Mental Arithmetic

"Are your experiences the same? Was it enriching for us?"

I am unpleasantly surprised when I see someone, whom I believe to be a functional adult, wrestle unsuccessfully with a simple math calculation. 7-1/2% sales tax on an item that has a price tag of $19.99. Change due the customer from a sale of $14.96 from a $50 dollar bill... pretty simple stuff. I think most CR4 members successfully calculated both of these examples before getting to the end of each sentence.

I don't really know if it from being too lazy to bother, or simple inability. I generally tend to believe laziness, but have been proven wrong a few times.

I know a little bit about being lazy... I once married a pregnant woman.

Another fine example of two countries separated by a common language.

I was with you until you got to pence, not to be confused with tuppence.

Form there on, there was no converting because I had no idea what, "crotchets and quavers etc all non decimal" means so how can I convert without a translator?

Give a thumbs up to ozzb for a great sentiment.

And as Doorman pointed out, making change is not a personal mental exercise for the clerk any more. "Count it back to you, are you kiddin' me you old geezer? Here, take it".

Actually, he missed minims and dotted minims.

Similar essentials except that I came to the States in 1966 so I went on to decimal money then. My problem came on visits there, after a day or two my mid-Atlantic accent slid into full English but I was not familiar with the money, old and new still being used. Bus conductors looked at ma as though I was nuts when I asked them to pick out the right coins.

Have a good friend in field services. He has traveled all over the world. He took a camel ride around the pyramids in Egypt that cost him some obscene amount of money because he counted the coins out to the camel driver himself.

Hemidemisemiquaver = 1/32 note (3 flags).

Come on! 0.03125 of a note!

Hemidemisemiquaver = ??/32 note...? Wiki ...

0.015625 then. Wow - that's 5 significant figures! Why so many for something so small, and why won't 0.016 do instead, like it says on this old slide rule?

"Almost" Off-Topic with these comments ... (how did 1/64th musical notes get into it anyway?)

1/64th notes are tricky-enough to play when done solo. How much individual (followed by GROUP) PRACTICE it must take for the whole orchestra to hit 'em all on-cue/in-sync, as in Beethoven's 8th (Symphony)...?

Right you are! I misremembered quaver as a quarter note. Binary point one place off!

It always comes down to the individual not the "generation".

Most of the people reading CR4 will be the sort that like doing stuff in their head and can do mental arithmetic, they want to understand basic physics, they want to use correct spelling, they like knowing what words actually mean etc.

Unfortunately most people in any generation aren't like that.

Do you need proof? Just look at the type of television watched by normal people.

p.s. Any spelling and grammatical errors in this post are deliberate.

I once won an award for math proficiency in college (over 50 years ago, when I was really good at using a slide rule), but I don't believe I was ever able to to convert 31/64 to decimal in my head, without considerable effort. I certainly couldn't do it now, without considerable effort.

I thought "crotchets" were different methods of tying knots using needles, and "quavers" were know as spelunkers.

I do commonly praise the occasional person at the cash register who realizes why I gave them a nickel and a penny in addition to a dollar for the item that cost 81¢.

dkwarner

31/64= 1/2 - 1/64

1/2= 0.5

1/4=0.25

1/8=0.125

1/16=0.0625

1/32=0.03124

1/64=0.015625

So 31.64 = 0.5 - 0.015625

OR 0.484375

QED and if you knew your 64ths which I used to but at 75 I have kinda forgotten those - it is even easier!

For crotchets ( a measure of length of a musical note) read crochet which is an art or skill that I never learned. But many of the girls of my generation did.

Sleepy

In fact I do know and use those decimal equivalents rather commonly, although I end the 1/32 equivalent with a 5 instead of a 4, and I have to think a bit for the 1/64. So far so good.

But now when it comes to multiplying 0.03125 by 5 or 7 in my head, forget it!

I do sometimes remember that 7 is 8-1, but again there is no way I can subtract 0.03125 from 0.5 in my head. Heck, I have to think about it to subtract on paper!

You are around three years my superior, so I'm having problems with the short-term too...

My mother taught piano and organ, so I do have some musical background, but I never heard those terms. I pretty much ended my musical efforts after about 3 years in college because I never developed the ability to read more than a single note at a time.

Not quite sure if the present younger generation doesn't know who to calculate mentally because they are lazy, but more so because that have not ever been placed in situations where they had to.

There was no requirement for me to memorize decimal equivalents down to the 64ths, I just did as the result of mechanical drafting on a board.

Likewise, there was never a requirement to be able to make change or calculate sales tax quickly in any class I ever took... but I learned to because as a youth, I worked as a waiter and everything was done by hand.

The members of this forum have learned how to do mental calculations and estimates as the result of repetitive-necessity.

