Differentiated Math Fact Practice

Making Ten

4th Grade Resources

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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