This is the third post of the problem solving series. Want to start at the beginning?

## The Separating Problems

Today we are going to dive right into the second group of problem-solving problems - the separating problems. Separating problems are the opposite of the joining problems that we discussed in the last post. In separating problems, there is an action--movement--taking away.

While this concept is generally introduced in kindergarten, it becomes a focus concept in first and second grade. By the time students leave second grade, they should have a very solid understanding of subtraction concepts both in equation and problem solving forms.

## Three Types of Separating Problems

- Difference Unknown
- Change Unknown
- Start Unknown

In this article, we are going to dig into each of these subtraction problem types and how you can teach them to your students.

## 1. Separating Problems with the Difference Unknown

When you think of traditional subtraction problems - this is it. You have the starting number and the change, but the goal is to determine the amount that remains or the difference (the answer to a subtraction problem). This type of problem is the most basic of all subtraction problems and should be your starting place when introducing the concept of separating.

As with any new concept, remember that it is important to start in the most concrete way possible. For young students, this means touching physical objects and completing the action of separating some from the group. It is this hands-on connection that helps students begin to internalize the concept of separating.

This is how I would introduce separating to my students. If you have a group set of manipulatives (they don't all have to be the same). I love to begin by having students build the starting set. At their desk on the carpet in front of them have students build the starting group for a problem.

I like to do this by reading the problem in parts. Using the image above, I would say "There are 7 slugs." (You can change this to match your manipulatives). Let's build a group of 7 to represent the 7 slugs." Then I'd give the students time to build. Then I would say something like "2 slugs crawled away. Let's make those slugs crawl away." I'd model taking 2 manipulatives and moving them away from the group, and then I'd have my students do the same. Then I'd finish by saying "How many slugs are left?" We would count the remaining manipulatives and conclude that there were 5 slugs left.

Next, I'd make the connection from what we did to the math concept of subtraction. I'd also show them how to write this problem using numbers instead of words and write the equation 7-2=5 on the board.

My goal during these introductory lessons would be:

- Students gaining an understanding of the concept of separating;
- Learning separating and subtraction related vocabulary;
- Converting to simple subtraction equations;

Let's take a look at another example:

This may seem like a very simple concept to us, but for our students, this is brand new. Fight the urge to rush through this most basic problem solving type until you are confident that your students have it! Trust me, when you move on to more complicated separating problems, you will be glad your students have a solid understanding of the basic problems.

## 2. Separating Problems with the Change Unknown

Once students are ready to move on, the next type of problem you want to introduce is separating problems with the change unknown. In these problems, you know the starting group and you know the ending group, but you don't know what happened (the change). Remember in joining and separating problems there is an action.

Again, I like to start with hands-on learning opportunities. With our manipulatives in hand I would begin reading the problem. "Clarence saw 8 butterflies. Let's build the group of butterflies." Then I wait for students to physically build this group. Next, I say "Some butterflies flew away (the action). What should we do now?" Usually, the students look at me a little confused because no number was provided. This is a good point to stop and have a little conversation and get the students to identify that while some butterflies flew away we don't know how many. There actually isn't anything we can do now. Then I read the final part of the problem, "there were 5 butterflies left." We talk about the fact that we know how many in the starting group are still there. This means they didn't fly away. So we count out 5 and move them over a little so that we can see two groups. As students do that, there are usually hands that start shooting up around the room. They already know that 3 flew away without me even asking the question. Why? Because they learned the concept of separating with the first problem type.

Here's another separating problem with the change unknown example. In this video, I show you some other strategies that I use with my students.

## 3. Separating Problem with the Start Unknown

The last and final type of separating problems is where the start is unknown. One of my favorite ways to introduce this problem type is with an exciting declaration, "It's Backwards Day!" Sometimes my students look at me rather curiously and other times they cheer because they know it's something I'm excited about. Inevitably, a sweet kiddo will raise their hand and ask "What's Backwards Day?"

That question allows me to introduce our next separating problem type where the start is unknown. Why do I introduce this problem type as Backwards Day? One of the easiest ways to solve these problems is by working backward. So that's just what we do.

For this problem-solving type, I read the first part of the problem "Amiyah had some pretzels." And then I make a big deal about how we are stuck. We don't know where to begin. Some is not a number so we don't know what to build. I pause and think . . . then I remember . . . it's Backwards Day! I share my idea that maybe we should work the problem backward.

This time we start again and I read the entire problem to the students. Then we start at the end and build the 8 pretzels that Amiyah had left. Then I go backward and read that Amiyah ate 9 pretzels. Here we stop and chat about what that means. Are these part of the 8 or are they different? We decide that if she ate them then they couldn't be the part that was left. So we build a group of 9 pretzels. Then we have to figure out what to do to find out how many there were at the beginning. Time to talk it through! With some prompting, we can usually make the connection to the other types of separating problems and the realization that we start with only one group. We try putting the two groups together to make one group.

At this point I love to teach my students how to check it to see if it works. So this time we start at the beginning with our new group and I read it like a separating problem with the difference unknown. We work through 17 - 9 and when the final remaining group is 8 (just like the original problem) we know we have successfully done it.

This separating problem type is the most complicated of the three. It generally takes the students longer to catch on to this type than the others. Just know this from the beginning and be prepared with lots of examples and practice opportunities.

Here are more suggestions when teaching separating with the start unknown:

- If your students are familiar with related facts, make the connection. For many students, related facts help this problem-solving type click!
- Teach the strategy this way--working backward is the opposite--so backward on a separating problem means to add the two groups together.

## Practice, Practice, Practice

After introducing and teaching each problem solving type it is crucial that students get lots of opportunities to practice. While we start as a group, much of our instruction time happens in small groups. Then students are sent to centers for those important chances to practice.

I also like to sneak some practice time in through morning work, early finisher activities and homework.

My go-to practice resource for all of the problem solving types is Mega Math Practice. This resource provides students with practice for every problem-solving type and guides them through using a variety of strategies. What I love about this is that each student really gets the opportunity to experiment with the strategies that work best for them. While one student may "get it" with a bar model, another student might have their light bulb moment using pictures or counters as models.

I know just how valuable a tool these practice activities are, which is why I put together a free set for you to use with your students. You can grab the freebie by clicking the image below.

You can find all of the separating problem-type practice sets in my store at Teachers Pay Teachers.

- Separating Problems - Difference Unknown
- Separating Problems - Change Unknown
- Separating Problems - Start Unknown

And if you are ready to have problem-solving practice right at your fingertips then you need to check out the Mega Math Practice Problem Solving Bundle! This bundle provides a plethora of problem-solving practice activities for all 11 problem-solving types. It will keep you covered all-year-long!

## Pin It So You Don't Forget It!

It's so easy to forget where you found something on the internet. Instead of trying to remember (you've got better things to use those brain cells for) pin it to your favorite classroom Pinterest board. Then you will be able to quickly come back when you need teaching tips for problem-solving and more fun math concepts.

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