2AgNO[sub]3(aq)[/sub] + Cusub[/sub] → Cu(NO[sub]3[/sub])[sub]2(aq)[/sub] + 2Agsub[/sub]
Could also be written
2Ag[sup]+1[/sup]sub[/sub] + 2NO[sub]3(aq)[/sub][sup]-1[/sup] + Cusub[/sub] → Cu[sup]+2[/sup]sub[/sub] + 2NO[sub]3(aq)[/sub][sup]-1[/sup] + 2Agsub[/sub]
to better illustrate Conservation of Charge.
Fractional coefficients are certainly acceptable in my view (Canadian university, many American textbooks, but many British profs) particularly in cases like these where they allow one to avoid using coefficients with all but one of the reagents or products. This tends to happen particularly often with oxygen and other diatomic or polyatomic gaseous products.
However, it may not be desirable for a high school student to use fractional coefficients. Once they understand how to balance equations (and it eventually becomes as easy as addition or subtraction), using fractional units can be helpful. But, when you’re just learning, it’s best to use whole numbers only.
About catalysts: A high school course will probably teach that a catalyst is something that increases the rate of the reaction without being involved in the reaction. In reality, many catalysts are involved in the reaction, and actually break down at some point to be regenerated later. This is especially true of acid- or base-catalyzed reactions. It’s better to think of a catalyst as something that allows the reaction to proceed by a different mechanism, though a high-school student should stick with the provided definition for now.
About charges on polyatomic ions: 666-in-base-2 has already mentioned this, but I’d strongly recommend writing the charges on everything when learning to balance equations. Besides being helpful in balancing equations, it’s helpful in learning the charges on all the polyatomic ions. (Even in high-level courses, I often was asked to write the charges to show that I understood what they were. Not on nitrate or phosphate – I mean things like showing the number of electrons on a iron atom while drawing a cycle for oxygen binding to hemoglobin.)
Actually, the catalytic reaction, leaving the poor lonely monatomic oxygen radical there to avoid complicating matters by balancing the equation with an O[sub]2[/sub] in place, is something like:
H[sub]2[/sub]O[sub]2[/sub] + MnO[sub]2[/sub] --> H[sub]2[/sub]O + MnO[sub]3[/sub] --> H[sub]2[/sub]O + MnO[sub]2[/sub] + O:
As I said earlier, the catalyst undergoes no permanent change in the reaction.