Secondary maths teachers - does this homework make sense?

[PointyChristmasFairyWand]

DD1 has just spent an hour working out the cube root of 426 to 2 decimal points. She's convinced that she has to do all the calculations by hand - no calculators allowed. She understands the method perfectly well - work out between which two whole numbers it falls, then work out between which two numbers to one decimal it falls, then down to the 2 decimal point numbers.

She has spent an hour on one sum and has 2 more to do. One of them isn't a cube, it's a fourth power.

I don't see the point of doing it all by hand. I really, really don't. Everything I have looked up suggests that 'doing it by hand' will still involve using a calculator for the individual sums. Am I being dim? Is there a possibility that my DD has misunderstood and that it's OK to use a calculator for the 7.52x7.52x7.52 etc.? I get that it's useful to practice multiplying decimals, but this?

It sounds like trial and improvement and yes she can use the calculator. I think that when they said dont use the calculayor they meant dont use the cube root button! This would always be a calculayor question on an exam.

Thanks, that sounds much more plausible. DD is going to check with her maths teacher just in case, but as a trial and improvement exercise it makes total sense. She feels better now, and has another day to finish the work so it will all be fine.

Successive approximation is a slow way to find roots by hand, too, which you would never use in reality.

The fastest way to do cube roots by hand is probably Newton Raphson.

Root 426, initial guess 7.

7, 7.56, 7.52, 7.52, we're done.

At each stage you have to work out x - (x3-426)/(3x2). I used a 1981 programmable calculator, but it's hardly much slower using an ordinary calculator. As others have said, doing it completely by hand is not quite the point...

Newton raphson is alevel syllabus sounds like the op dd is younger than that. She is correctly using the required method!

7.5245 7.5245 7.5245=426.022 .... GCSE GRADE E MATHS FOR YOU....

soul2000 you are missing the point - DD1 knows perfectly well how to do these multiplications, and was doing them. All of them. By hand, with no calculator because she thought she was not allowed to use the calculator. And she got the same answer you did too - it just took forever. And did you intend to shout? She absolutely understood the required method (though I like friday's method, much more elegant - but DD is yr8 so hasn't learned that one yet).

I told her that I very much doubted she was meant to do all of it by hand and that her teacher probably meant 'don't use the cube root button'. She's going to check today.

Ds is year 8 too.

I am sure it meant don't use the button!

Ds had similar homework. He used the "roots" button then tried to retrofit the working out. I have very rarely seen anyone get into such a muddle over anything. Lesson learned!

curlew!

DD's working out was scarily immaculate. Far more so than her usual standard, she had it all laid out beautifully. Which is why it was so very obvious that she knew what she was doing.

I reckon if she does it was way it's intended she will be able to do all three sums in about 25 minutes (her neat writing takes a little longer...)

was way the way. Need more coffee.

So what point are you making, soul2000? The cube root of 426 is irrational (it's algebraic, however), so any finite approximation of it will have an error, and the magnitude of the error in cubing that approximation will be cubic. You can only work accurately with irrational roots by treating them as surds, so any finite representation will have the property you give.

For example, to 100 significant figures, the cube root of 426 is approximately

7.524365203641101425512743481414002128348005126377745371287720456777361043416852334287501633996215274

The uncertainty is in the last digit, so all we know is that (assuming the arbitrary precision calculator I used is implemented competently) if we calculated one more digit, the last two digits would be in the interval (35, 45].

If you cube that, it has an error slightly earlier, and the last two digits are wrong, as you would expect: if you take an approximation and cube it, you are also cubing the error (or, more accurately, cubing the ratio between the true value and the approximate value).

425.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999930

To take the simplest example, if you approximate the square root of 2 as 1.41, which is correct to two decimal places, the error is about 0.3%. But if you multiply 1.41 by 1.41 and compare it with 2, the error is about 0.6%, enough to make it 1.99 rather than 2 to two decimal places.

The question as asked was "to two decimal places" which we can take as three significant figures, ie 7.52 (the correct rounding of the rather more accurate result I give above). If you multiply that out, it's 425.25, which is only accurate to two significant figures. 7.52 is, nonetheless, the right answer for "the cube root of 426 to 2 decimal places".

In order to get the result of cubing your answer to be accurate to two decimal places, as you appear to be asking for, you need a great deal more precision in your calculations: you're asking for five significant figures and the error term in the result will be cubicly greater than the error in the inputs.

With five significant figures:

7.52447.52447.5244
426.051 (426.05, five significant figures)

So you need to work to six significant figures:

7.524367.524367.52436
425.99909 (which rounds to 426.00).

and even then it's only "just" right to creep into the correct interval: with slightly different values you'd actually need seven significant digits in the root to get five in the result.

