I was reading several proofs of the CW approximation theorem. If $X$ is a space then the idea is to make $n$-equivalences $f_n:K_n \to X$ where $K_n$ is a $n$-dimensional CW-complex. This goes by induction. Given $f_n$, the first part consists of making $f_{n+1}'$ on $K_{n+1}'$ where $K_{n+1}'$ is obtained by attaching $(n+1)$-cells along representatives in the kernel of $f_{n*}:\pi_{n}(K_n) \to \pi_{n}(X)$.
For the second part, to obtain $K_{n+1}$, one again attaches $(n+1)$-cells along constant maps, which is the same as taking the wedge of $K_{n+1}'$ and a bunch of $S^{n+1}$. This is where I get confused. Reading http://www.math.cornell.edu/~hatcher/AT/AT-CWapprox.pdf for example, It seems that we attach for each generator of $\pi_{n+1}(X)$ a sphere $S^{n+1}$. In other proofs I read that we only attach for generators of the cokernel of $f_{n+1*}':\pi_{n+1}(K_{n+1}') \to \pi_{n+1}(X)$.
Does this make a difference ?