# Why the Properties of Addition ARE Useful to Students

When my oldest son was in seventh grade, he had trouble with a problem in his 7th-grade math book.  It concerned one of the Properties of Addition.  I couldn’t believe he didn’t know the answer!

It was a very straightforward problem with integers, and all it asked was how you could solve the problem using the Commutative Property of Addition.  He couldn’t remember what that property was!  And this was 7th grade.

## The Properties of Addition in Math Instruction

But as you can see with my son, the Properties of Addition did not go away.  Students will continue to add all through their schooling (and life!), but they may be adding fractions, decimals, or integers. It doesn’t matter! They need to know that the Properties of Addition are used in math because they are useful.

Those who use the Common Core State Standards for math know that the Properties of Addition are specific in grades 1, 2, and 3.

Way back in the 90s, when I had taught first grade, the idea of directly teaching the Properties of Addition was not even on my radar.  But instinctively, I did teach them. I was trained in Math Their Way, a philosophy of teaching math based on Piagetian principles of child development.

Students realized on their own that by flipping a Unifix® train of 4 red cubes and three blue cubes would result in the same sum:  4 + 3 = 3 + 4.

They learned that adding zero meant the other addend would be the resultant sum.  My first graders used the Commutative and Zero Property of Addition without mentioning their mathematical names.

Fast forward to the 2000s, and now we have a standards-based education.  However, the Properties of Addition first get mentioned in the third-grade California math standards.  But that was teaching them too late.  These properties are used even in beginning addition.

Having mostly taught third grade after a seven-year stint in first grade, I returned to second grade recently.  I am so glad I did! One of the biggest takeaways from teaching second-grade math is that the Properties of Addition are one of the keys to math fact fluency.

Fluency is not an automatic recall of facts or just fast and accurate. As defined by the National Council of Teachers of Mathematics, Fluency demonstrates flexibility in computational methods while understanding these methods to efficiently produce answers.

Flexibility in computational methods can mean many things.  As I began exploring number talks with my students, I saw this flexibility daily.  I saw students use compensation, breaking apart numbers, making a ten, or using friendly numbers.   And more to the point, students who used the Properties of Addition to add.

I believe that once the Properties of Addition are introduced in first grade, second-grade teachers need to design lessons to reinforce those properties as one of the flexible strategies to use when adding or even subtracting.

One of the most powerful ways for students to practice the Properties of Addition is through number talks.  But more importantly, when a student is explaining how he or she arrived at an answer, the teacher should help to label the strategy used, especially if it involved one of the Properties of Addition.  Many times, students explain a strategy without naming it.

We no longer work with simple sums when I teach the Properties of Addition in third grade.  We are now adding two and 3-digit numbers with and without regrouping. But is within these addition problems that we do practice the Properties of Addition.

In fact, within any addition column, we can use multiple strategies such as making a ten, compensating, decomposing, and of course, using one of the Properties of Addition.

Here’s one example of using the Associative Property of Addition with three addends.

But as third graders move from adding and subtracting to multiplication and division, they learn about the Properties of Multiplication.  Since addition and multiplication are related, it is important for students to point out similar properties (Commutative and Associative).

Where does this all lead? Students will now add, subtract, multiply, and divide fractions, decimals, and percentages when moving on to fourth and fifth grades.  Knowing how to use both the Properties of Addition and Multiplication as flexible strategies when computing will be invaluable.

Once students begin Algebra in middle school, these properties are a given.  Students must use them to solve expressions and balance equations.  Teaching the Properties of Addition is a useful and necessary strategy for working with more complex math.

## How Do I Teach the Properties of Addition?

When I first began teaching the properties, I created an interactive PowerPoint.  By interactive, I mean there are slides in the PowerPoint in which the students interact with the content on the slide (and it includes a lot of animation).

They interact by answering questions, discussing with a partner, or writing in a math journal (or any piece of paper).  I use both a math journal and printables.  I created printables to go along with the PowerPoint so that students would be even more engaged by practicing the Properties of Addition immediately.

You can see a preview of this PowerPoint on my YouTube channel.  Since this video was made, the PowerPoint has been updated and expanded, including new fonts and formatting.  But you’ll get an idea of how this PowerPoint works.