The problem today's youth face is a combination of both too much technology allowed in the classroom at the early grade levels, and enabling by adult peers.

Case-in-point, I was visiting a friend's house and his youngest daughter was sitting on the couch, listening to her iPod (rocking her head to the beat, and singing softly) while also switching between a text book, her notebook and an iPad. Looking closer, she had an iPad app that would show all the long-division steps of any entered problem; she was copying the steps to her notebook.

When I asked her dad what she was doing... "Her homework, they make her show all the steps so the iPad works better for her than a calculator."

Are your frigin' kidding me?!

Obviously the conversation continued, but I'm not going to rehash it here as I'm sure you all know how it went.

I occasionally tutor year 12 Maths.

Most kids are OK but some doing "vegie maths" are so far behind it makes me cry, I'll ask a question like "What's 10% of 500" and after some furious work with the calculator the answer comes back "234.545427876" or something equally stupid.

No idea + A flash graphics calculator = No idea

The teachers have pushed them through the grades but mostly it's the parents. Have they never asked their kids to solve a maths problem? Have they not noticed their little darlings never do any homework? It's too late after 12 years of schooling to blame others, they should have paid attention earlier.

Ummm,

part of the problem is down to parents and part of it is down to teachers and teaching skills.

It did not seem so bad when my children went through education, one ended up with a maths degree and a career in that subject.

But when it came to the grandchildren they seemed unteachable!

Why, because the teaching organisation had changed the way that (even) multiplication and division were done - i did not undestand the process that they were trying to comprehend, nor did their parents or ther siblings (uncles and aunts) and I just could not fathom what they were trying to do!

I have not asked how they do Square roots! That just sounded far too dangerous!

And they slipped backwards over a year or so but have now pulled themselves back through effort. Thank God! But there was no way that I could help them!

Sleepy

Could be right sleepy.

Tom Lehrer new this

http://www.youtube.com/watch?v=DfCJgC2zezw

Did they switch to the 'new math' in the UK too? Back when I was teaching (I retired from teaching in '95), I was aware that a change was occurring, but I never paid much attention. Obviously I never learned it...

It's enriching about 5/16 of the time.
But seriously, it's about using whatever system suits the application.
In my bow making I use fractions of inches, millimeters, feet per second, grains, pounds and string. no wonder I'm confused.
The important thing is to understand what they mean and their approximate magnitude so that if you convert with a calculator, computer, slide rule or mental arithmetic you have an idea if the answer is right.
I've seen online conversion programs that gave the wrong answer. Teaching how to do order of magnitude calculations is IMO more valuable than mental arithmetic, although if you can do 'em in your head, so much the better.
I'd rather have an approximate right answer than a wrong one to 3 decimal places
Del
(Now, what was the question again?)

Totally agree with that. I often sit in meetings here feeling stupid whilst my colleagues who have several doctorates do mental calculations using numbers to various powers of ten. I can still tell if the answer generated is in the right area without quite knowing why, it just feel right or wrong.

Del,

A GA from me, a very good answer in fact!

Use the System that works, for you especially.

I was in a meeting with my MD an a dozen or so other folk, we were exploring major options.- The MD asked a question and 10 people got out calculators and furiously worked out a new number, all were different, most were very wrong. I just wrote down the right answer as the effect of the question was to ask for the cost of doing the work for half the return , which was just plain obvious.

I did not get any brownie points from the guys!!

I have several stories similar to that!

On the other hand my Mental arithmetic is starting to get worse, slower and less accurate! I tried a short time ago to understand 1 oz in Grams as Prue Leith, bless her, had given two different answers to this question on the same page of one of her books. That took a lot longer than it should have! I have recently had dificulty in remebering which way round lbs and kgs are; you should have no difficulty with that from your archery!

Sleepy

=========

I grew up in the same era. I once had a school homework exercise to solve a few simple problems using a slide rule. The objective was to gain slide rule practice. When I handed the work in I was marked down because I had provided answers to three decimal places, and I was accused of using mental arithmetic. (3dp is a suspiciously accurate result on a standard 1 foot slide rule).

The following day I took in from home my father's 18" long 4" diameter Cylindrical Napier slide rule that I had used to complete the work. Because of the spiral scale it is the equivalent of a 41 foot straight slide rule. I had to show the teacher how to use it.

I have to disagree with your first paragraph! Although I now need a loupe to see it, I can still make 3dp calculations on my straight 6" Pickett, with a ±1 on the last digit. With a quality 12" straight or 4" circular rule, I can get 4dps on the lower half of the rule, again with a ±1 on the last digit. All it takes is practice reading scales and estimating the last digit. I still have all of these...

I continued teaching the use of the slide rule for several years after calculators became available, simply because it gave such good practice in reading scales with precision. Some, but certainly not all, students thought it was neat to be able to use one.

