T: What is another way to write 6 + 3 and still get the same answer?
T: That is a property in math called the commutative property. We can show it as an equation and a visual using integer tiles.
6 + 3 = 3 + 6
(++++++) + (+++) = (+++)(++++++)
T: How are they different? How are they the same?
S. They give you the same answer, 9. It doesn't matter which order you add in.
T: Right. So, the Commutative Property of addition states that you can add in any order, and still get the same answer. Commutative = different ORDER
T: Does this work for multiplication also?
T: Can you give me an example?
S: Well, 3 times 4 is the same as 4 times 3.
T: OK, what are some ways we can write that.
S: 3x4=4x3, 4*3=3*4. 3(4)=4(3).
T: All of those are good, remember to avoid using the x so we don't get it confused with a variable. Let's use the last way, with parentheses. How can we diagram the difference between those 2 expressions with integer tiles?
S: (May not know how to do this).
T: OK, multiplication is repeated addition so we can make three groups of 4 and 4 groups of 3.
3(4) = 4(3)
(pretend the parentheses are circles in the drawing)
T: Can someone summarize what this diagram means?
S: 3 times 4 is the same as 4 times 3. Both have a product of 12. So, the Commutative Property of multiplication says that you multiply in a different order and you still get the same answer.
T: Now we haven't talked about the other property. Which one have we not talked about and what do you think it means?
S: We haven't talked about the Associative Property.
T: Ok, what do we know about the order of operations?
S: (yelling) PEMDAS!! Parentheses first.
T: OK. So, which expression here is an easier computation: 19 + (36 + 4) or (19 + 36) + 4 and why?
S: The first one. 36 plus 4 makes 40 and I can add 40 and 19 in my head. 36 and 4 are friendly numbers.
T: How are the two expressions the same? How are they different?
S: The parentheses moved! That's a difference.... Something that's the same? Oh yeah, they get the same answer!
T: Are they in the same order?
S: Umm, yeah.
T: Right! So this demonstrates the Associative Property of Addition. Both expressions have the same answer and order, but different grouping.
T: Now for our final property... Same question! Which of these expressions is an easier computation, when following the order of operations: 2*(5*16) or (2*5)*16?
S: I would say the second one. Two times 5 is easy 10. And when you multiply 10 by 16 all your doing is putting a zero at the end of the 16. The answer is 160.
T: Awesome. You just demonstrated the Associative Property of Multiplication.
This is a dialogue that has happened in my 7th grade class last year, probably not as smoothly though. Any feedback or questions are appreciated.
A worksheet to practice this is on page 4 on this web site: