Calculating Square Root by Hand (Early Grades)
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- čas přidán 4. 01. 2021
- Watch this educational video from the Spirit of Math Curriculum, presented by Spirit of Math founder and CEO Kim Langen teaching methods of calculating square root by hand.
to find out more about Spirit of Math visit spiritofmath.com/
I learned this method 50 years ago from my Chemistry teacher. I later found out that this method is based on Bionomial Expansion (a+b)squared. Not many knew this long division method in school these days. Thanks to CZcams, this method has been revealed. I love this method.
It was not taught generally in class, but my primary school maths teacher taught me!
I too learned and later forgot this method years ago. I was amused that the presenter used chalk, which broke, while working the problem. That really brought back the 60s tome.
I really appreciate your clear and methodical procedure, and very pleasant ways.
Our math teacher showed us this method in extra classes. Everything was almost the same, except that she said that you can not only multiply by 2, but also add. For example, 48 * 2 = 96. But you can get 96 by adding 8x + x (88+8 = 96), which was usually intuitive, since we put two dots when we selected the number for multiplication.
Exactly the same in the second case: 487 * 2 = 974, but you can get the same thing if you add 7 to 967. Thus, 967 + 7 = 974. It always works.
That is, once again. When you have decided on a digit, multiplied, calculated the difference, and you need to multiply the top number by 2, we don’t have to do this. You can take the number that was the last one on your left and add it with the digit that you put the last (its own last digit).
We learned this when I was in 6th grade. Years later, I used Newton's method: start with any reasonable guess, then iterate: new guess = 1/2( guess + number/guess). It converges quite rapidly.
Heron's formula
@@SusanaSoltner I didn't know that - thanks.
Back when i was in grade six (in Canada), i went to Ecuador for the summer. I was bored as everyone was in school. So my mom enrolled me in school there for a couple of months. In that short period, my math skills jumped to a Canadian grade 8 level. I learned how to do square root by hand. When i got back to Canada, i went back to learning long division, and in grade eight, we learned to use calculators.
This is how I remember doing it in high school -- many moons, ago -- thanks for the refresher!
Feel same. Been a bit and I feel as you, just a little reminder to do elementary problems! Want a nice square concrete pad and although few concrete workers remember and quiet likely never did by the fun of me when I mentioned hypotenuse they get a big laugh at there 10th grade drop out lol. He who laughs first laughs last right lol. Bless their hearts lol. I always like the 3,4 and 5 or even double the number helps. What I really love is running a say three foot diameter pipe through a floor system lol. Usually take my measurements home lay out on piece of cardboard then bring in to work and always fits so nice nothing even gets mentioned lol but that’s fine huh. I will give anyone who may be interested the Pipe fitters hand book is small and like anything the more you do it you get even noticed less but who wants noticed if it all works nicely. I was a Union Ironworker and modest. Again thank you for the refresher, very nice ❤️. Calculators are very handy lol.. Left 4 men to form up for a metal building and wanted the exterior sheets to run down the side of the concrete pad to eliminate water 💦 running inside the building. Many ways of laying out and having one nice square corner sure simplifies ✌🏼.
Sincerely Grateful, HB
I learned this method sometime in middle school I believe. That would be in the 1960s. Thanks for the refresher
Wow! I never knew that calculating a square root could be so fun!
My schools never taught this, and I always wanted to know how to do it by hand.
I went through five bad videos before I found yours. One guy even helpfully blocked the view of the whiteboard while he explained what was on it.
It took about a minute to catch on watching you. Thank you!
It would be good if you would explain where this method is comming from. The binomial theorem. One can also use other algorithms as herons method.
technicaly, the source of this math is Euclid.
I agree. I'd love to see the proof behind this method.
@@tomvitale3555 a²+2ab+b²
* coming
I learned this from my 3rd grade (4th year) math teacher. He made math fun. Subsequent math teachers varied in quality, but I didn't have another who was that good until university.
Even if you pick a number that is too high for the next step, the algorithm is self-correcting.
I too leaned this in Grade 3, at age 9, from my Dad. His explanation wasn't quite as tight as one now finds on the internet, but was sufficient for me to have some fun.