I learned to use a slide rule in high school. My teacher taught the whole class how to use them. He made a deal with us that if he could beat us using the slide rule while we used our calculators we would learn to use one if he couldn't beat us then we wouldn't have to learn to use it. The students got to pick the 3 equations. Needless to say he finished way before we did so we all learned to use it. I can't use one today.

Are slide rules even available anymore?

Vedic mathematics is all mental calculation and quite easy to learn.

Some circular ones are still made, some special purpose. one guy has some collectibles including a 1947 Keuffel and Esser for $1650. "The box shows some wear."

Never mind working out how many Guinees 1 Gross 2 dozen and 3 eggs cost at a rate 3 1/2 pence each.

Or a more common and simpler example; two pounds and six ounces of apples at 11 1/2 pence per pound.

Books of tables were common place in warehouses and shipping agencies. Weighing scales had tables on the faces to convert weight to money.

All of sudden i feel old, we went decimal in 1966 ( money ) and weights and measures in 1970 ( i think ).

Jim

As the old saying goes,'we are all stupid, just on different subjects..But it is terrible to hire college age students for part time help and find they can't use a steel tape or break down inches in building anything or to understand wrench sizes that relate to the decible inch. in repairing most anything. I'm old enough that I'm having trouble with the metric system in wrench sizes. I too sat through an electronic school beside a french man who would let me work on three sheets of paper to work out a problem and he would tell me the answer and never make a mark.

If the electrical supply meter to the home is only accurate to 1% or so, why does the bill come in to 5 significant figures and a row develop unless the payment cheque is an exact match to the bill?

Thanks for the replies, I think the point I am trying to make is that we live in a base 10 world now whereas us ancients had to use many other systems. These relate directly to calculations using fractions which then leads to decimals and percentages. Also any engineer searching for a 5/16 drill would no doubt place it in his imperial micrometer (if he couldn't read the size directly) and read off 0.312 inches. He could of course use his Zeus table if he could bear the ongoing sarcasm from his mates and read its no doubt oil stained pages. Better to be on the ball. It seems we have lost this ability.

What do we all think of the numeracy test for all teachers not just maths which includes time limited mental arithmetic?

http://www.teacherstalk.co.uk/resources/qts-numeracy-test.php

I have a daughter that's about to graduate high school, and for the life of me I can't get her to visualize numbers. I don't know if it's because they're so dependent on calculators or what, but a question like how many eggs are in a half a dozen would stump her. I try and tell her that she almost has to see the numbers.

Some of you mentioned slide rules above... I think these tools go a long way to help with the visualization of math. I guess it was something along those lines that had me trying to master using an abacus in the past.

Guys

this thread has me hopping!!

I found a book in the local library which was tiny and was about understanding numbers. I bought one for every member of my family.children,grandchildren etc.

This was called "BE A WIZARD WITH NUMBERS" by a guy called ANDREW JEFFREY.

it is the sort of book which tells you how to quickly discern whether a number is divisible by 3, (2 and 5 are too easy), I thought that it had a rule to see if a number was divisible by 7 but I now cannot remember that! I used to practice these skills as a kid when being driven anywhere. ( Car Number Plates were the target random numbers for me to practice on.)

A lovely little book which i would have appreciated at age 9 or 10, Hope the family do!

For those that love their decimals, how did the Baylonians manage with their/64ths!

I keep meaning to spend some time doing calculations their way!

On Napier Slide Rules we used to use these before calculators in many Scientific Spheres; in one case we were designing and building a large variety of 4 pole Tchebbychef coaxial filters, all with similar characteristics, these required different cavity tuning for each filter ( Dishal's method) and different coupling loops for each frequency and bandwidth. Good fun this was. I had to work out the sizes of everything so that the production department could set up the filters; a benign hope as they were unable so I set out to tune each and every one in the production run. Much sleep was lost!

The one skill that I did learn was in setting up the coupling loops. These had to be set up to about 1 or 2 thousandths of an inch and were then soldered together. The filters were then assembled and then, invoking Dishal, tuned. these were in the days of Analog communication which required extraordinairily precise components but we did achieve the targets and the Microwave system was rolled out across a few countries. The coupling loops were set as accurately as stated though the use of large sets of Slip Gauges.

This would have been between 1954 and 1960 to the best of ny recollection.

Sleepy

As I recall the divisibility rules:

Divisible by 1: Yes, it is.

Divisible by 2: If even, it is.

Divisible by 3: If the sum of the digits is divisible by 3, the number is.

Divisible by 4: If the 10s digit is even, and the units is 0,4 or 8 it is. If the 10s digit is odd, and the units is 2 or 6, it is.

Divisible by 5: If last digit is 0 or 5 it is.

