confidence!