Divide total length by the length of the pieces to equal number of pieces. 
= 4 1/2 ÷ 1 1/4
convert to improper fractions 
= 9/2 ÷ 5/4
multiply by reciprocal of 5/4
= 9/2 * 4/5
= 36/10
simplify by 2
= 18/5
change to mixed number 
= 3 3/5 
ANSWER: 3 3/5 is the maximum number of pieces that can be cut, but if considering maximum number of FULL pieces that can be cut, it is 3 with 3/5 of a partial piece left. 
Hope this helps!  :)