It took me 3 years to pass HS algebra because the coaches/part-time math teachers didn’t like the way I solved problems. I got the right answers. But the way I got them was wrong apparently.
No no no 10+7 = 17 and 9 is always one less so 16
I would have done 10+6, but that’s effectively the same thing as the OP.
Aside from literally counting, what other way is there to arrive at 16? You either memorize it, batch the numbers into something else you have memorized, or you count.
Am I missing some obvious ‘natural’ way?
For my kids, apparently some kind of number line nonsense, which is counting with extra steps.
I just memorize it. When the numbers get big, I do it like you did. For example, my kid and I were converting miles to feet (bad idea) in the car, and I needed to calculate 2/3 mile to feet. So I took 1760 yards -> 1800 yards, divided by three (600), doubled it (1200), and multiplied by 3 to get feet (3600). Then I handled the 40, but did yards -> feet -> 2/3 (40 yards -> 120 ft -> 80 ft). So the final answer is 3520 ft (3600 - 80). I know the factors of 18, and I know what 2/3 of 12 is, so I was able to do it quickly in my head, despite the imperial system’s best efforts.
So yeah, cleaning up the numbers to make the calculation easier is absolutely the way to go.
A mile is 1760 yards, and there are three feet in a yard. Therefore, 1760 feet is 1/3 of a mile, and 2/3s of a mile is 3520 feet.
The imperial system is actually excellent for division and multiplication. All units are very composite, so you usually don’t need to worry about decimals.
Metric would be perfect if 10 wasn’t such a dog shit number to base our counting off of. Sure it works for dividing things in half, but how often do you need to break something down into fifths? Halves, thirds, and quarters are 90% of typical division people do, with tenths being most of the rest since 10 is that only number that our base system actually works with.
Yup. The reason I went with yards was because I knew 1760 was closer to a nice multiple of 3 than 5280 (neither 5200 or 5300 is a multiple of 3; I’d have to go to 5100 or 5400).
But yeah, imperial works pretty well for multiplication and division, it’s just not intuitive for figuring out the next denomination. Why is a mile 1760 yards instead of 1000 or 1200? Why is it 5280 feet instead of 6000? Why is a cup 8 oz instead of 6 (nicer factors) or 10? Why is a pound 16 oz instead of 8 oz like a cup would be (or are pints the “proper” larger unit for an oz)?
The system makes no sense as a tiered system, but it does make calculations a bit cleaner since there’s usually a whole number or reasonable fraction for common divisions. Base 10 sucks for that, but at least it’s intuitive.
As in, visualizing a number line in their heads? Or physically drawing one out?
I could see a visual method being very powerful if it deals in scale. Can you elaborate on that? Or, like try to understand what your kids’ ‘nonsense’ is?
I think my 7yo visualizes the number line in their head when there’s no paper around, but they draw it out in school. I personally don’t understand that method, because I always learned to do it like this:
7372 + 273 =====And add by columns. With a number line you add by places, so left to right (starting at 7372, jump 2 hundreds, 7 tens, and 3 ones), whereas with the above method, you’d go right to left, carrying as you go. The number line method gets you close to the number faster (so decent for mental estimates), but it requires counting at the end. The column method is harder for mental math, but it’s a lot closer to multiplication, so it’s good to get practice (IMO) with keeping intermediate calculations in your head.
I think it’s nonsense because it doesn’t scale to other types of math very well.
You still haven’t told me what the number line method actually is. I know how to add up the columns bud
Number line is something like this:
100 | 200 | 300 ... | 10 | 20 | 30 ... | 1 | 2 | 3 ==================================================You write out the numbers that are relevant and hop by those increments. So for 7372 + 273, you’d probably start at 7000, hop 100 x 5 (3 for 372 and 2 for 273), hop 10 x 14 (7 for 72 and 7 for 73), and so on. It’s basically teaching you to count in larger groups.
To multiply, you count by the multiple (so for 7 x 3, you’d jump in groups of 3).
This article seems to explain it. I didn’t learn it that way, so I could be getting it wrong, but it seems you do larger jumps and and the jumps get smaller as you go. I think it’s nonsense, but maybe it helps some kids. I was never a visual/graphical learner though.
So, are you just talking about number lines in general?
I learned how to use those in grade school too. 20+ years ago. But the way you phrased it made me think there was more to it. Calling it nonsense is… shocking.
I guess we used it for an exercise or something a couple times, but never for more than indicating how numbers work. They’ve taken that idea and kind of run with it, instead of leaving it behind once the basics of addition have been mastered. I learned multiplication as just repeated addition, and there’s no reason IMO to get a number line involved because addition should already be mastered.
This is a 2nd grade class, and I expect them to have long since mastered addition. At that point, a number line feels like a crutch more than a useful tool. Sure, use them in kindergarten and first grade to grasp how counting works (and counting by 2s and 10s), but that should honestly be as far as it goes. But they still use it for fractions and larger sums and products.