I learned this method in 7th grade, back in 1962 or -63. It wasn't part of the curriculum, but I asked our teacher, Mrs Galloway, if there were such a manual method, and she showed me after class. I'd long since forgotten it when I stumbled across this video. The Babylonian method is another way - much simpler to flowchart, but involves ever more lengthy long divisions.
What was part of the curriculum? Square root using a log book or square root using a calculator? Did you use a calculator in class in 1962?
@@johnchristian7788 Consumer-grade electronic calculators wouldn't be invented for another ten years. We were probably shown where to look up tabulated values in a handbook. Use of log tables wasn't introduced until high school (grade 9-10). My dad showed me how to use a slide rule at some point, but I don't remember when. Geez, this was over sixty years ago - I don't remember when they taught what.
@@lesnyk255 It's funny to think that even before calculators became popular, they didn't teach square root by pen and paper. They should really include in the curriculum in all countries.
I used to love using log tables.
@@johnchristian7788 Well, personally, I wouldn't go back to using log tables, slide rules, or manual typewriters except maybe at gunpoint. There are easier ways to get rough manual estimates of square roots if you've left your calculator or iPhone at home - polynomial approximation, for example, or the Babylonian method. This video was a bit of a nostalgia rush - 7th grade, Walpole NH JHS... long time ago....
thank you! well explained!
Just use Newton's method x=(x+a/x)/2 where a is the number whose sq root is to be found and x is the current approximation to the sq root. And iterate.
Superb! I took sometime shifting through many video clips to find out yours with simple explanation how to calculate the square root.
We're so glad to hear that! Thanks for sharing 🙌
It's obvious that the square root lies between 48 and 49, because 48^2 is 2304 and 49^2 is 2401. I can use a linear approximation to determine additional digits. 2376.6 - 2304 is 72.6, and the difference between 48^2 and 49^2 is 97, so 72.6/97 is my linear approximation, which gives me the next digit of 7. So far I have 48.7, and I can use linear approximations to double the number of significant digits with each iteration.
But everyone knows this.
I appreciate your great and simple explanation.
Very well explained, many thanks!
Good job!
Watched so many videos but this is the only one that helped me with this, thanks so much!
Glad it helped!
Amazing. Thank you, teacher!
The first rule of optimization is to identify the operations that take the most total time, and work on making those faster. If this is an infrequently used procedure, i.e. it won't represent a significant portion of a student's life, then why not teach the conceptually simpler approach of progressively refining an initial guess using a binary search?
I got 48.75 in about 20 seconds.
Divide 2376.592 by an approximate square root ie 50.
That gets you 47.53184.
Average 47.5 and 50 and you get 48.75
Trial and error can get you 3 significant figures very quickly by hand.
Your way of manually doing square roots is the way my 8th math teacher Mrs Wilker taught us how to do it . I will study this problem and do more problems like it. Lot of WHACK out ways of finding the square roots . They work, but very CONFUSING You is worth your weight in gold raised to 20^20 power . (HUNDRED QUINTILLION) Thank you.
I do this on my antique calculator by subtracting successive odd numbers. That could really lengthy on paper, though.
Thank you so much Madam ...
I learned this method long long time ago when there were no electronic calculators ,am now 70. y/o ,but instead of multipying by 2 we multiply by 20.Now a day they don't do this method any more.
No need to multiply the upper no by 2. Just add the upper no to the divisor, i.e., 4+4=8. Next time, add 8 to 88 and get 96. Either way.
I learned this from a high school classmate but I didn't get what he did. He wrote on paper so quickly. I didn't have time in class. I think if you're in an east or as Southeast Asian country or somewhere from South America they might have taught this. Asian countries taught tough stuff forvyoung kids that's not taught in the USA or Canada.
Great algorithm
Question for viewers
Can you derive such method for cube roots ?