The PowerPoint has gone through some revisions, but the format has remained the same.  Three properties (Commutative, Zero or Identity, and Associative) are all addressed as separate lessons, so it is easy to pause the PowerPoint after each lesson and continue the next day.

The PowerPoint also comes with the Presenter’s Notes, in which I give teaching suggestions while explaining when to advance the animation or slide.  I’ve added questions you can use to stimulate mathematical thinking and discussion. I’ve expanded the printables to include support for English Language Learners using sentence frames and a glossary.

You can find the Properties of Addition PowerPoint Lesson HERE.

## How do my Students Practice the Properties of Addition?

In addition to the PowerPoint, I created a hands-on center which I use whole class though it can be used in a small group or for intervention.  This center has a work mat and number tiles, but you would need to add unifix cubes, linking cubes, or some other type of manipulative.

It also has task cards which give the student a problem working out on the work mat.  In whole-class grouping, I read aloud the card and project on the screen.  Then I walk around as the students use their unifix cubes to build an equation.

Here’s a video showing one of my students building with unifix cubes.  The center has task cards for all three properties, but in this video, we worked on the Commutative Property of Addition.

Once the student has created a unifix train for the equation, they use the number tiles to build the matching equation.   From there, they build another unfix train, switch the addends around (commute the addends), and again use the number tiles to build the new equation.  We then discuss what they notice.  Here’s the same student continuing the work.

## Don’t Forget to Teach the Vocabulary!

This is a great opportunity to build and expand that math vocabulary:  addends, addition symbol, equal sign, commute,  sum, etc.  It is so important for students to learn and use that math vocabulary when discussing mathematical concepts (or else how are they ever going to understand word problems!! i.e., see my son above).

The second part of this center involves a Commutative Property Match-Up.  I took a video of my youngest son (who at the time was in third grade) to see if he could explain how to use the Commutative Property of Addition.

It was interesting to note his use of vocabulary (he said that 3 + 4 is the OPPOSITE of 4 + 3).  But he was able to show his understanding of this property.  Check out the video:

## Why is Commutative Property Useful?

The Commutative Property is a great strategy to use when adding multi-digit numbers.  When I taught first grade, I taught counting up as an addition and subtraction strategy.  But if students know that they can switch the order of the addends and start adding with the greater number FIRST, it makes counting up easier.

Also, back when I did teach first grade, I was using Math Their Way, so the students were constantly using manipulatives to build the concept of addition and would internalize this property.  They never saw 3 + 4 as just 3 + 4, but also as 4 + 3.  They knew they could switch the addends because they had done that with the manipulatives.  By learning this property, in addition, it will help immensely when we start learning the Commutative Property of Multiplication!

## What about the Zero Property of Identity Property?

Don’t forget to teach this one!  Most students know that when you add zero, the sum is the same addend or addends.  Why would this be important to know?  In third grade, students learn to round (3.NBT.A1).  When you round numbers, you end up with two and 3-digit numbers with zeroes in the ones and tens place.  Those numbers can easily be added using mental math.

We want students to be able to mentally round, add and give an estimate to check the reasonableness of an answer.  If they know that zero doesn’t affect a sum, then all they need to do mentally is concentrate on adding the tens or hundreds digits to get an estimate.  But read on.  The Zero Property is also useful when combined with the Associative Property.

## Why is the Associative Property Useful?

Do your students need to learn column addition?  Do they just go about adding a column of numbers in order?  STOP!  Teach them to use the Associative Property of Addition!  With this property, I teach students to find addends that make 10.

Ten is a great number because it has a zero!  And when you have the digit zero in an addend, you can mentally add those numbers faster, making adding more efficient and reliable.  I vividly remember my high school teacher (back in the 70s before math instruction was more than just arithmetic), telling us to use this shortcut:  when you add, find pairs of numbers that add up to 10 and cross them off as you add.

Can you imagine that I didn’t know this until HIGH SCHOOL?  If the students understand these properties well and you demonstrate how to use them in addition, I promise they will become better, faster, and more accurate with addition.

The best place to practice this property is with number talks.

## Continuing Throughout the Year

Don’t forget to continue to teach and practice these properties throughout the year.  We want the students to enter the upper elementary grades with an absolute understanding of these properties.