Were you taught divisibility rules/tricks?

[Farrar]

A number of the materials I've used with the kids have explicitly taught how to find if a larger number is divisible by 3,4,6,etc. The FAN Math books had a chapter about it which I used with one ds and the Indian math books and Beast Academy now have something about it which other ds is using. But I didn't see it in MM or MiF, though we didn't use all levels so it's possible we missed it.

 

Someone happened to tell me the 3's trick (where you add the digits) at some point in my education, but I know it was never taught as part of a class. And when I tried to explain it to dh, who is really good with numbers, he initially didn't believe me that it would work.

 

Is this something that is taught more clearly in other countries but not often in the US? Something that used to not be taught but increasingly is? Or something that dh and I just happened to miss?

[Ellie]

I only learned 2's and 5's.

[SparklyUnicorn]

I was not taught them  They are very useful though.  Magical really.

 

I'll ask DH abut his experiences.

 

[Paige]

I was taught them explicitly in an average American school in the 90s. It's in MM 4th grade too. My DD's doing those lessons now. 

[SparklyUnicorn]

I was taught them explicitly in an average American school in the 90s. It's in MM 4th grade too. My DD's doing those lessons now. 

 

What year did you graduate from high school? 

[justasque]

I was taught them. I make a point of teaching them to my students. I think the ability to see when things can be canceled is incredibly helpful, especially on standardized and other tests when you have to work quickly. Making a series of cancellations using small factors (2, 3, 5, 9, 10, etc.) can be much quicker than looking for a larger common divisor, and gives you somewhere to start when something initially looks so complex you can't quite see how to otherwise simplify it.

 

Some curricula seem to teach it, and some don't. I think there is quite a bit a good math teacher can add to a standard math program in the way of tips and tricks, ways to understand things, ways to approach writing down problems, problem solving approaches, etc. I wish there was more of this in math texts, in the margins if not in the main part of the text.

 

For those who don't know - to see if a number is divisible by three, add the digits and see if the resulting number is divisible by three - if so, then the original number is also. The same method works for nine - to see if a number is divisible by nine, add the digits and see if the resulting number is divisible by nine- if so, then the original number is also.

 

So - 36 --> 3+6=9, nine is divisible by 3, so 36 is divisible by 3.

[FairProspects]

No I was not taught anything at all like this, but I adore the 3s trick and the 2, 5, 10 rules laid out by Zacarro.

[8filltheheart]

My kids have learned them in Horizons.  

 

Here is a simple summation (skipping 2,5, 10 since I am assuming those are obvious)

 

3---sum of digits /3

4---last 2 digits  /4

6---2 and 3 rules apply

8--the last 3 digits  /8

9--sum of digits /9

 

There are rules for other numbers, but they are too complicated for me to remember.  I found this link: http://www.savory.de/maths1.htm

[Eagle]

I had never been taught these tricks and they seem like magic because of it. How could my school have missed something so useful? Dh was taught them and is convinced that I must have been too but have since forgotten. I am glad to hear I am not the only one that didn't know about these until teaching my own dc.

[Crimson Wife]

I definitely remember being taught the 9's trick. 2's, 3's, and 5's I knew but am not sure I was ever explicitly taught them.

[Farrar]

I was taught them explicitly in an average American school in the 90s. It's in MM 4th grade too. My DD's doing those lessons now. 

 

Weird. I did most of MM4 with one of my ds. Maybe we somehow skipped that part because I don't remember it being in there. The whole MM thing fell apart for us that year for various reasons, though we did get through most of it. Maybe I'm just forgetting.

[Farrar]

I had never been taught these tricks and they seem like magic because of it. How could my school have missed something so useful? Dh was taught them and is convinced that I must have been too but have since forgotten. I am glad to hear I am not the only one that didn't know about these until teaching my own dc.

 

I remember learning that 3's trick - it was a sub at the school who just told me as I was doing a worksheet or something with long division. And it felt like magic even as a kid. When I told dh about it and he didn't believe me, I went and looked up the proof for why it works and I remember reading it and understanding it, but it fell out of my brain because I couldn't explain it right now, that's for sure. So I'm totally going with magic.

[justasque]

There are also rules for 4 and 6.  There is a nice summary here, see below for an excerpt.


