Problem 61 - Algebra 1

Question

The area of a square is \(s^{2} .\) Show the relationship between the side length of a square and its area by copying and completing the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Side length, } s & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Area, } s^{2} & ? & ? & ? & ? & ? \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The completed table is: \( \begin{array}{|l|c|r|c|r|c|r|c|r|} \hline \text { Side length, } s & 1 & 2 & 3 & 4 & 5 \ \hline \text { Area, } s^{2} & 1 & 4 & 9 & 16 & 25 \ \hline \end{array}\)

Step 1: Compute the area for \(s = 1\)

The area of a square when side length \(s = 1\) is computed as \(s^{2} = 1^{2} = 1\).

Step 2: Compute the area for \(s = 2\)

The area of a square when side length \(s = 2\) is computed as \(s^{2} = 2^{2} = 4\).

Step 3: Compute the area for \(s = 3\)

The area of a square when side length \(s = 3\) is computed as \(s^{2} = 3^{2} = 9\).

Step 4: Compute the area for \(s = 4\)

The area of a square when side length \(s = 4\) is computed as \(s^{2} = 4^{2} = 16\).

Step 5: Compute the area for \(s = 5\)

The area of a square when side length \(s = 5\) is computed as \(s^{2} = 5^{2} = 25\).

Step 6: Complete the table

We substitute the calculated values into the table to get: \( \begin{array}{|l|c|r|c|r|c|r|c|r|} \hline \text { Side length, } s & 1 & 2 & 3 & 4 & 5 \ \hline \text { Area, } s^{2} & 1 & 4 & 9 & 16 & 25 \ \hline \end{array}\)

Key Concepts

Side Length and Area Relationship

Understanding the relationship between the side length of a square and its area is fundamental in geometry. A square is one of the simplest shapes, and its area is determined by the length of one of its sides. The formula to compute the area of a square is \( A = s^2 \), where \( A \) represents the area, and \( s \) is the length of a square's side. This formula implies that the area is equal to the side length raised to the power of two, which means side length is squared.

When we say the side length is squared, we mean that we multiply the side length by itself. Thus, if you know the side length of a square, you can easily find its area by squaring that length. Conversely, if you know the area, you can determine the side length by finding the square root of the area. This relationship is essential when dealing with not only squares but also other geometric figures and real-world problems involving area calculation.

Squared Numbers

Squared numbers are the result of multiplying a number by itself. In algebra, we represent this operation using an exponent of two; for example, \( 4^2 \) means \( 4 \times 4 \) which equals 16. The concept of squaring is critical in various areas of mathematics, including geometry, where squares of side lengths are used to calculate the areas of squares.

Examples of Squared Numbers

• \( 1^2 = 1 \times 1 = 1 \)
• \( 2^2 = 2 \times 2 = 4 \)
• \( 3^2 = 3 \times 3 = 9 \)

These results are known as perfect squares because they come from whole numbers. Knowing the squares of numbers allows pupils to understand patterns in mathematics and certainly facilitates quicker mental calculations. It is crucial for students to memorize some basic squared numbers to make learning algebra easier and more intuitive.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs. In the case of the area of a square, the algebraic expression is \( s^2 \), where \( s \) represents a variable standing for the side length of the square. Variables like \( s \) can take on different values, allowing algebraic expressions to describe a general rule or relationship.

Using Algebraic Expressions

Algebraic expressions become especially useful when solving problems that require general solutions or patterns. For instance, if we didn't know the specific side length of a square but we wanted a formula to calculate its area for any possible side length, \( s^2 \) gives us that flexibility. This expression showcases algebra's power: to express complex relationships simply and to solve problems in a generalized way. Students often work with algebraic expressions in various parts of math, developing the necessary skills to manipulate and use these expressions effectively in calculations and problem-solving.