Word problems can be such a struggle. Especially in grades 3rd and up when students have to work through all of the operations and multi-step word problems. Students often struggle to identify the operation presented, or miss part of the steps needed to solve. Explicitly teaching word problem types can help our students recognize the operations in context. The Common Core State Standards lay out the different multiplication & division word problem types that students should know. By teaching them explicitly, we help students understand multiplication and division in context to help them truly understand the operations. We connect how the math they’re learning is relevant to their real life. By focusing on the word problems and their types, we’re focusing on what makes multiplication and division what they are. An important note before we begin-throughout our exploration of both operations, we often return back to the total. Throughout our work, with arrays, area, and equal groups, we continue to come back to the concept of total to help students identify the operation needed. It is through identifying the total as known or unknown that students will truly understand the multiplication and division operations. As you will see, that’s a focus with each of the multiplication and division word problem types.

Equal Groups as Multiplication & Division

It’s crucial for students to understand that multiplication and division works with equal groups so it makes sense to begin here. When we first start working with multiplication and division word problems, I ask students to identify the items that are arranged in groups, and identify what that group looks like. For example, the group might be a box or a bag. Those objects could also just be objects in other scenarios. So, identifying that an object is grouped within something helps students to understand the word problem and that an equal group scenario is present.

MULTIPLICATION – Total Unknown

These are the word problems we typically think of with multiplication. It’s an arrangement of items into equal groups to find the total number of items. Identifying that the number of groups and the number of items into each group are known, while the total is unknown, helps students solve these word problems.

QUOTATIVE DIVISION – # of Groups Unknown

When the number of groups are unknown, it is a quotative division problem. Let me clear, I don’t teach my students the term quotative. Rather, I teach them that there are two types of division. When students are able to identify that the total number of objects is known, but the number of groups is unknown, they then know that they are dividing. It’s important that students have lots of practice with both types of division.

PARTITIVE DIVISION – # in each Group Unknown

When a word problem identifies the total number of items and how the items are arranged, but doesn’t give the number in each group, these are partitive division. Think of it as partitioning. The number of groups is known so the partitions are known. Again, I don’t teach this term with my students, but it’s important that students know that division is presented in two ways.

Arrays as Multiplication & Division

In conjunction with teaching equal group arrangements described above, arrays should also be included. Arrays are how students are first introduced to multiplication in 2nd grade. Students should have background knowledge with rows and columns vocabulary. Arrays are how I connect what we’re learning now, with what they learned before. Therefore, I teach arrays at the same time that I teach equal group arrangements. I correlate rows and the number of groups. I always emphasize that the columns are the same as the number in each row (or the group).

It’s important to note that the CCSS lay out two rigors of array word problems: ones where the items in rows are presented, and those that give the word problem in rows and columns. Through continued exposure and practice, students understand that these are interchangeable. Array word problems are not very common, but are definitely a problem type that students should have exposure and practice with. They also lay the foundation for students to work with our next word problem type: area.