If you really understand why this method works
you will be able to derive method for cube root yourself
I was taught this method in high school once we were solving quadratic equation
(to determine if discriminant is perfect square or to approximate roots)
and derived method for cube root myself
What country did you go to school that they just told you to find the method yourself? I'm suspecting that instead of multiplying by 2 we should multiply by 3 and use cubes instead of squares in the same method. Not sure if I should group by 3 digits 🤔
@@johnchristian7788 In Poland
I derived method for cube root for myself
and it was not homework
As soon as I understood why method for square root works I was able to derive method for cube root
Yes you group 3 digits
Yes you multiply by three but square of actual approximation not just actual approximation
Instead of appending last digit of next approximation you append square of last digit of next approximation
To number created in this way you add triple product of current approximation and last digit of next approximation shifted one position to the left
(10a+b)^3 = 1000a^3+300a^2b+30ab^2+b^3
(10a+b)^3 - 1000a^3 = 300a^2b+30ab^2+b^3
(10a+b)^3 - 1000a^3 = (300a^2 + 30ab + b^2)b
(10a+b)^3 - 1000a^3 = ((300a^2 + b^2) + 30ab)b
Very nice indeed. Maybe a smaller number would have made it clearer. But hey.
This is how I learned it many years ago when I was in 8th grade
My head exploded....
I do root calculations a different way ........try doing the 6 root od 41........and I'm 85.
I learned this method in school. Going forward I’m using a calculator.
As always, when teaching, start simple then use a complex
SQRT2500=50, 2376.592
They didn't teach this in school where I was. :(
I learned this as a kid, without explanation. I later proved to myself why it worked.
But truth is, I never use it. Newton's method converges faster.
Sqaure root of 20 is 5?
Thanks mami
Aren't you glad we now have calculators?
I felt bad for her as she got in over her head right quickly.
Are all square roots of non square numbers irrational?
No. You can convince yourself by looking at the problem from the opposite direction: if you take a rational number and square it, will you always get a square number? If it's an integer, yes (2*2 = 4; 3*3 = 9; etc), but if it's not an integer, then no: 0.5*0.5 = 0.25, so there exist non square numbers with rational square roots.
@@Merione thank you for taking my question seriously. I appreciate your response. Just like everything that is explained it seems obvious in hindsight and I probably should have just thought about it harder. That was a very satisfying and simple explanation.
All square roots of non-square integers are irrational.
Ah ok, thats probably what I was intuiting.@@robertveith6383
By "as close as possible" I assume it is, as you say in the first case, as close to but less than.
And the amazing statement at the end about square roots never repeat. Well, some certainly do, e.g. a square of a rational, such as 2.25, repeats with infinite 0s. So the divisor changing doesn't guarantee non-termination
😶
wheres the decimal point end up?
The decimal will never end since the square root of non perfect square is non terminating as well as non repeating. In otherwords they are irrational numbers.
It ends up between 8 on the left & 7 on the right -> 48.75
@@Matlockization it is simply a round off or we can say approximation
@@cbruata5198 Well, it depends on when you multiply the answer by itself how close you get to the original number. In this case, you can round the answer off to two decimal places, but as it stands the answer is not an approximation.
Step 1: Convert to binary. This avoids any need to guess.
Step 2: Apply the algorithm for binary numbers. Very fast.
Step 3: ( Optional ) Convert to base ten.
Final top digit , following YOUR rules, is 4 ,NOT 5.
WHATEVER YOU DO IN MATHEMATICS , YOU MUST EXPLAIN YOUR DEPARTURE FROM YOUR RULE!
{ESPECIALLY IF YOU REFUSE TO EXPLAIN EITHER YOUR RULE, OR YOUR UNEXPLAINED DEPARTURE.
YOU CANNOT CALL IT MATHEMATICS WHEN YOU ARBITRARILY MODIFY YOUR RULE WITHOUT EXPLANATION.
She doesn't explain everything. It would help if she could comment here to answer questions about her work.
I agree with why you're complaining (although lay off the ALL CAPS next time perhaps, eh?). But I think I have a plausible explanation for this departure from her previously stated rule...
She added a zero after the original number, so that she had a pair of digits at the end (instead of a single digit) to match the rest of her method. Given that the zero is technically not significant in relation to the original number, the use of 5 (instead of 4, as one would expect from her original rule) kind of acts as a rounding factor for the final answer.
