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Making Sense of Joining and Separating
Addition and subtraction story problems are not all the same. Even when two problems use the same numbers, the story can be very different depending on what is happening. In this topic, we look at four major types of addition and subtraction situations: join, separate, part-part-whole, and compare problems.
Join problems involve situations where something is added to a group. Separate problems involve situations where something is taken away from a group. However, it is important to notice that join problems are not always solved by addition, and separate problems are not always solved by subtraction. The operation depends on what information is known and what information is missing.
We will also study part-part-whole problems and compare problems. Part-part-whole problems describe two smaller groups that make up a larger group, such as red apples and green apples in one basket. Compare problems describe the relationship between two quantities, such as how many more or how many fewer. Unlike join and separate problems, these situations usually do not involve action. Instead, they describe a relationship between groups.
As you work through this topic, focus on understanding the structure of the story, not just looking for keywords. Future teachers need to help students think about what is happening in the problem, what is known, what is unknown, and which model or strategy makes sense.
Student Learning Goals
By the end of this topic, students should be able to:
Identify join, separate, part-part-whole, and compare problems.
Explain why keywords are not a reliable way to choose an operation.
Determine what information is known and what information is unknown.
Represent story problems using drawings, objects, equations, or diagrams.
Explain why a join problem may sometimes require subtraction.
Explain why a separate problem may sometimes require addition.
Describe how students might directly model different types of story problems.
Key Vocabulary
Join Problem - A story problem where something is added to a group.
Separate Problem - A story problem where something is removed from a group.
Part-Part-Whole Problem - A story problem with two parts that make one whole group.
Compare Problem - A story problem where two quantities are compared.
Unknown - The missing amount in the problem.
Direct Modeling - Acting out or drawing the problem based on the story.
Common Student Misunderstandings
If a student is looking only for keywords like “altogether” or “left,” then the student may be choosing operations without understanding the situation.
If a student assumes that all join problems use addition, then the student may not be thinking about which amount is missing.
If a student Assumes all separate problems use subtraction, then the student may not realize the starting amount could be unknown.
If a student confuses compare problems with take-away problems, then the student may need help seeing comparison as a relationship, not an action.
If a student is writing an equation before understanding the story, then the student may need to model the situation first.
Example Problem Types
Join - Aisha had 4 stickers. Her friend gave her 3 more. How many does she have now?
Separate - Milton had 7 stickers. He gave away 3. How many does he have left?
Part-Part-Whole - Franklin has 4 star stickers and 3 heart stickers. How many stickers does he have?
Compare - Amir has 7 stickers. Inaaya has 4 stickers. How many more stickers does Amir have?
Teacher Connection
This topic is especially important for future teachers because children often solve story problems by acting them out, drawing pictures, or using objects before they are ready for formal equations. A teacher’s job is to help students make sense of the story structure.
Instead of asking, “Do you add or subtract?” it can be more helpful to ask:
What is happening in the story?
What do we know?
What are we trying to find?
Could we act this out or draw it?
These questions help students build meaning before choosing an operation.
Quick Reflection Question
Why might the problem “Kadie had some stickers. Her friend gave her 3 more. Now she has 7. How many did she start with?” be a join problem even though it is solved with subtraction?