I think your teacher has failed you by telling you to memorize all of these formulas. All of it can derive from knowing a few fundamental concepts.
1. Time-value-Money:
It is the concept that a sum of money is worth more now than the same sum will be at a future date due to its earnings potential in the interim. So if you put $100 today (present value) in the bank and earn 5% of interest annually, then a year later you would get (future value) is
$100 + 5%*$100=$100(1+5%)=$105. So, in general, if interest compounds annually, then:
[math]A=P(1+r)^t \Lrarr \boxed{P=\frac{A}{(1+r)^t}}---(1)[/math]We extend this concept to multiple compound periods a year:
[math]A=P\left(1+\frac{r}{n}\right)^{nt} \Lrarr P=\frac{A}{\left(1+\frac{r}{n}\right)^{nt}}[/math]You only need to know equation (1), then adjust for compound periods and solve for P or A.
2. Ordinary Annuity aka Annuity Immediate (annually):
[math]P=m[(1+r)^{-1}+(1+r)^{-2}+\dots +(1+r)^{-t}]= m\cdot\frac{(1+r)^{-1}-(1+r)^{-(t+1)}}{1-(1+r)^{-1}}= \boxed{m\cdot\frac{1-(1+r)^{-t}}{r}}---(2)[/math]Again, extend equation (2) to n-compound periods, by adjusting your interest r.
[math]m\cdot\frac{1-(1+\frac{r}{n})^{-nt}}{r\div n}[/math]
3. Annuity Due: If you know the ordinary annuity, we can find the annuity due by discounting the ordinary annuity by 1 period. You don't need to memorize the annuity due formula.
[math]P^*=\frac{P}{1+r}[/math]Extend to multiple compound periods:
[math]P^*=\frac{P}{(1+\frac{r}{n})^{nt}}[/math]
4. Sinking Fund aka Accumulated value. Again, we need to know the ordinary annuity formula and adjust it by accumulating it in the future—no need to memorize the formula.
Assume compound annually,
[math]A=P(1+r)^t[/math]Extend it to multiply n-compounding periods
[math]A=P\left(1+\frac{r}{n}\right)^{nt}[/math]
5. Your example,
A=10,000, r=3%, n=4, t=4. Find m
First, find ordinary annuity (the only equation you need to know).
[math]P=m\cdot\frac{1-(1+\frac{r}{n})^{-nt}}{r\div n}= m\cdot\frac{1-(1+\frac{.03}{4})^{-4\cdot 4}}{.03\div 4}=m\cdot 15.02[/math]We want 10,000 in the future, so we accumulate the present value for 4 years (16 periods) and solve for m.
[math]\overbrace{\underbrace{10,000}_{\text{future value}}=
\underbrace{m\cdot15.02}_{\text{present value}}\cdot\underbrace{\left(1+\frac{r}{n}\right)^{nt}}_{\text{adjust for interest}}}^{Equation 1} =
m\cdot15.02\cdot\left(1+\frac{0.03}{4}\right)^{4\cdot 4}
\implies m=590.59[/math]The main thing I hope you can take away is just by knowing the ordinary annuity formula you can answer all of the other ones by adjusting the interest. No need to memorize all of them! That's how all of the other formulas are derived. Lastly, there really isn't an interpretation for why we divide certain things. They are just results of the sum of geometric series and adjusting interest.
