Why does the sum 7x8 catch people out? | Hacker News Reader

Why does the sum 7x8 catch people out?

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37 pointsschrofer61 comments

I honestly have not bothered memorizing times table. I always found it easier to compute correct answer each time I need it. I just memorized few anchors: any number times 10 is easy to compute, from that it is easy to compute any number times 5, so only thing I memorized is number times itself. After that you can turn almost any multiplication into simple subtraction or addition.

For example 7 * 8 = 7*7 + 7 = 49 + 7 = 56

How on earth did you manage to get out of grade school without having everything up to 12*12 burned into your brain? Where did you go to school?

Oh, how I envy you.

You just need to swap schools when you are beginning times tables. Learn the easy ones in the first school up to say 6, then the next school is already past ten, and assumes you know it. Its easier to work it out than learn it on your own. That's what happened with me.

How in the world do you justify the effort of swapping schools to get out of learning the times tables?

It sounds like you can swap school to avoid learning anything! Seriously, it sounds like a good way to avoid memorizing historical or political knowledge.

I was never taught politics in school. I learned Scottish history (where I am from) when I was way too young and have forgotten much of it, which is a shame when people ask me about it.

I'm pretty sure he isn't telling you how to get out of learning them, he's answering the OP about how it could have happened that a child wouldn't learn them and would instead have some other system.

Still it's pretty amusing to imagine him as a kid executing an elaborate plan to trick his parents into moving just to avoid learning parts of the multiplication table... ^^;

I got to 4th grade before I was finally forced to learn my times tables through 9x9.

Same situation, still don't know them.. 6*8? Calculator! Now chuck calculus and stats at me, easy. Guess my brain just doesn't like arithmetic.. The start of school was annoying, however late highschool was OK when they stopped asking you infront of the whole class.

Never learned division either.. In any form.

Auckland, New Zealand public schools.

Nobody ever taught me the number of days in each month. I was well into my 20s before I had it nailed

30 days hath September, April, June and November

That's pretty useless as a mnemonic. "30 days hath September, August, March and December" fits the pattern equally well.

Not if you remember the whole rhyme:

    Thirty days have September,
    April, June, and November.
    All the rest have 31,
    Except February alone,
    And that has 28 days clear,
    And 29 in a leap year.

My substitution works without contradictions there too.

This is my version: "30 days has September, all the rest I can't remember."

I still use my knuckles. Put your fists out in front of you, each knuckle, starting with the left hand, represents a month--the tall ones are long (31 days) the rest are short (28 or 30).

28, 29 or 30. Divide by 4, 100 and 400 to figure out 28 or 29.

I don't understand what you're saying.

I'm a fan of the knuckles mnemonic: Starting on the outside knuckle of either hand, a high knuckle is 31 days, and the valley between knuckles is less than that (generally 30; FU February), then when you get to the other end of the hand, you double-count the knuckle (or count the knuckle on the other hand) for July and August.

It's 3<cough cough h@ck COUGH>... unless it's February in which case 2@AHEM#

I use the same technique as TrainedMonkey and it has worked really well for me.

I actually have a great dislike for the rote memorization of multiplication tables. During school I had everything up to 12 * 12 burned into my brain, but it was as words rather than numbers and wholly sequential. So, when asked 7 * 8 I would have to run through the whole thing up to that point in my mind - "seven times one is one, seven times two is fourteen, ..., seven times eight is fiftysix" - before I reached the answer and could speak "fiftysix" to show I knew it.

I computed my multiplication tables this exact same way as the great-grandparent during grade and high school. I memorized the tables, but I had this same problem of having to sequentially run through them to compute anything. Eventually in college taking tons of math courses finally ingrained it in my head, but to this day I feel like I must have missed something when everyone else was doing their multiplication tables.

I basically got by without knowing times table and suffering for it until about grade 5 which was when I figured out the trick. After that rest of my grades picked up dramatically as well, as I had figured out that understanding how things work is way better than simple memorization.

So basically laziness carpe diem.

Your mention of anchors reminds me of Lucky Numbers:

http://www.ee.ryerson.ca/~elf/abacus/feynman.html

Aka Feynman vs the abacus.

I always just prime factorize and then repeatedly multiply. So 7 * 8 becomes 7 * 2 * 2 * 2 becomes 14 * 2 * 2 becomes 28 * 2 becomes 56.

>I honestly have not bothered memorizing times table. I always found it easier to compute correct answer each time I need it

Easier, but slower. It pays off to memorize and know instantly -- and of course you can use the partial multiplications from the times table for larger problems involving calculating the correct answer too.

Slower for what, exactly? How often do you need to do arithmetic - multiplication, in particular - in a hurry with no aids?

