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?

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

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.

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

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

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

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.

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.

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.

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!

1. The 11 Types of Addition and Subtraction Problems
2. How I Teach Problem Solving & The Joining Problems
3. The Separating Problems
4. Part-Part-Whole Problem Solving Group
5. Comparing Problems and Wrap-Up

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.