Which of the following is least?

1. Translate the problem requirements: We need to find which of the five negative fractions has the smallest value (furthest from zero on the number line)
2. Establish comparison benchmarks: Use \(-\frac{1}{2}\) and other simple fractions as reference points to quickly categorize the given fractions
3. Apply benchmark comparisons: Compare each fraction to our benchmarks to determine relative positions without extensive calculations
4. Identify the smallest value: Determine which fraction is furthest left on the number line (most negative)

Execution of Strategic Approach

1. Translate the problem requirements

When we're looking for the "least" among negative numbers, we need to think about what this means on a number line. Imagine a number line where zero is in the middle, positive numbers go to the right, and negative numbers go to the left.

For negative numbers, "least" means the number that's furthest to the left - the most negative value. For example, \(-10\) is less than \(-3\) because \(-10\) is further from zero.

So we need to find which of these five negative fractions: \(-\frac{32}{73}\), \(-\frac{13}{23}\), \(-\frac{12}{25}\), \(-\frac{8}{17}\), and \(-\frac{7}{16}\) is positioned furthest to the left on our number line.

Process Skill: TRANSLATE - Converting "least" for negative numbers to "most negative" or "furthest from zero"

2. Establish comparison benchmarks

Rather than doing complex decimal calculations, let's use simple benchmark fractions that we can easily work with. The most useful benchmark for negative fractions is \(-\frac{1}{2}\), since it's easy to compare other fractions against.

Let's see how each fraction compares to \(-\frac{1}{2}\):

• To compare a fraction like \(-\frac{13}{23}\) to \(-\frac{1}{2}\), we can cross-multiply: Is \(\frac{13}{23}\) greater or less than \(\frac{1}{2}\)?
• Cross-multiplying: \(13 \times 2 = 26\), and \(23 \times 1 = 23\)
• Since \(26 > 23\), we know \(\frac{13}{23} > \frac{1}{2}\)
• This means \(-\frac{13}{23} < -\frac{1}{2}\) (when we make both negative, the inequality flips)

Let's check each option:

• \(-\frac{32}{73}\): Is \(\frac{32}{73}\) compared to \(\frac{1}{2}\)? Cross-multiply: \(32 \times 2 = 64\), \(73 \times 1 = 73\). Since \(64 < 73\), then \(\frac{32}{73} < \frac{1}{2}\), so \(-\frac{32}{73} > -\frac{1}{2}\)
• \(-\frac{13}{23}\): As calculated above, \(-\frac{13}{23} < -\frac{1}{2}\)
• \(-\frac{12}{25}\): Cross-multiply: \(12 \times 2 = 24\), \(25 \times 1 = 25\). Since \(24 < 25\), then \(-\frac{12}{25} > -\frac{1}{2}\)
• \(-\frac{8}{17}\): Cross-multiply: \(8 \times 2 = 16\), \(17 \times 1 = 17\). Since \(16 < 17\), then \(-\frac{8}{17} > -\frac{1}{2}\)
• \(-\frac{7}{16}\): Cross-multiply: \(7 \times 2 = 14\), \(16 \times 1 = 16\). Since \(14 < 16\), then \(-\frac{7}{16} > -\frac{1}{2}\)

3. Apply benchmark comparisons

From our benchmark analysis, we found that only \(-\frac{13}{23}\) is less than \(-\frac{1}{2}\) (more negative than \(-\frac{1}{2}\)), while all others are greater than \(-\frac{1}{2}\).

This immediately tells us that \(-\frac{13}{23}\) is the smallest (most negative) value, since:

• \(-\frac{13}{23}\) is to the left of \(-\frac{1}{2}\) on the number line
• All other options are to the right of \(-\frac{1}{2}\) on the number line
• Therefore, \(-\frac{13}{23}\) must be smaller than all the other options

Process Skill: SIMPLIFY - Using benchmark comparisons to avoid complex calculations

4. Identify the smallest value

Since \(-\frac{13}{23}\) is the only fraction that's more negative than \(-\frac{1}{2}\), while all other fractions are closer to zero than \(-\frac{1}{2}\), we can confidently conclude that \(-\frac{13}{23}\) is the least value.

To verify: \(-\frac{13}{23} \approx -0.565\), while \(-\frac{1}{2} = -0.5\). Indeed, \(-\frac{13}{23}\) is more negative.

The answer is B: \(-\frac{13}{23}\).

Final Answer

The least value among the given options is B. \(-\frac{13}{23}\). This fraction is the only one that's more negative than the benchmark \(-\frac{1}{2}\), making it the smallest (furthest left on the number line) of all the choices.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "least" means for negative numbers: Students often think "least" means closest to zero (like with positive numbers), so they might look for the fraction with the smallest absolute value rather than the most negative value. For example, they might incorrectly choose \(-\frac{7}{16}\) because \(\frac{7}{16}\) has the smallest numerator, not realizing that "least" for negative numbers means furthest from zero on the left side of the number line.

2. Attempting to convert all fractions to decimals: Students may immediately try to divide each fraction to get decimal approximations, which is time-consuming and error-prone. This approach works but wastes valuable time that could be saved by using benchmark comparisons.

Errors while executing the approach

1. Cross-multiplication errors: When comparing fractions to the benchmark \(-\frac{1}{2}\), students frequently make mistakes in cross-multiplication. For example, when checking if \(\frac{13}{23} > \frac{1}{2}\), they might incorrectly calculate \(13 \times 2 = 24\) instead of \(26\), leading to wrong comparisons.

2. Inequality direction confusion with negative numbers: Students often forget that when comparing negative fractions, if \(\frac{a}{b} > \frac{c}{d}\) for positive fractions, then \(-\frac{a}{b} < -\frac{c}{d}\) for the corresponding negative fractions. They might correctly determine that \(\frac{13}{23} > \frac{1}{2}\) but then incorrectly conclude that \(-\frac{13}{23} > -\frac{1}{2}\).

3. Benchmark comparison errors: Students may incorrectly set up the comparison with \(-\frac{1}{2}\). For instance, when checking \(-\frac{12}{25}\) against \(-\frac{1}{2}\), they might compare \(\frac{12}{25}\) to \(\frac{2}{1}\) instead of \(\frac{1}{2}\), leading to wrong conclusions about which fractions are more or less negative than the benchmark.

Errors while selecting the answer

No likely faltering points - once students correctly identify that \(-\frac{13}{23}\) is more negative than \(-\frac{1}{2}\) while all others are less negative, the answer selection is straightforward.