Addition & Subtraction Problem Structures (Word Problems)

A common classification scheme of addition and subtraction word problems identifies four broad categories based on the type of action or relationship in the problems: join, separate, part-part-whole, and compare. 

1. Join/Separate

Join problems and separate problems involve actions that increase or decrease a quantity, respectively. In both categories, the change occurs over time. There is an initial quantity that is changed either by adding something to it or by removing something from it, resulting in a larger or smaller final quantity. There are three types of join/separate problems: Result Unknown, Change Unknown and Initial Quantity Unknown.

Result unknown problems are generally easiest for students. Change unknown and initial quantity unknown problems are more challenging. One reason is that these problems are often presented in language that suggests one action (e.g., separating) but require using the opposite action (e.g., joining) to find the answer. For instance, the separate - initial quantity unknown problem above asks Rodney had some cookies. He ate three cookies. Now he has seven cookies left. How many cookies did Rodney have to start with? There is a separating action in the problem, but a child can solve it using addition.

- 3 = 7 becomes 7 - 3 = 

2. Part-Part-Whole

Part-part-whole problems do not use action verbs—action neither occurs nor is implied. Instead, relationships between a particular whole and its two separate parts are established. There are two common types of part-part-whole problems and a third less common one. They are: Whole Unknown, One Part Unknown and Both Parts Unknown

Both parts unknown is less common as they generally lead to multiple solution possibilities. Sometimes teachers will use these problems to test a students ability to determine all possible answers. 

Part-part-whole problems involve a comparison of “parts” (subsets) with the “whole” (set).

3. Compare

Compare problems involve a comparison of two distinct, unconnected sets. Like part-part-whole problems, compare problems do not involve action. However, they differ from part-part-whole problems in that the relationship is not between sets and subsets but between two distinct sets. There are three types of compare problems, depending on which quantity is unknown: Difference Unknown (the quantity by which the larger set exceeds the smaller set), Larger Quantity Unknown and Smaller Quantity Unknown. 

Math Matters: Understanding The Math You Teach

Math Coach Corner: Problem Structures for Addition & Subtraction

Tales From Outside The Classroom: Addition & Subtraction Word Problem Types