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Problem Solving: 11 Types of Addition and Subtraction Problems

Getting students ready for the challenges and adventures of the 21st century is critical to their academic success. While we remember math vocabulary such as addition/subtraction and sum/difference, today’s students must know more than just vocabulary and keywords. Today's students must know how to think through a problem in order to figure out what is going on so they can decide the best way to solve it. It’s our job as educators to make sure our students have the resources in their learning toolbox to conquer these problem-solving skills. 

Help your students master problem solving by teaching them the 11 types of addition and subtraction problems and strategies and techniques for solving them.

When to Teach Problem Solving Skills

Real-life problem-solving skills are the keys to success for our students. But real-life problem-solving skills start in the classroom.  And . . . it's not something that we should wait in teaching.  Waiting until students have developed a solid number sense foundation is only hurting our students.  Instead, we need to have a mind shift as teachers.  We need to realize that learning how to problem-solve with numbers only helps that number sense foundation.
Teacher helping primary student with problem solving activity

Word problems and story problems have been around for decades.  I'm sure we've all seen a meme or two about some of the silly word problems that are out there.  While not all word problems are entirely realistic, the idea is a good one.  Help our students connect different ways to use numbers to real life by using things they already know about.  How boring would it be if we only taught math using math vocabulary?  Instead, we can make math concepts come alive, and show off how valuable they are, by connecting them with real life.  

It’s time to embrace the foundation of problem-solving strategies in the context of learning new math skills. I believe that even before a student can read, they must be exposed to problem-solving situations using math. Helping students develop problem-solving strategies they can refer back to as they continue to learn math skills will greatly improve their overall success in the future.  

Let's Take a Problem Solving Journey

If you are with me so far, then buckle your seatbelt and get ready.  I'm excited to dive deep into the world of problem-solving over the next few articles on this site.  If you stick with me you can expect to learn about
 the 11 types of addition and subtraction problems we should be teaching our students when it comes to problem-solving, teaching tips and ideas for teaching each of these problem types, and examples & resources you can use in your classroom.

In fact, it's going to be your own mini professional development on problem-solving.  I'm going to share the same information here that I would if I were presenting a problem-solving continuing education course for you and your fellow teachers.

Did you know there are 11 types of addition and subtraction problems?

If you answered that question 'no' then you are probably in the majority.  As I have been working on this series of articles I've been talking to teachers, lots of teachers.  And while every.single.one is familiar with problem-solving the vast majority did not realize that you could break addition and subtraction problem-solving problems into 11 types.  

So by now, you are probably wondering, "What are they?"  Let's jump right in!

11 Types of Addition and Subtraction Problems

Here are the 11 types of addition and subtraction problems divided into 4 groups:

Joining Problem Solving Group

  1. Sum Unknown
  2. Change Unknown
  3. Start Unknown

Separating Problem Solving Group

  1. Difference Unknown
  2. Change Unknown
  3. Start Unknown

Part - Part - Whole Problem Solving Group

  1. Whole Unknown
  2. Part Unknown

Comparing Problem Solving Group

  1. Difference Unknown
  2. Larger Part Unknown
  3. Smaller Part Unknown

I hope that as you see these broken down you realize that you are probably already teaching many, if not most of these problem-solving types to your students.  However, as teachers, these different problems often get lumped together into general and broad categories like addition problems and subtraction problems.  While this is absolutely a true classification, it does not differentiate these problems based on their differences.  And they each do have differences.  

These 11 problem solving types come from the First Grade Common Core Math Standard 1.OA.1 Which is: 
Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number.
Through this problem-solving series, we are going to break down the 1.OA.1 math standard and fill your teacher toolbox with everything you need to teach this standard to its fullest.  By breaking down these problem-solving types and teaching our students skills to recognize them we are strengthening their overall problem-solving abilities.

help students master problem solving with counters or manipulatives

Where should you begin?

Aside from understanding the different types of problems (which we are working on doing right now) the next beginning point is familiarizing yourself with strategies and techniques you can teach to your students.

There are a variety of different problem-solving strategies that you can teach your students.  The great thing about teaching them different strategies is that what clicks for one student might not for another.  By filling their toolbox with a variety of strategies, each student can find one or more that really works for them.  

Aside from the many strategies that students use to solve problems, there are also many ways to represent a problem before solving.  Through pictures and other types of models, our young learners begin to understand what is actually going on in the problem.  It is especially important to label the models.  

One of my favorite models that works great for visual learners is using a bar model (sometimes called a strip diagram or a tape diagram).  Using bar models students draw a visual representation of each part of the problem using rectangles called bars.  Not only does this provide a visual for the students but it makes the drawing quick and easy.  Pretty sure we've all had the students who want to draw a picture of the problem and it turns into the next Mona Lisa!  

I've put together this FREE resource for you that covers the 11 different bar models to work with each of the 11 problem types.  Just click the image below to grab your free problem-solving resource and start filling your teacher toolbox!


What's Next?

In the next article we are going to jump into the Joining Problem Solving Group.  We will dig into the three types of joining problems and how you can best teach them to your students.  If you do not receive emails from this site, you can sign-up here so that you don't miss the rest of the problem-solving articles when they are published.

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