A farmer has 12 feet of fencing. He constructs a rectangular garden whose sides are whole numbers. He uses all the fence to enclose the garden.

To solve the farmer's fencing problem, we'll start by establishing the relationship between the perimeter and the dimensions of the rectangular garden.

Given:
- The total length of fencing is 12 feet.
- The formula for the perimeter \(P\) of a rectangle is \(P = 2(w + l)\) where \(w\) is the width and \(l\) is the length.

Setting up the equation:
\[
2(w + l) = 12
\]
Dividing both sides by 2:
\[
w + l = 6
\]

### Part 1: How many possible shapes can the garden have?

Since both \(w\) and \(l\) must be whole numbers, we can list the pairs \((w, l)\) that satisfy \(w + l = 6\):

1. \(1 + 5 = 6\) (Garden dimensions: 1 ft x 5 ft)
2. \(2 + 4 = 6\) (Garden dimensions: 2 ft x 4 ft)
3. \(3 + 3 = 6\) (Garden dimensions: 3 ft x 3 ft)
4. \(4 + 2 = 6\) (Garden dimensions: 4 ft x 2 ft)
5. \(5 + 1 = 6\) (Garden dimensions: 5 ft x 1 ft)

This gives us the distinct pairs of dimensions:
- (1, 5)
- (2, 4)
- (3, 3)

Thus, the number of possible shapes is **3**, corresponding to:

- \(1 \times 5\)
- \(2 \times 4\)
- \(3 \times 3\)

**Answer (Part 1):** C. 1 times 5, 2 times 4, 3 times 3

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### Part 2: What are the areas of the possible gardens?

Now we compute the area for each garden shape using the formula \(A = w \times l\):

1. For \(1 \times 5\):
\[
A = 1 \cdot 5 = 5 \quad \text{sq ft}
\]

2. For \(2 \times 4\):
\[
A = 2 \cdot 4 = 8 \quad \text{sq ft}
\]

3. For \(3 \times 3\):
\[
A = 3 \cdot 3 = 9 \quad \text{sq ft}
\]

Therefore, the areas for the possible gardens are:
- 5 sq ft for \(1 \times 5\)
- 8 sq ft for \(2 \times 4\)
- 9 sq ft for \(3 \times 3\)

**Answer (Part 2):** The areas are: 5 sq ft, 8 sq ft, 9 sq ft.

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### Part 3: What are the length and width of the shape that has the largest area?

From the computed areas, we see that the maximum area is \(9 \, \text{sq ft}\) which corresponds to the dimensions of the shape:

**Answer (Part 3):** The shape with the largest area is \(3\) ft by \(3\) ft.

In conclusion, the farmer can create three distinct shapes, with areas being 5 sq ft, 8 sq ft, and 9 sq ft, and the largest area corresponds to a square garden measuring 3 ft by 3 ft.