(sqrt(2))/4 + 3/(2sqrt(2)) =

1. Translate the problem requirements: We need to add two fractions that both contain square roots: the first fraction has \(\sqrt{2}\) in the numerator, while the second has \(\sqrt{2}\) in the denominator. Our goal is to combine these into a single simplified expression.
2. Rationalize the second fraction: Transform the fraction with \(\sqrt{2}\) in the denominator by multiplying both numerator and denominator by \(\sqrt{2}\) to eliminate the radical from the denominator.
3. Find a common denominator: Once both fractions have rational denominators, identify the least common denominator so we can add the fractions together.
4. Combine and simplify: Add the numerators over the common denominator and simplify the resulting expression to match one of the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're working with here. We have two fractions that we need to add together: \(\frac{\sqrt{2}}{4} + \frac{3}{2\sqrt{2}}\).

The first fraction has a square root in the top (numerator) and a regular number in the bottom (denominator). The second fraction has a regular number in the top but a square root in the bottom. This is like trying to add apples and oranges - we need to get them in the same form first.

Our goal is to combine these into one simplified expression that matches one of the answer choices.

Process Skill: TRANSLATE - Converting the visual form of the fractions into an understanding of what mathematical operations we need to perform

2. Rationalize the second fraction

The tricky part here is the second fraction \(\frac{3}{2\sqrt{2}}\) because it has a square root in the denominator. In mathematics, we prefer not to have square roots in the bottom of fractions.

To fix this, we use a technique called "rationalizing the denominator." Think of it like this: if we multiply both the top and bottom of a fraction by the same number, we don't change the fraction's value - we're essentially multiplying by 1.

So let's multiply both top and bottom by \(\sqrt{2}\):
\(\frac{3}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2\sqrt{2} \times \sqrt{2}} = \frac{3\sqrt{2}}{2 \times 2} = \frac{3\sqrt{2}}{4}\)

Now our problem becomes: \(\frac{\sqrt{2}}{4} + \frac{3\sqrt{2}}{4}\)

3. Find a common denominator

This step is much easier now! Looking at our transformed fractions:
\(\frac{\sqrt{2}}{4} + \frac{3\sqrt{2}}{4}\)

Both fractions already have the same denominator: 4. This is like having \(\frac{1}{4}\) of a pizza plus \(\frac{3}{4}\) of a pizza - we can simply add the pieces together because they're the same size pieces.

Since both denominators are 4, our common denominator is 4.

4. Combine and simplify

Now we can add the fractions by adding the numerators and keeping the common denominator:
\(\frac{\sqrt{2}}{4} + \frac{3\sqrt{2}}{4} = \frac{\sqrt{2} + 3\sqrt{2}}{4}\)

In the numerator, we have \(\sqrt{2} + 3\sqrt{2}\). This is like having 1 apple plus 3 apples - we get 4 apples. Similarly, \(\sqrt{2} + 3\sqrt{2} = 4\sqrt{2}\).

So our fraction becomes: \(\frac{4\sqrt{2}}{4} = \sqrt{2}\)

Process Skill: SIMPLIFY - Recognizing that coefficients of like terms can be added, and that \(\frac{4\sqrt{2}}{4}\) simplifies to \(\sqrt{2}\)

Final Answer

Our final answer is \(\sqrt{2}\), which corresponds to answer choice C.

To verify: We started with \(\frac{\sqrt{2}}{4} + \frac{3}{2\sqrt{2}}\), rationalized the second fraction to get \(\frac{\sqrt{2}}{4} + \frac{3\sqrt{2}}{4}\), combined them to get \(\frac{4\sqrt{2}}{4}\), and simplified to \(\sqrt{2}\).

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Not recognizing the need to rationalize denominators

Many students see the expression \(\frac{\sqrt{2}}{4} + \frac{3}{2\sqrt{2}}\) and attempt to directly add the fractions without addressing the square root in the denominator of the second fraction. They might try to find a common denominator using 4 and \(2\sqrt{2}\) directly, leading to complex and incorrect calculations.

2. Confusing rationalization process

Students often struggle with what to multiply by when rationalizing \(\frac{3}{2\sqrt{2}}\). Some might multiply by \(\sqrt{2}\) only in the numerator or only in the denominator, rather than multiplying both by \(\frac{\sqrt{2}}{\sqrt{2}}\). This fundamental misunderstanding of maintaining equivalent fractions leads to incorrect transformations.

Errors while executing the approach

1. Arithmetic errors during rationalization

When rationalizing \(\frac{3}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\), students frequently make calculation errors. Common mistakes include: getting \(\frac{3\sqrt{2}}{2\sqrt{2}}\) instead of recognizing that \(\sqrt{2} \times \sqrt{2} = 2\), or incorrectly calculating \(2 \times 2 = 4\) in the denominator.

2. Incorrect combination of like terms

When adding \(\sqrt{2} + 3\sqrt{2}\) in the numerator, students sometimes treat these as unlike terms and leave them uncombined, or incorrectly add them as \(\sqrt{2 + 3} = \sqrt{5}\) instead of recognizing this as \((1 + 3)\sqrt{2} = 4\sqrt{2}\).

3. Simplification errors

Students often fail to recognize that \(\frac{4\sqrt{2}}{4}\) simplifies to \(\sqrt{2}\), either leaving the answer in unreduced form or making errors when canceling the common factor of 4.

Errors while selecting the answer

No likely faltering points - once students correctly arrive at \(\sqrt{2}\), this directly matches answer choice C without any ambiguity about units, decimal vs. integer forms, or selecting among multiple calculated values.