No Arguing
Children are fiercely protective of their mothers, so make sure your man keeps this in mind when spending time around you and your kids. One unkind word or mean look sent in your direction could anger and upset your children, forever changing the way they see him.
Tell your guy not to argue with you in front of them. If he needs to discuss something, ask him to save it for later, when the two of you are alone. Doing so in front of the kids will cause them to come to your defense or attack him. Neither of which are good for a budding relationship.
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Give Gifts
A little bribery never hurt anyone. Encourage your new guy to bring gifts or treats to your kids the first few times when meeting them. This will give them good memories of him, and make them want to see him again.
It could be a small toy they've been wanting, their favorite candy, or just buying them dessert at dinner. Anything that equates a "good" or "fun" feeling with your new guy is a smart move.