A common misconception about math, held by children and adults alike, is that it’s about knowing formulas, methods, and procedures. Indeed, these aspects of math are important, but what’s much more important – even essential – is an understanding of the meaning behind those formulas, methods, and procedures.

Regrouping and exchanging in two-digit addition and subtraction is one such procedure that is often used without understanding the underlying concepts. At Happy Numbers, we aim to resolve this misconception by designing activities that are visual, concrete, and coherent. Here, we would like to share with you some ideas we use to approach this challenging topic.

Exercises shown below represent several different lessons within our Grade 2 Addition & Subtraction domain. You can access these interactive exercises and more through your Happy Numbers account or our demo page.

## 1. Regrouping and Exchanging Using Objects in 10-Frames

Using real objects is always a great way to connect math to real life. For example, we use 10-frame boxes and apples to illustrate the need for regrouping in addition:

Students count and enter the number of packaged apples (a 2-digit number) as well as the number of loose apples (a single-digit number). They then drag the apples to the 10-frame and realize that it results in one more full box than they started with. This way, they “touch” what will later be called regrouping (or carrying over).

Exchanging for subtraction is illustrated in a similar way, using the same objects:

Students first count the total number of apples and are then asked to subtract a given number by “eating” the corresponding number of apples. At some point, students see that there is a need to get some apples from a full box, which perfectly illustrates what will later be called exchanging.

## 2. Using a Number Line to Explore Different Strategies of Addition and Subtraction

A great way to enrich students’ understanding of these operations is to present them with different methods of solving the same problem. This helps them to understand why those methods work and it builds mathematical reasoning and flexibility. And the number line is a perfect tool for that!

Here is an example of using the number line to model two different approaches to two-step addition:

In the first approach, students break up the second addend in order to reach a round number: the addend 9 is broken up into 6 and 3 so that students can “rest” on 40 as they solve the problem. The second approach illustrates how to add on a round number and subtract the difference: in this case, students easily add 10, and then correct for one less.

Both methods are great ways to build mental math ability with larger numbers, and use of a number line helps students understand the reasoning behind why these mental shortcuts work.

## 3. Regrouping and Exchanging Using Base-10 Blocks

Using base-10 blocks, which all students are familiar with, is a great way to continue learning the concepts of regrouping and exchanging.

For example, modeling 2-digit addition using cubes grouped into 10s and 1s will lead students to the point where they learn that they have enough ones to form a new ten and will move it to the tens column:

It is important for students to distinguish between problems that require regrouping and those that don’t. For that reason, we mix in some equations that do not call for that extra step, keeping students on their toes.

The same approach works perfectly for modeling exchanging for subtraction:

The students will figure out that in order to subtract cubes, they will need to exchange one ten for ten ones!

Again, it is important for students to distinguish between problems that require exchanging and those that don’t, so we use the same “trick” of mixing in some equations that don’t need that extra step.

We hope the ideas shared here will help you and your students through this potentially challenging topic. Please let us know how you approach regrouping and exchanging in your classroom. Do you use manipulatives, like base-10 blocks? Model with Happy Numbers on your IWB? How do you help your students to “see inside” this procedure?

Looking forward to hearing from you!

Yours,

Evgeny from Happy Numbers