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Division Worksheet

Two Models for Division

Division problems in daily living and arithmetic word problems are of two main types: grouping or sharing. The types are evident in the following two problems:

•xTwelve people, three to a car, how many cars?

•xTwelve people, three cars, each car with the same number of people, how many people in a car?

Both problems are solved by figuring 12÷3, but the answer 4 in each case means two very different things. For the first problem, it means four cars; for the second one, four people.

The first problem is a grouping problem. It is solved by asking how many 3s in 12 which can be figured by counting by 3s up to 12: 3-6-9-12 for four 3s. The second one is a sharing problem. It is solved like one about 12 cookies to be shared evenly with 3 children. Each child would get one, then another, and then another until all the cookies were gone. In this case, they would be gone after each child got their fourth cookie. Note that in either case, the 12 was “separated ‘neatly’” as explained for the ASMD Key.

No Need to Worry about which Model to Use

Interestingly, the mind tends to switch between the two models without us having to choose between them. What’s 100÷50? If you came up with 2 quickly, you made a grouping problem out of it. You answered how many 50s in 100. What’s 100÷4? If you answered 25 reasonably quick, you made a sharing problem out of it. You answered how much if 100 is split evenly four ways. Either problem can be thought of either way, but the mind naturally picks the easier model. (To think sharing for 100÷50, you’d have to picture something like handing out 100 cookies to 50 children twice around. To think grouping for 100÷4, you’d have to think of counting by 4s (4-8-12- …) all the way to 100 for 25 counts.)

In general, large divisors evoke the grouping model, small ones the sharing model. This matters in interpreting results, like four cars versus four people in each car, but it doesn’t matter in computing. Computing is just about getting the right answer. An algorithm may reflect a particular model or way of thinking, like how reflects exporting in Ten Land, or it may just be a matter of do this and that like with . A major advantage of an algorithm that portrays a natural way of thinking, though, is that it is more easily remembered and is more likely to lead to pressing its counterpart key on a calculator in problem solving.

Two New Division Algorithms

MOVE IT Math™ teaches two division algorithms unique to the program: for single-digit divisors and for multiple-digit divisors. They were named by the first group of children who used them: a class of second graders. Both algorithms employ the grouping model and use skip counting (2-4-6-...-20, 3-6-9-...-30, 4-8-12-...-40, ...) to answer the question about how many of one number are in another number. Only Chunk It is taught to mastery and is all that is needed to work the to receive the . Super Chunk It is taught for interest and empowerment, but division by double-digit divisors and up is relegated to the calculator. Long division is demonstrated and presented as another way to divide but is not taught to mastery.

The Division Worksheet

The Division Worksheet is used to introduce Chunk It. The worksheet may be used with any single-digit divisor and any dividend up to 999. To use it, have elementary school students work some division problems with it as shown in the example on the next page for 983÷4. Once they understand what is going on and can divide with the worksheet, they can then see that Chunk IT is just the worksheet without all the circling of 100s, 10s, and 1s. The skip counting in Chunk It effectively does the circling.