A certain rectangular photograph has an area of 61/2 square feet. What is the area of the photograph in square...

1. Translate the problem requirements: Convert the mixed number \(6\frac{1}{2}\) square feet to a decimal, then convert from square feet to square inches using the relationship that \(1 \text{ foot} = 12 \text{ inches}\).
2. Convert mixed number to decimal: Transform \(6\frac{1}{2}\) into decimal form to simplify calculations.
3. Apply area conversion logic: Since area involves two dimensions, when converting from feet to inches, we must square the linear conversion factor (12 inches per foot becomes 144 square inches per square foot).
4. Execute the conversion calculation: Multiply the area in square feet by 144 to get the area in square inches.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to do. We have a photograph with an area of \(6\frac{1}{2}\) square feet, and we need to find out what that same area would be when measured in square inches instead of square feet.

Think of it this way: imagine you have a rectangular piece of paper, and someone tells you it covers \(6\frac{1}{2}\) square feet of space. Now you want to know how many square inches that same piece of paper covers - it's the exact same physical area, just measured with different units.

The key information we have:

- Area of photograph = \(6\frac{1}{2}\) square feet
- Conversion factor = \(1 \text{ foot} = 12 \text{ inches}\)
- We need to find: Area in square inches

Process Skill: TRANSLATE

2. Convert mixed number to decimal

Before we can work with the area measurement, let's convert that mixed number \(6\frac{1}{2}\) into a form that's easier to calculate with.

A mixed number like \(6\frac{1}{2}\) means "6 and one-half." To convert this to a decimal:

- The whole number part is 6
- The fraction part is \(\frac{1}{2}\), which equals \(0.5\)
- So \(6\frac{1}{2} = 6 + 0.5 = 6.5\)

Therefore, our photograph has an area of 6.5 square feet.

3. Apply area conversion logic

Here's where many students make a mistake, so let's think carefully about this step.

When we convert length from feet to inches, we multiply by 12 (since \(1 \text{ foot} = 12 \text{ inches}\)). But area is different from length - area involves TWO dimensions.

Imagine a square that's 1 foot by 1 foot:

- In feet: \(1 \text{ foot} \times 1 \text{ foot} = 1 \text{ square foot}\)
- In inches: \(12 \text{ inches} \times 12 \text{ inches} = 144 \text{ square inches}\)

So \(1 \text{ square foot} = 144 \text{ square inches}\), not 12 square inches!

This means our conversion factor for area is:
\(1 \text{ square foot} = 12 \times 12 = 144 \text{ square inches}\)

Process Skill: INFER

4. Execute the conversion calculation

Now we can convert our photograph's area from square feet to square inches:

Area in square inches = Area in square feet × 144
Area in square inches = \(6.5 \times 144\)

Let's calculate \(6.5 \times 144\):

\(6.5 \times 144 = 6.5 \times (100 + 44)\)
\(= 6.5 \times 100 + 6.5 \times 44\)
\(= 650 + 286\)
\(= 936\)

Therefore, the photograph has an area of 936 square inches.

Final Answer

The area of the photograph is 936 square inches.

Looking at our answer choices:
A. 36 - Too small
B. 78 - Too small
C. 432 - This would be if we incorrectly used \(6 \times 72\)
D. 864 - This would be if we incorrectly calculated \(6 \times 144\)
E. 936 - This matches our calculation!

The correct answer is E.

Common Faltering Points

Errors while devising the approach

1. Linear conversion factor confusion

Many students incorrectly assume that since \(1 \text{ foot} = 12 \text{ inches}\), then \(1 \text{ square foot} = 12 \text{ square inches}\). They fail to recognize that area involves two dimensions, so the conversion factor must account for both length and width. This leads them to multiply \(6.5 \times 12 = 78\) instead of using the correct area conversion factor of 144.

2. Mixed number handling difficulty

Students may struggle with the mixed number \(6\frac{1}{2}\) and either:

- Leave it as a fraction throughout their calculations, making arithmetic more complex
- Convert it incorrectly (such as treating \(6\frac{1}{2}\) as 6.2 instead of 6.5)
- Skip the conversion step entirely and work with the fraction form, leading to calculation errors

Errors while executing the approach

1. Arithmetic calculation mistakes

Even with the correct approach, students often make errors when calculating \(6.5 \times 144\). Common mistakes include:

- Incorrectly calculating \(6.5 \times 44 = 264\) instead of 286
- Making errors in the distributive property breakdown: \(6.5 \times (100 + 44)\)
- Simple addition errors when combining \(650 + 286\)

2. Using wrong whole number in calculations

Some students may incorrectly use 6 instead of 6.5 in their calculations, either because they:

- Forget about the \(\frac{1}{2}\) portion entirely
- Decide to "round down" thinking it will be easier
- This leads to \(6 \times 144 = 864\), which appears as answer choice D

Errors while selecting the answer

No likely faltering points - once students have completed their calculations correctly, the final answer selection is straightforward as they simply need to match their computed result with the given answer choices.