時計算の基本【中学受験・SPI・公務員試験対策】（時計算1基本編)

中学受験のrestart

19 Oct 202108:07

Summary

TLDRThe video script introduces viewers to the concept of clock arithmetic and the fundamental approach to calculations involving time. It explains how to consider the speed of clock hands in terms of their movement, using the analogy of a race between a rabbit (long hand) and a tortoise (short hand). The script delves into the idea of angular velocity, where the long hand moves at a rate of 6 degrees per minute, while the short hand moves at 0.5 degrees per minute. By understanding these rates, viewers can solve problems related to the angles between the hands of a clock at any given time. The video also provides a method to visualize and calculate the angles between the hands for specific times, such as 4:35 and 11:25, using both the direct calculation and the race analogy. The summary emphasizes the importance of memorizing the speeds of the clock hands and applying the concept of angular velocity to solve various time-related problems.

Takeaways

🕒 **Understanding Clock Calculations**: The script introduces the concept of clock calculations and the basic way of thinking about them.
⏰ **Clock Speed as Angle**: It explains that the speed of the clock hands can be considered in terms of the angles they move, rather than the usual concept of speed.
📏 **Calculating Angles**: It demonstrates how to calculate the angles for the clock hands, with the long hand moving 6 degrees per minute and the short hand moving 0.5 degrees per minute.
🔄 **Clock Hands Movement**: The script clarifies that a clock's hands move at a constant rate of angles per minute, regardless of the clock's size.
📐 **Visualizing the Clock**: It suggests visualizing the clock face divided into 12 parts, with each part representing 30 degrees, to understand the movement of the hands.
🕰️ **Comparing Clock Sizes**: The size of the clock does not affect the rate at which the hands move in terms of degrees per minute.
📈 **Calculating Time Differences**: The method of calculating the angle between the long and short hands at any given time is explained, using the concept of angular velocity.
🏃‍♂️ **Rabbit and Turtle Analogy**: An analogy is used, comparing the long hand to a rabbit and the short hand to a turtle, to help understand the relative speeds at which they move.
✍️ **Practicing Clock Drawing**: The script encourages practicing drawing clocks to better visualize and solve problems related to the angles between the hands.
🤝 **Combining Hands' Movement**: It explains how to consider the combined movement of the long and short hands to solve for the angle between them at a specific time.
📘 **Memorising Key Figures**: Memorizing that the long hand moves 6 degrees and the short hand moves 0.5 degrees per minute can help in quickly solving clock problems.

Q & A

What is the main topic discussed in the transcript?
-The main topic discussed in the transcript is the concept of clock arithmetic and the basic way of thinking about calculations related to the movement of clock hands.
What is the term used to describe the speed at which the clock hands move?
-The term used to describe the speed at which the clock hands move is 'angular velocity'.
How is the angular velocity of the longer hand of a clock calculated?
-The angular velocity of the longer hand (minute hand) is calculated by the fact that it completes one full rotation (360 degrees) in 60 minutes, which means it moves at a rate of 6 degrees per minute.
How is the angular velocity of the shorter hand of a clock calculated?
-The angular velocity of the shorter hand (hour hand) is calculated by the fact that it moves 30 degrees in one hour, which means it moves at a rate of 0.5 degrees per minute.
What is the significance of understanding the difference in speed between the longer and shorter hands of a clock?
-Understanding the difference in speed between the longer and shorter hands is crucial for solving problems related to the angles between the hands at any given time, which can be used in various arithmetic and geometry problems.
How can one visualize the movement of the clock hands to solve problems?
-One can visualize the movement of the clock hands by considering the race between the 'Rabbit' (longer hand) and the 'Tortoise' (shorter hand), where the Rabbit moves 6 degrees and the Tortoise moves 0.5 degrees per minute.
What is the key to writing the clock face neatly when solving problems?
-The key to writing the clock face neatly is to divide the 12-hour clock into three equal parts for each hour and carefully plot the position of the hands based on the minutes passed.
How does the size of a clock affect the perceived speed of its hands?
-The size of a clock does not affect the actual speed of its hands because the angular velocity remains the same; however, it may affect the perceived speed due to the difference in the length of the hands and the scale of the clock face.
What is the method to find the angle between the longer and shorter hands at 4:35?
-To find the angle at 4:35, you would note that the longer hand is at the 7.5-minute mark (since it moves 0.5 degrees per minute) and the shorter hand is just past the 4, making the angle approximately 72.5 degrees.
How can you use the concept of angular velocity to solve for the angle between the hands at 11:25?
-At 11:25, you would calculate the angle for the longer hand from 11 o'clock (180 degrees) and subtract the distance the shorter hand has moved in 25 minutes (12.5 degrees), resulting in an angle of 167.5 degrees.
What is the advantage of using the 'Rabbit and Tortoise' analogy when solving clock arithmetic problems?
-The 'Rabbit and Tortoise' analogy helps visualize the relative speeds of the clock hands and simplifies the process of finding the angle between them, especially for times that are not on the hour or half-hour.
Why is it important to remember the angular velocities of the longer and shorter hands?
-Remembering the angular velocities of the hands (6 degrees per minute for the longer hand and 0.5 degrees per minute for the shorter hand) is important as it allows for quick and accurate calculations of the angles between the hands at any given time.

