Using 9 unit squares, solve the following. 1. What is the smallest perimeter possible? 2. What is the largest perimeter possible? 3. Make a figure with a perimeter of 18 units. 4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Understand the Question A unit square has a side length of and a perimeter of . When unit squares are placed together edge-to-edge, each shared side removes from the total outer perimeter. The perimeter formula is: where is the number of shared edges between adjacent squares. Smallest perimeter: Make the shape as compact as possible to maximize shared edges. Largest perimeter: Make the shape as elongated as possible to minimize shared edges. Part 1. What is the smallest perimeter possible? Step 1 · Find the Smallest Perimeter A square is the most compact arrangement of unit squares, giving the maximum number of shared sides. Total sides of separate squares . In a square, there are horizontal and vertical shared sides: Each shared side removes units from the perimeter: Answer > 1. Part 2. What is the largest perimeter possible? Step 1 · Find the Largest Perimeter Arranging the unit squares in a single row ( strip) creates the minimum number of shared sides, giving the maximum perimeter. In a strip, there are shared sides between the squares: Answer > 2. Part 3. Make a figure with a perimeter of 18 units. Step 1 · Construct a Figure with Perimeter of 18 Units To get a perimeter of : Therefore, the figure must have exactly shared sides. A C-shaped arrangement of unit squares has shared sides, giving a perimeter of . Answer > 3. A C-shaped figure made of unit squares has a perimeter of . Part 4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning? Step 1 · Analyze Shapes for Each Perimeter For 12 units: Only one shape is possible (the square). To have a perimeter of , the figure requires shared sides, which is only possible in a grid. For 20 units: Multiple shapes are possible. Any open, non-looping arrangement of squares (like an L-shape or T-shape) has exactly shared sides, resulting in a perimeter of . For 18 units: Multiple shapes are possible. Different arrangements having exactly shared sides can be formed by rearranging the squares. Answer > 4. There is only one shape for (a square). For perimeters of and , other shapes can be made. Common Mistakes Counting Shared Sides Twice: Forgetting that one shared side reduces the perimeter by (one side from each joining square), not . Including Internal Edges: Measuring internal boundary lines as part of the perimeter instead of only the outer boundary.

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Question 13

Using 9 unit squares, solve the following.

What is the smallest perimeter possible?
What is the largest perimeter possible?
Make a figure with a perimeter of 18 units.
Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Solution

Understand the Question

A unit square has a side length of $1\text{ unit}$ and a perimeter of $4\text{ units}$ .
When $9$ unit squares are placed together edge-to-edge, each shared side removes $2\text{ units}$ from the total outer perimeter.
The perimeter formula is:

P = (9 \times 4) - 2 \times (\text{number of shared sides}) = 36 - 2k

where $k$ is the number of shared edges between adjacent squares.

Smallest perimeter: Make the shape as compact as possible to maximize shared edges.
Largest perimeter: Make the shape as elongated as possible to minimize shared edges.

1. What is the smallest perimeter possible?

Step 1 · Find the Smallest Perimeter

A $3 \times 3$ square is the most compact arrangement of $9$ unit squares, giving the maximum number of shared sides. Diagram 1

Total sides of $9$ separate squares $= 9 \times 4 = 36$ .

In a $3 \times 3$ square, there are $6$ horizontal and $6$ vertical shared sides:

\text{Total shared sides} = 6 + 6 = 12

Each shared side removes $2$ units from the perimeter:

\begin{aligned} P &= 36 - (2 \times 12) \\[0.6em] &= 36 - 24 \\[0.6em] &= 12\text{ units} \end{aligned}

Answer

1. $12\text{ units}$

2. What is the largest perimeter possible?

Step 1 · Find the Largest Perimeter

Arranging the $9$ unit squares in a single row ( $1 \times 9$ strip) creates the minimum number of shared sides, giving the maximum perimeter. Diagram 2

In a $1 \times 9$ strip, there are $8$ shared sides between the $9$ squares:

\begin{aligned} P &= 36 - (2 \times 8) \\[0.6em] &= 36 - 16 \\[0.6em] &= 20\text{ units} \end{aligned}

Answer

2. $20\text{ units}$

3. Make a figure with a perimeter of 18 units.

Step 1 · Construct a Figure with Perimeter of 18 Units

To get a perimeter of $18\text{ units}$ :

\begin{aligned} 36 - 2k &= 18 \\[0.6em] 2k &= 18 \\[0.6em] k &= 9 \end{aligned}

Therefore, the figure must have exactly $9$ shared sides. Diagram 3

A C-shaped arrangement of $9$ unit squares has $9$ shared sides, giving a perimeter of $18\text{ units}$ .

Answer

3. A C-shaped figure made of $9$ unit squares has a perimeter of $18\text{ units}$ .

4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Step 1 · Analyze Shapes for Each Perimeter

For 12 units: Only one shape is possible (the $3 \times 3$ square). To have a perimeter of $12$ , the figure requires $12$ shared sides, which is only possible in a $3 \times 3$ grid.
For 20 units: Multiple shapes are possible. Any open, non-looping arrangement of $9$ squares (like an L-shape or T-shape) has exactly $8$ shared sides, resulting in a perimeter of $20\text{ units}$ .
For 18 units: Multiple shapes are possible. Different arrangements having exactly $9$ shared sides can be formed by rearranging the squares.

Answer

4. There is only one shape for $12\text{ units}$ (a $3 \times 3$ square). For perimeters of $18\text{ units}$ and $20\text{ units}$ , other shapes can be made.

Common Mistakes

Counting Shared Sides Twice: Forgetting that one shared side reduces the perimeter by $2\text{ units}$ (one side from each joining square), not $1\text{ unit}$ .
Including Internal Edges: Measuring internal boundary lines as part of the perimeter instead of only the outer boundary.

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