Also known as "grouping" or "partitioning" problems, the division story problems provided below can be used as a math skills assessment to gain a better understanding of what mathematical knowledge your students already possess, and what strategies they employ when solving problems.

## Assessments for Designing Instruction

One of the major educational goals of Cognitively Guided Instruction is that instruction should be designed to facilitate growth at each child's individual level of understanding. Assessing what prior knowledge your students have about solving division problems is, therefore, an essential component.

In this article, you will find a series of problems that represent Measurement Division and Partitive Division Problems. It is important to note the distinctions between the two, because children employ different strategies to solve them based on the information provided in the problem.

In **Measurement Division** problems, the total number of objects and the number of objects in each group is provided, while the number of groups remains unknown. In **Partitive Division** problems, the exact number of objects and the number of groups is known, while the number of groups is the unknown. These problems types are detailed in the findings of Fennema, Carpenter, et al, and may be used for math skills assessment or as a springboard for instruction.

## Sample Problems

Measurement Division

- Mary had 9 cookies to put on plates for a party. If she puts 3 cookies on each plate, how many plates will she need?
- Alex has 48 gummy worms. He puts 4 in each Halloween treat bag. How many bags did he fill?
- Room 124 is going on a hayride. There are 27 kids in the class. 9 kids can fit on a wagon. How many wagons must they take?
- The hens laid 20 eggs. There are 4 eggs under each hen. How many nests are there in the hen house?

Partitive Division

- Megan has 15 cookies. She puts the cookies into 5 bags, with the same number of cookies in each bag. How many cookies are in each bag?
- Riley has 16 caterpillars. She puts the caterpillars into 4 jars, with the same number of caterpillars in each jar. How many caterpillars are in each jar?
- Mom has 12 marshmallows. She puts the marshmallows on 4 sticks to roast. How many marshmallows are on each stick?
- Mrs. Wistrom has 20 stickers. She gives them to 10 children, so that they each have the same amount. How many stickers did each child get?

## Strategies for Solutions

When students are first beginning to solve Measurement Division problems, they typically use direct modeling as a solution strategy. They will create sets containing the given number until they have used all of the objects to directly represent the action in the problem. Next, they will count how many sets they made to find the solution. This is called the *Measurement strategy*.

Partitive Division problems can also be directly modeled. Instead of determining how many groups there are, the student must find the number of objects in each group. This number is the unknown, and the typical strategy used to find the solution is Dealing* –* the student counts out the total number of objects and then deals them out to each of the stated groups until all of the objects are gone. S/He then counts the number of objects in each group to find the answer. This strategy is called the *Partitive strategy*.

As the students' mathematical understanding progresses, these direct modeling strategies are replaced by counting strategies. For Measurement Division, this typically involves skip counting. Since the number of objects in each group is unknown in Partitive Division problems, the student must use trial and error to determine what number to skip count by.

As is the case with addition, subtraction and multiplication problems, children then progress to using derived and known facts as their standard solution strategy. Using a math skills assessment before beginning instruction will help you determine what level each student is at in their mathematical understanding.

## Related Problem Types

There are other types of problems which utilize division skills. These related problems are basically variations of Multiplication, Measurement Division and Partitive Division problems. They involve:

- rate
- grouping
- partitioning
- price
- Scaler Multiple

By utilizing these and other math skills assessment in the classroom, you will easily determine what knowledge your students possess when they enter the classroom. From there, you may make instructional decisions that will allow each child to grow and succeed.

For more information about using Cognitively Guided Instruction in your classroom, or for math skills assesment samples for other problem types, please refer to other articles in this series.

## This post is part of the series: CGI Math In Your Classroom

- CGI – An Approach to Teaching Mathematics
- Teaching Different Types of Math Story Problems
- Using Cognitively Guided Instruction for Teaching and Assessment
- Math Skills Assessment for Division Problems Using Cognitively Guided Instruction
- Elementary Math – Assessment of Strategies for Solving Math Problems