## Overview

In addition and subtraction story problems, we aim to ensure that** children develop their understanding of addition and subtraction by making sense of relationships among quantities**. Working with story problems provides students with a context to make sense of these relationships and develop an understanding of the operations.

In this series, we:

- discuss the story problem situations that students learn in grades K-1
- provide a learning progression that supports students to develop a strong conceptual understanding of the addition and subtraction operations
- model ways to represent the story problem situations through boardwork
- identify and address challenges and misconceptions that students often experience

In part 1, we introduce the four problem situations and describe what students might understand about addition and subtraction from the types of problems learned in Kindergarten.

In parts 2 and 3 of this series, we discuss how student understanding of the operations is extended through solving for more challenging positions of unknown quantities in Grade 1.

**The Four Problem Situations**

As described in the CCSS, there are four problem situations students learn in grades K-1 that inform their understanding of addition and subtraction.

The four situations are:

- Add-to
- Take-from
- Put-together/take-apart**
- Compare

** We refer to put-together and take-apart together, for reasons we will discuss later on.

Familiarity with each of the problem situations allows teachers to intentionally sequence the problems they present to students rather than randomly presenting students with the problems.

The table below illustrates the four problem situations using typical problems that students are presented with in Kindergarten.

Reproduced with permission from Tokyo Shoseki publishing company, Japan. Mathematics International Grade 1, pp. 44, 42, 52, 55

Although the story details may differ for the problems that you utilize in your classroom, each story problem can be categorized into one of four story situations (add-to, take-from, put-together/take-apart, or compare) based on what is happening in the story problem, or the relationship among the quantities. For example, the take-from situation is characterized by removing some part of an initial quantity, resulting in a new quantity. This relationship can happen in many different contexts. In the table above, we see this situation happening with a child removing fish from a tank. However, this relationship among quantities (ie situation) can happen in infinite other contexts — e.g. money that is spent, cookies that are eaten, etc.

Each story situation involves multiple quantities, one of which is unknown. Whether addition or subtraction is used to solve a problem depends on which of those quantities is unknown. For example, consider the add-to problem in the table above.

Because we are solving for the total amount of fish that results from the 3 fish being added to the initial 5 fish, one way of solving is to use addition (5 + 3 = 8). On the other hand, if we did not know how many fish we had to begin with (but knew that 3 were added to the tank, resulting in 8 fish total) we would need to use subtraction to solve (8 – 3 = 5). Thus, it is not the story situation that determines which operation is used to solve. Instead, this is determined by which quantity is unknown and what the relationship is among quantities in the problem.

By the end of grade 1, students will need to have learned how to solve for all unknown quantities in all situations. A table with all these types of problems can be viewed in the 2013 California Math Framework for Kindergarten (p. 67) and 1st grade (pp. 92-94).

It’s important to note that while students explore the relationships among quantities in story problems, they should be building fluency of composing and decomposing numbers within ten at the same time.

You can read more about building fluency and find suggestions for how to do so in our resource on early number sense, <Eventually link resource on early number sense here>. As students’ fluency develops, they should be encouraged to apply these skills in solving story problems. After all — if they always count to solve problems, they will not necessarily see the value of the operations.

As we take a look at each situation below, we will name the quantities and discuss why certain ones are easier to solve for, and why. This is key to consider when planning or studying your students’ learning progression. Knowing what is new and difficult about each problem situation (and solving for certain quantities within each situation) is key to supporting student understanding of addition and subtraction.

In the remaining section of this resource we will discuss types of problems that typically inform students’ initial understanding of addition and subtraction in Kindergarten: add-to and take-from with the result (the sum or difference) unknown and put-together/take-apart.

**Students’ Initial Understanding of the Operations**

Some situations are easier for young students to understand than others. Furthermore, within each situation some missing quantities are more straightforward for students to solve for than others. Here, we review the problem situations students first learn about.

**Add-to Situation**

Below is a typical add-to problem that students learn to solve early on:

Reproduced with permission from Tokyo Shoseki publishing company, Japan. Mathematics International Grade 1, p. 44

In an add-to situation, there is a sequence of events. There is an initial quantity, which the CCSS labels as ** start** (in the example above, 5 goldfish). Then, some action occurs which leads to another quantity being added to the initial quantity. This second quantity is called

**(in this case, the change is the 3 goldfish that are added). The addition of the second quantity creates the total amount called**

*change***(in our example, we solve to discover that it is 8 total goldfish). Thus, in an add-to situation, start is increased by change to create result. The result is the easiest quantity to solve for in the add-to situation, making this type of problem the typical entry point for addition story problems. (In part 3 of this series, we discuss solving for unknown start and change quantities).**

*result*Below, we represent the add-to with result unknown problem as a diagram. We don’t recommend introducing this as a model to students, but it be may be helpful as teachers to see the problem type represented visually.

