A grocer stacked oranges in a pile. The bottom layer was rectangular with 3 rows of 5 oranges each. In...

1. Translate the problem requirements: Understand the 3D stacking pattern where bottom layer is \(3 \times 5\) rectangle (15 oranges), each orange in layer 2 sits on exactly 4 oranges from layer 1, and each orange in layer 3 sits on exactly 4 oranges from layer 2. Find maximum possible oranges in layer 3.
2. Analyze the geometric constraints: Determine how oranges can be positioned when each must rest on exactly 4 oranges from the layer below, focusing on what this means for spacing and arrangement.
3. Map out layer 2 possibilities: Identify where oranges can be placed in the second layer given the \(3 \times 5\) bottom arrangement, considering that each needs 4 supporting oranges beneath it.
4. Determine layer 3 maximum: Based on the actual positions possible in layer 2, calculate how many oranges can fit in layer 3 while maintaining the 4-orange support requirement.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• Bottom layer: A rectangular arrangement with 3 rows and 5 columns of oranges. Think of this like a \(3 \times 5\) grid, so we have 15 oranges total on the bottom.

• Second layer rule: Each orange in the second layer must rest on exactly 4 oranges from the bottom layer. This means the orange sits at the intersection where 4 bottom oranges meet.

• Third layer rule: Each orange in the third layer must rest on exactly 4 oranges from the second layer.

• Goal: Find the maximum number of oranges possible in the third layer.

Process Skill: TRANSLATE - Converting the 3D stacking description into clear geometric constraints

2. Analyze the geometric constraints

Now let's think about what it means for an orange to "rest on 4 oranges" below it:

Imagine looking down from above at a layer of oranges arranged in a grid. For an orange to sit on exactly 4 oranges below, it must be positioned at the corner where 4 oranges meet - like sitting in the intersection of a tic-tac-toe grid.

In a \(3 \times 5\) rectangular arrangement, if we label positions as coordinates, an orange that rests on 4 oranges would sit at positions between the grid points. For example, if oranges in layer 1 are at positions \((1,1)\), \((1,2)\), \((1,3)\)... then an orange in layer 2 could sit at position \((1.5, 1.5)\) - right between oranges at \((1,1)\), \((1,2)\), \((2,1)\), and \((2,2)\).

Process Skill: VISUALIZE - Understanding the 3D spatial relationship through 2D coordinate thinking

3. Map out layer 2 possibilities

Let's figure out exactly where oranges can be placed in the second layer:

With a \(3 \times 5\) bottom layer, we can think of the oranges as occupying a \(3 \times 5\) grid of positions. The "intersection points" where 4 oranges meet form a smaller grid.

If the bottom layer has 3 rows and 5 columns, then:

• Between 3 rows, there are 2 "between-row" positions

• Between 5 columns, there are 4 "between-column" positions

So the second layer can have at most \(2 \times 4 = 8\) oranges arranged in a \(2 \times 4\) rectangular pattern.

However, we need to be careful - each of these 8 positions does indeed rest on exactly 4 oranges from the layer below, so all 8 positions are valid for layer 2.

4. Determine layer 3 maximum

Now we apply the same logic to find layer 3:

Layer 2 has oranges arranged in a \(2 \times 4\) pattern. Using the same "intersection" reasoning:

• Between 2 rows, there is 1 "between-row" position

• Between 4 columns, there are 3 "between-column" positions

Therefore, layer 3 can have at most \(1 \times 3 = 3\) oranges.

Each of these 3 oranges in layer 3 would rest on exactly 4 oranges from layer 2, satisfying our constraint.

Process Skill: APPLY CONSTRAINTS - Systematically applying the same geometric rule to each successive layer

5. Final Answer

The maximum number of oranges that could be in the third layer is 3.

Verification:

• Layer 1: \(3 \times 5 = 15\) oranges

• Layer 2: \(2 \times 4 = 8\) oranges (each resting on 4 from layer 1)

• Layer 3: \(1 \times 3 = 3\) oranges (each resting on 4 from layer 2)

The answer is (C) 3.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the stacking constraint

Students often misunderstand what "each orange rests on 4 oranges" means. They might think it means each orange simply touches 4 oranges, rather than understanding that the orange must be positioned at the intersection point where exactly 4 oranges meet. This leads to incorrect visualization of how the layers are arranged.

2. Confusing the rectangular arrangement

Students may incorrectly interpret "3 rows of 5 oranges each" and get confused about which dimension represents rows versus columns. This confusion can lead them to set up the wrong initial grid (\(5 \times 3\) instead of \(3 \times 5\)), affecting all subsequent calculations.

3. Not recognizing the pattern for successive layers

Students often fail to see that if layer n has dimensions m×k, then layer n+1 will have dimensions \((m-1) \times (k-1)\). They might think each layer simply removes one orange from each side randomly, rather than understanding the systematic geometric constraint.

Errors while executing the approach

1. Miscounting intersection points

When determining how many "between-positions" exist, students often make off-by-one errors. For example, with 3 rows, they might think there are 3 between-row positions instead of 2, or with 5 columns, they might count 5 between-column positions instead of 4.

2. Incorrect application of the constraint to layer 3

Students may correctly find that layer 2 has 8 oranges in a \(2 \times 4\) arrangement, but then incorrectly apply the intersection logic to find layer 3. They might calculate \(2 \times 4 = 8\) again, or make arithmetic errors when computing \((2-1) \times (4-1) = 1 \times 3 = 3\).

Errors while selecting the answer

No likely faltering points - once students correctly execute the approach and find that layer 3 can have 3 oranges, the answer selection is straightforward as this directly corresponds to choice (C).