# Pre-Algebra 5 – Commutative & Associative Properties of Addition

Hello. I’m Professor Von Schmohawk
and welcome to Why U. In the first lecture, we explored
the origins of the first number systems. We also saw how the people
on my primitive island of Cocoloco first learned about the decimal number system. Once the Cocoloconians
discovered decimal numbers we could do much more than count coconuts. We could do arithmetic calculations with coconuts! The first arithmetic operations we invented
were addition and subtraction which came in very handy
when dealing with coconuts. For instance, if you have three coconuts and then your neighbor gives you five more you will have eight coconuts. Interestingly, if you start out with
five coconuts and then your neighbor gives you three more you will also have eight coconuts. For some reason, five plus three
gives you the same answer as three plus five. Eventually we figured out that it doesn’t matter in which order
you acquire your coconuts. You still end up with the same number of coconuts. Since the two numbers
on either side of the addition symbol can switch positions
without changing the answer we say that they “commute”
since commute means to travel back and forth. Mathematical operations in which
the numbers operated on can be switched without affecting the result
are said to be “commutative”. This illustrates what the Cocoloconians call The Commutative Property of Addition of Coconuts. Apparently, this property applies
to adding anything so we will just call it
like three and five if we call these two quantities A and B then we can write this property
in a more general way. Although addition is a commutative operation,
subtraction is not. For example, four minus three is one. However, if you switch the order of the three
and the four, the result will not be the same. Three minus four is negative one. As we saw in the previous lecture we can write four minus three as an
addition of positive four plus negative three. Now, since addition is commutative we can switch the four and the negative three
without changing the result. This is a trick which can come in handy
in algebra problems. Addition is a “binary operation”. Binary operations are mathematical calculations
which involve two numbers. These numbers are called “operands”
to produce a result called the “sum”. In addition operations, the operands are sometimes
a binary operation you may often see additions
involving more than two operands. This is possible because pairs of operands
can be added one at a time with each sum replacing the pair. In this way, an unlimited number of operands
can be added sequentially. The commutative property can be applied
to addition operations involving more than two numbers. By switching adjacent pairs of numbers,
operands can be reordered in any way we please. For instance, in this addition
involving four operands the two at the end
could be moved up to the front. Or the five could be moved to the back. In addition to the commutative property here is another interesting property of addition
that we discovered. Let’s say that you have five coconuts and your neighbor on the left has three. Both of you get together and pool your coconuts
into one group of eight. Then your neighbor on the right
gives you four more. Now you will have your group of eight
plus four more, for a total of twelve coconuts. On the other hand, let’s say you started out by pooling your five coconuts with your neighbor
on the right who had four coconuts. So you start out with a group of nine coconuts. Then your neighbor on the left
gives you his three so you end up with three
plus your group of nine or a total of twelve coconuts. You still end up with the same number of coconuts. In these two scenarios the coconuts were grouped
in different ways before they were added. However, we ended up with the same number. This illustrates what is called
The Associative Property of Addition because it doesn’t matter in which way
the coconuts are grouped or associated with each other
before they are added. In the end, they all add up to the same number. If we call these quantities A, B, and C then we can write this property
in a more general way. The commutative property of addition involves
moving around the numbers to be added whereas the associative property of addition
involves grouping them differently. Using the associative and commutative properties,
we can rearrange groups of numbers. Let’s see what this looks like
on a number line. As an example, we will take an addition problem
involving positive and negative numbers. Let’s start at the origin
and add positive two plus positive three plus negative six plus negative two plus positive four. This all totals up to one. However, because of the commutative property we are free to rearrange this sequence of
numbers in any order we like. For instance, we could add all the
negative numbers first and we will still get the same result. We could also use the associative property
to group some of the numbers to be added. For instance, the negative two
and the positive two could be grouped. Since this group adds up to zero,
we could replace it with a zero or eliminate it altogether. Having a familiarity with the
properties of addition allows us to start building a tool chest
of mathematical tricks which we can use later
to simplify complicated problems. In the next few lectures, we will explore
the properties of more arithmetic operations such as multiplication and division.

## 49 thoughts on “Pre-Algebra 5 – Commutative & Associative Properties of Addition”

1. nice coat and pipe

2. coconut math algebra

3. thats good…………..

4. this amuses me

5. Hilarious

6. Nice pipe

7. I will use the video for my bilingual class of maths. Thanx

8. A look behind the fundamental properties of the most basic arithmetic operation, addition.

9. Thank you soo much I finally get it!

10. Thanks this will be great to show my kids.

11. vids are so addicting it's like two AM

12. Who would have thought the most basic and simple rules of math can be so entertaining 🙂

13. These videos are awesome!!!

14. this is so cool !

15. Awesome…

18. This helped a lot thanks

19. Thanks

20. This was so incredibly helpful! My girl did not understand these concepts until she watched this video. Please keep making them because they are SUPERB!

21. Thank you !! This helped loads with my homework.

22. good job

23. I'm doing this for school:-(

24. Thx for the help though!:o3

25. Amazinggggg! Thaank youu

26. Blah, blah, blah

27. Thnx Sir for such an easy & helpful presentation.

28. Awesome! The 5th grade girls couldn't stop giggling about the coconut bras and they all learned the associative property! Thanks for the high quality lesson!!

29. Cool!

30. I LIKE IT

31. I love his "primitive island of Cocoloco". It stinks that they stop using it later on.

32. Wish you could do one for second graders! I teach them number strings–commutative and associative property.

33. thanks

34. I love your video series.

35. helps so much!

36. lol I love these

37.  I like how this generalizes to group theory. You actually thought group theory to children without noticing it, but then applied on familiar numbers. But all terms here can be used when talking about matrices or other stuff!

38. Oh, professor! A slip of the tongue! So near to perfect yet human after all – thank goodness! Only one letter in one word in the 83 videos I have downloaded – incredible! Check it out at 6:33. What a brilliant series – I look forward to many more episodes in the decades to come. Thank you, thank you, thank you!

40. Reminds me of the Kingdom of Mocha.

41. Why the fuck wasn't this taught like this in middle/high school? I got some garbled "process oriented" memorize this crap instead of actual theory.

42. Coconuts

43. He maintains that base ten number systems were "discovered." Mathematics is a social construction, base ten number systems and mathematics as a whole are not discovered, they are culturally agreed upon what some philosophers would call an artifact. Overall though, I will use this video to supplement my math class, thanks MuMathU.

44. Do one with multipication

45. What a beautiful work. Thank you very much sir.

46. This is a perfect video, like how it ties together at end!

47. I can follow this…a miracle has occurred.

48. cooooocoooonuuuuts

49. now I understand how coconut works.