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 word problem types that students should know. By teaching them explicitly, we help students understand multiplication 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. 

Equal Groups- Total Unknown

It’s crucial for students to understand that multiplication works with equal groups so it makes sense to begin here. When we first start working with multiplication 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. 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.

Array

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.

Area

Area is a critical component of our multiplication work. It’s a major part of 3rd grade math instruction and it’s important that students work with many word problems where they have to recognize area in context and solve. This is especially important to learn how to differentiate between area and perimeter by making connections between arrays and what is known about multiplication. Area should be introduced with arrays and connected to an area model. This concrete, visual example helps students connect what they already know (arrays) with their new learning (area). Unit squares are written in the standards to be this bridge. Area models continue on as you work with more the more complex rectilinear figures, and to help students understand the distributive property of multiplication. The CCSS don’t list area problems as their own line, rather, they’re included with arrays. While this certainly makes sense since arrays can be created inside the rectangle, and that’s how area should be introduced, most area word problems don’t use rows or columns when providing side lengths. I think it’s an important distinction to look at area word problems separate from arrays. Solving area problems again connects back to total and known/unknown values. When students are trying to identify the total area of the shape, and are given both side lengths, students need to multiply to find the area.

Comparison

The most complex multiplication word problem types are comparison word problems, in my opinion. Just as they are in addition & subtraction, the way these problems are structured, cause them to be more complex than the others. For these, it’s important to look at each of the different problem types and digest what’s being given. The multiplication examples have very similar wording since they both have the comparison statement: “6 times as much”. This is a description of the relationship between two groups-the total, and the comparison group.  The key to understanding multiplicative comparison problems is that the comparison is the repeating groups, whereas the constant is the group size. In a multiplication word problem, you are told the amount in the comparison group, 6, and the constant, 4. Each bag of apples is $4, so if it’s 6 times as much, it’s the same as having 6 of the bags. Thus, 4 x 6.