The Partial Sums Algorithm (a Step-by-Step Teaching Guide)

The partial sums algorithm is one of my absolute FAVORITE addition strategies to teach each year.

This place valued based strategy is a key support in helping younger elementary students truly understand adding two-digit and three-digit numbers.

The partial sums algorithm is also a HUGE help in illustrating the concept of regrouping.

In fact, I use this strategy alongside the standard algorithm to help my upper-elementary students make the bridge to the more advanced method.

You can read more about this here: The Power of Partial Sums (Especially for Upper-Elementary!)

It took me several years to refine how we tackle partial sums…and I want to help save you time right now with this comprehensive look at teaching the partial sums algorithm.

Start with Two-Digit Numbers

Whether I am introducing the partial sums algorithm to a second grader when adding two-digit numbers or a fourth grader to help support the standard algorithm, I always begin with two-digit addition first:

This makes it much easier to focus on the algorithm itself, without the challenge of adding larger, multi-digit numbers.

Add Without Regrouping First

As we take on those two-digit addition problems with partial sums, I am mindful of choosing problems without regrouping.

Just as we were discussing above, this makes it easier for students to focus on the algorithm:

Make the Concept Concrete

Ok, now that we have the types of problems we’ll present to our students established, let’s talk about something very important:

Using math manipulatives to make the concept concrete.

Using manipulatives to represent the addends is such a great help as students learn to add ones, tens, and hundreds.

The partial sums algorithm can be modeled with base ten blocks…but I love to use place value disks instead.

These little disks are the PERFECT visual in representing the tens and ones for each digit. As I introduce the algorithm to my students, we line up the disks right next to those addends:

(if you are just starting to use place value disks with your students, I have a post with a free resource for you right here: Introducing Place Value Disks with 4 Guiding Questions)

The place value disks are a key support as we move to the next step…

Set up the Partial Sums

I have seen so many variations on setting up the partial sums algorithm. And there really is no one correct way to do so…it’s a matter of adding each part, then combining the partial sums.

And while there are several different methods to set up partial sums, I love to introduce this algorithm in the same way for all of my students:

Starting with the ones place first, then moving to the tens.

I know many will believe it is easier to add starting with the greater value first and moving down in place value…but there is a reason for the order I choose!

Though I’m not a huge fan of standard algorithms, they are a necessary tool my students will all eventually learn.

And as they learn to add partial sums moving from ones, to tens, and so on…it will be easier for them to make the transfer to the standard algorithm.

Add the Partial Sums

Once students have set up the addends on the side of the equation, they are ready to put everything together.

We add the ones, then the tens, then we find the sum:

Scaffold the Steps

As students become comfortable with the algorithm, they can move to addition with regrouping.

I do this in a scaffolded way, starting with parters of 10 first:

Then we move to more complex addends:

To help save time, I have a huge set of scaffolded practice sheets featuring two-digit addition with regrouping, using partial sums: Two-Digit Addition with Regrouping

These sheets feature built-in scaffolds and leveled pages with problems that increase in difficulty:

Move to Three-Digit Numbers

Once my students are comfortable with regrouping two-digit numbers with partial sums, we move to three-digit addition:

(this is from another set of leveled practice pages: Three-Digit Addition with Regrouping)

Release the Scaffolds

As your students become more fluent with the partial sums algorithm, it’s the perfect time to release those scaffolds.

I encourage my students to work efficiently by simply adding below the equation (when they are ready!).

They are always so proud when they add with fluency using this algorithm! And I am delighted because I know this is paving the way to addition with the standard algorithm.

Get Started Now!

I have been working hard to save us time! And this series of partial sums practice sheets does just the trick.

You will have everything you need for your students to acquire, understand, and master the partial sums algorithm:

I hope these ideas will be as helpful for your students as they have been for mine!

Let me know if you’d be interested in more algorithm posts like these…I have LOTS of ideas brewing…