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Problem Solving: Comparing Problems

Welcome back to the final post in this problem solving series.  In this post we are going to talk about the last group of addition and subtraction problem solving problem types - the comparing problems.  Before you jump in, I would suggest that you take a few minutes and start at the beginning.  Each post in this series builds upon the last so, while you can absolutely start here, starting at the beginning lays the foundation.  Get ready to take your problem solving instruction to the next level!

What are Comparing Problems?

teach students a variety of different problem types to help them master problem solving
In math, when we hear the word comparing, we automatically jump to comparing numbers with greater than and less than.  Comparing numbers is the foundational skill to the comparing problem solving type.  While we don't use the > and < symbols or the vocabulary "greater than", "less than" or equal to, the math concept is the same.  

However, one main difference between comparing numbers and the comparing problem solving type is that these problems often go one step further.  Instead of just comparing the numbers, these problems add in an addition or subtraction component and ask "how many more" or "how many fewer".

It's this combination of these two skills that make this the most difficult of all the problem solving types.  

Three Types of Comparing Problems

comparing problems are the fourth problem solving type
There are three different types of comparing problems that we need to introduce our students to:

  1. Comparing when the difference is unknown
  2. Comparing when the smaller part is unknown
  3. Comparing when the larger part is unknown

We are going to dive into each of these problem types and how I'd teach them in my classroom.

Comparing When the Difference is Unknown

Comparing with the difference unknown is the most common and most basic of all of the comparing problems.  These problems often introduce students to two groups and ask how many more or how many fewer one of the groups has than the other.  These problems will have students subtracting or counting up to solve the problem.

As you teach this problem type, it is important to focus on what is happening.  As we've talked about throughout this series, it is important that students see the context of the problem.  One main goal when teaching this problem type is that students see the difference between these and the separating problems.  While both will use subtraction concepts, one involves and action of separation and one is merely a comparison of two groups.

Here's a comparing when the difference is unknown problem and some strategies I'd teach my students to use with it.



As you can see, there are many great tools and strategies students can use to solve these problems.  I love teaching my students ALL the different strategies.  Since each student is unique and students have a variety of learning styles, you never know which strategy will be the one that gives them the "ah-ha" lightbulb moment we all love to see. 

Comparing When a Part is Unknown 

This problem solving type introduces students to one group and then provides a clue to how many more or fewer are in the second group.  With these two problem types, I love to make my students math detectives.  I start by explaining what a detective is.  I teach students that detectives look for clues in order to figure out what is happening. Then I explain that sometimes in math, we have to use clues to answer problems.


As I present students with their first sample problem, I model in my best detective ways how I look for clues.  If you want to really peak student interest, then wear your favorite trench coat and a large magnifying glass as you teach this lesson.  I start by reading the problem aloud and then giving my best thinking pose.  Then I start to read again, and after I read the first part of the problem I give a big "Ah-Ha!  I found a clue!"  I write the first clue on the board and have students help me determine what the clue means.  This first clue is often the statement that there are two groups.  I draw two large circles or lines on the board and we label each one with the group name.

Become math detectives as you search for problem solving clues

Next, I keep reading as I look for another clue.  Once again, I use my best award-winning acting skills to make a big deal about finding a clue.  This next clue is often the known group so we draw it in the correct place so that we connect clue 1 and clue 2.  

Finally we find our last clue.  This time, the clue tells how many more or fewer we will find in the next group.  I start by writing this out with just a couple words, like "Black 2 fewer" or "Green Apples 3 more."  Then I focus in on the important word (more, fewer, less) and ask students to think about when we've heard that before.  There might be a some guesses, but usually within just a minute someone will make the connection to addition or subtraction.  Once that connection is made, we go back and write our clue using the addition and subtraction sign.  It might look like: "Black -2" or "Green Apples +3."  

This process, while it might seem repetitive to us as teachers, is what helps build the bridge in the minds of our students.  And it is this that is so much more important than just teaching keywords.  By going through this process we are really teaching our students how to visualize the problem and identify the context.  This visualization is so much more important than knowing keywords.  When students can see what is happening, they can use their reasoning skills to solve the problem no matter what words are used.  But when students are taught only keywords, they get easily stumped when a problem is worded in a different way.

Larger Part Unknown

Students can learn to understand abstract problems by drawing what is happening

In these problems, the small group is known and students will be determining the size of the larger group using addition or counting on.  Here's an example of some strategies I use in my instruction.


Smaller Part Unknown

bar models are a great problem solving strategy to help students visualize

In this problem type, students are provided with the larger group and asked to determine how many fewer are in the other group.  Here, students will use subtraction concepts to find the answer.  However, setting the problem up as a subtraction equation is just one of many strategies.  Check out the strategies I use with my students in this short video.


