Properties of Operations are those three pesky words that reoccur throughout the common core standards. Some of you may be wondering exactly what they mean. Properties of Operations are the foundation of arithmetic; we use them when performing computations and recalling basic facts.

In this post, I will focus on the following 3 properties that are used with addition and multiplication:

**Commutative Property****Associative Property****Distributive Property**

**8 x 2 = 2 x 8.**Although both equations represent the same amount or product, representing them with a visual model looks different (eight groups of two versus two groups of eight). See below.

An idea to help with the commutative property may be:

- Providing students with tiles or counters and asking them to model 8 x 2 & 2 x 8; 3 x 4 & 4 x 3, 5 x 3 & 3 x5. Then ask the students to compare the product of each factor pair (ie. 8 x 2 and 2 x 8). Students should be able to explain that the products are identical. To sum it up, students should model, model, model multiplication equations in any order to see if the products are always identical.

Next, the associative property states that changing the grouping of the factors does not change the product. This property works closely with the commutative property because we often change the order of groupings of factors when multiplying numbers to make it easier to solve problems.

Using 3 x 2 x 2 as the basis of our groupings, below you will see a visual model of how the associative property works.

- Providing the students with counters or tiles and asking them to model (3 x 2) x 2 and then 3 x (2 x 2); like the example above. From this point allow the students to determine patterns they notice. Consider asking the students:
*What is the product of both expressions? Why do they think the product is the same for both?*Try this investigation with different expressions including: (5 x 2) x 2 and 5 x (2 x 2), (3 x 4) x 2 and 3 x (4 x 2). Students should begin to generalize that changing the groupings does not change the product.

- Asking students to model 2 groups of 6 (2 x 6) using connecting cubes and find two hidden facts inside of 2 x 6 like 2 x 3 and 2 x 3. Allow the students time to discover as many hidden facts inside of the 2 x 6 that they can. Try this out with different multiplication facts through 10 x 10. Do not forget to ask students what they notice as they are doing this investigation.

Jonathan Crabtree says

Hi Greg. This is great info.

In my experience, teachers often miss the 'second' Distributive Property. Let me explain… Teachers sometimes say integer multiplication is repeated addition because multiplication distributes over addition. For example:

2 × (+4) = 2 × (0 + 2 + 2) = 2 × (0 + 1 + 1 + 1 + 1) = 0 + 2 + 2 + 2 + 2 = 0 + 8 = 8

Yet multiplication also distributes over subtraction. For example:

2 × (-4) = 2 × (0 – 2 – 2) = 2 × (0 – 1 – 1 – 1 – 1) = 0 – 2 – 2 – 2 – 2 = 0 – 8 = -8

So integral multiplication of 'a multiplied by b' written as a × b involves EITHER a added to zero b times in succession OR a subtracted from zero b times in succession, according to the sign of the multiplier.

If you and your readers would like to know more about multiplication, you can watch my 'Lost Logic of Elementary Mathematics' slideshow, (online via Microsoft OneDrive), at:

https://1drv.ms/p/s!AiiJ6XgphELidETf6CoiWWpuGec

My elementary math paper exploring multiplication is at http://bit.ly/LostLogicOfMath

Best wishes to you and your readers for a happy and healthy 2017!

Jonathan Crabtree

http://www.jonathancrabtree.com/mathematics

Claire Corroon says

While I love your area models I often find the younger children can better grasp the commutative property via a group model e.g 2 plates of 4 cookies = 2 x 4 whereas 4 plates of two cookies looks much different, 4 x 2. Still 8 cookies in total though! I use the phrase "same value, different appearance" to help explain it.

See more here: https://sites.google.com/site/primarycpd/latest-news/multipleways

Claire

Greg says

Thanks for your comment Claire. It love the phrase same value, different appearance.

Greg

kbenedick says

When I downloaded your Quick Reference Sheet for Properties, I get a file 404 error message. Am I too late?

Greg says

I am sorry to hear that you are having difficulty downloading the Quick Sheet. Don't worry you are not to late, it is free FOREVER : ) I just checked the link myself and I was able to downloaded with no problem. Please email me at mrelementarymath@gmail.com and I will ensure that you receive it.

Greg

angie clark says

This activity covering properties is a great way to get student to learn the difference between commutative, associative and distributive properties. I especially love the idea of the visuals, which gives them a go-to on what is correct. Great Activity!

TheElementary MathManiac says

As usual, your models look great!

Tara

The Math Maniac

Greg says

Thanks Tara! It took a while, but I really wanted to represent the properties with clear images because I am a visual person. I appreciate your comment. It makes the work worth it!

Greg