Divisible by 6: If Divisible by 3 and even.

Divisible by 7 or 8: I don't recall if there were easy ones for these.

Divisible by 9: If the sum of the digits is divisible by 9, the number is also.

Not always obvious: With the 3 and 9 rules, if the sum of digits is a multiple digit number, those digits could also be added together, essentially reapplying the same rule.

I probably learned these around age 10.

There is no rule available for 7, 13, or 17 (At least. As the rule sought grows, more of the divisors become prime, which seems to be the prerequisite for the "no rule" rule. Note that not all primes have no rule. 11 is an example.)

The rule for 8 is basically if it is divisible by 2 and still even, it's divisible by four. If it's divisible by 4 and still even, it's divisible by 8. Another way to arrive at the same decision, though, is to factor the value being divided. It is divisible by any "Power of two" (or binary) number up to the value of all the 2's in it multiplied together (Two taken to the Nth power, hence 24 being factored as 2*2*2*3) yields the decision that 24 is divisible by 8, which is, in turn, 2 raised to the third power, due to the three 2s among its factors. Any number thus factored is also divisible by any lower power of two.

The rule for 9 is as stated, but note that this also relies on the "Powers of X" rule. That also implies that if the sum of all the digits is large enough on the first summation to require a two digit sum (i.e all the digits sum up to 12 or more) and this points to summation (I refer to this as a second order summation) again, to further reduce the sum, yielding a "divisible by 9" rule, then if the sum of the digits is large enough that the second summation AGAIN yield a two digit number, summing them for the third time and getting an answer divisible by three indicates the original number is divisible by 27, or three raised to the third power. In fact, it works. The assumption (Never tested by me because of the sheer size of the original number. I just never needed to do so.) is that it would work for three to the fourth, or fifth, or, really, ANY power.

And since this is just application of the factoring rule a la the rule for twos, above, it makes sense it would continue add infinitum.

Sorry! It's probably obvious I enjoy playing math games in my head. My wife says that makes it a scary place for anyone more than 12 or so years old, and that it's only a welcoming place for "child-geeks" and other "Enginerds" (which is what my family all call me).

I can't stand calculators, and while I can use a slide rule (learned it in High School) I'd rather calculate values to the limits of reasonable accuracy in my head, since I can't estimate/extrapolate on a slide rule to the 4th or 5th decimal, no matter how precise and accurate the input values are. But that spiral slide rule sounds like a real gem. I never heard of nor saw one of those, and I'll bet it would be of huge value to a collector, now.

Good Answer,

I've sent you a GA! Arithmetic is fun!

It's my memory that lets me down, I really thought that I had read the rule for 7s!

I knew that I had not used it but there we go!

And even had the book reference to quote! But having had a brief look at the book again I cannot see it.

Sleepy

I find there is a much more important issue than just doing arithmetic in one's head and that is; it serves as a mental exercise. at the age of 77, I use a calculator, but I also do numbers in my head. I also strive to use correct English and spelling; not because I'm trying to show others how smart I am, but to keep my brain functioning as well as possible. The result of people not using their brain is the loss of function over the years. The brain must be constantly exercised or it will die. Although I am retired, I keep active; maybe not physically, but certainly mentally. I spend a lot of time designing on Autocad and learning something new every day.

Mental arithmetic comes in handy when shopping in the supermarket. I have to compare prices of items that come in various container sizes and priced at so much per number. I also have to compare the nutritional numbers. A calculator would be quicker, but I usually don't carry one around with me.

i am also from a generation that uses the calculator to much, i wished we had more mental arithmetic at school, i noticed it is not about just crunching numbers, but smart shortcuts to make it easier to get to the solution.

Anyway it's about standardization in my book, making it easier for the (lazy) common folk, never understood much of the whole UK measurement, currency, weight system anyway

We haven't covered some of the multiplication tricks. I remember being impressed with the method of multiplying two numbers in the nineties, such as 91x98. Actually, they don't have to be in the nineties but it is easier if one is close to one hundred. I don't know why it works but since it is easier to multiply by 100, you add 2 to the 98 and subtract 2 from 91 giving 8900. You subtract each of the original numbers from 100 and multiply the results and add to the 8900.

100x89+(100-98)(100-91)=8918

With a little practice you can read the answer directly.

I have no idea why it works either, and for that very reason I wouldn't know when to use it.

At my age, I don't think I could do that in my head either, and if I did, I certainly would not have any confidence in the correctness of the answer.

Whether its a machine, a circuit, or a process (including a mathematical process), I'm never comfortable unless I pretty much understand how it works. Perhaps that's why I got in trouble with differential calculus...

i normally just do 91x100=9100 -(2x91=182)=8918

Me, also. Seems simpler to me. AND I know how it works!!