# 4th Grade Math: Intro to Division- Different, but the Same

Dear Families,

Last week, the fourth graders began a problem about water and juice machines. The water machine holds 156 bottles and the waters are packed in 6-packs. The question was, “how many packs fit in the machine?” The juice machine holds 156 cans and there are 6 flavors in the machine. The question was, “how many of each flavor?”

Some students noticed that these are really division problems, and can be represented by the expression 156 divided by 6. While many did not realize it was division, they were still able to solve it, since multiplication and division are inverse operations. It was great to see the variety of multiplication strategies that students used. We saw ratio tables using double or multiplying by ten; some students used landmarks such as multiplying by 10 or 20; while others broke up the 156 into parts that were divisible by 6. See below for some of the student work.

An interesting thing that came up in the discussion, however, was how the questions were actually different, even if they can both be represented by 156 divided by 6. That’s what makes division challenging. The 6 can mean two different things. In the water problem, the 6 is how many in a group and we need to find how many groups. This is called quotative division. The juice machine provides the number of groups and you need to find how many in each group. This is called partitive division. Don’t worry. The kids don’t need to know those names, but some of them liked using them anyway!

We will be continuing division in the upcoming weeks. Enjoy the student work.

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