# Distributing the Law!

## Objective

SWBAT multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

#### Big Idea

Using distributive property and base-ten materials, students learn a simple way to multiply two -digit numbers by two-digit numbers.

## Warm Up!

20 minutes

In this lesson I want my students to use distributive property to better understand multiplication.  To begin, I ask students to join me on the carpet.  Since, I want them to look for structure within the problem; I write 33 X 46 on the board.  I say, we have been working on two digit multiplication for about three days now. I see some of you are composing and decomposing number situations and relationships through observed patterns in order to make the equation simple.  I am going to demonstrate how to make this problem simple using the distributive law.  For instance,

33 X 46 = 33 x (40 + 6)

You guys have pretty much know what distributive property is, can someone explain how we can use it to make this multiplication problem simpler? No one seems to understand how to apply distributive property to simplifying multiplication.  So, I began to explain step by step!  First, I explain that if I added 40 + 6 it would give me 46, therefore it would equal the same amount as the equation on the left of the equal sign. I write it just to make sure they understand.

33 X 46 = 33 X (40+6) 40 + 6 = 46 so then, 33 X 46 = 33 X 46

I ask what numbers are in the ones, and tens place so that students can recognize the value of each digit before I move forward.

Notice that I wrote the original problem to the left of the equal sign and the even though the problem to the right appears to be different it is not. I made it simple because now I can arrange the numbers by the last digit, like this:

= 33 X 40 + 33 X 6

Multiply 33 X 40! Can someone tell me the product of these two numbers?  1,320

Now multiply 33 X 6! Can someone tell me the product of these two numbers? 198

Ok! Now we need to add the two products together to get our answer.

1,320 + 198 = 1,518

Now I show students how to solve this problem the long way, so they can see the difference.  I want students to see the patterns used so that they can simplify their solutions.

I go through the steps a couple of more times just to point out why the structure of distributive law makes multiplying two-digit numbers by two-digit number a lot simpler.

MP7-Look and make use of structure.

20 minutes

I explain that this lesson will not be a computation task, I want you all to work together, and share ideas!

Now that students have had time to practice a bit, I want them to work in groups for a little while. I ask students to move into their assigned groups. I tell them they are going to be using the distributive property to make multiplying two-digit numbers  simple!  To engage students is ask, who is tired of doing the long way of multiplying? Nearly the entire class raise their hand.

I tell students this time we are going to use base-tens. Students are eager to know how base-tens will apply to solving multiplication problems. So I explain and demonstrate at the same time. I briefly go over the amount for each base-ten just to make sure students can relate there value to the place of each number.  For instance, I write 34 on the board, and I ask students what number is in the tens/ones place. Students understand that 3 is in the tens place. So, I ask what base-ten can I use to represent 3 tens. Students can also explain why three ten-rods are needed to represent 3 tens.  I say, we hopefully can understand the value of each digit within a number, so that we can apply the distributive property with ease.  We are going to repeat what we did in the first part of the lesson.  I point back to the work on the board.

= 33 X 40 + 33 X 6

Multiply 33 X 40! Can someone tell me the product of these two numbers?  1,320

Now multiply 33 X 6! Can someone tell me the product of these two numbers? 198

Ok! Now we need to add the two products together to get our answer.

1,320 + 198 = 1,518

As I review, I make sure to observe how students respond to each question, so that I can kind of gauge where they are in their learning. I place two sets of two digit numbers in each group, and set the timer for about 30 minutes to see how students are thinking their way through problem-solving.  As students are working, I chime in a time or two to check for understanding.  For instance, I might ask what number is in the ones/tens place. How do you know? How can base-tens help you understand multiplying two-digit numbers? Explain. As students respond and explain I notice that some students are basically recalling number in the ones/tens place, but they can not explain the difference in value between each digit. To help them gain a deeper understanding I apply a place value chart and ask them to insert the right digits under the right column.  As students insert digits, I ask them to represent the digits using the base-ten materials.  I repeat this a few more times just to make sure students understand why each digit value is important when multiplying.

When students time is up, I invite different groups to share what they learned with the rest of the class. I encourage students to ask questions if they do not understand.

## Deliberate You Conclusion!

15 minutes

During this part of the lesson, I invite students back to the carpet so we can discuss students strengths and weakness. I call on student volunteers to share first.  For instance, Peter explain that he did no like to use the distributive property to solve two-digit multiplication problems.

I asked him to explain why. He said, "Even though, the activity was fun, he took less time to do it the long way!" I asked him to illustrate it on the board. He began to solve the problem using distributive property. He could explain each step, and he did note the value of each digit, and why it was important to align the numbers correctly when multiplying.  However, he illustrated the same problem using the long method.  Using another strategy to solve this problem he still could explain why he align numbers according to their value. I ask the rest of the class to tell me the product of the problem Peter solved using long/distributive property. Students notice that even though Peter used two methods to solve the problem, he came up with the same answer. I ask students does it matter which method you use. Some students still wonder if using different numbers would they get the same results. I explain Yes, you would get the same results; however, some students might find it easier to solve this problem using the long method, or distributive property. I ask students which method is shorter. They all say distributive property.