"A number is evenly divisible by 2 if the last digit is 0, 2, 4, 6 or 8. 168 is evenly divisible by 2 since the last digit is 8.

A number is evenly divisible by 3 if the sum of the digits is evenly divisible by 3. 168 is evenly divisible by 3 since the sum of the digits is 15 (1+6+8=15), and 15 is evenly divisible by 3.

A number is evenly divisible by 4 if the number formed by the last two digits is evenly divisible by 4. 316 is evenly divisible by 4 since 16 is evenly divisible by 4.

A number is evenly divisible by 5 if the last digit is either 0 or 5. 195 is evenly divisible by 5 since the last digit is 5.

A number is evenly divisible by 6 if it is evenly divisible by 2 AND it is evenly divisible by 3. 168 is evenly divisible by 6 since it is evenly divisible by 2 AND it is evenly divisible by 3.

A number is evenly divisible by 9 if the sum of the digits is evenly divisible by 9. 549 is evenly divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is evenly divisible by 9.

A number is evenly divisible by 10 if the last digit is 0. 1,470 is evenly divisible by 10 since the last digit is 0."

 

I have a divisibility handout based on the website linked above.  I've added:    

Prime Numbers are only evenly divisible by 1 and themselves. Put another way, prime numbers have only two factors, 1 and themselves.

Composite numbers have more than two factors. They are evenly divisible by at least number other than one and themselves.

1 is neither prime nor composite.
 

[Paige]

What year did you graduate from high school? 

 

1997. 

[SKL]

I was taught this in elementary school.  (Some are more obvious than others.)

 

2 = even numbers

3 = the digits add to a multiple of 3

4 = the last 2 digits are divisible by 4

5 = ends with 0 or 5

6 = divisible by both 2 and 3

7 = I think there was a complicated trick, but I have forgotten it

8 = the last 3 digits are divisible by 8

9 = the digits add to a multiple of 9

 

 

ETA I learned this in school around 1975 in the USA.

[Tanaqui]

Note, these tricks only work if you're operating in base 10. Which I assume we all are.

 

It's interesting to realize that the 9 trick always works for the number that's one less than the base. So if you're working in octal, for some reason, anything divisible by 7 will add up to 7. Or if you're using a duodecimal system, then the trick works for 11. Of course, now that I've explained it it should be obvious why that is the case.

[Roadrunner]

Aops preA has a chapter on it that explains the why behind it. It's a thing of beauty.

[desertflower]

I was taught these helpful tricks as well here in America. I graduated 1994. Perhaps SparkyUnicorn and I went to the same schools. Lol

[Eagle]

Dh remembers having to derive the proofs of why each trick works. He thinks it was grade 5 but for sure somewhere between 5 and 7. I am always amazed at what his school taught for math. I feel shortchanged by my own math education in comparison. I have degrees in engineering and the math was always a challenge for me. If I had been taught some of the things that my ds is learning now, it would have made things so much easier.

 

Dh knows his math facts instantly as they were drilled endlessly. He knows all the primes up to 200-ish. He knows the factoring tricks.

 

Interestingly, he also did Making Mathematics Meaningful (the original c-rod set) at home as a child. He does arithmatic using the make-10 method.

[kiwik]

I was taught 1950/60's new math using textbooks that by the time I used them were in the wrong (pre decimal) currency. My teachers were heavy on algorithms but not much else. Honestly I was good at maths and never noticed that the line in a fraction was a division. Only think I remember is the twos one.

[Jane in NC]

This Well Trained Elder Stateswoman ;) was taught these quick algorithmic tricks.  I was raised on Dolciani.

[Soror]

I don't remember learning them but am teaching them to ds.

[kiana]

I did learn them but I'm not sure they were part of any curriculum.

 

The 3, 9, 11 tricks also work to find the remainders on division.

 

For example -- 1234567 -- 1+2+3+4+5+6+7 = 28, 28/3 has a remainder of 1, so 1234567/3 has a remainder of 1.

 

I don't think I've seen the 11 trick explicitly mentioned, so it's when the alternating sum of the digits is divisible by 11. For example: 1-2+3-4+5-6+7 = 4, so 1234567 is not divisible by 11.

[Sherry in OH]

I was taught and remember rules for 2, 3, 5, 6, 9, and 10.  Rules for 4, 7, and 8 might have been taught, but I do not remember them.  I graduated in the 1980s.

