King School has an enrollment of 900 students. The school day consists of 6 class periods during which each class...

1. Translate the problem requirements: Understand what we know about the school structure - 900 students total, 6 class periods per day, 30 students per class, each teacher teaches 5 periods and has 1 free period, no student has free periods
2. Calculate total class slots needed: Determine how many individual class sessions must occur across all 6 periods to accommodate all students
3. Calculate total teacher-periods available: Figure out how many teaching slots we need filled, knowing that teachers work 5 out of 6 periods
4. Solve for number of teachers: Use the relationship between teaching demand and teacher availability to find the total number of teachers

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by organizing what we know about King School in everyday terms:

Think of this school like a big scheduling puzzle. We have:

• 900 students who all need to be in class
• The school day is divided into 6 time slots (periods)
• Every class has exactly 30 students
• Teachers work 5 out of 6 periods (they get 1 period off)
• Students don't get any periods off - they're in class all 6 periods

Process Skill: TRANSLATE - Converting the word problem into clear, manageable pieces of information

2. Calculate total class slots needed

Now let's figure out how many individual classes the school needs to run.

Think about it this way: if we have 900 students and each class holds 30 students, then during any single period, we need:

\(900 \text{ students} ÷ 30 \text{ students per class} = 30 \text{ classes}\)

Since there are 6 periods in a day, and we need 30 classes during each period:

Total class sessions per day = \(30 \text{ classes} × 6 \text{ periods} = 180 \text{ class sessions}\)

This means 180 individual classes must be taught throughout the entire school day.

3. Calculate total teacher-periods available

Now here's the key insight: each teacher only works 5 periods out of 6.

Let's say we have T teachers total. Since each teacher teaches 5 periods:

Total teaching periods available = \(T \text{ teachers} × 5 \text{ periods each} = 5T \text{ teaching periods}\)

Think of it like this: if you have 10 teachers and each works 5 periods, you have 50 teacher-periods of work available.

4. Solve for number of teachers

Here's where we connect our two calculations:

The school needs 180 class sessions to be taught (from step 2)

The teachers can provide 5T teaching periods (from step 3)

For the school to function, these must be equal:

\(5T = 180\)

Solving for T:

\(T = 180 ÷ 5 = 36\)

Process Skill: INFER - Recognizing that teacher availability must exactly match teaching demand

Final Answer

The school has 36 teachers.

Let's verify: \(36 \text{ teachers} × 5 \text{ periods each} = 180 \text{ teaching periods}\), which matches exactly the 180 class sessions needed.

The answer is C. 36

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint that teachers work 5 out of 6 periods

Students often overlook this crucial constraint and assume teachers work all 6 periods. This leads them to think they need fewer teachers since each teacher would handle more classes. They might calculate: \(180 \text{ total class sessions} ÷ 6 \text{ periods per teacher} = 30 \text{ teachers}\), leading to answer choice B.

2. Confusing students and classes when calculating capacity

Students may incorrectly think about "30 students per class" and get confused about whether they should divide 900 by 30 or multiply. Some students calculate total classes needed incorrectly, thinking they need 900 classes total rather than understanding that with 30 students per class, they need \(900 ÷ 30 = 30\) classes running simultaneously during each period.

3. Not recognizing this as a capacity-matching problem

Students may not realize they need to match "teaching demand" (180 class sessions needed) with "teaching supply" (teachers × periods worked). Instead, they might try other approaches like dividing total students by teachers directly, leading to incorrect setups.

Errors while executing the approach

1. Arithmetic errors in basic calculations

Students make simple calculation mistakes such as: \(900 ÷ 30 = 25\) instead of 30, or \(30 × 6 = 150\) instead of 180, or \(180 ÷ 5 = 30\) instead of 36. These arithmetic errors lead directly to wrong answer choices.

2. Using 6 instead of 5 when calculating teacher capacity

Even if students understand the constraint conceptually, they may accidentally use 6 periods per teacher in their calculation: \(6T = 180\), giving \(T = 30\). This gives answer choice B instead of the correct answer C.

Errors while selecting the answer

No likely faltering points