Me, bad at math: yeah they taught us that as a way to do in grade school.
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.
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If your teacher gets mad about breaking an addition problem into easier problems, then that teacher should be fired. Phony tale.
If anything, these are exactly the techniques that “New Math” was supposed to teach. Your brain doesn’t work math the same way as a computer. People who are good at math tend to break the whole thing down into simple pieces like this. New Math was developed by studying what they did and then teaching that to everyone.
I tend to add 9 to things by bumping the tens digit up by one (7 becomes 17) and then subtracting 1 (17 becomes 16).
Most of the arguments against New Math tended to prove the point; our mathematical education was in dire need of fixing.
The second method is very chemistry-like. I do that too naturally
I thought that too, 9 is like a halogen, it wants to resolve to 10 anyway it can like fluorine wants one last electron. So allow the 9 to rip one off of the neighboring numbers and then perform the calculation.
I’ve never really liked the anthropomorphic description of chemical bonding, but maybe it’s actually similar to the addition thing. On the one hand, we can say 9 wants to resolve to 10 and takes a 1, and on the other hand we could say there are a bunch of different ways we could rearrange these numbers but the end result is the same as if we resolve 9 to 10 first. Maybe chemical reactions are similar, so there’s a bunch of configurations that could have happened, but the end result is the same as if we had said fluorine wants that last electron
Although, electron affinity is a thing… so the analogy does break down pretty quick. Rip
Yo but hear me out. Because 7 ate(8) 9, 7 + 9 = 7
What does adhd have to do with anything?
ADHD is sometimes used as a catchall to mean a set of behaviors that does not coincide with the majority at school or work. Ive met a bunch of people on ADHD medicine, but it was usually because they wanted to force themselves to be good at or like something they didnt want to do normally.
In this case its called ADHD because the student has found their own way to solve it despite the method the teacher is teaching and that the rest of the class uses.
It’s because it’s stupid. The bottom answer is at least sort of similar to a simple rule for adding 9s. But the op is just so incredibly specific that it won’t help most of the time.
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?
My mental image is squishing the 7 into the 9 but only 1 is able to be squished in, leaving 6 overflowing
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.
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.
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.
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’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
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.
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!
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”?
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.
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
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.
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.
Yes, but physics is math with more variables.
But there’s plenty of math related to quantum states that can make sense, such as if you know a given machine will give the right answer 51% of the time, and you want to know how many iterations you’ll need to get a certain confidence that you are seeing the correct answer. That’s basic statistics, which is also math, but it’s relevant to quantum states in that you’re evaluating a computing system based on qubits.
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…
Has nothing to do with ADHD.
Wouldn’t say nothing to do with.
Many neurodivergent students find themselves in situations where they haven’t fully absorbed the taught material. Many of them end up figuring problems out themselves, with varying degrees of creativity and successNeurotypical students do the same thing. It’s not like every neurotypical will internalize every piece of material they are taught.
Yup, I’m most likely neurotypical (never been diagnosed either way, just never had issues w/ traditional learning), and I generally ignored the teacher and did things my own way. I was always really good at math, so the teacher’s way was usually less efficient for me, so once I understood the operation, I’d create shortcuts.
We’d go over the same material a lot, so I’d usually just do homework while the teacher taught some new way to do the same operation. I’d get marked down for doing it differently from the instructions, but I’d get the answer right.
Whatever number is closest to 10 steals enough to make itself 10. Same goes for hundreds, thousands, whatever. Get your round numbers first, add in the others later. All numbers must become 10. In a pinch, a number may become a 5, but if so, it’s really just become a half-10, and it should feel bad about itself that isn’t a full 10 yet.
Let’s make that 9 a 10 because it’s good enough, it’s smart enough, and goshdarnit people like it. Also, I don’t wanna add with a 9. So 10 + 7 would be 17, but we added 1 to the 9 to make it 10 so now we take 1 away, 17 - 1 = 16.
ezpz
9 plus a number? No. 10 plus a number, minus 1. Yis.
I just memorized any addition with 9 adds a 1 in front while reducing the other number by one. Same general step, but there’s no 10 in my head, just 9+7 -> 16. Basically, promote the tens column while demoting the ones column. I think of it more like a mechanical scoreboard (flip one up, flip the other down) than an operation involving a 10.
If it’s anything other than 9, I fall back to rote memorization, unless the number is big, in which case I’ll do the rounding to a multiple/power of 10.
Yeah that’s a more accurate description of what i actually do in my head to. I’m not “adding 10”, because I already would use a short hand method for adding 10 anyway to promoting the tens place or flipping the score card, as you said.
8+8 and 8X2 are literally the exact same thing, why did they feel the need to make that an extra step?
Probably because they were forced to memorize times tables, but not arithmetic so they wanted to show where they are leveraging that memorization from
9 is 3+3+3, 7+3 is 10, 3+3 is 6, 6+10 is 16. I’m also a fucking heathen.
What the fuck
Might as well do:
9 is 1+1+1+1+1+1+1+1+1, 7 is 1+1+1+1+1+1+1 therefore 9+7 is 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 which is 16.
lol, it is pretty bizarre I know. I just know 9 breaks into 3+3+3 because it’s a square number, and adding one of those 3s to 7 makes it 10, which is easier to add stuff to, then I just get rid of the remaining 3s by adding them to 6, then 10+6 is a very easy equation to intuitively add, because you just replace the “0” with “6” to get “16” and you’re done.
That’s pretty much exactly how you first learn to do arithmetic. Break the (whole) numbers down into their smallest (whole) part and count em up. That’s what number lines do, or using colored blocks to visually do the addition, or any number of other techniques you use when you’re first learning arithmetic. I mean, even just knowing 3 + 3 = 6 is memorization, a shortcut form of 1 + 1 + 1 + 1 + 1 + 1 = 6.
I explained to a teacher one time this as my method, the get to ten version, and she looked confused as hell like why would anyone do that. She was cool with it though, gave me a whatever works for you kind of response.
