Division

Here are examples of 4 different types of division problems.

Easy Measurement: Trixie has 12 fish. She has 3 of those fish in each of her fish tanks. How many fish tanks does Trixie have?

Easy Partition: Trixie has 12 fish and 3 fish tanks. She has the same number of fish in each fish tank. How many fish are there in each fish tank?

Hard Measurement: Rose has 12 fish. Trixie has 3 fish. Rose has how many times as many fish as Trixie has?

Hard Partition: Rose has 12 fish. She has 3 times as many fish as Trixie has. How many fish does Trixie have?

Easy Measurement

1. What Is The Model For Easy Measurement?

Here is the simplest way to model the Easy Measurement problem above:

Count out 12 objects to represent the fish.
Separate out 3 of those objects to represent the fish in the first fish tank.
Continue separating out 3 objects at a time - with each group of 3 representing the fish in another tank.
Count the number of groups of 3.

2. How Do You Teach Easy Measurement?

Here is Trixie at 3 years and 6 montbs. And here is Rose at 5 years and 8 months. Both Trixie and Rose needed explicit direction from me. Their difficulties had nothing to do with their ability to think logically. They simply did not completely understand what the problems meant - and I needed to explain it to them.

Easy Partition

1. What Is The Model For Easy Partition?

Here is the simplest way to model the Easy Partition problem above:

Count out 12 objects to represent the fish.
Deal out those 12 "fish," one at a time, into 3 "fish tanks."
Make sure that each "tank" has the same number of "fish."
Count the number of "fish" in any one of the "tanks."

2. How Do You Teach Easy Partition?

The only teaching that I do here is to pose an Easy Partition problem for Rose to solve. She know exactly what to do.

Hard Measurement and Hard Partition

1. What Is The Model For Hard Measurement?

Here is the simplest way to model the Hard Measurement problem above:

Count out 12 objects to represent Rose's 12 fish.
Count out 3 objects to represent Trixie's 3 fish.
Separate out 3 of Rose's "fish" to match Trixie's 3 "fish."
Continue separating out 3 of Rose's "fish" - each group of 3 to match Trixie's 3 "fish."
Count the number of groups of 3 in Rose's set of 12 "fish."

2. What Is The Model For Hard Partition?

Here is the simplest way to model the Hard Partition problem above:

Count out 12 objects to represent Rose's 12 fish.
Deal out those 12 "fish" into 3 groups, making sure that each group has the same number (4) of "fish."
Count out 4 objects (the same number as in each of Rose's groups) to represent Trixie's fish.

Summary and Additional Comments

Click here to learn more about the relationships among Easy Multiplication, Easy Measurement and Easy Partition."

Click here to learn more about the relationships among Hard Multiplication, Hard Measurement and Hard Partition."

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