We always do a lot of projects around our house. My oldest daughter Emma loves to do anything that involves beading. She works a lot with clear elastic and makes bracelets to give her friends. Instead of tying knots at the end, try this technique. Attach a heavy duty chip clip to the end of the elastic. I like the ones with the magnets because they are heavier and have the extra weight which is nice. Make sure that elastic is secure and is in the teeth of the clip before you start beading. Just give it a good yank and then start beading. It makes for a good anchor too and is much for manageable for kids.
Here's one of her bracelets with the clip attached.
You can also attach another one at the end if you need to stop before the bracelet is finished. Happy Beading!