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I'm trying to teach math to my 7 year old daughter. I'm teaching following type of equations. $$\cdots - x = y$$

I'm able to explain her the rule that:

when $\cdots- x = y$, we can always take $x$ (value on the left of equation) to the other side of $=$ sign and flip( $-$ to $+$ and vice versa), $-$ to $+$ and get the answer.

Meaning when $\cdots - x = y$, we can always do $\cdots = y + x$ and get the answer.

This rule works for \begin{align*} x + \cdots &= y \\ \cdots + x &= y\\ \cdots - x &= y \\ \end{align*}

But it doesn't work for $x - \cdots = y$. Because if you apply the rule, you get - (answer) and not just ( answer )

My question is given that I'm trying to teach this to 7 year old, is there any better method where one rule would cover all 4 cases? Any ideas, thoughts...

\begin{align*} - x + \cdots &= y\\ -\cdots + x &= y\\
- \cdots - x &= y\\
- x - \cdots &= y \end{align*}

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I always tro to avoid that rule (moving over and replacing sign) as it doesn't give any insight in the problem. If you have, for example, 3+x=5, I always say we add -3 to both sides (because that doesn't change equality), and we get -3+3+x=-3+5, hence x=8. – Fredrik Meyer Jan 31 '11 at 1:55
I agree with Fredrik that thinking in terms of doing the same thing to both sides is a better approach. However, I disagree with his arithmetic... :-) – Jesse Madnick Jan 31 '11 at 2:04
I've to come up with examples where i avoid negative numbers, they haven't reached there yet. – zobars Jan 31 '11 at 2:18
up vote 5 down vote accepted

You shouldn't be using "rules" at all; that is not what mathematics is about, at any level. (This is an enormous pet peeve of mine. There is a commercial floating around Hulu about some kind of online tutoring program where a woman describes to a girl the rule for computing the area of a triangle given its base and height, and then completely fails to draw the diagram that explains why this rule works. It annoys me to no end. (This particular example is also brought up in the infamous Lockhart's lament.))

I have some amount of money in my bank account. When I withdraw $x$ dollars, I have $y$ dollars left. How much money did I have originally? $x + y$. How much money do I have now? $(x + y) - x = y$.

I have some amount of money in my bank account. When I deposit $x$ dollars, I now have $y$ dollars. How much money did I have originally? $y - x$. How much money do I have now? $(y - x) + x = y$.

At some point it is probably a good idea to mention that $x + y$ is the same as $y + x$ (that is, depositing $x$ dollars and then depositing $y$ dollars is the same as depositing $y$ dollars and then depositing $x$ dollars). Then you've covered all of the "cases."

Alternately, a physical analogy ought to work well. I am some distance away from a wall. When I move $x$ feet towards the wall, I am $y$ feet away from the wall. How far was I originally away from the wall? $x + y$. How far am I away from the wall now? $(x + y) - x = y$.

I am some distance away from a wall. When I move $x$ feet away from the wall, I am $y$ feet away from the wall. How far was I originally away from the wall? $y - x$. How far am I away from the wall now? $(y - x) + x = y$.

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Hmm.. Why do you say math is not about rules ? I agree that i would really like them to learn what's actually going on when i'm teaching them x + .. = y type of equations and for single digit integers she's able to do mental math(similar to your bank deposit examples) and come up with answers, it's when i reached to double digit numbers i had to use something more than mental math. What would that be ? Am i just trying to teach something that has to wait ?? – zobars Jan 31 '11 at 2:07
@zobars: mathematics is as much about following rules as literature is about writing words. If you'd like a thorough discussion of this, you can read Lockhart's lament, which I've linked to above. – Qiaochu Yuan Jan 31 '11 at 2:12
Alright Qiaochu, i get the point. I would personally not like to go by rules. But i'm trying to find out best way to teach an elementary level kid and i get the idea that i could still do that without forming rules, they would get better idea with physical analogy. I'll give it a try and see how it goes.. Thanks for your answer and time. – zobars Jan 31 '11 at 2:15
@zobars: There's a place for rules and memorization, and a place for understanding. I would say everyone should memorize the multiplication tables, simply because having them at your fingertips is so much more useful than trying to figure them out from scratch every time; but understanding what multiplication is will be more useful than not understanding it. But once you get past the very basics, memorization just tends to get in the way of both understanding and ability to use the material. Some memorization is still useful (e.g., $(\sin x)'=\cos x$), but much less than people think. – Arturo Magidin Jan 31 '11 at 3:09
@Arturo, can't agree more with you. Just want to tailor this to elementary kids. – zobars Jan 31 '11 at 19:10

I'm not clear on what the "rule" you say "doesn't work" is... Still...

As Qiaochu says, don't do "rules". The key to all of these manipulations is:

If two things are equal, and you do the same thing to each of them, the results will also be equal.

So, if $A$ is equal to $B$, then adding $2$ to $A$ will result in the same thing as adding $2$ to $B$: if $A=B$, then $A+2 = B+2$.

If you have $\cdots - x = y$, then you have two things that are equal. Adding $x$ to both will still give you equal things, so $$(\cdots - x) + x = y + x.$$ Then using the fact that $-x+x = 0$, you get $\cdots = y+x$.

All of the manipulations you propose are instances of this: if you have two equal things, and you do the same thing to both, the results are still equal.

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Yes Arturo, i'm realizing exactly how i should go about this now. I was just presuming that it would be hard for them to really understand equality, but i guess i don't know until i've tried. Thanks. – zobars Jan 31 '11 at 2:16
@zobars: You may want to go over the points of equality; they are intuitive enough, so perhaps they won't have any trouble with them. Everything is equal to itself; if $A$ is equal to $B$, then $B$ is equal to $A$; and if $A$ is equal to $B$, and $B$ is equal to $C$, then $A$ is equal to $C$. Also, you may want to delete one of the two comments above. – Arturo Magidin Jan 31 '11 at 2:55
Yes that makes sense. Thanks. i finally figured out how to delete a comment... – zobars Jan 31 '11 at 3:01

"Do the same thing to both sides" also works for other operators - multiply, divide, powers, roots ... you could say it's the master/meta rule of all the others.

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