This is a constant issue with kids. It would be rare to find one single event that caused someone to have integrity or to not have integrity. What really happens is that over the course of life thousands of little events push us one direction or another. As parents it is important for us to make sure that every time we have the opportunity to push them toward integrity we do it, and that is a matter of integrity on our part.
We have dealt with this with both of our older children recently. My wife gave one a certain time limit to finish some school work and then asked to see it when the time was up. The child said the work was done, but it wasn’t. Not getting it done in the time allotted is not good, but lying about it is unacceptable. Just today I sent the other to do some school work on the computer only to come in minutes later to find a game being played instead. This event also involved punishment.
There are accidents that do happen, sometimes we really do forget things and when we do we need to admit it and not make some other excuse. Sometimes we say the wrong things believing them to be true at the time. When we realize it we need to clear the air. If we walk out of a store and find that the dog food bag on the bottom of the cart didn’t get scanned, then we should go back and pay for it. Let me promise you that people will think there is something wrong with you and give you strange looks when you do the right thing, but it will help your kids learn that integrity matters.
The Bible says, "LORD, who shall abide in thy tabernacle? who shall dwell in thy holy hill? (2) He that WALKETH UPRIGHTLY, and worketh righteousness, and SPEAKETH THE TRUTH IN HIS HEART. (3) He that BACKBITETH NOT with his tongue, NOR DOETH EVIL TO HIS NEIGHBOUR, NOR TAKETH UP A REPROACH AGAINST HIS NEIGHBOUR. (4) In whose eyes a vile person is contemned; but he honoureth them that fear the LORD. He that SWEARETH TO HIS OWN HURT, AND CHANGETH NOT. (5) He that PUTTETH NOT OUT HIS MONEY TO USURY, NOR TAKETH REWARD AGAINST THE INNOCENT. He that doeth these things shall never be moved." Psalm 15:1-5. If that doesn’t just scream integrity I don’t know what does, unless it is Job. He said, "All the while my breath is in me, and the spirit of God is in my nostrils; (4) My lips shall not speak wickedness, nor my tongue utter deceit. (5) God forbid that I should justify you: till I die I will not remove mine integrity from me. (6) My righteousness I hold fast, and will not let it go: my heart shall not reproach me so long as I live." Job 27:3-6. Job was not bragging, he was stating his determination to live right. God confirmed that Job was a man of integrity in Job 2:3. How often people use their situation or certain circumstances as a reason to do something dishonest, but God doesn’t accept those excuses. Job was a man of integrity in spite of the situation he was facing, I want a son like him. I wonder why there aren’t any Jobs in the world today. People say "I had to" do what ever, and that is a lie as well. Will we be men who like Job will say, "my heart will not reproach me as long as I live"? Will we train our children to be those who can dwell on the Lord’s holy hill?