Outlines

00:00

🕒 Introduction to Clock Calculations

The first paragraph introduces the concept of clock calculations and the basic approach to solving problems related to timekeeping. The speaker, タカシ (Takashi), discusses how to approach speed problems using clock arithmetic, emphasizing that the size of the clock affects the speed at which the hands move. He explains the concept of angular velocity, which is the rate at which the clock hands move in degrees per minute. Takashi provides the angular velocity for both the longer hand (6 degrees per minute) and the shorter hand (0.5 degrees per minute), and then uses these concepts to solve a problem involving finding the angle between the long and short hands at a specific time (4:35). The explanation includes a step-by-step method to visualize and calculate the angles on a clock face.

05:03

🏃‍♂️ The Race of Clock Hands

The second paragraph elaborates on a different method to solve clock problems by visualizing the hands of the clock as characters in a race. The longer hand is likened to a rabbit, and the shorter hand to a turtle, with the rabbit moving faster (6 degrees per minute) compared to the turtle (0.5 degrees per minute). This metaphorical approach helps to understand the relative speeds and how the gap between the hands changes over time. The speaker uses this method to calculate the angle between the hands at 11:25, considering the starting positions and the progress made by each hand over 25 minutes. The summary concludes by emphasizing the utility of this approach for solving problems involving the relative positions of the clock hands at odd times.

Mindmap

Keywords

💡Clock arithmetic

Clock arithmetic refers to the method of solving problems related to the movement of clock hands. In the video, it is the central theme, where the presenter introduces how to approach speed-related problems by considering the angles and rotations of clock hands. It is used to calculate angles between the hour and minute hands at specific times.

💡Angular velocity

Angular velocity is the rate of change of angular displacement. The video explains that in the context of clock hands, angular velocity is used to describe how fast each hand moves in terms of degrees per minute. It is crucial for understanding how to calculate the angles between clock hands.

💡Hour hand

The hour hand is the longer hand on a clock that indicates the hours. The video discusses its speed in relation to the minute hand, noting that it moves 30 degrees per hour, which is equivalent to 0.5 degrees per minute.

💡Minute hand

The minute hand is the shorter hand on a clock that indicates the minutes. It is mentioned in the video that this hand moves faster than the hour hand, covering 6 degrees per minute, which is a key concept for solving the problems presented.

💡Angles on a clock

Angles on a clock are used to determine the positions of the clock hands. The video explains that there are 12 divisions on a clock, each representing 30 degrees, making a full circle 360 degrees. Understanding these angles is essential for calculating the relative positions of the hands.

💡Speed of clock hands

The speed of clock hands is different for the hour and minute hands. The video clarifies that the minute hand moves at a speed of 6 degrees per minute, while the hour hand moves at 0.5 degrees per minute. This difference in speed is important for determining the angle between the hands at any given time.

💡Time calculation

Time calculation is the process of determining the angles between the clock hands at specific times. The video provides examples of how to calculate these angles, such as at 4:35 and 11:25, using the concept of angular velocity and the speeds of the clock hands.

💡Clock face

The clock face is the surface of a clock where the hands move. The video script discusses the importance of visualizing the clock face when solving problems, particularly when dividing it into sections to determine the position of the hands more accurately.

💡Practice

Practice is emphasized in the video as a means to improve one's ability to solve clock arithmetic problems. The presenter suggests that viewers should practice drawing the clock face and calculating the angles to become more proficient in clock arithmetic.

💡Turtle and rabbit analogy

The turtle and rabbit analogy is used in the video to illustrate the relative speeds of the minute and hour hands. The rabbit represents the faster minute hand, while the turtle represents the slower hour hand. This analogy helps viewers visualize the concept of angular velocity and the movement of the clock hands.

💡Problem-solving

Problem-solving is the ultimate goal of the video, where the presenter aims to teach viewers how to solve clock arithmetic problems. The video provides methods and examples of how to calculate the angles between the clock hands at specific times, which is a form of mathematical problem-solving.

Highlights

Introduction to clock arithmetic and basic calculation concepts

Explains how clock speed can be considered in terms of angles and degrees

A clock's minute hand completes one rotation in exactly 1 hour, or 60 minutes

The minute hand moves at a speed of 6 degrees per minute

The hour hand moves more slowly, at a speed of 0.5 degrees per minute

By remembering the speeds of the hour and minute hands, you can solve problems more quickly

Example problem: Calculate the angle between the hour and minute hands at 4:35

Method 1: Calculate the angle from the 4 o'clock position considering the minute hand has moved 35 minutes

Method 2: Visualize the hour and minute hands as a rabbit (fast) and turtle (slow) race

The difference in angles between the hands increases by 5.5 degrees every minute

Another example: Calculate the angle at 11:25 using the race concept

At 11:25, the rabbit (minute hand) is 30 degrees ahead of the turtle (hour hand)

The angle difference increases by 5.5 degrees for every minute from 11 o'clock

Final answer for the 11:25 example is an angle of 167.5 degrees

Practical application: This method is convenient for solving problems with in-between times

Summary: Memorize the speeds (6 degrees/min for minute hand, 0.5 degrees/min for hour hand)

Consider the relative movement of the hour and minute hands to solve problems efficiently

This approach helps to visualize and solve clock arithmetic problems in a unique and engaging way