List K consists of 12 consecutive integers. If -{4} is the least integer in list K, what is the range...

1. Translate the problem requirements: We need to understand what "12 consecutive integers starting from -4" means, identify which of these are positive, and find the range (highest minus lowest) of only the positive integers.
2. List out the consecutive integers: Starting from -4, write out all 12 consecutive integers to visualize the complete set.
3. Identify the positive integers: Separate out only those integers from our list that are greater than zero.
4. Calculate the range of positive integers: Find the difference between the largest and smallest positive integers in our identified subset.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is asking in plain English. We have a list K with 12 consecutive integers, which means 12 numbers that follow each other in order without any gaps (like 5, 6, 7, 8...). The smallest number in this list is -4.

The question asks for the "range of the positive integers" in list K. This means:

• We need to find which numbers in our list are positive (greater than zero)
• Then calculate the range of just those positive numbers
• Range means: largest positive integer minus smallest positive integer

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. List out the consecutive integers

Since we start at -4 and need 12 consecutive integers, let's write them all out:

Starting from -4: \(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7\)

Let me verify this is correct:

• We start at -4 (the least integer given)
• We count 12 numbers total: -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7
• These are consecutive (each number is exactly 1 more than the previous)
• Count check: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ✓

So our complete list K is: \(\{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7\}\)

3. Identify the positive integers

Now we need to pick out only the positive integers from our list. Remember, positive integers are numbers greater than zero.

Looking at our list: \(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7\)

The positive integers are: \(1, 2, 3, 4, 5, 6, 7\)

Note that:

• Negative numbers (-4, -3, -2, -1) are not positive
• Zero (0) is neither positive nor negative
• Only 1, 2, 3, 4, 5, 6, 7 are positive

Process Skill: APPLY CONSTRAINTS - Correctly identifying which integers meet the "positive" requirement

4. Calculate the range of positive integers

Now we find the range of our positive integers: \(1, 2, 3, 4, 5, 6, 7\)

Range = Largest positive integer - Smallest positive integer
Range = \(7 - 1 = 6\)

Let's double-check this makes sense:

• Smallest positive integer in our list: 1
• Largest positive integer in our list: 7
• The difference (range): \(7 - 1 = 6\)

Final Answer

The range of the positive integers in list K is 6.

This matches answer choice B. 6

Verification: Our list K contains the positive integers \(\{1, 2, 3, 4, 5, 6, 7\}\), and the range \(7 - 1 = 6\) is correct.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "range of positive integers" means

Students often confuse this with either: (a) the range of ALL integers in the list, or (b) simply counting how many positive integers there are. The question specifically asks for the range (max - min) of ONLY the positive integers, not the entire list.

2. Forgetting that zero is neither positive nor negative

When identifying positive integers from the list, students may incorrectly include zero as a positive integer. This would make their smallest positive integer 0 instead of 1, leading to an incorrect range calculation.

3. Misinterpreting "12 consecutive integers" starting from -4

Some students might think the list goes from -4 to +12, or they might incorrectly count backwards from -4. The correct interpretation is that -4 is the smallest value, and we count 12 consecutive integers forward from there.

Errors while executing the approach

1. Arithmetic errors when listing consecutive integers

Students may make simple counting mistakes when writing out the 12 consecutive integers starting from -4, potentially missing a number or adding an extra one, which would shift all subsequent calculations.

2. Incorrectly identifying which integers are positive

Even with the correct list, students might mistakenly include 0 as positive or exclude 1, leading to an incorrect set of positive integers (e.g., thinking the positive integers are \(\{0,1,2,3,4,5,6,7\}\) instead of \(\{1,2,3,4,5,6,7\}\)).

Errors while selecting the answer

No likely faltering points

Once students have correctly identified the positive integers as \(\{1,2,3,4,5,6,7\}\), the range calculation \((7-1=6)\) is straightforward and the answer choice B is clearly 6. There are no common errors in this final selection phase.