# How To Break Down the Distributive Property of Multiplication

Did you ever think that as a third-grade teacher or even as 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.  When standards were introduced at the state level in the late 1990s and early 2000s, the Distributive Property was still relegated to middle school math for the most part.  But over 3 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 do you teach the Distributive Property of Multiplication?  After 3 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.

Normally, I use two approaches to teaching math based on my current group.  I either use Lesson Inquiry or Direct Instruction.  As I wrote in my previous post, math is a subject area that one-third of my class struggles with daily.  So for this lesson, I decided on a hybrid approach.  I would definitely use a hands-on, inquiry, guided questioning approach COMBINED with some direct instruction with steps.

I designed my own two-day lesson with my own resources.  When I create lessons or think about how I’m going to teach a concept or standard, I try to think like a student (especially if I am going to use an inquiry approach).  What part or parts of the Distributive Property of Multiplication (DPM) do students have a hard time 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 DPM Present the Most Difficulties?

That’s an easy question to answer.  ALL OF IT.  Think about it.  It is unlike any of other Properties of Multiplication, so there’s no building on that.  It involves notation that they are usually unfamiliar with or rarely use:  mixed operations and parentheses in the same number sentence.  There are many steps in the process and each step can lead to an error.  We all know how difficult multi-step problems are for students!

What is the answer then?  Break it down into steps.  Don’t rush to get to that DPM sentence on the fist 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 models or large numbers because they have done this using addition with the Break Apart Strategy.  But first,  let’s start with breaking apart an array.

### Breaking Apart Arrays

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.  But then, 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 would I want them to know this?  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 subsequent 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.  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 practice this several times and name the two new arrays with multiplication sentences.

The next step 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.

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 why?  So how do you expect third graders to explain or understand why there is an ADDITION sign in a DPM 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 if you have the manipulatives out, and the students are composing matching multiplication sentences, they naturally conclude that you would have to ADD both products to get the final product!  So in our first day’s lesson, I only did this:

Notice that I have NOT introduced the DPM sentence yet.  That, I believe, was my mistake several years ago when I started teaching this concept.  The students could NOT understand why the array was broken apart or why we were adding.  By using manipulatives and just slowing down, this made those two concepts clear and comprehensible.  You’re probably asking, how do I know my students got it?  I was checking for understanding by circulating and looking at whiteboards.

### Day TWO,  Introducing the Steps

On day two, I began by reviewing what we learned the day before.  Just a quick warm up as had an anchor chart partially prepared. Using a piece of yarn, I moved the yarn around the array splitting it different ways, until we agreed that splitting it at the five mark was the best solution.  We would come back to the anchor chart at the end of the lesson to reflect on what we learned.

Now, it’s time for the Distributive Ninjas to take over!  I really enjoy using technology and using PowerPoint.  I created a PowerPoint with Ninja Theme.  It has animation, sounds, and printables for the students to follow along and practice.  I used this Distributive Property of Multiplication PowerPoint as a Guided Practice.  This time, however, the students were going to learn steps to writing a DPM sentence, because that is where most of the 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

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) just indicate what to do first.

Here are the steps the students learn:

In direct instruction, steps are important.  You want to make sure the students do each step one at a time.  Once you know they can do each step, then 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.  Sort of like 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 in which they had to draw an array for a multiplication sentence first, then follow the steps.

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 just 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 I make time for them.  If I had an extra day to just to focus on the DPM, I would definitely put out this center and games for the day.  But as teachers know, the pacing guide does to 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!!!!

I just recently created a new Pinterest Board for the Properties of Multiplication.  Consider following it for more ideas, resources, and tips!

If you have other ways of teaching the DPM, please share!

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