[Luckymama]

I was taught them sometime during the mid-late 1970s in one of the schools I attended (we moved several times k-8).

[Lolly]

Nope. Learned them alongside my kids. Only ones I was taught were 2,5, and 10. 

[regentrude]

I was taught them explicitly in math in public school in Germany.

[SparklyUnicorn]

1997. 

 

I graduated five years earlier. 

[Farrar]

This is interesting. I graduated high school in '94 but I didn't get taught this stuff.

 

I sometimes think my math education was just terrible though. I had a good 1st-3rd grade math teacher and then... a good algebra I teacher, but I missed the whole first quarter of algebra because I was in the remedial class initially and the remedial teacher was like, gee you see bored and I was like, in my best jaded thirteen year old voice with the nastiest look possible, "We're. Reviewing. Addition." And she switched me. But then none of my high school math teachers were very good. Also we moved a few times in my school career. I guess it's possible I just missed this stuff, though dh never moved. They just didn't teach it.

[SparklyUnicorn]

This is interesting. I graduated high school in '94 but I didn't get taught this stuff.

 

I sometimes think my math education was just terrible though. I had a good 1st-3rd grade math teacher and then... a good algebra I teacher, but I missed the whole first quarter of algebra because I was in the remedial class initially and the remedial teacher was like, gee you see bored and I was like, in my best jaded thirteen year old voice with the nastiest look possible, "We're. Reviewing. Addition." And she switched me. But then none of my high school math teachers were very good. Also we moved a few times in my school career. I guess it's possible I just missed this stuff, though dh never moved. They just didn't teach it.

 

Mine was terrible. 

 

I even recall a teacher mentioning that math was not her strong suit.  Super...

[RootAnn]

I don't remember learning them prior to 7th grade, but they were definitely part of my 7th/8th grade experience -- possibly Math Counts extra helps?

I learned a ton of cool math stuff from my 7th/8th grade math teacher - either as part of regular class or Math Counts (which I took instead of one of our Home Ec rotations). 

DH is smarter than I, but I was faster & more able to do the 'competition math' type problems that dd#1 brought home from math camp last summer.

Thank you, Mr. Thronson (whose own daughter homeschools her kids, I found out a few years ago)!

[Jackie]

I graduated in 1995 and was taught the tricks for 2, 3, 5, 9, and 10 explicitly during elementary. Not sure, but I'm thinking it may have been while working on simplifying fractions.

 

ETA: Except for one amazing Algebra teacher/MathCounts leader in 8th grade, my math education was average.

[Orthodox6]

Born in 1955. Was taught those rules in public school.

[Clear Creek]

I learned them during my elementary school years when I was being homeschooled with Abeka. Rod & Staff teaches them, so my oldest learned them before she began prealgebra.

[Guest]

Graduated in 2000 and don't remember learning them. I'm teaching them to ds this year and am in awe!

[wapiti]

I did not learn them but that doesn't mean I wasn't taught them... (I think there may have been a lot that was taught around grades 6/7/8 that I didn't learn because I was daydreaming/bored).

 

I dont't know where the topic is in the new MM but I think it's in the old MM6. I too enjoyed the coverage in AoPS Prealgebra.

[ErinE]

I was explicitly taught them in middle school. My district had a very strong math program and a serious violence problem. For high school, we moved to a safer district, but the math program was much weaker.

[LucyStoner]

I found 2, 5 and 10 to be obvious and recall knowing them without being taught them directly. 3 someone taught me when I was about 9 or 10 but not in school. I extrapolated that into the rule for 9 and later figured out the rule for 6 because of 2 and 3. 4 I didn't learn until I was in high school and a friend showed me. I graduated in 1998, I attended school in several different states and districts and it never really came up in school if I recall correctly. I wasn't awesome about paying attention in school a lot of the time.

[Mom22ns]

I was taught them way back in the 70s. The only ones I remembered were 2s, 5s, and 10s (which were obvious and didn't need to be taught) until I taught them to my kids. Now as teenagers, my kids only remember the same ones I did. I never found them magical or even interesting really, just another thing to memorize which I do easily, but I also let go of non-essential stuff easily too :).