Perhaps if you were in a cash business, or a trade involving lots of measurement, but there's not much call for it for a software engineer.

When I need arithmetic, I'm usually sitting down in front of a computer, and almost everything I use can do arithmetic - the bash shell, pry repl, Javascript console, emacs buffer, Google or Wolfram Alpha - there's hardly a piece of software I use that can't do arithmetic.

I regularly do arithmetic when planning meals, playing video games, analyzing my favorite sports team's contract situation, etc. Being able to do it in my head without breaking workflow (to bring up a calculator or even just to type in an equation) is a positive.

> How often do you need to do arithmetic - multiplication, in particular - in a hurry with no aids?

Every couple of minutes during my day I do some kind of mental calculation relating to my software development work. Anything from sense checking calculations when debugging, estimating file sizes/memory usage, worrying about when numeric overflow/underflow might be a problem. There are countless situations when being able to do quick calculations comes in handy. Of course I rarely multiply two numbers like 7x8 in isolation any more, but I am immensely grateful how Mrs El Guindi, my fifth grade teacher, drummed into us all fast, accurate mental arithmetic. Thanks to her, I can now look at a spreadsheet of numbers and immediately spot which ones are out of place.

Estimation is subtly different skill from exact arithmetic, IMO.

>Slower for what, exactly? How often do you need to do arithmetic - multiplication, in particular - in a hurry with no aids?

Almost everyday. Besides being a computer programmer, it's obviously used everytime I buy stuff in multiples (and want to know what I'll pay), everytime I build something (and want to know its dimensions), everytime I cook, etc.

I see memorizing the times table as a form of caching. I use it so frequently and it's so 'cheap' to keep this information in my head that any kind of calculation just isn't worth it.

Plus, it's pretty easy to combine this information to multiply above 12.

As a non-native speaker, is “sum 7x8” an error? Shouldn’t it be “product 7x8”?

American vs. British English usage. sum is an acceptable general term in British usage. In American English, it would indeed need to be product.

In (British) English, equations are sometimes called 'sums'. It's a confusing colloquialism, and you're right to be confused by it.

I spent the last year tutoring math full time (approximately 28 fourth grade students per day in small groups).

At one point we did an exercise with a blank 12x12 multiplication table. I had students fill in what they already knew. Ones, fives, tens, elevens up to 11x10, and twos were easiest. Most of them could figure out threes and fours pretty quickly. I taught them the finger trick for nines (9x6... hold up your fingers, drop the 6th finger, you have 5 on the left and 4 on the right for 54.) Most of them also had 6x6, 7x7, and 8x8 memorized.

This left 6x8, 7x8, 12x6 through 12x9, 11x11, 11x12, and 12x12. Not surprisingly, these are also the products adults tend to have the most trouble with. I had students select some problems they didn't have memorized, and commit to memorize them over the coming week.

Lest you think it was all about memorization, I also taught a number of techniques for quick computation -- often taking the form of number splitting (6x8 = 5x8 + 1x8) or quick drawings (draw 6 horizontal lines, cross them with 8 vertical lines, count the intersections for 6x8 -- this can be combined with the previous technique by counting by 5s and then counting the remaining intersections.)

Those students who gained proficiency in multiplication also improved quickly in division, fractions, and various other parts of the curriculum.

The "times table difficulty chart" is sort of fascinating.

I've always thought that 7x8 would be the hardest, but their sample set says 6x8 is.

Looks like these kids hadn't been taught the 9s-table trick, nor the "6-times even-X always ends in X mod 10" trick. OK, I made that last one up...but 6-times even-X often rhymes, which I remember finding quite useful for memorization.

Separately, I was always taught that multiplication produces "products", and addition produces "sums". Perhaps the Brits disagree.

The word "sum" is used here to mean an arithmetic expression or math in general. For example, a parent might ask their child, "Have you done your sums?" to mean "Have you done your maths homework?"

The 6-times-rhyme was a plot device in a short story I read once, where a witness remembers a license plate because it ended in 47 - which they remember, because 6x7 is obviously 47.

The thing I found interesting about the chart is that it's not 100% reflective along the diagonal. The 7x8 error rate is different from 8x7.

I'm surprised there's not higher accuracy along the diagonal. I learned the squares as a special case.

In my case 6x8 is exactly the one I have trouble with. 7x8 is easy for me because in dutch it's pronounced as 'seven times eight is six five' (as we say six and fifty, like the germans). So it's basically just sequential numbers.

But for some reason, in my late twenties, I still go 'six times eight is five times eight is forty plus eight is forty-eight'. Every time!

