A common misconception among students is that, when dividing, you always put the larger number first. When this concept “sticks”, it is extremely difficult to undo later when students learn that you can, in fact, divide a smaller number by a larger one. Some examples that young students can understand are having 2 pounds of dog food that has to be split among the 6 dogs in the pet shop, or 4 candy bars that have to be split among 10 children. Working on equal sharing situations like this helps students develop the number sense to ultimately handle operations with fractions.
One way to avoid the misconception of putting the larger number first is to focus instead on the parts of an equal sharing problem. These problems involve three countable quantities: the number of groups, the size of each group, and the whole. Have students model and solve the three problems below:
1) Eva has 3 plates. She puts 5 cookies on each plate. How many cookies does Eva have?
2) Eva has 15 cookies. She wants to put 5 cookies on each plate. How many plates will Eva need?
3) Eva has 15 cookies. She wants to put the same number of cookies on 3 plates. How many cookies can Eva put on each plate?
Have the students compare the three problems, noting what is different about each situation. Focus on what the numbers represent, on which quantities are known and which are unknown. Develop the concept that an unknown whole calls for multiplication, while an unknown number of groups or size of group calls for division of the whole by either the size of the group or the number of groups. Emphasize that it is the whole that is being divided – which might usually be the larger number but does not have to be.
Next have the students work the following problems:
Tell what is unknown in each problem. Then solve the problem.
1) Amy has 16 quarters. If each video game machine takes 2 quarters, how many games can she play?
__________________ is unknown. She can play ___________ games.
2) Sue has 4 bags of marbles. There are 5 marbles in each bag. How many marbles does Sue have all together?
____________ is unknown. Sue has _____________ marbles all together.
3) Sue’s class has 21 students. There are 7 students at each center. How many centers are there?
____________________ is unknown. There are _____________ centers.
4) Sue has 24 candies. She wants to share them equally among 4 friends. How many candies will each friend receive?
_____________ is unknown. Each friend will receive __________ candies.
5) There are 15 fish in the pet store that are divided equally into 3 fish tanks. How many fish are in each tank?
_______________ is unknown. There are ____________ fish in each tank.
Finally, have students write and solve their own problems. Ask that they compose one problem of each type.
1) Multiplication (whole is unknown)
2) How many groups? (Number of groups is unknown)
3) How many in each group? (Size of the group is unknown)