The mean of \((54,820)^2\) and \((54,822)^2\) =

1. Translate the problem requirements: We need to find the mean (average) of \((54,820)^2\) and \((54,822)^2\), which means we calculate \([(54,820)^2 + (54,822)^2] ÷ 2\)
2. Recognize the symmetric pattern: Notice that 54,820 and 54,822 are consecutive even numbers that differ by 2, positioned symmetrically around 54,821
3. Apply algebraic expansion using the middle value: Express both numbers in terms of 54,821 (let's call it 'a'), so we have \((\mathrm{a}-1)^2\) and \((\mathrm{a}+1)^2\)
4. Simplify using difference of squares concepts: Calculate the mean by expanding \((\mathrm{a}-1)^2 + (\mathrm{a}+1)^2\) and dividing by 2 to find the relationship to \(\mathrm{a}^2\)
5. Match result to answer choices: Compare our simplified expression to the given options to identify the correct form

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking for. We need to find the mean (or average) of two squared numbers: \((54,820)^2\) and \((54,822)^2\).

The mean of two numbers is simply their sum divided by 2. So we need to calculate:

\([(54,820)^2 + (54,822)^2] ÷ 2\)

Instead of jumping straight into calculations with these large numbers, let's look for patterns that might make this easier.

Process Skill: TRANSLATE - Converting the problem into a clear mathematical expression

2. Recognize the symmetric pattern

Let's look at our two numbers: 54,820 and 54,822.

These numbers are very close to each other - they differ by exactly 2. More importantly, they sit symmetrically around the middle value 54,821.

54,820 is 1 less than 54,821
54,822 is 1 more than 54,821

This symmetric positioning around a middle value is key to solving this problem efficiently. When we have two numbers that are equally spaced around a center point, we can use this pattern to our advantage.

Process Skill: INFER - Recognizing the symmetric relationship that will simplify our work

3. Apply algebraic expansion using the middle value

Since 54,821 is our middle value, let's call it 'a' to make our work cleaner.

So: \(\mathrm{a} = 54,821\)

This means:

• \(54,820 = \mathrm{a} - 1\)
• \(54,822 = \mathrm{a} + 1\)

Our problem becomes finding the mean of \((\mathrm{a}-1)^2\) and \((\mathrm{a}+1)^2\).

Mean = \([(\mathrm{a}-1)^2 + (\mathrm{a}+1)^2] ÷ 2\)

Let's expand these squares:

\((\mathrm{a}-1)^2 = \mathrm{a}^2 - 2\mathrm{a} + 1\)
\((\mathrm{a}+1)^2 = \mathrm{a}^2 + 2\mathrm{a} + 1\)

4. Simplify using difference of squares concepts

Now let's add these expanded forms:

\((\mathrm{a}-1)^2 + (\mathrm{a}+1)^2 = (\mathrm{a}^2 - 2\mathrm{a} + 1) + (\mathrm{a}^2 + 2\mathrm{a} + 1)\)

Notice how the middle terms cancel out:

\(= \mathrm{a}^2 - 2\mathrm{a} + 1 + \mathrm{a}^2 + 2\mathrm{a} + 1\)
\(= 2\mathrm{a}^2 + 2\)

So our mean becomes:

\(\text{Mean} = (2\mathrm{a}^2 + 2) ÷ 2 = \mathrm{a}^2 + 1\)

Substituting back \(\mathrm{a} = 54,821\):

\(\text{Mean} = (54,821)^2 + 1\)

Process Skill: SIMPLIFY - Using algebraic expansion and cancellation to reach an elegant result

5. Match result to answer choices

Our calculated result is \((54,821)^2 + 1\).

Looking at the answer choices:

1. \((54,821)^2\)
2. \((54,821.5)^2\)
3. \((54,820.5)^2\)
4. \((54,821)^2 + 1\)
5. \((54,821)^2 - 1\)

Our result exactly matches choice (D): \((54,821)^2 + 1\).

Final Answer

The mean of \((54,820)^2\) and \((54,822)^2\) = \((54,821)^2 + 1\)

The correct answer is (D).

This result makes intuitive sense: the mean of the squares of two numbers positioned symmetrically around a center value is slightly larger than the square of that center value - specifically, it's larger by 1 when the numbers differ from the center by 1 each.

Common Faltering Points

Errors while devising the approach

1. Attempting brute force calculation without recognizing the pattern

Many students immediately try to calculate \((54,820)^2\) and \((54,822)^2\) directly, leading to unwieldy arithmetic with large numbers. They miss the key insight that these numbers are symmetrically positioned around 54,821, which would allow for a much more elegant algebraic approach.

2. Not recognizing the symmetric relationship

Students may notice that 54,820 and 54,822 are close to each other but fail to identify that they are exactly equidistant from 54,821 (differing by 1 on each side). This symmetric positioning is crucial for applying the algebraic substitution method that makes the problem manageable.

Errors while executing the approach

1. Algebraic expansion errors

When expanding \((\mathrm{a}-1)^2\) and \((\mathrm{a}+1)^2\), students commonly make sign errors or forget terms. For example, they might write \((\mathrm{a}-1)^2 = \mathrm{a}^2 - 1\) instead of the correct \(\mathrm{a}^2 - 2\mathrm{a} + 1\), or similarly mishandle the \((\mathrm{a}+1)^2\) expansion.

2. Incorrect cancellation of terms

After expanding both expressions, students may incorrectly cancel terms when adding \((\mathrm{a}^2 - 2\mathrm{a} + 1) + (\mathrm{a}^2 + 2\mathrm{a} + 1)\). They might cancel the wrong terms or fail to recognize that \(-2\mathrm{a}\) and \(+2\mathrm{a}\) cancel perfectly, leading to computational errors in the final simplification.

Errors while selecting the answer

1. Substitution error when matching to answer choices

After correctly deriving \(\mathrm{a}^2 + 1\), students may forget that \(\mathrm{a} = 54,821\) and either leave their answer in terms of 'a' or incorrectly substitute a different value. This leads them to select an incorrect answer choice even though their algebraic work was correct.