Problem 3 - Thinking Mathematically

Question

The hour hand of a clock moves from 1 to 4 o'clock. Through how many degrees does it move?

Step-by-Step Solution

Answer

The hour hand moves through 90 degrees.

Step 1: Understand the Structure of the Clock

In a clock, there are 12 numbers indicating the hours. Each pair of adjacent numbers forms an angle of \( \frac{360}{12}=30 \) degrees.

Step 2: Find the Number of Hours Passed

According to the exercise, the hand of the clock travels from 1 o'clock to 4 o'clock. This means a total of 4-1=3 hours has passed.

Step 3: Calculate the Movement in Degrees

Multiply the number of hours that have passed by the degrees each hour represents on the clock face. The calculation is as follows: 3 hours * 30 degrees/hour = 90 degrees.

Key Concepts

Degrees Calculation

When calculating the degrees moved by the hour hand on a clock, it's essential to consider the circular nature of the clock face. A clock is a circle, and a full circle is 360 degrees. The clock is divided into 12 equal sections, each representing an hour. Therefore, each section—or each hour—forms an angle of \( \frac{360}{12} = 30 \) degrees. By understanding this division, we can calculate the exact movement of the clock hands during a given period. If you need to find out how far an hour hand has moved, you simply count the number of hourly sections it has crossed and multiply by 30 degrees. This approach simplifies the process of determining the degrees moved during a given period of time.

Hour Hand Movement

The movement of the hour hand on a clock is not merely about hours but about angles and precision. With each passing hour, the hour hand moves 30 degrees. This movement is uniform and occurs regardless of the speed at which the minutes or seconds move. Understanding the hour hand's movement is key when solving problems related to clocks. For instance, when the hour hand moves from the 1 o'clock position to the 4 o'clock position, it passes over three hour markers. Since each hour marker transition equals 30 degrees, moving between markers 1 to 4 results in a movement of \( 3 \times 30 = 90 \) degrees. This consistency allows us to predict clock hand positions accurately at various times without visual reference.

Math Problem Solving

Clock angle problems are a great way to exercise your problem-solving skills in mathematics. They combine basic geometry with real-world applications, making them both practical and engaging. When solving such problems, it's important to approach them logically and step-by-step.

• First, understand the context and details. Recognize that the clock is a circle and the hour hand travels 30 degrees per hour.
• Determine the period in question—how many hours are involved in the movement?
• Use multiplication to calculate the angle: multiply the number of hour intervals by the degree rotation for each hour.

These problems help in honing analytical skills by requiring clear calculation processes and logical reasoning. Practice with clock angle problems can improve one's ability to tackle a variety of mathematical challenges.