Critical Thinking Worksheets - 15 Worksheets Library - Free Printable

Problem:

The task is to determine how many squares are in the given image and explain the solution.

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Step-by-Step Solution:

#### 1. Understand the Task
We need to count all the squares in the provided grid, including:
- Individual small squares.
- Larger squares formed by combining smaller squares.
- Any other squares that can be identified within the grid.

#### 2. Analyze the Grid
The grid consists of a 4x4 arrangement of smaller squares. This means there are 16 individual small squares in total. However, we must also consider larger squares formed by combining these smaller squares.

#### 3. Count Squares of Different Sizes
To systematically count all the squares, we will categorize them by size:
- 1x1 squares: These are the smallest individual squares.
- 2x2 squares: Squares formed by combining 4 smaller squares.
- 3x3 squares: Squares formed by combining 9 smaller squares.
- 4x4 squares: The entire grid itself.

##### Step 3.1: Count 1x1 Squares
There are 16 individual 1x1 squares in the grid.

##### Step 3.2: Count 2x2 Squares
A 2x2 square is formed by combining 4 smaller squares. In a 4x4 grid, we can fit:
- Horizontally: 3 positions (since 4 - 2 + 1 = 3).
- Vertically: 3 positions (since 4 - 2 + 1 = 3).

Thus, the total number of 2x2 squares is:
\[
3 \times 3 = 9
\]

##### Step 3.3: Count 3x3 Squares
A 3x3 square is formed by combining 9 smaller squares. In a 4x4 grid, we can fit:
- Horizontally: 2 positions (since 4 - 3 + 1 = 2).
- Vertically: 2 positions (since 4 - 3 + 1 = 2).

Thus, the total number of 3x3 squares is:
\[
2 \times 2 = 4
\]

##### Step 3.4: Count 4x4 Squares
The entire grid itself is one large 4x4 square. There is only:
\[
1 \text{ square}
\]

#### 4. Sum Up All Squares
Now, we add up all the squares of different sizes:
\[
\text{Total squares} = (\text{1x1 squares}) + (\text{2x2 squares}) + (\text{3x3 squares}) + (\text{4x4 squares})
\]
\[
\text{Total squares} = 16 + 9 + 4 + 1 = 30
\]

#### 5. Consider Overlapping or Unique Squares
Upon closer inspection of the grid, there are no additional unique squares formed by irregular shapes or overlapping patterns. All squares are standard and fit within the grid's structure.

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Final Answer:

\[
\boxed{30}
\]

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Explanation:

The solution involves systematically counting squares of different sizes within the grid. By breaking down the problem into manageable parts (1x1, 2x2, 3x3, and 4x4 squares), we ensure no squares are missed and avoid double-counting. This methodical approach guarantees accuracy in determining the total number of squares.