Broken rod tips can be useful to some folks. The broken tip can be used for repair purposes where a splice repair is used to mend a broken rod. Longer broken pieces can be used to make ice fishing rods. Rodbuilders collect them for repair purposes but most folks just toss it away. As for your broken rods, breaking the tip down to the first rod guide will alter the action, but slightly, therefore the reason why you do not notice the difference with your casts.
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David Parker
Guild Certified Professional Rod Builder