> "It's those numbers near the middle that kids find the hardest - the sixes, sevens, eights and nines," says Flurrish's director

One problem with getting kids to memorize times table the English way is that those numbers are near in the middle, not at the top. The 11 and 12 times tables can be worked out by the same process as for multiplying by 13, so why memorize for 12 but not for 13 ? Children should finish memorizing at the easy 10 times, so they have a sense of having nailed the numbers, instead of a sense of dangling at the difficult 12 times tables and it can only get harder.

Also, when reciting tables, why should children recite both 7 x 8 and 8 x 7 when they could be practising the commutative rule. They should only memorize multiplying when, say, the second number is higher than the first, so memorize 7 x 8 but not 8 x 7. Children would then feel like there's less needing to be tackled, the only really hard ones being 4 x 8, 6 x 7, 6 x 8, 7 x 8, and some squares.

I suggested the second number being higher than the first for memorizing multiplication instead of vice versa because children also need to memorize addition tables up to 9 + 8, and because they'd do that before times tables, it's better to for the second number to be lower than the first for that. Another benefit is kids wouldn't even need to say "plus" or "times" when reciting, just the two single-digit numbers because whether it's addition or multiplication is implied by which number is greater, e.g. "3, 2 is 5", "7, 4 is 11", "2, 7 is 14", "7, 8 is 56", "6, 6 is 36".

I was with you up to the last sentence. Dropping the operator just seems to obfuscate things unnecessarily.

You're right, perhaps...

  3 and 2 is 5

  7 and 4 is 11

  2 seven's are 14

  7 eight's are 56

  6 by 6 is 36

to get rid of mathy words "plus" and "times".

The trick is to remember the product of every number multiplied by itself and go from there.

It's easier for me to know that 7x7 is 49 or that 8x8 is 64. Then I either go up or go down. If asked what the sum of 7x8 is I remember that 7x7 is 49 and add 7 to it or remember that 8x8 is 64 and substract 8 from 64.

I thought I was the only one. Since grade school, I've pre-cached that as "The one I always get wrong is 56."

Me as well! Amazing.

Alternative reading: Why the hell is an Eton educated Chancellor of the Exchequer so unable to field questions on primary school arithmetic, that the news is now seriously trying to float the possibility that 7x8 is in some way particularly tricky?

This is hilarious and brings back memories.

Being lazy, I didn't actually memorize times tables combos knowing I could just figure it out. For example, the 9x series is just 10 times minus the number (9x4? just 10x4 = 40-4= 36). Or I would just sum up the numbers in my head. To this day I still haven't memorized the full times tables and I'm not sorry. Although, it retrospect it might have actually been faster to memorize, it probably contributed to mental development thinking about the problem each time rather than spitting out a burned in answer.

But there was one combo that didn't have a quick cheat and I couldn't do easily by simply adding. 7*8. So I flat out memorized that one because I didn't have an option. I think it's one of only a few table combos I actually have "memorized".

The teachers were never the wiser or if they were they didn't let on.

I find 7x8 one of the easiest to remember, because I was taught to remember 56=7x8 (5-6-7-8)

As a homework problem (University), we once had to find all such equations, in all possible bases. Turns out there is only two, and they are both in the decimal system: 56=7x8, and 12=3x4.

I think there are patterns that help for most of the facts; the 2s, 5s, 9s, and 10s are dead easy. Then there are facts involving powers of twos, squares, the 3s are easy too, really 7 is the first prime that isn't hard at all, and 8 is the hardest fact.

After coaching my kid through Common Core, It recently hit me to think of the progression

14 x 2 -> 28 x 2 -> 56

When I was in middle school, I had a book (in Chinese), teaching children how to carry out some simple calculations fast. It had a back story, with village boys trying to help out the adults when they're stuck with such simple calculations.

It mentioned some interesting facts and I can still remember them today. Like when you divide anything by 7, you get a permutation of the string 142857 as the fractional part. Also notice how 14, 28 and 57 (yes, I mean it) are multiples of 7.

57 isn't a multiple of 7?

One of my grade school teachers taught me a trick for 7x8. She would tell us to think of the grades one would have to pass before reaching the 7th and 8th grades of school... 5th and 6th. Thus, I will always remember that 7x8=56.

I am glad I learned my times tables and would encourage everyone to do it. It's not that hard and it pays dividends for life. Having complete familiarity with human-scale numbers is part of being a numerate human.

As I read the headline the answer popped into my head instantly. Mind you, I'm only good at multiplicating one digit numbers. If it's more than 12*12, I'm toast.

I hopefully read the article to finally learn why some products are harder to remember than others...but no why in there.

Why does the word "product" catch people out?

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