This is only an explanation I've come up with, after the fact. Interestingly, using 4 (going by her original rule) gives a resulting divisor of 9744, which ends up giving 38976 for the drop-down subtraction, and then an answer of 10044 ... implying a very very scary iteration of the algorithm if there was another pair of digits in the original number!! 😨
Or you could use logarithm tables…
Looks like a neat method, but frankly you lost me and I have a strong background in mathematics. May I suggest you redo this video? Writing out a script with queue cards may help. Citing a published source for this trick would be great. Other commentators suggest it is a reorganized Binomial expansion....I tend to agree, though more background would be nice .
4:04: Where did you get the 6 ??? Can't stand bad teachers.
Evidently you never paid attention when multiplication was being taught in school, or paid attention when she was explaining the very step you're complaining about. The 6 comes from the first 8*8 (=64), she put the 4 down and carried the 6 to the tens column for the next step. Pay attention in future, champ. 🤡
This is hideous
It is cumbersome to use.
@@robertveith6383 Indeed. However, you would think that the whole point is not the mechanics, which a calculator will do right quickly, but to explain why the calculator and this method come up with (approximately) the same answer. She might also show us what 48.75 x 48.75 equals. It's 2376.5625. She says you could go on and on, but she doesn't say that you would get closer and closer to the target square, 2376.592 if you did.
It's weird, I know, but this mechanics for the sake of mechanics reminds me of filling out the capital gains page of an IRS form. You know, you've put in the amount you paid for 200 shares of Zockman Birtwistle Corp. stock and the amount you sold it for and subtracted the first number from the amount you sold it for. Then the instructions to me just get silly. Something like:
Take the total on line 3 and multiply by .15
Take the total on line 1 and multiply by .35 if you bought the shares more than two years ago. If less than two years ago, multiply by .28. Write this number in on line 4.
Take the total on line 2 and add it to the number on line 4. Write this number on line 5.
Subtract line 5 from line 3.
I expect it to continue with: then sing the 4th verse of The Star Spangled Banner and write the number of words in it on line 6. Count contractions as one word.
Every time I had to fill out such a form, I had no idea why I was merrily multiplying by, adding to, subtracting from those numbers from the top to the bottom of the page, and somewhere in the middle, I would start giggling because I had no idea WHY I was doing those particular calculations with those particular numbers. It was like being given a set of 10 algebra problems that had no relationship to the real world, just to practice the mechanical steps to the solution. OK, now that I got the solution, what's the point? There seems to be no point. You got seven correct, so you get a 70% score. Oh, now I get it. The point of learning math is to learn math. You don't actually use it for anything. Well, at least tell students that. The idea is to train your brain to think in a variety of ways so that it is functional to its full potential by the time you're 18.
Well, that's what it seems like in too many math classes. I'm not against getting the right answer. That is, of course, important. I'm not against showing the steps to the teacher, so she knows you didn't come up with a lucky guess. But not often enough do we hear why we would want the right answer in the first place except to please the teacher.
PS I cannot believe that this teacher cannot subtract 7259 minus 6769 in her head. I'm hoping she's putting on a little act for young students, who might be struggling to remember what you do to subtract 6 from 5 and 7 from 2. But how young a student would be trying to find the square root of 2376.592 and why on earth would they want to?
Yeah you didn’t explain this well and the loss of confidence lost me
I will use a calculator..🤬
So what does this work? Doesn't just reinforce to students that maths is confusing and opaque? Waste of a video.
Not a "waste of a video" for people curious enough about maths. Somehow I doubt that students who already think "that maths is confusing and opaque" are her target audience, champ. Waste of a comment. 🤡
I learned this algorithm 50 years ago, and I still remember we had to find the root of 5 in a test back then using this " pedestrian" method. It's not bad to know that this method exists and what it is based on.
cut your hair!
Few times she says to use calculator to find the number you put on right hand side of two digit, three digit, .... to multiply the whole thing by ...... my question is if I'm using a calculator to find the next digit, why not just use it to find the square root to begin with. I wonder.