In many Japanese curricula, initial problems in each situation are illustrated and represented with math blocks, as seen in the textbook image above. This helps students understand that objects in story problems can be represented more abstractly as blocks. Additionally, the action associated with each situation is often emphasized (in the image above, we see how a hand is pushing three blocks, representing the change, towards a group of 5 blocks, representing the start). Showing the action in story problems helps students generalize what is happening in each situation. This can be captured in boardwork — after students show their classmates how they pushed one group towards the other to solve the problem, you can represent their strategy as we show below.

The linear nature of the math blocks is also a strategic choice of the curriculum writers — this subtly prepares students to use a powerful modeling tool later on, the tape diagram. We write about this modeling tool in depth in our resource, <Modeling Addition and Subtraction Situations>.

**Put-together Situation**

**Put-together Situation**

The other situation that often informs students’ initial understandings of addition is put-together. An example of this follows.

Reproduced with permission from Tokyo Shoseki publishing company, Japan. Mathematics International Grade 1, p. 42

In a put-together situation, there are two separate quantities that are both referred to as ** addends** in the CCSS (in our example, 3 goldfish and 2 goldfish). Then, when these two quantities are considered altogether, they become a new, single quantity, called

**(in this case, 5 goldfish). Thus, in a put-together situation, we are changing the way we see those two quantities.**

*total*Below, we represent this situation visually (again, this is to better understand the situation as teachers, rather than to use as an instructional model with students).

In the curriculum example above we see the put-together action associated with the story situation shown by two hands pushing the addends towards each other. This can be incorporated into boardwork as shown below:

It’s important to note that action may be emphasized in initial put-together problems to help students generalize and recognize the situation, but that this situation is technically considered “static”. By this, we mean that all quantities are present at the same time — the total can be simultaneously considered as the two addends (e.g. the girl’s goldfish and the boy’s goldfish), or as its own quantity (all the goldfish). Consequently, this situation is more abstract than the add-to situation. The add-to situation can be thought of as changing one quantity so that it becomes another. (In the add-to situation example, we added 3 goldfish to the initial 5, making it 8 goldfish).

### Take-from Situation

Most curricula introduce subtraction in a take-from (or take-away) situation. A typical problem may resemble the following:

Reproduced with permission from Tokyo Shoseki publishing company, Japan. Mathematics International Grade 1, p. 52

The take-from situation is very similar to the add-to situation — they both consist of a sequence of events involving an action. Take-from problems begin with an initial quantity that is also labelled as ** start** in the CCSS (in the example above, the start is the original 5 goldfish). However, instead of change being an additional amount added to it, it is some amount removed from it (in this case, 2 goldfish). Finally, this removal of

**creates the new quantity, which is called**

*change***(in our example, this is the quantity we solve for in the problem above, discovering it to be 3 goldfish). Similar to the add-to situation, students almost always first experience this situation by solving for an unknown result.**

*result*This problem type (take-from with result unknown) is represented in the model below.

In the curriculum example above, the action associated with take-from situations is the change being pushed away from the result. This can be shown on the board as the following:

### Take-apart Situation

Parallel to the put-together situation is another subtraction situation students typically encounter in their initial study of the operation: the take-apart situation. In the put-together situation we solved for the unknown total, but in the take-apart situation we solve for an unknown addend. We show this in an example below.

Reproduced with permission from Tokyo Shoseki publishing company, Japan. Mathematics International Grade 1, p. 55

As we see in the example problem above, in the take-apart situation we solve for a missing addend (the number of white rabbits) given the total (8 rabbits) and the other addend (5 grey rabbits).

You may have noticed that the action associated with this situation (a hand pushing away one quantity) is identical to what is shown for the take-from situation. No quantity is actually removed in this situation, but it can help students to visualize taking away the known addend (the white rabbits) from the total (8 rabbits in all) to find the missing addend (the number of grey rabbits). This helps students reason that the operation to find the missing part is subtraction. We see how this can be shown on the board below.

It may be helpful to take a closer look at the relationship between put-together and take-apart situations. We compare them below.

In the Cognitively Guided Instruction (CGI) research, both put-together and take-apart situations are considered as part-part-whole situations. Put-together situations are part-part-whole situations when the whole is unknown, and take-apart situations are those in which a part is unknown. It can be useful to think of these situations together as part-part-whole situations, because it helps us see that addition is called for when the whole is unknown, and subtraction is called for when a part is unknown.

Like the put-together situation, the take-apart situation is considered “static” because nothing from the total is actually removed (although, as we noted earlier, you may pretend to remove an addend to solve the problem).

**Discussion Questions**

After reading this resource, it may be helpful to discuss the following questions as a team.

- What ideas about the teaching-learning of addition and subtraction discussed in this essay strike you as important in your setting?
- How is addition and subtraction introduced in your curricula? What do you like about it and what do you see as opportunity for improvement?
- What experiences have informed your students’ understanding of addition and subtraction? What are they successful with? What are they challenged by?
- What do you want to learn more about?

### Next Steps

This concludes part 1 of our series on addition and subtraction story problems. We recommend next reading part 2 of the series to learn how students extend their understandings of the operations by learning the compare situation.