Help with the Price equation
Here is a simple example using your data in which both terms of the Price equation are needed, since the value of the character for $z_2 $ changes in the second generation. I used your suggested change $z_2'=(9\cdot 2 + 1\cdot 3)/10 = 2.1$. 

The Price equation or theorem is:
$$(1)\hspace{10mm}w\cdot \Delta z =  \text{cov}(z_i,w_i) + E(w_i\Delta z_i)  $$

While the idea here is just to give a credible calculation using both terms in the right side of the equation, and while the intuition behind the equation is essentially mathematical (or is it? $^1$), Steven Franks' paper, 'George Price's Contribution to Evolutionary Genetics', J. Theor. Biol. 175 (1995), 373-88, is a good introduction. His characterization of the right side may be the best one can do: 


  "The two terms may be thought of as changes due to selection and transmission, respectively. The covariance between fitness and character value gives the change in the character caused by differential reproductive success. The expectation term is a fitness weighted measure of the change in character values between ancestor and descendant. The full equation describes both selective changes within a generation and the response to selection..." p. 376. 


We work with reference to the following data and will define variables below. 

$$\begin{array}{c | c | c | }  n_i &amp; 3 &amp; 2  \\ \hline
z_i &amp;1  &amp; 2  \\ \hline
w_i &amp;1&amp; 5   \\ \hline
n_i' &amp; 3 &amp; 10  \\ \hline 
z_i' &amp; 1 &amp; 2.1  \\ \hline \end{array}$$

All cites WK are to the Wiki page on the Price equation: 



(1) We can write $\text{cov} (w_i, z_i)$ as $E(w_iz_i) - wz.\hspace{10mm}$ WK eq. (7).

Explicitly: 

$$ \text{Cov}(w_i,z_i) = E(w_iz_i)- wz = \frac{ w_1z_1n_1 + w_2z_2n_2}{n_1+n_2} - wz $$

$$ = \frac{1\cdot1\cdot3 + 5\cdot2\cdot2}{3+2} - (2.6)(1.4) =\frac{23}{5} - (2.6)(1.4)= 0.96$$



(2) We can write $E(w_i\Delta z_i) = E(w_i z_i') - E(w_i z_i).\hspace{10mm}$ WK eq. (8).

Now $E(w_i z_i') = \frac{1}{n}\sum w_iz_i'n_i\hspace{40mm} $ WK eq. (9a). 

$ = \frac{1}{2 + 3}(1\cdot 1\cdot 3 + 5\cdot(2.1)\cdot 2) = 24/5.$

$E(w_iz_i) = 23/5$ from (1) above.

So  $ E(w_i z_i') - E(w_i z_i) = \frac{24}{5} - \frac{23}{5} = 1/5$



(3) Finally we have $\Delta z =  z' - z = \frac{2.1(10) + 3(1)}{13}-\frac{3\cdot 1+2\cdot 2}{5} = \frac{29}{65} $

and $w = \frac{3\cdot 1+ 2\cdot 5}{3+2} = \frac{13}{5} = 2.6$

So $$w \Delta z = (2.6) \frac{29}{65} = 1.16 $$

$$= \text{cov}(z_i,w_i) + E(w_i \Delta z_i) = 0.96 + .2 = 1.16 $$


  $n_i$ is the number of elements in group $i.$ 
  
  $n_i'$ is the same for the filial generation.
  
  $z_i$ is the value of a character for group $i.$
  
  $z_i'$ is the same for offspring of $z_i.$
  
  $w_i$ is a fitness weight, often the number of offspring an element $z_i$ will have. 
  
  $w$ is the average fitness for the parental generation.
  
  $z$ is the average value of z for parental generation;
  
  $z'$ is the average of z for the filial generation.
  
  $z_2' - z_2$ (for example) is the average value of $z_2$ in the second filial 
  generation minus the average of $z_2$ in the parent's generation.  


$^1$ This site condenses some critical discussion of the Price equation. I have not worked through it yet but it addresses some lingering doubts I have about just what this equation means. Strongly recommended.