Distributive Property of Multiplication – How To Break It Down

Teaching the Distributive Property in 3rd grade? Read on to see how I go about teaching this challenging math concept!

Did you ever think that as a third-grade teacher or even an elementary teacher, you would be teaching the Distributive Property of Multiplication?  When I started teaching over 30 years ago, there weren’t even any standards.  Teachers just taught what was in the textbook. Breaking apart multiplication facts was just not on my radar.

When standards were introduced at the state level in the late 1990s and early 2000s, the Distributive Property of Multiplication was still relegated to middle school math for the most part. But several years ago, California adopted the Common Core State Standards. And there it is. Right there.  Yes, I have to teach it. Most importantly, my students have to learn it and use it.

So how did I go about teaching the Distributive Property of Multiplication?

Don’t Listen to the Textbook Publisher!

After many years of figuring that out, I’ve got some ideas and tips to share.  First of all, contrary to the math textbook publisher’s opinion, this is not just ONE lesson taught in ONE day.  If you can, don’t even use the textbook on this one.  Teachers know better.

I might add too, that the publisher’s explanation is more suited to high school students than to elementary students.

How Did I Teach the Distributive Property of Multiplication?

Usually, I use a mix of approaches to teaching math.  Sometimes I use Lesson Inquiry.  Sometimes I use Direct Instruction.  So for this lesson, I decided on a hybrid approach.  I would teach the Distributive Property of Multiplication using a hands-on, inquiry, guided questioning approach COMBINED with some direct instruction with steps.

I designed my two-day lesson with my resources to teach the Distributive Property of Multiplication.  When I create lessons or think about how I teach a concept or standard, I try to think like a student. Especially if I am going to use an inquiry approach.

• Which part or parts of the Distributive Property of Multiplication (DPM) do students have difficulty comprehending or learning?
• What prerequisite skills do they need to use the DPM?
• What can I use to make the DPM comprehensible?

So, let’s start with the first question.

Which Parts of the Distributive Property of Multiplication Present the Most Difficulties?

That’s an easy question to answer.  ALL OF IT.  Think about it.  It is unlike any other Property of Multiplication, so there’s no building on that.  It involves notation they are usually unfamiliar with or rarely use:  mixed operations and parentheses in the same number sentence.

However, now that students have been instructed with the Common Core State Standards for Mathematics, students know how to decompose a number, be flexible with numbers, and can use the Properties of Addition.  These are all helpful when connecting to the DPM.

There are many steps in the process, and each step can lead to an error.  We all know how complex multi-step problems are for students!

What is the Answer, Then?

Break it down into steps.  Don’t rush to teach the Distributive Property of Multiplication number sentences on the first day!  Slow it down, so the students understand WHY we break apart an array, then ADD the two parts back to get a final product.

Students can relate to breaking apart complex representations or large numbers because they have done this using addition with the Break Apart Strategy.  Students are already familiar with building arrays to represent a multiplication sentence.

But first,  let’s start with breaking apart an array.

Breaking Apart Arrays for the Distributive Property of Multiplication

You would think that breaking apart an array is an easy step.  Not really.  Arrays can be broken apart in many ways:  vertically or horizontally.  But is there a way to break apart an array to make the process more efficient or easier?  YES!

One thing I do with students is practice breaking apart arrays at strategic points.  First, I would have them create an array and then let them explore how many ways they could break apart the array.  We would share ideas, solutions, etc.  I would pick at least three students to share how they broke apart the arrays. I purposely pick students who have the least efficient way, a sort of efficient way and the most efficient way to break apart an array.  Why? I want students to see that mathematicians want to find a solution and work efficiently!

So, I’d pose a question?  Where could you break apart the array to make it easier to find the total?  These are two ideas I wanted the students to discover:  break apart an array at five, or if it’s an even number across, break apart the array in half.  With guided questions, the students could discover this on their own.

Why Is This Important to Know?

Breaking apart an array at five means I will eventually multiply by five and almost all students can count by fives or know their five facts.  Breaking apart an array in half means both later arrays will be the same!  For third graders, if you teach them these two fine points of breaking apart an array, you’ve taken some of the difficulty out of the process.

How do you practice this?  With manipulatives because they make the concept real.  Students need to see and touch math for it to make sense!  I have my students build an array with foam tiles.  Then they use their pencil (or ruler) to show where the array will be broken apart.  We practiced this several times and named the two new arrays with multiplication sentences.

The next step in teaching the Distributive Property is to connect symbols and numbers.  On whiteboards or paper, students practice writing multiplication sentences for the broken-apart arrays.  Once they get the hang of that, it’s time to move on to the next step.

Adding the Products in the Distributive Property of Multiplication

If you were to ask students about long division and why do they bring down the next number or why do you multiply or why do you subtract, how many could explain the reason?  So how do you expect third graders to explain or understand why there is an ADDITION sign in a Distributive Property of Multiplication sentence?

Note:  yes, there are two ways to write DPM sentences, such as (7×5)+(7×2) or 7(5+2), but both ways do involve the use of addition.  They probably couldn’t even tell you why, even though they might compose the DPM sentences correctly.

