Nobody decides the kids' chores should split. It happens quietly. At your house the dishwasher gets emptied on Thursdays. At your mom's, the rule is that the kids clear the table after supper. Then one child starts asking which rule applies, and the answer becomes whatever house you happen to be at, and the chores start falling out of the week.
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What goes wrong first
The first sign is not a fight. It is a missed job that everyone assumes the other house handled. A load of laundry sits in the hamper, the bathroom towel goes unchanged, the table goes uncleared two evenings in a row. Each house thinks the other one has it. The kids notice before you do, and they learn that the job is optional.
How to catch it early
Once a week, ask each child to name three jobs they did at each house. If a child can only name jobs from one house, the list has gone lopsided. When you arrive at either house, look at the sink, the hamper, the trash and the shoes by the door. Those four spots tell you more about the week than anything said at dinner.
Make one list, and keep a copy at each house
Write the chores on one card and keep the same card at each house, in the same order. Keep the jobs identical where you can, even if the names change. The dishwasher is still the dishwasher. If your mom's kitchen has no dishwasher, write rinse and stack so the kids know the job is the same in spirit.
Agree on what counts as done
Most drift is a difference in standards, not effort. Decide with your mom or her helper what done means in her house, and decide with your kids what done means in yours. Write the two definitions side by side on the card. When a child asks why the rule changed between houses, you will have an answer that is the same at both.
Name who checks the card
Give one adult the job of checking the card every week. Keep it simple: who asks the kids, who looks at the sink, and who makes the call if a job slides for two weeks in a row. A card with no checker is just a wish list.