My mental image is squishing the 7 into the 9 but only 1 is able to be squished in, leaving 6 overflowing
I’d argue memorizing it is the natural way, at least if you work with numbers a lot. Think about how a typist can type a seven letter word faster than a string of seven random characters. Is that not good proof that we have pathways in our brain that short circuit simpler procedural steps?
Theres more complicated ways for sure, but I think we have identified all the simple ones. Could break it into twos I guess.
I’m also in 10+6 gang, and it’s more universal, as in a decimal system you will always have a 10 or 100 to add up to, and a “pretty” 8+8 is less usual
Okay this is nice and all but how do people do 3974* 438 mentally, without paper? And bigger and some outright freaks seem to do it in an instant
For me:
3974 * 438 -> 4000 * 438 - 26*438 -> 4000 * 438 - 26*440 - 26*2 -> 4000 * 438 - 20*440 - 6*440 - 26*2And so on, and I’d do some of the intermediate calculations as I go (e.g.
20*440and6*440).But that’s only really needed if I need a precise answer. If I can get away with an estimate, I’ll simplify it even more:
4000 * 430 ~= 43 * 4 * 10000 = 86 * 2 + 10000 = 1,720,000Actual answer:
1,740,612.4000 * 440would be easier (I like multiplying 4s), but I know it would overshoot, so I round one up and the other down. Close enough for something like estimating how much a large quantity of something kind of expensive would cost (i.e. if my company gave everyone a hot tub or something).Not any great easy way I can think of to do that one but I would attempt to do 400 by 3974 and then add chunks of 438 x 10 or x5 until I got really close and then add individual blocks.
So like 400 by 3974, you can round to 4000 and remove 4 x 26 = 104 after doubling 4000 twice. So we have 4000 to 8000 to 16000 remove 104 is 15896, add zeros is 1,589,600. Forget all other numbers but this one.
We are missing 38 x 3974. We can do the same round and remove trick to add 10 x 3974 by changing it to 10 x 4000 - 10 x 26. We need four of those though, so we can double it and turn from 40000 - 260 to 80000 - 520 and then 160,000 - 1040 or 158,960. Need to remove 2 x 3974 though, so remove 8000 and add 52 so 151,012.
Hopefully ive been able to keep that first number fresh in my head this whole time, which involves repeating it for me, and I’d add 1,589,600 and 151,012. Add 150000 and then 1,012 so 1,739,600 and then 1,740,612.
That all said, I make way more mistakes than a calculator, and I was off by 400 or so on my first run through. Also its really easy to forget big numbers like that for me. I’d say if you gave me ten of these to do mentally I’d get maybe 2 correct.
That’s great but this is juggling numbers in memory and I simply cannot do this reliably. I will have this one current operation and put the other ones into the mental basket so to say and it evaporates and blurs as I calculate the other thing right so I wonder how these folks can do this and really fast. Not that I ever seriously tried other than some rare bored moments so maybe it is simply a matter of training?
Its very impressive though when you give these ppl two big numbers and they say result nearly in an instant
Over time those bigger numbers become more common too. Someone who can mentally do the type of problem I just did and get it right quickly likely have a ton of practice and will know quicker tricks, and be able to simplify it in a way.
Another part is they would be able to recognize a wrong answer more accurately as well. I didnt realize my answer was off by a lot until I put it in a calculator, but someone with more practice might know intuitively they were wrong.
I just don’t consistently do this type of math, I used to be good at it in school but its become mostly irrelevant for me outside impressing someone a slight bit. It is helpful to have the ability to do things manually but it just rarely comes up.
Mental arithmetic is all little tricks and shortcuts. If the answer is right then there’s no wrong way to do it, and maths is one of the few places where answers are right or wrong with no damn maybes!
Unsolved problems do not all fall into binary outcomes. They can be independent of axioms (the set of assumptions used to construct a proof).
I like your funny words, mathemagic man
Hmm, you seem to be completely discounting calculus, where a given problem may have 0, 1, 2, or infinite solutions. Or math involving quantum states.
In math, an answer is either right, wrong, or partially right (but incomplete).
Quantum states is physics, not math.
And mathematically a probabilistic theorem is still a theorem.
Those are quite far from mental arithmetic though
Calculus is generally pretty easy to do mental arithmetic on, especially when talking about real-world situations, like estimating the acceleration of a car or something. Those could have multiple answers, but one won’t apply (i.e. cars are assumed to be going forward, so negative speed/acceleration doesn’t make much sense, unless braking).
Math w/ quantum states is a bit less applicable, but doing some statics in your head for determining how many samples you need for a given confidence in a quantum calculation (essentially just some stats and an integral) could fit as mental math if it’s your job to estimate costs. Quantum capacity is expensive, after all…
Unless you consider probabilities. That’s a very strange field—you can’t objectively verify it.
You can’t objectively verify anything in mathematics. It’s a formal system.
Once you start talking about objective verification, you’re talking about science not math.
It is actually the opposite, since it is purely abstract everything in math is objective. There is literally no subjectivity possible in something that isn’t in the real world.
