Problem 2 - Thinking Mathematically

Question

The hour hand of a clock moves from 12 to 4 o'clock. Through how many degrees does it move?

Step-by-Step Solution

Answer

The hour hand moves through 120 degrees.

Step 1: Determine the Degrees per Hour

Understand that a full loop of the clock (12 hours) equals 360 degrees. Thus, each hour represents a movement of \( \frac{360}{12} = 30 \) degrees.

Step 2: Calculate the Degrees Moved

Calculate the degrees moved from 12 to 4 o'clock. This is a time period of 4 hours. So, the movement in degrees is 4 hours times 30 degrees per hour, equating to \( 4 * 30 = 120 \) degrees.

Key Concepts

Degrees in a Circle

To comprehend how the hour hand of a clock moves, we need to start by understanding the concept of a circle and how degrees are distributed across it. A circle is essentially a loop with no ends, and it is commonly divided into 360 equal parts, known as degrees. This division helps us measure angles efficiently.

Since a clock face is circular, it follows this concept. Calculating how much the hour hand moves involves understanding how these 360 degrees are spread around the clock. Imagine the clock as a pie, each hour from 1 to 12 represents a slice of this pie, with each slice equaling 30 degrees (\( \frac{360}{12} = 30 \)). This is because a clock has 12 numbers around it, and dividing the full circle of 360 degrees by 12 gives us 30 degrees per hour. This concept is crucial for analyzing any movement on the clock face.

Clock Geometry

The geometry of a clock face is a practical application of circular angle measurement. A clock is not just an instrument for telling time, but also a wonderful illustration of how geometric principles work in everyday life.

Each number on the clock represents a position in a 360-degree circle, divided into 12 sections of 30 degrees each. Therefore, the interval between any two consecutive numbers, such as from 12 to 1 or from 3 to 4, represents a movement of 30 degrees.

• Moving from 12 to 1 is a movement of 30 degrees.
• To reach from 12 to 3, the hand moves a total of 60 degrees (2 intervals of 30 degrees each).

In the problem, the hour hand moves from the number 12 to number 4. This involves moving through 4 intervals, each contributing 30 degrees. Thus, the total movement is 120 degrees (\( 4 \times 30 = 120 \)). Understanding these basic principles of clock geometry is essential for solving problems related to clock movements.

Mathematical Problem-Solving

Mathematical problem-solving involves applying concepts systematically to find solutions. When dealing with problems like the movement of a clock hand, it's important to break down the problem and solve it in steps.

First, identify the relationship between time and angle. Recognize that each hour on a clock corresponds to 30 degrees. The problem can be solved by calculating how many degrees are between the given times.

• Step 1: Compute the degree associated with one hour, which is 30 degrees.
• Step 2: Count how many intervals (or hours) the hand moves, in this case, 4 intervals from 12 to 4.
• Step 3: Multiply the number of intervals by the degrees per interval: \( 4 \times 30 = 120 \) degrees.

By breaking it into these manageable steps, mathematical problems become more solvable. Practice and understanding of foundational concepts are key, as they enable you to apply knowledge to a range of problems effectively.