For a certain set of numbers, if x is in the set, then x - 3 is also in the...

1. Translate the problem requirements: We have a set with a specific rule - whenever \(\mathrm{x}\) is in the set, then \((\mathrm{x} - 3)\) must also be in the set. Given that 1 is in the set, we need to determine which of the three given numbers (4, -1, -5) must also be in the set.
2. Apply the rule forward from the given number: Starting with 1 in the set, repeatedly apply the rule \((\mathrm{x} - 3)\) to generate all numbers that must be in the set.
3. Check if we can work backward to reach any option: For any number to be guaranteed in the set, we must be able to reach it by starting from 1 and applying the rule some number of times.
4. Evaluate each option against our findings: Compare each of the three options (4, -1, -5) with the numbers we've determined must be in the set.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're told in everyday language:

We have a special set of numbers that follows one simple rule: whenever any number is in the set, we must also include that number minus 3 in the set.

Think of it like a chain reaction - if we put a number in, we automatically have to put in a smaller number (3 less than it). We're told that the number 1 is definitely in this set.

Our job is to figure out which of these three numbers MUST also be in the set: 4, -1, and -5.

Process Skill: TRANSLATE - Converting the rule "if \(\mathrm{x}\) is in the set, then \(\mathrm{x} - 3\) is also in the set" into a clear understanding of how numbers get added to our set.

2. Apply the rule forward from the given number

Since we know 1 is in the set, let's see what other numbers we can generate by following the rule.

Starting with 1:

• 1 is in the set (given)
• Since 1 is in the set, then \(1 - 3 = -2\) must also be in the set
• Since -2 is in the set, then \(-2 - 3 = -5\) must also be in the set
• Since -5 is in the set, then \(-5 - 3 = -8\) must also be in the set
• Since -8 is in the set, then \(-8 - 3 = -11\) must also be in the set
• And so on...

So our set definitely contains: 1, -2, -5, -8, -11, -14, ...

This creates a pattern where we keep subtracting 3 from our starting number of 1.

3. Check if we can work backward to reach any option

Now let's think about this differently. Could any of our three options (4, -1, -5) be the starting point that eventually leads to 1?

For a number to MUST be in the set, we need to be able to reach it by starting from 1 and applying our rule (subtract 3) some number of times.

Let's check each option:

• Option I: 4 - Can we get from 1 to 4 by subtracting 3? No, subtracting 3 from 1 gives us smaller numbers, not larger ones.
• Option II: -1 - Can we get from 1 to -1 by subtracting 3? Let's see: \(1 \rightarrow -2 \rightarrow -5 \rightarrow -8\)... We never hit -1.
• Option III: -5 - Can we get from 1 to -5 by subtracting 3? Yes! \(1 \rightarrow -2 \rightarrow -5\). We found it!

4. Evaluate each option against our findings

Let's verify our reasoning by checking each option systematically:

Option I (4): For 4 to be in the set, we would need to start from 1 and somehow reach 4. But our rule only lets us subtract 3, which takes us away from 4, not toward it. So 4 does NOT have to be in the set.

Option II (-1): Starting from 1, we get: \(1 - 3 = -2\), then \(-2 - 3 = -5\), then \(-5 - 3 = -8\), etc. We get the sequence 1, -2, -5, -8, -11, ... Notice that -1 is not in this sequence. So -1 does NOT have to be in the set.

Option III (-5): From our sequence above, we can see that -5 appears when we apply the rule twice: \(1 \rightarrow -2 \rightarrow -5\). So -5 MUST be in the set.

Process Skill: APPLY CONSTRAINTS - Recognizing that the rule only works in one direction (subtracting 3) limits which numbers can be reached from our starting point.

Final Answer

Only option III (-5) must be in the set. We can reach -5 from our starting number 1 by applying the rule exactly twice: \(1 - 3 = -2\), then \(-2 - 3 = -5\).

Therefore, the answer is C. III only.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the direction of the rule: Students often think the rule works both ways - that if \(\mathrm{x}\) is in the set, then both \((\mathrm{x}-3)\) AND \((\mathrm{x}+3)\) are in the set. They miss that the rule only goes one direction: from \(\mathrm{x}\) to \((\mathrm{x}-3)\). This leads them to incorrectly assume that since 1 is in the set, then \(1+3=4\) must also be in the set.

2. Confusing "must be in the set" with "could be in the set": Students may think that if a number COULD potentially be in the set (like if we had additional starting numbers), then it MUST be in the set. The question asks what MUST be in the set given only that 1 is in the set, not what might be possible under other circumstances.

3. Not recognizing the arithmetic sequence pattern: Students may fail to see that starting from 1 and repeatedly subtracting 3 creates the sequence 1, -2, -5, -8, -11... They might try to work backwards or use trial-and-error instead of systematically applying the rule.

Errors while executing the approach

1. Arithmetic errors in the sequence: When generating the sequence by repeatedly subtracting 3, students may make simple calculation mistakes. For example, calculating \(1-3=-2\), then \(-2-3=-6\) instead of -5, which would cause them to miss that -5 is in the sequence.

2. Stopping the sequence too early: Students might only apply the rule once \((1 \rightarrow -2)\) and conclude that only -2 must be in the set, missing that -5 also appears when they continue the pattern one more step.

3. Incorrectly checking if numbers fit the pattern: When testing if -1 belongs to the sequence, students might incorrectly think that since -1 is "close" to -2, it must be reachable, without properly verifying that -1 can be generated by starting from 1 and subtracting 3 repeatedly.

Errors while selecting the answer

1. Selecting multiple options when only one is correct: After correctly identifying that -5 must be in the set, students might second-guess themselves and also select other options, thinking "Well, if -5 is in there, maybe -1 is close enough to also be correct." They fail to stick with their systematic analysis.

2. Choosing based on intuition rather than analysis: Students might look at the answer choices and think "4 seems reasonable since it's close to 1" or "both -1 and -5 are negative like what we'd get from subtracting" without actually checking their work against the sequence they generated.