One benefit to teaching the different solving strategies that I've shown here is that many times, a student struggling with addition or subtraction concepts makes the connection to these strategies and is able to help them with basic equations too!  That's a win-win!

Practice Comparing Problems

Using a variety of problem solving strategies helps students better understand problem context.
If you've been through the series then you know I follow the same practice procedures any time I introduce a new skill or concept.  With our young learners, it is important to always start with hands-on manipulatives that help make the abstract concepts of math more concrete.  Hands-on learning also helps students to develop that important skill of visualizing what is happening.  

From there, we move to paper practice using a variety of different strategies.  These two can also be combined in centers where students start with hands-on building and then transfer what they built to paper using drawing and/or words.  

My go-to practice for problem solving is Mega Math Practice because it includes a variety of practice activities using all the strategies I've taught here and more!  It's a great way to expose students to lots of different problem solving methods without adding more time or work to your schedule.  We all know there's very few teachers who have time for that!

Here are the Comparing Problem practice sets I use in my math classroom.

Mega Math Practice problem solving bundle will help you provide your students with quality problem solving practice and save you time!

Mega Math Practice problem solving bundle will help you provide your students with quality problem solving practice and save you time!

Mega Math Practice problem solving bundle will help you provide your students with quality problem solving practice and save you time!

You can try these comparing problems practice activities with your students by grabbing this FREE Resource!  It will give you a look at all the great practice your students will get from Mega Math Practice.

Grab this free problem solving resource for your classroom.

Wrapping Up . . .

After 5 weeks, I can't believe that we have come to the end of our problem solving journey.  I have loved digging in with you and helping unpack the ins and outs of addition and subtraction problem solving types.  I hope that these posts and videos have provided you with some new ideas that you can add to your problem solving toolbox. 


Here's a few of the most important concepts to remember when teaching your young students problem solving:
  • Teach students to VISUALIZE what is happening
  • Provide students with a variety of STRATEGIES to use when solving
  • Celebrate that we can solve problems in DIFFERENT ways and still get the same answer
  • Focus on the CONTEXT
  • PRACTICE, practice, practice by starting with concrete, hands-on opportunities and slowly moving more abstract.
And . . . if you want to save some time and fill your toolbox with ready to use practice problems for all 11 problem then grab the Mega Math Practice Problem Solving Bundle from Teachers Pay Teachers.

Mega Math Practice problem solving bundle will help you provide your students with quality problem solving practice and save you time!

The Problem Solving Series

I want to make sure that you can quickly and easily access any of the posts from this problem solving series.  While I feel starting from the beginning is best in order to get the foundation, you can also jump right to the post that will help with your current teaching needs.  And . . . don't forget to grab the freebie in each post!

Save these Problem Solving Tips and Strategies

Problem solving is something you teach year after year.  And while we can use some of the same strategies you never know when you'll need to pull something new from your teacher toolbox.  Pin this to your favorite classroom Pinterest board so you can quickly come back for tips, ideas and problem solving strategies. 

Problem Solving: Part - Part - Whole Problem Group

Everyday, real-world math is problem solving.  That's why it is so important that our students understand the problem solving process.  I hope that you are enjoying this blog series and have gotten some helpful tips and ideas for teaching problem solving that you can use in your classroom.  Today we are moving to our next problem type, the Part-Part-Whole problem.  


Problem Solving in the Primary Grades

As teachers it is imperative that we give our students a solid foundation in problem solving.  This means moving beyond teaching key words and teaching our students how to think through what is happening in a problem in order to determine the appropriate math steps to take.  

If you missed the first few posts in this series, I highly recommend that you go back because there is so much great information to help you teach problem solving skills in your classroom.  If this is the first post you've seen, here's what you've missed:

What are Part-Part-Whole Problems?

Part-part-whole problems are those addition and subtraction problems where you have two parts that together make up a whole.  You might be wondering how this is different from the joining and separating problems...  Well, there's one important difference - CONTEXT!


In the joining and separating problems ,we focused on getting students to see the action that was happening.  One group joins another group to make a bigger group.  Or, some of a group leaves to make a smaller group. There is a start, a change, and an end.  There is action in these problem types, and we want our students to see that.

In the part-part-whole problem group, there is no action. The start and the end are the same because nothing is joining or leaving.  Instead, we focused on two different parts of a group that make up the whole.  


Here's an example:  

Look at this group of pumpkins.  The group is already formed.  No more pumpkins are being added and none will be taken away.  Instead of having one of those actions, we want to break down our information by the color of the pumpkins.   We have two orange pumpkins and one white pumpkin.  Now our focus will be on the total number of pumpkins.

