Properties of Addition and Multiplication

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SWBAT explain these properties in their own words and identify representative equations.

Big Idea

An early introduction to and individualized understanding these properties of multiplication and addition sets a great foundation for higher math!


3 minutes

I tell the students that today we are going to look at some different sets of equations to determine if there is a rule that consistently applies to them, and if so, what is the rule?  At the end of today I tell them that I expect them to be able to describe the rules in their own words and to also (possibly) explain how an awareness of this rules is mathematically useful.

Guided Discussion of Definition of Properties

15 minutes

In third grade, it is not necessary for students to know the names of these properties.  In fact, it can inhibit learning if the memorization of the names is emphasized over the deep understanding.  I have included the names in this lesson because my students, in my school district, are assessed on district and state tests that have not been updated to truly reflect the Common Core and that require students to identify the properties by name.  

I teach this lesson mid-year when students have already used many of these properties but there is no reason that it could not be taught earlier than that.

I project these examples of the commutative property and ask students to look and think (silent - I actually will time them for 60 seconds) about what patterns they see in the examples, what differences they see between the examples and non-examples, and anything else they notice that they think is significant.  I have intentionally included equations with the same answers so that they don't think that having the same end sum or product alone makes 2 equations representative of the commutative property.

I go through the same process with examples of the associative property and examples of distributive property.

In this lesson, after the discussion of each property, I also explicitly taught the vocabulary to provide ELL Vocabulary Support.

If students are struggling with the vocabulary I have found that side vocabulary assignments, such as creating a drawn or online animated vocabulary cartoon, can help them practice the word they need to know.

Is This Always True? (Testing the Properties)

25 minutes

Prior to asking students to work independently, I keep them at the carpet and go through an example from each of these sets with the whole class.  I always begin by Testing the Theories -On Level Equations  because that's the target level.  After I go through the first equation there and we talk about it, I will move to trying out one equation from Testing the Theories -Extra Support Equations.  

Students who choose to work on one of these two sets can come get an equation page and go to their seats to get started.  They know I will be coming around shortly after I get the third group started.  They are the students who have self-selected to try something more challenging because they are confident in their understanding of the process.  For this group, I go through the first equation from Testing the Theories Enrichment Equations. Also, I model how to show the work for each of the problems so that, for example, they see that I add the quantities within the parenthesis prior to adding on the 3rd amount in an equation that shows the commutative property of addition. 

When I first introduced these properties earlier in the year, I did not use any of the specific names and would not necessarily be doing so now were it not for the knowledge that my students will need these specific terms on the test.  For that reason, I've also created On Level EquationsExtra Support Equations, and Enrichment Equations pages that do not have the terms associative, commutative or distributive in them. 


20 minutes

I ask students to pretend to be a journalist reporting a breaking news story about a new property that has been discovered for addition/multiplication stories.  They get 5 minutes to jot down key words and decide what news station they are from and how they will introduce themselves and all those fun details.  Then they and their partner either present their breaking news story (about the commutative property/ about how addition equations in which the addends are flipped retain the same sum) to another pair of children or to the class, depending on available time.

I have the partnerships created prior to the activity to ensure maximum engagement and to give students the ability to develop their social skills while working with someone they may not ordinarily choose.