Well, there are certainly wrong ways to arrive at the answer, e.g. calculating 2+2 by multiplying both numbers still gets you 4 but that is the wrong way to get there. That doesn’t apply to any of the methods in the post though.
That’s also all common core is. Instead of teaching the line up method which requires paper and is generally impractical in the real world, they teach ways to do math in your head efficiently.
What is “common core” and what is the “line up method”?
This kind of feels like how I constantly get the “which word/shape/number (etc.) in this series is incorrect” questions on tests wrong. I severely overthink it. “Well, these four all have chloroplasts and this other one gains energy from photosynthesis via a symbiotic relationship with another organism, so it must be that one.”
*Gets test back*
“Oh, it was the one that didn’t live in a rain forest.”
I think that third one down is actually how they’re teaching it at my kids’ schools now. It’s called “making a 10” I think, basically that same idea, add up to 10 first, then do the rest.
this is a false story. everybody knows 7 ate 9
I realized something. I relate so much to ADHD memes not because i have it but because they simply do a lot of things that they think only people with ADHD do. In my school they encouraged you to come up with techniques like this. Often 9 is hungry in different ways. Another exmple is multiplication. 5099 is 50100-50 which is much easier to calculate.
This isn’t even an ADHD thing.
In my school they encouraged you to come up with techniques like this.
You’re either very lucky and were in a school that went against established norms, or you’re young enough that you were taught the “new” math that boomers hate. Because this is the new math.
Boomers, GenX, and elder millennials were primarily taught via rote memorization. You simply memorized the times tables, and committed “8*3=24” to memory. You didn’t calculate it every time. You just memorized the tables, regurgitated them ad nauseam to appease the teachers, and then referred to those memorized tables for any multiplication you needed to do.
For reference, this is the times table I’m referring to. Our quizzes/tests required you to fill out the entire thing in less than 5 minutes:
We had to fill this out multiple times per week. The goal of the time limit was to force you to memorize it, instead of calculating it out every time. You simply didn’t have time to calculate each one out. Then once you had it memorized, if you ever had to do 8*3, you would just refer to your memorized times tables for it.
But the issue with this is that it doesn’t teach you how to actually do the math in your head, it just teaches you the times tables. You aren’t calculating it out each time, so you don’t develop any shortcuts or methods to make it easier. If a teacher ever saw you turn 9+7 into 10+6, they would bust out the red pen and start slashing. Even though 10+6 is undeniably easier to do in your head, the teachers weren’t concerned with that; They wanted to know that you had memorized what 9+7 is. These memes are primarily aimed at the millennials and GenX with ADHD, because they were the ones who got bored of rote memorization and started coming up with shortcuts (which then got docked points on their quizzes.)
Why are these posts always shitting on teachers? I don’t know what teachers you’re seeing, but I’ve never seen any teacher of any subject / age-group ever discourage anyone for thinking about something a different way. Quite the contrary, different ways of approaching problems are always encouraged.
My math teacher (at a private school) was just a random students’ mom. She had no higher degree and only taught the book. If you got the right answer by using a method not included in the book, it was marked half-credit because she didn’t understand and wasn’t interested in hearing your logic, because “that’s not what the book says”.
Being taught by people who have no drive for knowledge and just want to teach the standardized test answers SUCKS.
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Why is everything ADHD?
Yeah, this has nothing to do with ADHD.
9+7=10+7-1=16
I like the way the second ADHD method is also a flip on the “seven ate nine” dad joke
The “ADHD way” is literally what they are teaching in school.
Yup, this is what parents are complaining about when they say math has changed. Before, math was primarily about rote memorization. You just memorized that 9+7 is 16. There were multiplication tables you were expected to memorize and regurgitate ad nauseam. Sure you could count it out on your fingers, but that only works for numbers under 11. For anything above that, you just referred to your memorized addition, subtraction, multiplication, or division tables. But this also meant that numbers outside of those tables were really difficult to do in your head, because you were poorly equipped to actually calculate them out.
Common core math is attempting to make math easier to do in your head, by teaching the concepts (rather than promoting rote memorization) and helping students learn shortcuts to avoid getting lost. 9+7 is 16, but it’s also 10+6 or 8*2, which are much easier to visualize in your head without counting on your fingers.
Yep, and what happens is that when kids need help they can’t explain the “new” way from the beginning and only half remember stuff which is extremely confusing to hear as a parent so then the parents get mad at the method.
Admittedly I was in school multiple decades ago, but our teachers wanted us to memorize addition and multiplication tables. Which of course made anything outside the tables hard to do. I (and others apparently) thought it would be a great idea to use shortcuts like this.
So many failed tests. So many. When teachers saw us write down that we took the 21 apples multiplied by 7 bushels and just did 2x7, and tack a 7 on the end, they broke out the red pen.
“Show your work!”
“How? You taught me to memorize, and I did it from memory…”
No no no. Adding nine is just subtracting one, but adding to the front digit. 9 + 7 is actually 7 - 1=6, then add that 1 to the front. 16. Let’s not make more complicated than it needs to be.
Holy shit! That’s how I do it. Caught so much crap for it when I was a kid.
