71

My daughter in grade 3 is learning about telling time at her school. She eagerly showed me this method she has discovered on her own to tell the minutes part of the time on an analogue clock. I wasn't sure at first because I have never heard about it before but it works really well.

Here is her method:

  • Look at the minute hand and see what number it is pointing to, let's say it's $3$.

  • Add a zero at the end to make it $30$.

  • Halve it to get the minutes, so $3$ becomes $30$ and halving it gives $15$, $6$ becomes $60$ and halving it gives $30$ and so on.

It works well but I am not sure why. What mathematically justifies the method used?

amWhy
  • 204,278
  • 154
  • 264
  • 488
Sabeen Malik
  • 665
  • 1
  • 5
  • 7
  • 7
    How about 12 AM/PM? – Rafee Mar 21 '18 at 02:39
  • 115
    @Rafee This is a known defect in many clocks – pjs36 Mar 21 '18 at 02:48
  • 7
    This has me curious how you interpret a clock. This is more or less how I was taught as well and I can't imagine another way to learn it. – user2752467 Mar 21 '18 at 06:24
  • 4
    quite a handy method indeed when just telling the time, but I'd say that it's more useful to reason with "quarter past X, quarter to Y, half past X" when waiting for time or when rushing for an appointment, you kinda know it better when you're in advance or late, and this is something that's easy to grasp on analog watches/clocks – gl_prout Mar 21 '18 at 07:39
  • 10
    @JustinLardinois I imagine many people just learn it by rote - the same way many people learn their times tables. "6 7s are 42" is just a fact that they know to be true in isolation from any interpretation of how that works, and so is "the long hand pointing at the 4 is '20 past'". – Brondahl Mar 21 '18 at 09:24
  • 3
    @JustinLardinois Not sure how kids who are just learning to count would handle it, but knowing that all the way around is 60 minutes makes it pretty straightforward imo. I can't remember ever using the hour numbers to count minutes. – JollyJoker Mar 21 '18 at 10:01
  • @Rafee Unfortunately, I don't think many clocks are made with a 0 at the top. – Monty Harder Mar 21 '18 at 15:09
  • 9
    @MontyHarder, that's how you can tell a clock was made by a mathematician! – G Tony Jacobs Mar 21 '18 at 15:51
  • 5
    I think I first learned to do it counting by 5's, and then when I got more sophisticated, I counted by 15's to get close, and then by 5's, and eventually I just remembered them. I suspect the clock was more of a mnemonic for my multiplication facts involving 5 than the other way around. Not 100% sure on that, because memory gets hazy. – G Tony Jacobs Mar 21 '18 at 16:11
  • @Justin Fractions and rote. Half circle is 30 minutes, quarter is 15, third is 20, third of a quarter is 5, 1 third of a quarter and a little past half of another is 5 + 3 = 8, and so on. That's how I read analogue clocks. I don't remember ever paying attention to hour numbers when counting minutes. I find it interesting that others do it differently. – JoL Mar 21 '18 at 17:15
  • 1
    You should be proud your 3 year old can take 11 and solve for 110 / 2 = 55. Smart kid. – cbmeeks Mar 21 '18 at 20:17
  • 9
    "it works really well" ? You mean it work when the minute hand is exactly on an hour (demarker). I would not call that "really well", as it eliminates all other possible times. You also say "I am not sure why? " the equation you describe is (x * 10)/2 = 5X (as described below). Your profile indicates that you are a Business analyst(?) are you just trying to get up votes on your question? – DaniDev Mar 21 '18 at 20:36
  • @Rafee, it seems to work just fine for 12. OP's daugther just needs to understand what "mod 60" means. :D – Andrea Lazzarotto Mar 22 '18 at 00:19
  • She was very clever to notice that at 3 years old. – Felix Marin Mar 22 '18 at 03:11
  • 11
    @FelixMarin She isn't 3 years old; she's in grade 3. – Your IDE Mar 22 '18 at 08:26
  • 3
    It is interesting to note that how questions starting with my daughter has done this/that get an enormous no. of upvotes (this is the second time this month), but actual interesting questions hardly get any or are often overlooked. – Your IDE Mar 22 '18 at 08:31
  • 5
    It's a good idea to give the age of the child rather than using terms like "grade 3" which is likely to have different meanings in different countries. – Michael Kay Mar 22 '18 at 08:40
  • @Rafee nobody in their right mind (=nobody who wouldn't also use imperial measures) uses that format. and on a mod12 clock face you normally can still tell if you already had lunch :P – nonchip Mar 22 '18 at 10:40
  • @MontyHarder, see https://www.instrmnt.co.uk/01-a – Joel Reyes Noche Mar 22 '18 at 11:00
  • @YourIDE I'm sorry. I guess I didn't read the post very carefully. – Felix Marin Mar 22 '18 at 17:58
  • 6
    @DaniDev it works well for her and makes her life easy, I am not going to make things complicated for her. As for your comment on my qualification/intention, for someone with Physics background, you seem to be making subjective statements about the psychology at work. I am pretty okay with asking dumb questions if I don't understand things, without shame, even if people like yourself discourage it. – Sabeen Malik Mar 23 '18 at 00:19
  • Someone else here also seemed suspicious of the intention. @YourIDE I am happy to delete my own question if you guys think it's not worthy of the forum, I got what I came here for. – Sabeen Malik Mar 23 '18 at 00:34
  • 1
    I am not disparaging your daughter's ingenuity or problem solving skills in the least. Kudos to your daughter for figuring out a clever way of multiplying by five. I was just questioning of the phrasing of your question "It works well but I am not sure why. " Peace to you, my friend. – DaniDev Mar 23 '18 at 01:00
  • 5
    @DaniDev frankly, I think in my "proud parent moment" and not paying attention to the solution myself, I got carried away and missed the obvious. I wasn't sure why it worked or rather how to explain it to her in a way where she can understand it and in fairness, I am not a very mathematically minded person (and that's where you guys come in). I went home and showed her the answers here and how it's exactly like how we learned the 5 times table and she was happy with the fact that I was able to find her a satisfying answer. Peace to you too! – Sabeen Malik Mar 23 '18 at 01:23
  • Interestingly, this is also one way to tell the time in spoken Cantonese. So, for example, it would be common to say the equivalent of "7 o'clock 7" to mean "7:35" because the little hand points at the 7 when it is 35 minutes past the hour. – Flounderer Mar 22 '18 at 13:12