You see, in the real world there aren't always actions happening.  Sometimes we just need to take inventory of what we have, and this often includes breaking down into "sub-groups" that we call parts.

While this example is really simple, a real world application could include much larger numbers or a group with more than 2 parts.

While the math concepts of addition and subtraction will remain the same, it is important that students see the different contexts in which the mathematical operations can be used.

Part-Part-Whole Problem Types

Help your students master part part whole problems with these fun and engaging Mega Math Practice problem solving resources
There are two types of problems that fall into this problem solving group.  The addition problems are where the whole is unknown and the subtraction problems are where one part is unknown.  Let's dig into each of these problem types and how you might teach them in your classroom.

1. Whole Unknown

When you have a problem where you know the two parts of a group but you don't know the total group, then you have a Whole Unknown problem type.  These problems are addition problems.  I actually like to teach these problems along with the Joining Problems since both use the the same addition concepts.  

Once students are comfortable with the joining process, they are ready to use addition to put two parts together without there being an actual joining action.  As you introduce these problems, make sure to have students visualize the problem.  It's important for them to see that addition is used in different situations.

Here's an example of how I might teach a Whole Unknown problem in my classroom:


2. Part Unknown

Teach students important problem solving skills and the part part whole problem solving type with these Mega Math Practice resources
The second type of problem in this group is when one of the parts is the unknown.  These problems connect to your subtraction concepts and are great to teach along side the Separating problems.  Before introducing students to the fact that there are different types of problems, make sure that they are understanding the subtraction concepts first.  When we throw too many different things at them, we can cause confusion and frustration. So make sure to lay the groundwork first, then show them the different times that subtraction can be used.

When looking at part unknown problems you can find the missing part in both the first and second positions.  However, the math students will use will be the same. 

Here's an example of how I would teach this type of problem to my students and the strategies that I would use:


Helping Students Master Part-Part-Whole

Now that we have understanding of this problem type, it's important to provide students with the instruction and practice opportunities to help them reach mastery.  In my math classroom, 90% of my math instruction happens in the small group setting.  I love this because I can really tune into each student, their understanding and their needs.  

Start with Hands-On Learning

I always start instruction and practice with physical objects that the students can move.  For these problem types, using one manipulative that is available in two colors is perfect.  The goal is for the two groups to have something in common (the common group) but to also be separated in some way to make the different parts.  Another option is two different parts with a more common generality like two types of animals or two types of food.

Here's some hands-on manipulatives that are perfect for introducing part-part-whole problems:
  • colored counting cubes
  • buttons
  • pom poms
  • counting bears
  • colored clothes pins 
These are all fairly easy to find and make for a great hands-on lesson.   As you introduce these, have students create the entire group first.  This is really important to help them differentiate these problems from a joining or separating problem.  If we have them add one color and then the next, they are likely to connect that to the joining action.  

A fun way to do this is to have some small paper or plastic cups with the manipulatives already inside.  Then have students dump them all on the table at one time to create the whole group.  Help students connect this as the whole.  Then have students separate the group into the two parts and help them connect this to the vocabulary term 'part'.  

Move into Paper Practice

Part Part Whole Problem Solving practice problems in print and digitalformats
Once students have an understanding of the concepts then they are ready to move into paper and pencil practice.  My favorite practice resource is Mega Math Practice because it is jam-packed with ready to use problems for each problem type.  I don't know about you, but sometimes I dream about word problems, but then when it's time to make one up while I'm teaching I draw a total blank!  That's why I love having these already made problems that are ready to go!  

When putting together Mega Math Practice, I also made sure to include a variety of different math strategies.  This way you can be confident knowing that your students are being exposed to numerous strategies and practice for each of them.  Students will work on drawing a picture, creating bar models, using ten frames and writing about the problem.  An easy way to create a well-rounded math program that will support students no matter their learning style.

Try Your Free Mega Math Practice for Part-Part-Whole Problems

I know how helpful these resources can be for teachers and students alike.  That's why I want you to try them out in your classroom for free.  These Free Part-Part-Whole resources will help your students practice these important problem solving skills.



You can also find the full Mega Math Practice in both printable and digital formats. You can find the Part-Part-Whole problem solving practice in my store at Teachers Pay Teachers.



And . . . if you are ready to fill your teacher toolbox with ready to use problems for all 11 types of problem solving problems, you can grab the Mega Math Practice Problem Solving Bundle and be ready for the entire year!


Don't miss the final post in this problem solving series!  Sign-up to get notified by email with the last post is published.  

Save These Ideas

Pin this to your favorite classroom Pinterest board so that you can quickly come back for more problem solving and math tips and ideas.


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