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Problem Solving: Separating Problem Group

I hope you are enjoying this series on the 11 different types of addition and subtraction problem solving.  Today we are jumping into the second group of problems, the separating problems.  If you are looking specifically for subtraction problem solving information, then absolutely keep reading because you are in the right place.  But . . . if you have time I'd highly recommend that you start with the overview of the 11 problem solving types and how I teach problem solving & the joining problem solving group.  These first two posts of this series lay the foundation for the remaining posts. And . . .  you don't want to miss the free resources that are waiting for you in each post.  

3 of the 11 types of problem solving problems are separating problems for subtraction

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

  1. The 11 Types of Addition and Subtraction Problems
  2. How I Teach Problem Solving & The Joining Problems

The Separating Problems

subtraction problem solving is the separating group

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 last post.  Separating problems are used to teach subtraction.

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


There are three distinct types of separating problems:

  1. Difference Unknown
  2. Change Unknown
  3. 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 subtraction concepts.

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 subtraction 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:
  1. students gaining an understanding of the concept of separating
  2. learning separating and subtraction related vocabulary 
  3. converting to simple subtraction equations
Let's take a look at another example:


While to us as teachers, this seems like a simple concept, for our students this is brand new.  Fight the urge to rush through this most basic problem solving type until you are confident that you 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). 


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.  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 of 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 subtraction and 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 the separating problems is where the start is unknown.  One of my favorite ways to introduce this problem type is with an excited declaration "It's Backwards Day!"  Sometimes my students look at me like I'm crazy 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 Backward's Day?  One of the easiest way to solve these problems is by working backwards.  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 backwards.  

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 backwards 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 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.  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.

separating problems with the start unknown is the most difficult of the three separating problem types

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

Here's a couple more suggestions when teaching separating with the start unknown:
 
  1. If your students are familiar with fact families, make the connection.  For many students fact families help this problem solving type click!
  2. Teach the strategy that working backwards is the opposite - so backwards on a separating problem means add the two groups together.

Practice, Practice, Practice

subtraction problems fall under the separating group of problem solving.  Teach students how to think though these problems in order to build a solid foundation.
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 with pictures.  

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.
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!

Mega Math Practice is a resource filled with lots of no prep problem solving practice.  Students are guided through multiple strategies and work on solving the 11 types of addition and subtraction problems.



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Learn about the different types of addition and subtraction problem solving types.   This article focuses on the separating problems for subtraction.  Learn about each type and how you can teach it in your classroom.

Problem Solving: Joining Problems


Welcome back for Part 2 of our Problem Solving series.  In the first post we focused on the importance of problem solving and identifying the 11 different type of addition and subtraction problems.  If you missed that post, I highly recommend that you jump over there and read it because it is the foundation for all the other posts.  It's a quick read and you'll be back before you know it.  


How I Teach Problem Solving

As with all math concepts, I am very strategic in how I teach problem solving.  I always begin new concepts in a concrete way.  The math concepts can be very abstract so giving our students something real they can count, see, touch and move, really helps to make those abstract concepts make sense in their little brains.  As students develop an understanding, we gradually move into a more abstract approach.  Here's the progression I use to teach starting with the most concrete and moving more abstract.

Joining Problem Solving problems teach students to think through addition problems.
  1. Hands-On Manipulatives
  2. Pictures
  3. Number Representations (ten frames, tally marks)
  4. Bar Models
  5. Numbers
It's really important to move through these different steps because each creates a mental building block for the next.  If there's a block missing, then students often struggle.  

When working with struggling students I suggest starting from the beginning to get an idea of what the students understands and where the disconnect might be happening.  This could be as quick as a couple of hands-on problems and then moving to drawing or using pictures.  This quick informal assessment will help you identify where a student needs to begin in their focused work.

Incorporating Small Group Instruction

It is important for students to see addition problems in different ways
In my classroom small group instruction is a very big part of our math block.  While I will introduce and review concepts as a class, the bulk of the direct instruction happens in small groups.  This allows me to meet each student at their point of learning and move them forward.  

One great way to form your groups is based on where they are in the progression from concrete to abstract.  At any give time, I may have a small group working on joining problems using manipulatives while another group is working on number representations and yet another using bar models.  

Practice, Practice, Practice

Students needs lots of practice opportunities when learning a new skill or concept.  We learn and practice together, we learn and practice in small groups, and their math centers are filled with more practice opportunities.  

The science of learning is pretty fascinating.  One study I read said that it takes 17 exposures to a new word, skill or concept to learn it.  But then it went on to say that at least 30 exposures were needed in order to recall it.  And,  to truly develop a new skill takes any where from 6 months to years!  Don't underestimate the power of repeated practice for our students.

give students a variety of problem types in order to help them master problem solving and addition concepts

The Joining Problems

Let's jump right in to the first problem solving group - the Joining Problems.  Joining problems are used when teaching addition.  They are the most basic of all the problem solving types are generally introduced in kindergarten.  However, students will continue to develop their understanding of these problems through first and second grade too.  

It's important that even at this young age, our students begin to develop a strong foundation and understanding of joining. This foundation must include different ways that joining problems can present themselves, vocabulary, and conceptualizing the joining action.