But suppose you have the manipulatives while the students compose matching multiplication sentences. They naturally conclude that you would have to ADD both products to get the final product! Here’s a recap of the first day’s lesson.

Notice that I have NOT introduced the DPM sentence yet.  That, I believe, was my mistake several years ago when I started teaching Distributive Property.  The students could NOT understand why the array was broken apart or what we were adding.  Using manipulatives and just slowing down made those two concepts clear and comprehensible.

Day TWO,  Introducing the Steps

On day two, I reviewed what we had learned the day before.  We started with a quick warmup with an anchor chart partially prepared. Using a piece of yarn, I moved the yarn around the array splitting it in different ways, until we agreed that splitting it at the five mark was the best solution.  We would return to the anchor chart at the end of the lesson to reflect on what we learned.

The Distributive Property of Multiplication Ninjas!

Now, it’s time for the Distributive Ninjas to take over!  I enjoy using technology and using PowerPoint. I created a PowerPoint with Ninja Theme.  It has animation, sounds, and printables or worksheets for the students to follow along and practice.

I used this Distributive Property of Multiplication PowerPoint as a Guided Practice in my teaching the Distributive Property.  This time, however, the students were going to learn the steps to writing a DPM sentence because that is where most errors occur.

The first part of the DPM PowerPoint focuses on breaking apart an array, writing multiplication sentences, and then adding the two products to the total product.  With two printables that go along with the slides, my students practiced breaking apart the same array in two different ways.

On the printable, I have these four steps:

• draw a vertical line to split the array 
• write a multiplication sentence below each array
• solve each multiplication sentence
• add the two products

Again, I am trying to cement the concept of breaking apart, multiplying, and then adding which are all parts of a DPM sentence.

The second part of the DPM PowerPoint now introduces the DMP sentence with parentheses and the addition symbol.  Students already know why we add, so the addition symbol is not a mystery.  I explain that the parentheses (like the ones we learned about in the Associative Property of Addition) show what to do first.

Time for Some Direct Instruction on the Steps

In direct instruction, steps are essential.  You want to make sure the students do each step one at a time. Once you know they can do each step, give them two steps at a time to follow.  Then let them follow all the steps in a guided practice problem.  If they can do all the steps successfully, then it’s time for partners to explain the steps to each other, taking turns.   Notice that this practice of procedure comes AFTER I try to build a conceptual understanding of this property. The first lessons on teaching the Distributive Property must focus on conceptual understanding.

Essentially, each partner has to teach the other partner the steps.  If you can teach it, then you know it! From there, it was time for independent practice.  I gave students a simple worksheet where they had to draw an array for a multiplication sentence first, then follow the steps.

Resources for the Distributive Property of Multiplication

I’ve also created a DPM center and games to go along with the DPM.  The DPM center is also great for small groups for those students who are still not getting it or need more practice understanding the process of breaking apart and adding, matching multiplication sentences, or writing DPM sentences.

The DPM games are great to have out during the entire multiplication unit so that students continue to get some practice with the DPM.  I sneak them in when we have extra time or make time for them.  If I had an extra day to focus on the DPM, I would put out this center and games for the day.

Another resource I created to help practice this critical property are games for the Distributive Property.  In this resource, there are four games for the students to play. Some games can be played individually, in pairs, in a small group, or even with the entire class!

Recently, I added a new addition to the DPM resources: The Distributive Property of Multiplication on Google Slides®.  Students can practice this property on a Chromebook, tablet, or desktop computer.

There are 26 slides ranging in Depth of Knowledge levels 1, 2, and 3. There are 5 problems for each DOK level for a total of 15 problems.   It’s great preparation for online state testing. Here are some more highlights about this digital interactive notebook for the Distributive Property of Multiplication.

• All the slides provide more instructions and information to the student in the SPEAKER NOTES section of each slide (similar to the Presenter’s Notes area in PowerPoint).
• Each section has a slide that prepares the student for work in the section with ideas, tips, or strategies to use.
• Part 1 and Part 2 each have a Reflection slide at the end for student reflection on what was learned.
• The question stems for Part 3 are modeled after the sample questions for the Smarter Balanced Assessment Consortium assessment given to third-grade students.

But as teachers know, the pacing guide doesn’t wait for you, so I have to keep going to stay on track and meet district guidelines for assessment.  Click HERE to see all my TpT resources for the Distributive Property of Multiplication, including this BUNDLE, and save, save, save!!!!

If you’re looking for more ideas for multiplication, check out my Pinterest Boards.  I have several boards related to multiplication, including the Distributive Property of Multiplication.  Consider following it for more ideas, resources, and tips!

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Click below for more articles on teaching multiplication.

• Frustrated Students Don’t Know the Multiplication Facts?
• 3 Tried and True Ways to Teach Multiplication
• 5 Helpful Multiplication Videos.

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What are some ways you teach your students about the Distributive Property of Multiplication?

Share your ideas in the comments!