-9^2 = -81

[insufferablefury]

I need a help from maths teachers.

My dc says her teacher taught the children the answer is 81. And when she has done the work on online maths programme, it's wrong obviously.

I have explained, it's -81, since it's really - (92). And to get answer as 81, the question need to be (-9)2, not -9^2. But being not a maths specialist, cannot explain why in a logical way , and the child blatantly believes the teacher, which is a good thing, but not really if the teacher is wrong.

How do you maths teachers explain this to your students?

[DadDadDad]

"Yeet bish" - nice! Although if I tried to use it, I can picture my DD putting on this face --> "not cool, Dad, not cool..."

[noblegiraffe]

@DadDadDad

Just saw this on twitter and it reminded me of this thread.

[DadDadDad]

Ha, noble, so this moment was the biggest mic drop in the history of maths:

[Andrew Wiles] gave a lecture a day on Monday, Tuesday and Wednesday with the title "Modular Forms, Elliptic Curves and Galois Representations". There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said. ... Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D.

[noblegiraffe]

I found myself describing the tombstone at the end of a proof as a ‘mic drop’ to Y12 the other day.

[DadDadDad]

When I was an undergrad, writing out mathematical proofs, I think writing QED at the end was (in my mind) a bit pretentious, and the standard notation by lecturers was to draw a little box at the end. I tended to draw a smiling face in my notes, so I was using smileys in the early nineties before they became a thing. I was a bit of a trendsetter now I think about it.

[noblegiraffe]

Is the sad face standard mathematical notation?

[TeenPlusTwenties]

super

[DadDadDad]

Yes, Teen - non-integer exponents only really work (if you're sticking with real numbers) with positive numbers in the base. If you define (16)(1/2) as -4 then you have to decide if (16)(1/4) is 2 or -2 and you run into an inconsistency. See my jottings in the picture.

[TeenPlusTwenties]

This is interesting.
Is ^1/2 then the same as the square root sign in assuming the positive?

[DadDadDad]

I'm not sure where you are getting your definition from, impossible, but the Wikipedia article starts as follows and it reads perfectly in line with how I've always understood the definition:

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x.[1] For example, 4 and −4 are square roots of 16 because 42 = (−4)^2 = 16. Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by √x, where the symbol √ is called the radical sign or radix.

The "cube roots of 1" is a topic when studying complex numbers, where one looks at the properties of the three cube roots. It's natural to talk about numbers having three cube roots, and just as natural to talk about them having two square roots.

[Teachermaths]

@impossibletoday

There was an exam question squareroot symbol 64 which had 2 answer slots and it expected - 8 and 8. I'm struggling to find it now but it was on the new spec somewhere!

[impossibletoday]

Definition: The square root of a number is the positive number which if you multiply it by itself gives you than number. But as I said above the negative of that number may well be the root of an equation

[insufferablefury]

Thank you, impossibletoday and DadDadDad. I need to read them, though most goes over my head I assume. But I will definitely try to get it.
I enjoy learning maths, I really wish I paid more attention when I was younger, but still nice to learn something I didn't know.

[DadDadDad]

impossible - I don't think that's quite right either. 3 and -3 are both square roots of 9. You're right that the symbol √ refers to the positive (real) square root, so √9 = 3, but -√9 = -3 is also a square root.

en.wikipedia.org/wiki/Square_root

If we allow ourselves complex numbers, every number apart from zero has two distinct square roots (and three distinct cube roots etc).

[impossibletoday]

@insufferablefury

Teachermaths, she knows √ 9 = +/- 3

I couldn't let this pass.
The square root of any (positive) number is positive. The square root of 9 is 3 and nothing else.
However, the roots of the equation x^2 = 9 are x=3 and x= -3 (the negative of the square root of 9)

[noblegiraffe]

cant What?!! surely there was not someone more qualified...

[cantkeepawayforever]

Noble,

Not until recently (long story)...

[DadDadDad]

Ha, just saw your post @insufferablefury as I was helping my DD with her maths homework, and she was telling me "you explained that so much better than my maths teacher!" I'm on a roll.

For balance, I should point out the many times my children have told me that my maths explanations make no sense and I obviously don't have a clue what I am talking about!

[insufferablefury]

@DadDadDad,, substituting 9 with x really made sense to me. Great explanation. Thank you.

[insufferablefury]

@StealthPolarBear, thank you for being kind.

And thank you everyone, although a lot of things went over my head tbh, it was great that I got an answer and also created great discussion among maths specialists. It was very interesting to read, and good to know that there are so many passionate teachers and maths specialist on MN who are willing to help us parents.

[TeenPlusTwenties]

If we're now talking about writing things out clearly, my bugbear is I spend so much time trying to get DD2 to lay things out clearly (to help her get things right), and then on exam papers there's no room to do this so it positively encourages them to cut corners and make mistakes.
Plus in maths lessons they have squared books (I presume this is pretty standard) and again in an exam they are suddenly expected to do stuff on blank paper. No other subject expects this.

[noblegiraffe]

Are you maths lead, cant?

[cantkeepawayforever]

I don’t know if your other primary colleagues would have a mathematician’s natural visceral reaction to seeing that sort of monstrosity?

I don't know. If I put it this way - that type of recording is rare in pupils in my year who have come up through my school (suggesting that we all care about it!) but it is very common in children arriving as 'in year admissions' into my year group (suggesting that as a school we are stricter on it than others are).

[Sammy867]

I suppose it depends on how you interpret the number line for a child.

As a student you are usually shown a number line when first learning maths and -9 represents a physical point within that number line and not a number plus an operation.

Following this line of thought -9 would not be perceived as (0-9) as this is not a point on the number line and so the equation would then become (-9)^2. I guess that’s where confusion can come into it

I would also like to add another thought that
-92 = -1 * (92)
Which also equals -81

[noblegiraffe]

1a might be technically correct but that doesn’t mean I’m not going to cry when they write it instead of a!