Question

How many \(1-\mathrm{cm}\) squares would it take to construct a square that is \(1 \mathrm{~m}\) on each side? MISSED THIS? Read Section 1.6

Short answer

10,000 1-cm squares are needed to construct a 1-m square.

Step-by-step solution

Understand the problem

We are asked to determine how many 1 cm x 1 cm squares are needed to cover a square that is 1 meter x 1 meter. Since 1 meter is equal to 100 centimeters, we can convert the dimensions of the larger square into centimeters to be consistent with the unit for the smaller squares.

Calculate the area of the large square

Calculate the area of the 1 m x 1 m square in centimeters. The area A of a square is given by the formula A = a^2, where 'a' is the length of the side. Since 1 m = 100 cm, the area of the large square is A = (100 cm)^2.

Calculate the number of small squares

Each 1 cm x 1 cm square has an area of 1 cm^2. The number of small squares needed is equal to the area of the large square divided by the area of one small square.

Perform the calculation

Divide the area of the large square by the area of a small square to find the number of small squares needed to cover the larger square: Number of squares = (100 cm x 100 cm) / (1 cm x 1 cm).

Key concepts

Unit Conversion

When working with mathematical problems that involve measurements, it's common to encounter different units that need to be converted to make calculations easier or to present the answer in the requested unit. Unit conversion is the process of converting a given unit of measurement into another unit of measurement without changing the actual amount or value. This concept is particularly important in the calculation of areas, as seen in the exercise involving the conversion from meters to centimeters.

For example, knowing that \(1 \text{m} = 100 \text{cm}\) allows you to convert the measurement of a square from meters to centimeters, which is crucial to find out how many \(1-\text{cm}\) squares fit into a \(1 \text{m}\) square. This step of unit conversion simplifies our calculations and ensures we're comparing like with like. Converting units is a fundamental skill in math and science and is often one of the first steps in solving measurement-related problems.

Metric Units

The metric system is an international system of measurement based on multiples of ten. It's used by most countries for everyday measurements and is the standard in scientific measurements worldwide. The basic units of length in the metric system are millimeters (mm), centimeters (cm), meters (m), and kilometers (km), with 10 millimeters in a centimeter, 100 centimeters in a meter, and 1,000 meters in a kilometer.

Understanding these metric units is essential when dealing with problems such as area calculations. As seen in the exercise, recognizing that a meter consists of 100 centimeters allows students to work with units consistently across different measurements and to comprehend the scale of objects they're working with. Familiarity with this system simplifies the process of calculating areas, volumes, and lengths in a wide variety of contexts.

Area of a Square

The area of a square is a key concept in geometry defining the size of a two-dimensional space enclosed by the square. The formula to calculate the area \(A\) of a square is \(A = a^2\), where \(a\) is the length of one side of the square. For instance, in the exercise provided, since the side of the large square converts to 100 cm, the area becomes \(100 \text{cm} \times 100 \text{cm} = 10,000 \text{cm}^2\).

This simple formula illustrates that the area is proportional to the square of the side length, meaning that even small increases in side length can significantly increase the enclosed area. The concept becomes practical when calculating space for construction, analyzing land area in geography, or solving problems in mathematics. By understanding how to calculate the area of a square, students are equipped to tackle various applications in both academic and real-world scenarios.