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)
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
🕒 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.
🏃♂️ 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
💡Angular velocity
💡Hour hand
💡Minute hand
💡Angles on a clock
💡Speed of clock hands
💡Time calculation
💡Clock face
💡Practice
💡Turtle and rabbit analogy
💡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
Transcripts
中学受験の湯スタートたかしです今日は
時計算の1回目と計算の基本的な考え方を
紹介しようと思います
去年は内容欄に張り付いてあります使って
みてくださいじゃあいメリーのエアーさあ
時計算これは速さの単元なんだけど時計
どうやって速さの問題になるのかっていう
ところんだけどこのタイプの時は時計算で
は使いません布袋何で出てくるのはこっち
根張りがぐるぐる回る方の穴どこの時を
考えていきますそしてこの針が動く速さを
考えよっていうことになるわけ
さあ
そそれ無理じゃない同時の中設定ねあの速
そう考えどう
こんな大きな時計もある4腕時計は小さい
大
速さが違うのではああああ
あーなるほどね大きさ違ったらね針が動く
速さ確かに違う気がするよねそうと計算を
考えるときの速さっていうのは普通の速さ
とちょっと違います
時計をこう見てみようかこの目角度を
考えるということです例えばこの角度何度
がすぐわかるかなそうこれはグーリ1周が
360度でこれが12個に分かれてる
でしょ
そうすると360はる12でこの12から
位置までの角度は36ということになり
ます
こんな風に針が動く角度で考えていくと
こういう考え方ができる
角速度ということですつまり普通の速さと
いうのは例えば自足っていうと1時間
あたりに何ちろメートル進むかっていう
ことだよね目と型の場合は例えばある時間
で何度進みますかっていうこの角速度を
考えていくということになりますこうする
と大きな時計小さな時計も時間毎に動く
角度は一緒だからね同じように考えること
ができるっていう事
さあそうするとまず長い針長身の速さを
考えてみよっか時計を動かしていくと
こんな風に動いていくねつまりこの長い針
っていうのは一回転するのにちょうど1
時間で回ってきますつまり
360度60分で回るわけ
ということは360は60分で1本あたり
6どう動くことになりますつまりこの
小さなメモリー一つが6両あるんだよなん
か意外とパクドがあるようなちゃうよね
じゃあ今度は短い針を考えてみましょう
溶け動かしてみるよこうするともちろん
長い春よりずいぶん遅いね
1時間60分でこの人メモリ文などだ
30度米30度を1時間で回っていきます
つまり30 a 60で分速0.5度これ
が短い針の速さです長い針が6度短いあり
が0.5度これはね覚えてしまった方が
早いと思います
じゃあこれを使って問題解いてみようか次
の時刻に長針と短針の間の角度で少ない方
の角度を求めてくださいということです
例えばこの4:35なんだけどこれねと
景山は自分で時計を書いてくださいで
そこそこ綺麗に書くコツはマンかでしょう
ねまずこれをジョンと思います12登録と
キュートさねそしてそれぞれを3等分して
いくと比較的きれいに時計を書くことが
出来ますこれ練習してみてねっそうすると
まず一つの解き方は4:35でしょう
長い針はなのところ俺を間違えようがない
よねで問題は短い有田短い針は読んじゃ
ないよ4のところからちょっと進んでる
でしょ
だから求めたい角度はそこになります
そうすると問題はこの短い針だよね
この短い針は4時からちょっと動いている
理由つまり4時の線から考えるとなったら
4までの角度は俺は州中重度だよね30度
が3つですじゃここの角度はとこれは短い
針が35分で動いた距離だよねつまり1分
に0.5度35分では17.5度動くこと
になります
ということで90から17.5を引いた
72.5っていうのはを耐えになる理由
サージは次にバンね
11:25の時を考えます長い針は大きく
右替針は11と12の間ですそうすると
11時から考えと長身と11時の間の角度
は不調の180
ここからここの角度を引くんだけどこれは
短いマリが25分で進んだ距離ですつまり
10 d 点ほどっていうことになるわけ
ということで180から12.5度を引い
て
167.5というのが答えになる理由
でもう一つ解き方がありますこれはね時計
の捉え方なんだけどこれ高針が回っていく
よねそうすると長い針と短い針が追いかけ
愛好しているというような考え方をする
わけ
つまりイメージとしたらじゃ外張りのほう
が早いんだよねだから長い有賀ウサギさん
です短い針はを添えカメさんということに
しておきましょうそうすると今うさぎと
かめが同じとこからスタートして用意ドン
で進んでいきますするとウサギさんは1分
に6ドッカメさんは1分に0.5のしか
進みませんということは1分あたり
5.5度物さがきれい駅ことになるよね
これを使う理由
そうするとさっきの例えばにば
11:25の時のあくどなんだけど
11時から考えていきます
この場合はカメさんよりもウサギさんが
30度だけ進んだ状態だよね
ここから要因とんですそうするとウサギ
さん早いからどんどん差が開いていくこと
になるよねどれくらい開くのかっていうと
いっに5.5度ずつ者が開いて言って25
分間ずっと開き続けるでしょ
最初30度だった性豪轟天号かけ25だけ
さらに開くことになりますということで
167年っていうのが出てくるわけでこの
やり方ねこういう中途半端な時間の時に
便利です
4:14の時の2つの針の間
こういうのもさっきのやり方を使えば4時
よスタートとして考えます
そうするとこの前はカメさんよりウサギ
さんが120度後ろからスタートする理由
そしてウサギさんがカメさんを追い詰めて
いきます14分間追い詰める分け1分に
5.5度物さが詰まっていきますという
ことは最初120度も開けたんだけども
轟天号各14分分だけさが短くなるつまり
答えは4サンドっていうふうに出てくる
わけではまとめますと景山は角速度1分
あたりの速さを考えていきます
長い針は1分26度短い針は1分に
0.5度この数字は覚えておいたほうが
早いですそして虎渓山は長い有富短い針の
長走って考えてください
異空間に5.5度物さがつまったり差が
開いたりする超そうしているいうふうに
考えるとうまく解けます
今日はこれで終わりにしますありがとう
ございました
ブーフ
ん
ん
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