[amiesmom]

I was never taught explicitly, but being the math nerd that I am, I guess I figured them out on my own. One of my favorite math activities to do with my kids is to use a 100s chart and circle all the multiples of a certain number then look for patterns. I think it's fun when they discover the 'rules' themselves.

[Kuovonne]

I was taught them because I remember using them in high school. But I don't know when I was taught them.

 

I was taught the tricks for 2, 3, 5, 6, 9, and 10.

I never bothered with tricks for other numbers.

 

I am informally teaching them to my kids, but focus on 2, 3, and 5, because they are prime.

Those three tricks, plus memorizing 7x7=49 and 7x13=91, make it easy to find the prime

factors for any number under 100. And I never needed to quickly find the factors of any

number over 100.

[Lecka]

Yes, I was taught them more than one year.  We learned them when we were learning to factor in pre-Algebra, and we learned them again when we needed to factor to solve binomial equations in Algebra.  

 

We learned a ton of them, and would have quizzes just over them (not the rules -- but just over factoring, and basically the kind of thing where you would want to use the rules).  

 

I think it was just part of the curriculum.  It came across like -- you need to just go ahead and use these, so you can factor and solve binomial equations.  

 

I am pretty sure I did it in all 3 years of middle school (6th, 7th, and 8th) and I took Algebra I in 8th grade.  It was more of a review in 8th grade, but it came across like the teacher thought it was important, she was going to keep giving quizzes until she thought everyone in the class was in a good place (using the quizzes more just to see how people were doing -- it was not a big part of our grade).  

 

I learned them for every number through 10 (except 7 I guess).  

 

They were all either "add up the numerals and see if they are divisible by 3" or "if it is an even number, look at the last two digits, and see if they are divisible by 4" and things like that I think (I do not remember them so well, but I know I used to know them).  I don't quite remember them now, but I do remember all the factors for the numbers through 100 and the tricky ones that would be divisible by 17 or 19 or something like that.  I know 17 x 3 =51 is one that was popular for trying to "trick" us (not really tricking though, since 5+1=6 so you can see it is divisible by 3 that way).  

 

It was a major part of learning about factoring, though.  Definitely it was in our math books, or I think it was.  I think we would have homework that was just writing "yes" or "no" to whether a number was divisible by 3 or not.  It seemed like a big deal.  I had an impression like I was not going to be able to solve binomial equations without learning them, and they were a big thing for Algebra I.  

 

(I graduated high school in 1996.)

[Spring Flower]

I taught them in my pre-algebra class when I was a PS teacher. This was about 8-10 years ago. 

[Lecka]

I remember the divisibility rules being such a big deal in Algebra I, part of me is like, wow, how did you get through Algebra I without them?  It seems like it would be really hard to not know them.  

 

My teacher was very "I am going to keep giving this same kind of quiz b/c it is IMPORTANT and you will need to learn it to PASS the CLASS."  It was that kind of vibe.  I got a C- one quarter.  There were a lot of kids accustomed to getting As, not getting As, and it was like -- we thought we had better listen to her.  

 

She was also the type to go back and complain to the 6th and 7th grade math teachers.  I think that is why I had them the previous two years, maybe.  I think the teacher was going back to them and saying "I have kids dropping like flies, they do not know how to factor," or things like that, it is the kind of thing it seems like she would do.  

 

She had a strong personality and was also the cheerleading coach.  She was a little like the cheerleading coach from Glee.  

[MiMi 4under3]

I never learned them until I saw them in DD's Saxon book!

[deerforest]

I learned them. Graduated high school in '87 so probably learned them in the late 70s.

[HS Mom in NC]

I graduated from high school in 1991. I was not taught them.

[Btervet]

I learned 2,3,4,5,6,9,11 in middle school.

 

I'm confused how people can do any math past pre-algebra without knowing them. How do you factor? Simplify? I'd be lost simplifying a fraction like 426/591 without knowing both numerator and denominator were divisible by 3. How would you approach it otherwise?

[ByGrace3]

I never learned it as a child, but yes we covered it in MM 4B (revised version)

[ByGrace3]

I learned 2,3,4,5,6,9,11 in middle school.

 

I'm confused how people can do any math past pre-algebra without knowing them. How do you factor? Simplify? I'd be lost simplifying a fraction like 426/591 without knowing both numerator and denominator were divisible by 3. How would you approach it otherwise?

I just divided to see if the same number could go into both...