Three Types of Joining Problems

There are three distinct types of joining problems:
  1. Sum Unknown
  2. Change Unknown
  3. Start Unknown
In this article we are going to dig deep into each of these addition problem types, what makes them unique and how you can teach them to your students.

1.  Joining Problems with the Sum Unknown

When you think of basic addition problems this is what you think of, problems where the sum (the answer to an addition problem) is the unknown factor.  This is the starting place for teaching addition and until students understand this most basic problem type it will be difficult to move on to the next two.

Helps understand the concepts of addition with joining problems

Here's an example of a joining problem with the sum unknown and how I would teach it in my classroom.


As you can see this is the most basic type of joining problem.  My focus when I'm teaching this is to make sure that students develop an understanding of the action of joining.  I want them to see the joining happening in their minds and then transfer that to their paper.  This visualization is really important to understanding what is happening.  It's the same visualization skill that also helps students with reading comprehension.
Problem solving with addition begins by understanding the concepts of joining

Math Talks are a wonderful way help students build this understanding.  Over the years I've learned that rushing to get to numbers and equations usually only leads to confused students.  Take the time to help them understand what is happening and you've given them a foundation to build on.

Some ways that you can help students visualize the joining is by:
  • modeling it with actual items and physically moving them together
  • using students as the manipulatives 
  • asking students to model the joining action using their fingers (Yes! fingers are good for our young learners to use!)
Once students have a good understanding of the action taking place in these joining problems then they are ready to move on to the next two problem types.

2. Joining Problems with the Change Unknown

In this type of joining problem, students know what they are starting with and what they are ending with but they are not told about the change that happens in the middle.  Students must understand that another group joins the first group - that is the change.  {See why it is important to spend time on the first type before moving on}

Students should learn how to solve addition problems with different parts missing

Let's jump into an example problem to see how I teach this to my students.


With this problem type, students must determine the change.  There are many different ways students can see this.  One of my favorite things to do in my classroom is to celebrate all the different ways.  It's important for students to know that in math there is often more than one way to get to the correct answer.  By showing students all the ways, you increase the likelihood that each and every student will find at least one way that makes sense to them.
addition problems with the change unknown help students develop a deeper understanding of addition concepts

Here's some of the ways students can solve this type of joining problem:
  1. Counting On
  2. Counting Backwards
  3. Using Learned Math Facts
All of these will get the student to the right place.  By giving students the freedom to choose the method that works best for them we set them up for success.  BUT . . . I don't just let my students stop at their method of choice.  I encourage them to learn and practice the other methods too.  As the numbers get more complex, students might find that another method becomes easier for them to use.  

3. Joining Problems with the Start Unknown

Similar to the second joining problem, joining problems with the start unknown are a higher level joining type of problem.  The problem provides students with the change that happens and the sum at the end but doesn't tell them how the story starts.  I like to tell my students it is like reading a book but starting in the middle.  When you read from the middle to the end you are left wondering what happened at the beginning!  That summarizes this problem-solving type best.


Here's an example of a joining problem with the start unknown and how I model it to my students.


Again, you can see the importance of students understanding the joining process in order to figure out what happens.  The thinking process behind this type of problem is very similar to the last one.  Therefore, it can really help students to make that mental connection.

As students get into second grade and are learning about the commutative property of addition, it becomes something that is very easy to pick up and understand when they have already been exposed to these different types of joining problems.  Similarly, as younger students are learning about fact families, they can make the connection to these types of problems.

Variety Leads to Victory

When it comes to teaching the joining problems it is important to make sure that you expose students to all three problem types.  It's also important to understand that this doesn't happen in one lesson.  In fact, the process of getting through all three problem types could easily take weeks.

the three types of joining problems to help students master addition problem solving
Don't just teach each problem type but give students lots and lots of opportunities to practice each problem type.  And make sure they are practicing in a variety of ways too.  My go-to resource for practice activities is Mega Math Practice!  

Sure, coming up with additional story problems isn't crazy difficult.  And as a primary teacher I know that you are more than capable of creating some quality problems.  But I also know that you are busy - really busy.  That there are more pulls on your time and mental energy than ever before.  That's why I put all my problems together into one ready-to-use resource for you.  

These are my go-to practice activities for my students.  There are practice problems that start with pictures and move through to more abstract problem-solving strategies.  They can be easily added to a math center, or used in a daily problem-solving journal.  I also like to keep them handy so that I have a variety of problems ready to use during my small group teaching. 

I've put together an exclusive freebie that you can only get here so that you can try out these Mega Math Practice Problem Solving Practice Activities!  Click the image below to grab your free Joining Type Problem Solving practice.


You can also find a full set for each of the Joining Problem Types in my store at Teachers Pay Teachers.  Use the links below to see each of these no-prep problem-solving practice sets.
And when you are ready to make problem-solving a priority in your classroom all year long - I've got you covered with the Mega Math Practice Problem Solving Bundle.  This bundle includes everything you need to help your students practice all eleven types of problem-solving problems.  If the thought of never having to make up your own story problem again makes you smile then this bundle is for you! 

We've Only Just Begun!

There are still three more problem-solving articles headed your way.  Make sure to sign-up for the email notifications so you will know when each one is published.  You can sign-up here and I'll pop into your inbox with each new article.

Save these Addition Problem Solving Ideas

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