Have you ever heard of the CPA model for math instruction? For those who aren’t familiar, this refers to Concrete-Pictorial-Abstract instruction and is the hallmark of Singapore Math.

But what exactly is the CPA model?

Of course, the “hands on” component is the Concrete Stage of math instruction…which just so happens to be my favorite part of math! This stage involves lots of manipulatives. Here are some of my favorite math manipulatives to use:

Which manipulatives do you prefer to use in your math lessons? Grab some nifty FREE labels for organizing all of your manipulatives {HERE}.

To walk you through the CPA math model, I think it’s best to explore a specific topic so you can get the whole picture. 🙂 Let’s explore: Two Digit Addition. In first grade, my students begin two digit addition in January. Our old method would have involved simply writing two digit problems and instructing students to add the ones column first and then add the tens column. But introducing this traditional algorithm before we make sure students have a firm grasp on place value is a HUGE mistake!

So how can we make two digit addition concrete? Since the concrete stage is the “Tools and Touch” stage, we begin with the best tools possible for exploring place value…unifix cubes. Why unifix cubes and not base ten blocks? Because having students actually *build* a tower of ten will better reinforce the concept that a ten is made up of ten individual units, or ones. Count and Bundle is a fun game to help students realize that any quantity of items (above 9) can be grouped into sets of tens and ones. A recording page makes this a great independent center activity. Simply have students select a numeral card and count out that many individual linking or unifix cubes. Then have students see how many bundles of ten they can build. Students record how many tens and how many ones are in the number they made. This is a super easy *yet super effective* way for students to see place value in a *CONCRETE* way!

Another fun and concrete place value activity is Race to 100 with dry beans. Give each student a bowl of 100+ dry beans. have students roll two dice, add them, and count out that many beans. This is fun subitizing and addition practice, too. I love when I can *sneak* in more than one skill! 😉 As students build groups of ten, they can place them on their bean board. The first student to reach 100 beans wins. Periodically ask students questions like: How many beans do you have? Some students will start trying to count EVERY SINGLE bean individually, so this is a great time to remind them that their beans are grouped in tens and some leftover ones. CONCRETE PLACE VALUE!

What about linking place value to dimes and pennies? Another CONCRETE manipulative with real world implications! In “Race to $1.00”, students build groups of ten pennies that they trade for dimes to build their tens. This is also a great base for when students begin to regroup!

But how we can translate these concrete experiences with place value into two digit addition *concrete* experiences? Why, let students add using their linking cubes of course! If students are ready, you may move on to base ten blocks in which the ten trains are already assembled. Otherwise, feel free to keep letting students assemble their ten trains.

Present a double digit addition problem to students and have them use base ten blocks to act out the problem. For instance, when presented with the problem of 23 + 15, students would build a model that looks like the following using their unifix cubes:

Then tell students that to find the sum of these two addends, we’re going to combine the ones first. Then combine the tens. (ALWAYS have them combine the ones first…even in these early concrete stages. If we can have them develop this habit it’ll make life so much easier when they move on to the traditional algorithm for addition! We don’t want to unlearn bad habits!)

Highlight the steps. Be specific and let each student have the opportunity to manipulate the materials and make connections with the concepts. Say: Let’s combine the ones. Slide all of your ones together and see how many ones you have now. (8) Now, let’s combine the tens. Slide all of your tens together and see how many tens you have now. (3). So you now have 3 tens and 8 ones? What number is that? When we combined 23 with 15, we ended up with a sum of 38. *That*, my friends, is how we introduce two digit addition in a CONCRETE way.

Now let’s move along to the PICTORIAL portion of our two digit addition lessons.

Now we will help students relinquish their use of manipulatives in favor of pictures. In the pictorial component, first introduce images of base ten blocks or unifix cubes for students to work with in solving their problems. For example, they would solve the previous problem (23 + 15) using a picture that looks like this.

Students may wish to circle the ones and record the number of ones, then circle the tens and record the total number of tens. I also like to show students a quick and easy way to draw tens and ones. I like to use straight sticks (like tally marks) for my tens and small circles for my ones. By now, students have had enough hands on practice building their ten trains with linking cubes that they realize that stick is actually made up of ten individual units. Write out some problems and have students draw models of the numbers to help them add. Remind them to always add the ones first! “Build a Sum” is a fun game for students to play in centers. Provide some pre-programmed 2 digit addition problem cards and a recording page. Have students draw models to add. (Always provide the problems for addition without regrouping because we SURE don’t want a student to end up with a problem that requires regrouping until they *know* how to handle it!) In the following problem, the student had the equation 33 + 25.

Picture models like this are also great for building an understanding of odd and even numbers! Have students focus on the ones and circle the pairs. Are there any poor little “odd balls” left over without a partner? If not, the number is even! Can students find any patterns when they add an Even + Even, Odd + Odd, or Even + Odd? Encourage them to make connections!

Even word problems can be solved using this pictorial model!

Now it’s time to move on to the Abstract phase of two digit addition.

In the abstract phase, we want to *wean our babies* from using models and pictures and guide them in using numbers and symbols. Since we’ve established solid place value sense in our kiddos, and sense we’ve drilled them on starting with the ones first, moving on to the traditional algorithm is generally a piece of cake at this point.

Of course, a fun little addition song always makes things more memorable. Am I right, or am I right?

(The full words to this song are found in this unit!)

I also love providing gridded sheets that help students keep their tens and ones lined up. Graph paper works great for this as well!

Want to help guide YOUR students through a CPA model of two digit addition? Deanna Jump and I have teamed up to provide you a FABULOUS unit with everything you need. In the unit, you’ll also be able to teach students about expanded form, using expanded form to add 2 digit numbers, and even and odd numbers! The unit is engaging, hands on, and FUN. And it even includes a few crafts. 😉

Addition fluency, addition strategies, and problem solving practice are so *critical* in first grade that we’ve created a growing bundle with fabulous monthly units similar to this one. All include hands on practice! Check out the growing bundle below!

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