4 Answers4

244

Each number on the clock face is worth five minutes. One good way to multiply by $5$ is to first multiply by $10$, and then divide by $2$. This works because $5=10\div 2$.

G Tony Jacobs
  • 29,851
  • 4
  • 46
  • 100
  • 46
    You can also divide by $5$ in a similar fashion: Multiply by $2$ and then divide by $10$. – G Tony Jacobs Mar 21 '18 at 02:54
  • 77
    You should encourage your daughter to come up with as many mathematical tricks like this as possible. As she gets older, she should be able to explain more clearly why it always works (or when a trick shouldn't be used). Not only will it give her more insight into math and be more efficient, but she might have more fun at the same time. As she gets better at multiplying (sounds like she is ready to start learning more), see if you can get her to reason out how to quickly multiply by 9 or 99, etc. – Nick Brown Mar 21 '18 at 13:30
  • 3
    @NickBrown This is the sort of thing my dad taught me long ago. He worked as an auditor for A&P stores long before computers or even electronic calculators, and he took inventories often enough to give Count von Count a challenge for best counter. He learned/created a lot of tricks to be good at it. – Monty Harder Mar 21 '18 at 15:14
  • 38
    You can also divide by $2$ in a similar fashion: Multiply by $216$ and then divide by $432$. :-) – Asaf Karagila Mar 21 '18 at 17:24
  • 35
    @AsafKaragila , I always do that when working in base $432$ – G Tony Jacobs Mar 21 '18 at 19:05
  • @SabeenMalik To clarify, the issue is that you did not see adding a 0 at the end as multiplying by 10? – BCLC Mar 21 '18 at 20:00
  • 2
    @GTonyJacobs That trick is especially useful in calculating tips at a restaurant – Harrison Paine Mar 21 '18 at 20:50
  • 2
    @nickbrown we have a large portable whiteboard in our living room where we just try to explain things. I am encouraging pattern finding and making sense by drawing their thoughts out and finding issues. We have already done the 9 fun exercise where she figured out the pattern. I have learned so much from my kids, it's amazing. – Sabeen Malik Mar 23 '18 at 00:08
  • 2
    @BCLC, yes, I didn't think of adding the zero at the end as multiplying by 10 but when I saw this answer, I had the "duh!" moment and realised how I already do it like that in my head. I am glad I asked cause it just proves further to me that we all miss some obvious things and to be gentle with the ones that don't "get" things rather quickly. BTW as a newcomer here, I really like this community :) – Sabeen Malik Mar 23 '18 at 00:10
  • @SabeenMalik Ok, that's good. Was just wondering with the whole 10k rep on SO – BCLC Mar 23 '18 at 02:52
  • @GTonyJacobs Nice job Tony, glad to be here with you - Jonathan Haack ... it is just multiplying by 5 ... – oemb1905 Oct 19 '18 at 22:05
32

The minute hand passes over $12$ numbers in $60$ minutes.

That is $5$ minutes for each number.

Note that multiplying by $5$ is the same as multiplying by $10$ and dividing by $2$

Thus $3$ translates into $15$ which is $3(10)/2$.

Similarly $6$ translates into $30$ which is $6(10)/2$.

Mohammad Riazi-Kermani
  • 67,745
  • 4
  • 36
  • 86
  • +1 for explaining the significance of 12 and 60, which we need to explain how come each "number" represents 5 minutes rather than any other number. – Rosie F May 27 '18 at 06:13
5

The minutes are reckoned as

(Minutes pointing numeral N)(number of minutes in one hour =60)/(maximum minutes digits available on dial= 12) $= 5 N $

The procedure your daughter gave has effect of multiplying pointed figure $N$ by $5$; .. so it works.

Narasimham
  • 36,354
  • 7
  • 34
  • 88
3

From the perspective of the minute hand, the numbers mark the 1/12ths of an hour. But we are usually more interested in the 1/60ths of an hour: the minutes.

Since 12 and 60 differ by a factor of 5 (i.e. $12*5=60$), converting between the two is quite easy. We just multiply by 5.

Or as your daughter prefers, you could multiply by 10 (by adding a 0) and then divide by 2. Since $10/2=5$, it amounts to the same thing.

Matthew Matic
  • 425
  • 2
  • 12