Friday, July 17, 2015

Commutative versus Associative Property...

These properties with rather lengthy words are often mixed up. I'd like to invite teachers to comment or tweet me with how they would approach this and what they would or wouldn't do. This is a dialogue with T meaning teacher and S meaning student. I think this could be integrated into a multiplication number talk when it comes to the end of the dialogue.

T: What is another way to write 6 + 3 and still get the same answer?
S: (3+6)
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?
S: Yes.

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.

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