Interacting through the Commutative and Distributive Properties

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We all know that multiplication can be learned in various ways whether it be by solving it visually through array models or equal groups, or by simply using repeated addition. Teaching math to our students with different strategies has endless permutations. Let us truly remember that teaching math to our students should always be engaging and interactive. This is primarily because we want to let our students feel that there should be nothing to worry about in learning mathematical concepts. Teaching math in an engaging and interactive way promotes the mindset of embracing math in our everyday experiences. What if your students still get confused? Let this simple activity I’ll share with you help you teach multiplication through the spirit of teamwork. Besides, if one student still does not get the lesson, there are other people including you and their classmates to help each other out. 

Commuting and Distributing

A great way to start teaching the commutative and distributive properties of multiplication are by telling your students that numbers are just like the people we see and meet in our daily life. Numbers interact with other numbers and operations in different scenarios. They react differently based on where they are located or how they relate to the other values and operations in the equation. Let your students imagine an equation like a train. A train is something that contains people sitting in different seats but leads them to the same destination. Similarly, the equation is something that contains numbers and operations for them to reach a specific destination, the final answer. 

When teaching commutative and distributive properties, the example of a train is a good one: it is something that contains people sitting in different seats but leads them to the same destination.

Once you’ve established this metaphor to your students, this is when you can insert your lesson about the commutative property of multiplication. Numbers are commuters! They can sit anywhere in the train and still arrive at the same destination. The commutative property is simply the property that assures the same value of the product even if the order of factors in the expression is interchanged. 

An image of a train can help explain the commutative and distributive properties of multiplication.

So, if you write the expression 4 x 6, it will have the same answer as the expression written as 6 x 4. Both expressions produce the answer 24. The order of the numbers inside a multiplication expression does not change the final answer. Similarly, if a commuter sits at the front part of the train, and another sits at the back, they will still arrive at the same stop even if they exchange seats. 

Moving on to the next property, the distributive property is about sharing for us to know the answer easier. The most common number to be shared is the number 5. For example, if the expression is 8 x 4, we know that the bigger number 8, can be made into 2 smaller numbers of 5 + 3. Now that smaller numbers are present, we can share 4 to 5 and to 3 as well. This will make the expression as (5 x 4) + (3 x 4). It’s as if 8 passengers in the train grouped themselves into 5 and 3 respectively. Then, each shared 4 snacks they brought to each other in their respective groups. 

When using a train, students can see how the product is the same with different commutative and distributive properties of multiplication.

With these metaphors, your students can appreciate the relationship of numbers in an equation. It is a great way to acknowledge the interactions and relations found in simple equations if we imagine our own simple interactions with people. 

Pass it on!

To apply the commutative property, you can try this simple activity with your students by just gathering different objects and passing it to each other. You can ask your students to bring any object they want to share to the class. The activity is simple: let your students pass their objects to each other in any order they want until the last person puts all the passed objects inside a big container. 

Once they pass it to each other in any order they want, ask them: Even though you passed the objects to each other in different orders, will the container have the same final objects inside? The answer is clear: yes! Just like interchanging the order of passed objects in the activity, interchanging the order of numbers in a multiplication problem will always have the same answer. This is the essence of the commutative property. No matter how the numbers travel or where the numbers are located inside a multiplication equation, the product remains the same. 

Once your students get the flow of the activity, you can incorporate number cards. You may give each of your students random single digit numbers. Ask them to find a pair to pass their number card to. With this, they will understand that numbers interact with each other commutatively in a multiplication equation no matter what order or placement they are in. For example, one student has a number card containing 4. If that student partners with another peer having a number card containing 6, they can understand that reading it as 4 x 6 or 6 x 4 does not change that the answer they will produce is still the same.

Now, you can also incorporate the distributive property using the same number cards. Give each student number cards of the same value. Then, ask one group to group themselves by 5 while the rest group themselves into a smaller group. Now, the interaction is distributed into two groups. The group of 5 students will share the same value of the number card and produce a certain product. Similarly, the smaller group will produce a certain product. Once each has a product, they can add their products to each other and get the final product. For example, you give number cards containing the value 6. Every student will receive cards of the same value. If you ask them to group by 5 and by 4. One group will answer 5 x 6 and 4 x 6 respectively. Then once they have the products, they can answer 9 x 6 by adding their respective products!

To end, there are many ways to represent and teach math. It can be a tool for camaraderie and relationships between peers. Numbers are not as different from the way we interact with each other. Inspire your students that they can learn math without the technical aspect of it. They can learn math by interacting with the things and people around us!

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