Commutative and Associative Properties

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Objective

SWBAT distinguish between the commutative and associative properties of math.

Big Idea

Students will learn about the commutative and associative properties.

Do Now

6 minutes

For a quick assessment of prior topics, I will post two Do_Now problems on the board for students to complete.

Write the number below in words and in expanded form:

3,572,924,000

Use parentheses to make the equation true.  (Remember order of operations)

16 - 10 + 2 x 4 = 44

Students will have 5 minutes to work on the problems independently in their notebooks.  Then, we will come together as a class to go over the answers.  For each question I will have a student come up to the board to share their answer.  If a student disagrees, I will ask him to explain his answer (MP3).  At the end of the discussions, I will share the correct answers with the class and clarify any confusion.



Hook

5 minutes

During today's hook, I will share with students a story about how I commute to work:

Most adults commute to work everyday.  They take the same route everyday.  I commute to work using the bus.  I travel from my home to school and then back home from school.

Then, I share with students the equation version of my commuting story, which leads to an informal introduction of the commutative property: 

home + school = school + home

Next, I share a story about my childhood to motivate our learning of the associative property:

Do you have cliques at school?  I recall from my childhood that some girls would only associate with certain other girls. In middle school, I was good friends with two girls, Jennifer and Elisabeth.  Unfortunately, Jennifer and Elisabeth were not friends with each other. They were very different.  Jennifer was very studious and loved to read. Elisabeth was very outgoing and loved to talk on the phone and gossip.  They didn't have much in common and they didn't get along well.  As a result, I had sometimes had to choose who I was going to associate with. Sometimes I would associate with Jennifer, which left Elisabeth by herself. Sometimes I would hang out with Elisabeth, but Jennifer was by herself.

I share with students the equation version of this story as well:

(Jennifer + Ursula) + Elisabeth = Jennifer + (Ursula + Elisabeth)

Mini Lesson

32 minutes

I will use my Smartboard to display the work for today's mini lesson. Here are images of two important slides:

Discussing the Commutative Property

Both addition and multiplication are commutative.  This means that the order doesn't change the sum or the product.

Example 1:  5 + 3 = 3 + 5

I will ask the class if this equation is true.  Do both sides of the equation equal 8?  Students should realize that the order of the numbers does not change the answer.

Example 2:  7 x 4 = 4 x 7

I will ask the class if this equation is true.  Do both sides of the equation equal 28?  Students should realize that the order of the numbers does not change the answer.

I will randomly select 4-5 students to share their own examples of the Commutative Property. Then I will randomly select 8 students to come up to the SmartBoard and practice with some examples.  This will give me a quick assessment of students' understanding of the topic.

Associative Property

Both addition and multiplication are associative.  This means that grouping addends or factors will not change the sum or product.

Example 3:  15 + (85 + 24) = (15 + 85) + 24

I will ask the class if this equation is true.  Do both sides of the equation, following the order of operations, equal 124?  Students should realize that the grouping of the numbers does not change the answer.

Example  4:  (3 x 2) x 4 = 3 x (2 x 4)

I will ask the class if this equation is true.  Do both sides of the equation, following the order of operations, equal 24?  Students should realize that the grouping of the numbers does not change the answer.

I will randomly select 4-5 students to share their own examples of the associative property.  Then I will randomly select 8 students to come up to the Smartboard and practice with some examples. This will give me a quick assessment of students' understanding of the topic. 

 

Compatible Numbers

In my class I treat Compatible Numbers as an extension of the Associative Property. Most of the time, my students demonstrate enough understanding of the Property that we can quickly transition to Compatible Numbers.

The major point that I try to get across is that by applying the Associative Property intelligently, calculations can be made easier by grouping compatible numbers.

Example 5:  15 + (85 + 24) =

I'll ask my students, "How can we re-group this to make it easier to solve?" My students should realize that grouping the 15 and 85 which sum to 100 will make the problem easier to solve.

Example 6:  Calculate 60 + (40 + 83)  mentally

Example 7:   Calculate 25 + (25 + 41)  mentally

After a few problems like these, I will ask my students to come up with their own definition of compatible numbers, as well as an explanation of why compatible numbers are useful.

Homework

2 minutes

Students will be assigned a series of problems from their textbook. Like yesterday's assignment students will complete a series of problems where they are asked to identify the relevant property by looking at an example:

9 + 6 = 6 + 9     is the Commutative Property for Addition