-{(sqrt(1/3))/(1/3)}, -{(1/5)/(sqrt(1/5))}, -{(1/3)/(sqrt(1/3))}, -{(sqrt(1/5))/(1/5)}, -1 If the 5 numbers listed above are denoted x_1, x_2, x_3, x...

1. Translate the problem requirements: We have 5 negative expressions involving fractions and square roots that need to be simplified and ordered from smallest to largest (most negative to least negative). We need to find which expression corresponds to \(x_4\) (the second largest value).
2. Simplify each complex fraction: Convert each expression to a simpler form by applying the rule that dividing by a fraction is equivalent to multiplying by its reciprocal, and use the property that \(\sqrt{\frac{1}{n}} = \frac{1}{\sqrt{n}}\).
3. Compare the simplified values: Since all values are negative, determine their relative sizes by comparing the positive portions, keeping in mind that larger positive values correspond to smaller (more negative) negative values.
4. Order and identify \(x_4\): Arrange the five simplified values from smallest to largest and identify which original expression represents the fourth value in this ordering.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're being asked to do here. We have 5 expressions that all look quite complex, involving fractions and square roots, and they're all negative numbers. We need to arrange them from smallest to largest (remember, smallest means most negative, like -10 is smaller than -2), then find which expression is the fourth one in this order.

Think of it like arranging temperatures: -10°F is colder (smaller) than -2°F. So when we order our 5 negative numbers, we're looking for the one that's fourth from the most negative.

The five expressions are:

• First: \(-\sqrt{\frac{1}{3}} \div \frac{1}{3}\)
• Second: \(-\frac{1}{5} \div \sqrt{\frac{1}{5}}\)
• Third: \(-\frac{1}{3} \div \sqrt{\frac{1}{3}}\)
• Fourth: \(-\sqrt{\frac{1}{5}} \div \frac{1}{5}\)
• Fifth: \(-1\)

Process Skill: TRANSLATE

2. Simplify each complex fraction

Now let's make these expressions easier to work with by simplifying them one by one. Remember that dividing by a fraction is the same as multiplying by its reciprocal (flipping the fraction).

Let's work through each one:

First expression: \(-\sqrt{\frac{1}{3}} \div \frac{1}{3}\)
Dividing by \(\frac{1}{3}\) is the same as multiplying by 3.
So: \(-\sqrt{\frac{1}{3}} \times 3 = -3\sqrt{\frac{1}{3}}\)
Since \(\sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}\), we get: \(-\frac{3}{\sqrt{3}} = -\frac{3\sqrt{3}}{3} = -\sqrt{3} \approx -1.732\)

Second expression: \(-\frac{1}{5} \div \sqrt{\frac{1}{5}}\)
This becomes: \(-\frac{1}{5} \times \frac{1}{\sqrt{\frac{1}{5}}} = -\frac{1}{5} \times \sqrt{5} = -\frac{\sqrt{5}}{5} \approx -0.447\)

Third expression: \(-\frac{1}{3} \div \sqrt{\frac{1}{3}}\)
This becomes: \(-\frac{1}{3} \times \frac{1}{\sqrt{\frac{1}{3}}} = -\frac{1}{3} \times \sqrt{3} = -\frac{\sqrt{3}}{3} \approx -0.577\)

Fourth expression: \(-\sqrt{\frac{1}{5}} \div \frac{1}{5}\)
Dividing by \(\frac{1}{5}\) is the same as multiplying by 5.
So: \(-\sqrt{\frac{1}{5}} \times 5 = -5\sqrt{\frac{1}{5}} = -\frac{5}{\sqrt{5}} = -\sqrt{5} \approx -2.236\)

Fifth expression: \(-1\)
This is already simplified: \(-1\)

Process Skill: SIMPLIFY

3. Compare the simplified values

Now let's look at our simplified values and their approximate decimal equivalents:

• First: \(-\sqrt{3} \approx -1.732\)
• Second: \(-\frac{\sqrt{5}}{5} \approx -0.447\)
• Third: \(-\frac{\sqrt{3}}{3} \approx -0.577\)
• Fourth: \(-\sqrt{5} \approx -2.236\)
• Fifth: \(-1\)

Since these are all negative numbers, the one with the largest absolute value is actually the smallest number. Think of it on a number line: -2.236 is furthest to the left (smallest), then -1.732, then -1, then -0.577, and finally -0.447 is closest to zero (largest).

4. Order and identify \(x_4\)

Arranging from smallest to largest (most negative to least negative):

• \(x_1 = -\sqrt{5} \approx -2.236\) (Fourth original expression)
• \(x_2 = -\sqrt{3} \approx -1.732\) (First original expression)
• \(x_3 = -1\) (Fifth original expression)
• \(x_4 = -\frac{\sqrt{3}}{3} \approx -0.577\) (Third original expression)
• \(x_5 = -\frac{\sqrt{5}}{5} \approx -0.447\) (Second original expression)

Therefore, \(x_4\) corresponds to the third original expression: \(-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}\)

Final Answer

The value of \(x_4\) is \(-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}\), which corresponds to answer choice C.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the ordering of negative numbers

Students often forget that for negative numbers, the one with the larger absolute value is actually smaller. For example, -3 < -1 because -3 is further from zero on the number line. This is counterintuitive since for positive numbers, 3 > 1. Students may incorrectly think that since |-3| > |-1|, then -3 > -1.

2. Getting overwhelmed by complex fraction notation

The expressions like \(-\sqrt{\frac{1}{3}}/\frac{1}{3}\) and \(-\frac{1}{5}/\sqrt{\frac{1}{5}}\) can appear intimidating. Students might abandon a systematic approach and try to guess or use rough estimates without proper simplification, leading to incorrect comparisons.

Errors while executing the approach

1. Incorrectly simplifying division by fractions

When converting \(-\sqrt{\frac{1}{3}} \div \frac{1}{3}\), students often make errors with the rule "dividing by a fraction equals multiplying by its reciprocal." They might forget to flip the fraction or apply the multiplication incorrectly, getting \(-\sqrt{\frac{1}{3}}/3\) instead of \(-3\sqrt{\frac{1}{3}}\).

2. Making errors while rationalizing square root expressions

When simplifying expressions like \(-\frac{3}{\sqrt{3}}\) to \(-\sqrt{3}\), students frequently make algebraic mistakes. They might incorrectly rationalize the denominator or make sign errors during the manipulation of square root expressions.

3. Computational errors with decimal approximations

When calculating approximate values like \(\sqrt{3} \approx 1.732\) or \(\sqrt{5} \approx 2.236\), students may use incorrect values or make arithmetic mistakes that throw off their entire comparison. Small errors here lead to incorrect ordering of the final values.

Errors while selecting the answer

1. Confusing the position in the ordered sequence

After correctly determining the order from smallest to largest, students might select \(x_2\) or \(x_3\) instead of \(x_4\). This happens because they lose track of which position corresponds to which original expression, especially when the ordering differs significantly from the original presentation order.

2. Selecting based on the simplified form rather than original expression

Students might correctly identify that \(x_4 = -\frac{\sqrt{3}}{3}\), but then look for this simplified form among the answer choices instead of recognizing that it corresponds to the original expression \(-\frac{1}{3}/\sqrt{\frac{1}{3}}\). They need to match their result back to the original notation given in the choices.