Are US high-school graduates still supposed to know their times tables? How many do?
If you gave 100 teenagers pencil and paper and asked them whether 29/67 was more or less than 16/37, how many of them could answer in 2-3 minutes like us older folks? How many could answer at all, assuming no cheating?
It occurred to me to wonder when I was at the market yesterday. My bill was $5.39; I handed the young cashier $6 and she got the 61 cents change out of the register, then I changed my mind and said, here, I’ve got 40 cents, and gave her that. She got her phone out to figure what to do.
Yeah dont do that. In larger amounts that is what “shortchange artists” do.
Kids should be able to do simple addition and subtraction, and sure the times tables. But the crap that was dinned into us- multiplying and diving fractions, etc- that is useless now.
And when IRL would you need to know those kinds of fractions in the OP?
The problem with doing this is: you have turned a trivial calculation ($5.40 - $5.31 = .1¢) into something that requires much more concentration.
Now the cashier has to decide how to interrupt the transaction already in progress. Who gets what change back and what new calculations to make.
As mentioned, this is a well-know scam used to confuse cashiers into returning more money than they were given.
Yep - and the former cashier in me wonders why people don’t just give the five dollars and the 40 cents at the same time instead of handing me $6 , then wanting a dollar back and then giving me 40 cents. Because when you do that, I can count back the change, but I’m wondering if you are trying to shortchange me. . Just say “wait a second, I have the change” before handing me anything.
Don’t assume older folk could answer it- I haven’t worked as a cashier in 40 years and I still remember multiple cashiers getting terrified when they accidentally keyed in $100 for the amount tendered instead of $10. They wanted the manager to void the sale because they didn’t understand that if you keyed in $90 too much as the amount tendered, you gave $90 less change than the machine displayed.
Never - now, if the question is whether 3/4 is more or less than 1/2, that might be useful.
I can tell you about my kid’s recent experience. They are in seventh grade now, and what they were taught in first through fifth grade math a few years ago was very similar to what I was taught in the late 70s to early 80s.
They were supposed to memorize the times tables, learn how to multiply large numbers, do long division, and all of the fraction stuff. Sometimes the exact methods were different than the way I was taught, but they all get to the same answer.
Seventh grade was the first year my kid had a calculator as part of their school supplies, and I think it was the same for me 45 years ago.
So to answer the question, there will be individual differences in that some kids, like mine, never did memorize their times tables, and have suffered because of it. There will also be other kids who did learn them, and practiced sufficiently at unaided arithmetic that they’ll be able to handle the OP’s problems. Both sets of kids will pull out their phones to do long division or multi-step arithmetic.
Back in my day, I only learned about half my times tables, and I know how to do all of the pencil and paper stuff, I’m just likely to make a mistake. I don’t see how kids today are any different, except we all have calculators in our pockets now.
We had to calculate square roots by hand when I was in jr. high. I must have missed the method, so I just used trial-and-error until I came up with the answer. (ISTR it looked something like long division.)
Yup. But a lot of parents will look at their children’s homework and say “What is this nonsense? Math didn’t work this way when I was in school! Why can’t they just do the math, like we did, without all of this nonsense?”. Even though, as you said, it’s still the same methods, and the only reason they think it’s different is because they don’t actually remember how to do it themselves.
Of course, the fact that so many adults don’t remember how to do it is maybe a sign that those methods we’ve been using since the 70s aren’t actually all that great, and that maybe we should be looking for new ways to teach math.
Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that:
Three from two is nine, carry the one, and if you’re under 35 or went to a private school, you say seven from three is six, but if you’re over 35 and went to a public school, you say eight from four is six …and carry the one, so we have 169.
It’s just a couple of multiplications. 16*67 is the same as 16*64 + 16*3, or 1024 + 48, or 1072. 29*37 is a little trickier, but it’s the same as 33^2 - 4^2. 33 is almost 1/3 of 100, so 33 squared is almost 1/9 of 10,000, which would be 1111… but it has to end in a 9, and be a multiple of 9, so it’s clearly 1089. 1089 - 16 is 1073. Since 37 * 29 > 16*67, it follows that 29/67 > 16/37.
Although, those numbers are so close that I don’t think that’s coincidence, so that probably was the way that @Timz set up the problem.
Hm, I was pretty good in arithmetic and math, but I don’t think I ever did learn to calculate square roots by hand, or if I did learn it, I immediately forgot it through lack of use. Is that something a slide rule would have been able to do? If so, that’s probably what I did (of course, I have completely forgotten how to use a slide rule too, by now.)
I roughed it in my head, based on how much less than 1/2 was 29/67 compared to 16/37. I concluded that 16/37 was smaller, but not by much. I tested it on a calculator, and I was right, but it’s a lot closer than I thought.
As has been mentioned in other threads about how math is being taught, some of the things I learned as advanced tricks to solve problems quickly are what is being taught as standard. Others, like the fraction stuff, I’m pretty sure was exactly the same.
The only real substantive difference is that my worksheets were blurry purple text from a spirit duplicator, while my kid’s were in cheaply printed workbooks with tear out pages. Still a page with twenty 2 and 3 digit multiplication problems and an insistence on showing your work.
About converting it to a multiplication problem, or about how to do the multiplications mentally? I’ll grant that most students wouldn’t do the math mentally (and realistically, there’s no reason for them to do so), but the conversion to multiplication is something that is taught, and that students who pay attention should be able to do.
Well, no, not at all; a slide rule does not give you arbitrary precision, not the way you are supposed to use it, anyway. Maybe that is the point (and why a slide rule was more useful in general than knowing how to compute square roots via long division)
Maybe my memory is faulty (again), and I did used to know how to calculate square roots by hand. Calculators came in fairly common use when I was in high school, if I am remembering that correctly, and I would have been likely to be an early adopter, if I could have afforded it.
The slide rule gives you, let’s call it three digits; the pocket calculator gives you 10. An HP-35 was about $395 in 1970s’ dollars so not many high-schoolers were buying them.
By hand, as many digits as you have patience for… (better not make any mistakes!)
How many digits do you need, is a question the student needs to ask themselves.
I was also taught the method, but let us be honest, I would not say kids are supposed to be able to do longhand square roots.
To be clear, a slide rule can’t do anything to arbitrary precision, but it can do square roots, and to a precision that’s good enough for almost any practical purpose. It can also do logarithms, exponentials, trig functions, and anything else you could do with a scientific calculator.
And a greater proportion of teens I know than adults I know can use a slide rule, but that’s an unfair comparison, because I’m the one who taught them how to use it.
My father was very proud of having spend $300 on an electronic four-function calculator in the late 70s, because the mechanical equivalent would have cost him $500, so he was saving $200. Never mind that, not many years after that, you could get an electronic calculator with the same capabilities for $10.