Error analysis of numerical methods: there's a good reason they're an undergraduate topic.

friday I'm feeling uncharitable here so I am assuming that soul was saying something along the lines of 'oh any idiot could do this, it's only GCSE maths'. Hence completely missing the point of my question, which was 'Is my Yr8 DD right in thinking she has to do all the calculations involving this approximation without a calculator?', to which the answer seems to be a unanimous 'no'.

And thank you for working all this out, it's fascinating stuff!

Pointy. I was only having a joke... Laughing at myself Actually, Not your DD...

SORRY FOR UPSETTING YOU.

I Know many people who have passed GCSE Maths who could not do it even with a Calculator.

Soul- please could you stop using capital letters? Please?

Ok. I am sorry to upset anyone, i was laughing at my own inadequacies.

I think its very good that a Yr8 can work that out.

'Is my Yr8 DD right in thinking she has to do all the calculations involving this approximation without a calculator?'

As you say, the answer is "no". It just means don't use the root key (or some other route your calculator offers).

Of course, a generation ago, working out a cube root without using your calculator at all would have easy, provided you had a book of log tables handy (those much under 50, ask your dad).

How many times do you have to divide 426 by to get a number between 0 and 10? Twice. Write down 2.

Look up 4260 in the table of logarithms in www.girishgovindan.com/uploads/mt/Log_Antilog.pdf the book, and write down the answer: 6294

Now we've got 2.6294

Divide by 3, giving 0.8765.

Keep the zero in mind, and look up .876 in the book: 7516

Find the mean difference for the 5 from the last columns if you can be bothered which is 9; add it on, so you have 7525.

How many times do you need to multiply that by ten? 0, because the first digit was zero, so the answer is 7.525. Not bad: two look ups, one division by three, one single-digit addition and an answer good to three significant figures

It won't take any long to find fourth roots (just divide by 4 instead of 3).

This is the sort of stuff that nostalgics bang on about when talking about how much "harder" maths was in the past. But it's pointlessly harder: people were taught to use logs in late primary or early secondary school, because they were the only practical way to do a wide range of tasks (there were books of logarithms of sines and logarithms of tangents and so on, so you could do O Level trig). But they were taught as a black box: I suspect the workings above trigger dim memories for some readers, but I bet that of those that knew how to use log tables effectively, only a tiny proportion knew how and why they worked, or could have explained it. Whereas now, logarithms are (properly) mostly in A Level maths, where they're used for much more useful and interesting purposes than extracting roots and doing multiplication of four-digit numbers.

How many times do you have to divide 426 by 10 to get a number between 0 and 10, of course.

Friday you are on my Christmas car list for thinking I am young. I had log tables in school .

soul

Update - DD1 asked the teacher about use of a calculator for the interim calculations and the teacher said no. I am .

So I am going to implement plan B - which will still be time consuming but less prolonged and utterly pointless than plan A. I am also going to raise this at parents' evening, because this teacher has now made a serious dent in my respect for her.

Pointy - that's ridiculous, and not at all the point of trial and improvement anyway. Check out the school's homework policy so you know what the maximum homework time per subject should be. Then write or email to say that dd has spent more than long enough on this.

I will be emailing the teacher. I will be very, very polite.

DD will also be handing in perfectly worked-out homework. In her own handwriting. The teacher will have nothing to complain about. I will make sure that DD gets through the homework as efficiently as possible, and just in case DD's teacher reads MN that is all I am going to say.

mine I will be checking the homework policy, but I doubt there is a maximum. They've gone Academy anyway so will do what thehell they want. I'm very disillusioned.

Update: Have checked the homework policy. For the upper sets it says 45 minute per subject. We're into our third hour on this. Hurray, ammo!

Am gong to contact the school tomorrow and find out the best way to get in touch with the teacher.

Told you so! That's why I didn't say IF it says it's taken too long etc etc. this teacher needs a talking to.

I have a feeling she thinks most parents won't realise what a bad piece of homework this is. A friend of mine who is a parent governor and whose DD is my DD's best friend told me she just 'looked and saw a whole page of calculations and didn't have a clue what they were about'.

Well, DD's teacher has got hold of one parent who knows what trial and improvement is, who knows the GCSE syllabus, who has a master's degree and several training qualifications and who is going to challenge her.

Politely, of course, always.

Thanks for the support!

And actually, now that I've had time to sit and think... What really upsets me about all this is that I am one of those parents who has faith in teachers. That faith has always, always been justified so far. DD's maths teacher is a good teacher, DD really rates her and now it feels like a let down. I understand that anyone can have an off lesson, but today the teacher had the chance to say 'Hey, actually I made a mistake, what I meant was 'don't use the cube root button, but you can use a calculator for all the decimal multiplication'. She didn't do that, and she must know that the homework as set and reiterated is a meaningless piece of busywork. I'm disappointed in her and that's sad.