Which of the numbers (sqrt(5))/2, sqrt(5/2), 2/(sqrt(5)) sqrt(2/5), and 1/(sqrt(5)) is the greatest?

1. Translate the problem requirements: We need to determine which of five given expressions involving square roots and fractions has the largest numerical value.
2. Convert expressions to comparable decimal forms: Use the approximation \(\sqrt{5} \approx 2.24\) to convert each expression into decimal form for easy comparison.
3. Calculate approximate values systematically: Work through each expression one by one, performing the necessary arithmetic to get decimal approximations.
4. Compare and identify the maximum: Rank the calculated values to determine which expression yields the greatest result.

Execution of Strategic Approach

1. Translate the problem requirements

We have five different expressions that all involve square roots and fractions, and we need to figure out which one gives us the biggest number when we calculate it out. Think of it like having five different recipes that each give you a certain amount of cake - we want to know which recipe gives you the most cake!

The five expressions we're comparing are:

• \(\frac{\sqrt{5}}{2}\) (square root of 5, divided by 2)
• \(\sqrt{\frac{5}{2}}\) (square root of the fraction 5/2)
• \(\frac{2}{\sqrt{5}}\) (2 divided by square root of 5)
• \(\sqrt{\frac{2}{5}}\) (square root of the fraction 2/5)
• \(\frac{1}{\sqrt{5}}\) (1 divided by square root of 5)

Process Skill: TRANSLATE - Converting these mathematical expressions into language we can work with systematically

2. Convert expressions to comparable decimal forms

To compare these expressions easily, we need to turn them all into regular decimal numbers. The key is to use a good approximation for \(\sqrt{5}\).

Since we know that \(2^2 = 4\) and \(3^2 = 9\), the square root of 5 must be somewhere between 2 and 3. A bit closer to 2, actually. Let's use \(\sqrt{5} \approx 2.24\), which is quite accurate for our purposes.

Now we can substitute this value into each expression to get decimal approximations that we can easily compare - just like converting different units of measurement to the same unit so we can see which is bigger.

3. Calculate approximate values systematically

Let's work through each expression one by one, substituting \(\sqrt{5} \approx 2.24\):

Expression A: \(\frac{\sqrt{5}}{2}\)
= 2.24/2 = 1.12

Expression B: \(\sqrt{\frac{5}{2}}\)
= \(\sqrt{2.5}\)
Since \(1.5^2 = 2.25\) and \(1.6^2 = 2.56\), we know \(\sqrt{2.5}\) is between 1.5 and 1.6
More precisely: \(\sqrt{2.5} \approx 1.58\)

Expression C: \(\frac{2}{\sqrt{5}}\)
= 2/2.24 ≈ 0.89

Expression D: \(\sqrt{\frac{2}{5}}\)
= \(\sqrt{0.4}\)
Since \(0.6^2 = 0.36\) and \(0.7^2 = 0.49\), we know \(\sqrt{0.4}\) is between 0.6 and 0.7
More precisely: \(\sqrt{0.4} \approx 0.63\)

Expression E: \(\frac{1}{\sqrt{5}}\)
= 1/2.24 ≈ 0.45

4. Compare and identify the maximum

Now let's line up our calculated values to see which is greatest:

1. \(\frac{\sqrt{5}}{2} \approx 1.12\)
2. \(\sqrt{\frac{5}{2}} \approx 1.58\)
3. \(\frac{2}{\sqrt{5}} \approx 0.89\)
4. \(\sqrt{\frac{2}{5}} \approx 0.63\)
5. \(\frac{1}{\sqrt{5}} \approx 0.45\)

Looking at these values, we can clearly see that Expression B gives us the largest result at approximately 1.58.

This makes intuitive sense: \(\sqrt{\frac{5}{2}} = \sqrt{2.5}\), and since 2.5 is greater than 1, its square root will be greater than 1. Meanwhile, expressions C, D, and E all involve either taking square roots of numbers less than 1 or dividing by numbers greater than 1, which tends to make the results smaller.

Final Answer

The greatest value among the five expressions is \(\sqrt{\frac{5}{2}}\), which corresponds to Answer Choice B.

Common Faltering Points

Errors while devising the approach

1. Choosing an inefficient comparison method: Students may attempt to cross-multiply or find common denominators instead of using decimal approximations, leading to unnecessarily complex algebra that increases error probability.

2. Using an inaccurate approximation for \(\sqrt{5}\): Students might use \(\sqrt{5} \approx 2.2\) or \(\sqrt{5} \approx 2.3\), which could lead to incorrect relative rankings, especially when comparing values that are close to each other.

3. Misinterpreting the radical notation: Students may confuse \(\sqrt{\frac{5}{2}}\) with \(\frac{\sqrt{5}}{\sqrt{2}}\), not recognizing that \(\sqrt{\frac{5}{2}}\) means taking the square root of the entire fraction 5/2.

Errors while executing the approach

1. Arithmetic errors in division: When calculating expressions like \(\frac{2}{\sqrt{5}} = \frac{2}{2.24}\), students frequently make division errors, potentially getting 0.98 instead of 0.89, which could affect the final ranking.

2. Incorrectly estimating square roots: Students may struggle to approximate \(\sqrt{2.5}\) or \(\sqrt{0.4}\) accurately, potentially thinking \(\sqrt{2.5} \approx 1.25\) (by incorrectly using \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) logic) instead of the correct 1.58.

3. Calculation sequence errors: Students might calculate \(\sqrt{\frac{5}{2}}\) as \(\frac{\sqrt{5}}{2}\) instead of \(\sqrt{2.5}\), essentially computing the wrong expression due to order of operations confusion.

4. Errors while selecting the answer

1. Mismatching calculated values to answer choices: After correctly calculating that the largest value is approximately 1.58, students might incorrectly associate this with the wrong answer choice, perhaps confusing \(\sqrt{\frac{5}{2}}\) with \(\frac{\sqrt{5}}{2}\).

2. Selecting based on intuition rather than calculation: Students might override their calculated results with incorrect intuitive reasoning, such as thinking 'fractions are always smaller' and selecting an expression that looks larger superficially.