Kimberly Crowther
*The following is taken from “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics” by Randall I. Charles, page 8. I found this document on Blackboard.
BIG IDEA #6 PROPERTIES: For a given set of numbers there are relationships that are always true, and these are the rules that govern arithmetic and algebra.
Examples of Mathematical Understandings:
Properties of Operations
• Properties of whole numbers apply to certain operations but not others (e.g., The commutative property applies to addition
and multiplication but not subtraction and division.).
• Two numbers can be added in any order; two numbers can be multiplied in any order.
• The sum of a number and zero is the number; the product of any nonzero number and one is the number.
• Three or more numbers can be grouped and added (or multiplied) in any order.
Properties of Equality
• If the same real number is added or subtracted to both sides of an equation, equality is maintained.
• If both sides of an equation are multiplied or divided by the same real number (not dividing by 0), equality is maintained.
• Two quantities equal to the same third quantity are equal to each other.
Three Basic Mathematical Properties:
1. Associative
2. Commutative
3. Distributive
A solid understanding of all three of these properties is necessary for every level of math. A young child learns them and continues to use them throughout the rest of their mathematical education and life. Therefore, if any of them are incorrectly taught or learned at any level, it hinders a student’s progression to further mathematical understanding and application. To see the importance of these properties, we will look at each of them at different levels of mathematical understanding.
I found a website, http://www.onlinemathlearning.com/numberproperties.html, that I thought really explained these properties very well. On this website, there is also a video available that shows a teacher teaching these properties to his class using a smartboard. The following three explanations of these properties are taken directly from that website.
Associative
An operation is associative if a change in grouping does not change the results. This means the parenthesis (or brackets) can be moved.
Numbers that are added can be grouped in any order.
For example: 
(4 + 5) + 6 = 5 + (4 + 6) 

(x + y) + z = x + (y + z) 
Numbers that are multiplied can be grouped in any order.
For example: 
(4 × 5) × 6 = 5 × (4 × 6) 

(x × y) × z = x × (y × z) 
Numbers that are subtracted are NOT associative.
For example: 
(4 – 5) – 6 ≠ 4 – (5– 6) 

(x – y) – z ≠ x – (y – z) 
Numbers that are divided are NOT associative.
For example: 
(4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6) 

(x ÷ y ) ÷ z ≠ x ÷ ( y ÷ z) 
Commutative
An operation is commutative if a change in the order of the numbers does not change the results. This means the numbers can be swapped.
Numbers can be added in any order.
For example: 
4 + 5 = 5 + 4 

x + y = y + x 
Numbers can be multiplied in any order.
For example: 
5 × 3 = 3 × 5 

a × b = b × a 
Numbers that are subtracted are NOT commutative.
For example: 
4 – 5 ≠ 5 – 4 

x – y ≠ y –x 
Numbers that are divided are NOT commutative.
For example: 
4 ÷ 5 ≠ 5 ÷ 4 

x ÷ y ≠ y ÷ x 
Distributive
Distributive property allows you to remove the parenthesis (or brackets) in an expression. Multiply the value outside the brackets with each of the terms in the brackets.
For example: 
4(a + b) = 4a + 4b 

7(2c – 3d + 5) = 14c – 21d + 35 
What happens if you need to multiply (a – 3)(b + 4)?
You do the same thing but with one value at a time.
For example:
Multiply a with each term to get a × b + 4 × a = ab + 4a
Then, multiply 3 with each term to get “ –3b – 12” (take note of the sign operations).
Put the two results together to get “ab + 4a – 3b – 12”
Therefore, (a – 3)(b + 4) = ab + 4a – 3b – 12
Citation: http://www.onlinemathlearning.com/numberproperties.html
Lesson Plans:
1^{st} grade from UEN website
http://www.uen.org/Lessonplan/preview?LPid=21440
2^{nd} grade from UEN website
http://www.uen.org/Lessonplan/preview?LPid=18872
4^{th} grade from UEN website
http://www.uen.org/Lessonplan/preview.cgi?LPid=21642
6^{th} grade from UEN website
http://www.uen.org/Lessonplan/preview.cgi?LPid=6392
Great website with fun activities and technology instruction:
Games for practice and understanding:
1. Powerpoint to practice with the whole class (you can make one that includes all three properties or just whatever one you’re focusing on at the time).
2. Flash cards for partners to play against each other and see who can match number sentences together the quickest (could also be a memory game, but not as effective for actually teaching math).
As we can see, knowing the properties of math is important for every grade level and is significant in every part of mathematical understanding. It is vital for students to understand the basics in order to continue strengthening their math knowledge and success.