Consider the differences between sharing and grouping problems. Sharing involves partitioning a known quantity into equal parts of unknown amount, such as deciding how to split 12 cookies evenly among six people. In grouping, the problem solver must split a known quantity, such as the 12 cookies, into equal sets of a known quantity, such as two per bag, and determine how many groups there are. Marilyn Burns, author of many books concerning elementary math, says that children should learn how to solve problems of both types.
Introduce a partitioning problem. Tell a story such as the one Burns suggests, a tale about four friends who find a five-dollar bill on the way to school. They turn it in to the office at school. When nobody claims the money, they have to decide how to divide it equally among themselves. Ask the children how much money each child should receive. Provide manipulatives, such as play money, or paper and pencils to aid their thinking. Ask them how they arrived at their answers.
With help from the children, show how to use the play money or pictures to solve the problem. Then demonstrate a formula for quicker calculation, such as long division, the teaching of which is detailed at Burns' Math Solutions Web site. Point out how the formula is organized: The total number of cookies --- the dividend --- is inside the bracket. The number of children --- the divisor --- is to the left of the bracket. The unknown equal share each child receives is the quotient that will be written on top of the bracket.
Begin a grouping problem, such as this one from Burns: There are two cartons of eggs in the refrigerator plus three extra eggs. A mother likes to eat a two-egg omelette each day. Ask children to think about how many omelettes she can make. Collect answers.
Use manipulatives to solve the problem in step 4. Pretend that counters represent eggs. Place one in each compartment of two egg cartons and set aside three extra. Move the counters until there are 24 "eggs" in one carton --- two in each compartment. Place the remaining three in the other carton, with two in one compartment and a remainder of one in another.
Give each child a piece of paper and a pencil. Tell them to draw 27 eggs quickly, circle groups of two and name how many groups this makes. Compare answers to the ones from step 4. Then use long division to solve 27 divided by 2 (group size). The answer, or quotient, will be the number of groups. Ask what the answer "13 remainder of 1" means.