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Order is essential in the definition of multiplication because not all forms of multiplication are commutative, such…

Order is essential in the definition of multiplication because not all forms of multiplication are commutative, such as matrix multiplication. This is why it is taught as a separate property.

Originally shared by Science on Google+

Why Was 5 x 3 = 5 + 5 + 5 Marked Wrong?

It seems absurd at first glance: we all know that 5 x 3 is equal to 3 x 5, which is 15. But check out the formal definition of multiplication:

The multiplication of two whole numbers, when thinking of multiplication as repeated addition, is equivalent to adding as many copies of one of them (multiplicand, written second) as the value of the other one (multiplier, written first).

In other words, 5 x 3 = 3 + 3 + 3 + 3 + 3

Why should this matter? It matters because the term equal is not the same as equivalent. Although 5 x 3 is equal to 5 + 5 + 5 it is not equivalent to 3 + 3 + 3 + 3 + 3. Suppose you were buying chocolates for your sweethearts on Valentine’s Day. You would have 3 boxes of 5 chocolates each in one case, and 5 boxes of 3 chocolates in the other case. What you choose to buy depends on how many sweethearts you are trying to impress, right? 

Perhaps more importantly, the difference is also a fundamental concept in computer science. 

Notice that the second problem is marked incorrect as well. That’s because keeping rows and columns straight in matrix multiplication is important. As explained in the link below: “Order is essential in the definition of multiplication because not all forms of multiplication are commutative, such as matrix multiplication. This is why it is taught as a separate property.”

So, what do you think? Do you agree with the teacher or not?


Join the Conversation


  1. C Ames the point that the poster is making is that the distinction between 5 x 3 and 3 x 5, i.e. that of non-equivalence, becomes important later in computer science. That’s why the concept is important early on. The second link has a longer explanation. 


  2. At the primary school level, marking wrong for order is blithering idiocy. You’re only dealing with scalars at this point, and for scalars multiplication is commutative. In fact, that’s one of the ways that kids figure out what commutative is when doing scalar arithmetic: multiplication and addition are, subtraction and division aren’t.

    There are also overloaded uses of the + operator where A+B B+A, but we don’t mark kids down for not knowing that until we’ve taught them the different operations in the first place.


  3. WTF!?!?!?!  Equivalence has absolutely nothing to do with either of those two questions.  The teacher really is an overly pedantic idiot.  Whomever wrote the attempt at justification above is way off base too.


  4. Sorry, but in software development we do optimizations like that all the time.  If you can refactor a an algorithm to use fewer CPU cyles and still come out with the same answer, by all means do it!  (Assuming it doesn’t impact program maintainability etc.)


  5. These concepts are taught in Grade 3 as described in Common Core standards:

    Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

    Check it out:


  6. You’re not reading that correctly Science on Google+ 

    Note also w.r.t. q#2 on the paper this from the page you refer to:  _”Students understand that rectangular arrays can be decomposed into identical rows or into identical columns”_


  7. Why negate ? psychology studies again and again say that “reward” is a much better motivator. If I were the teacher, I’ll give those two questions +1 each, and if they write down both 5+5+5 and 3+3+3+3+3, I’ll give them +2 points.


  8. I think this is overly pedantic and is against any the associative property, giving further confusion down the line while learning mathematics. With such strict rules about what multiplication is, how do you go about teaching kids how to solve for x?

    5x = 15

    “You can’t just move numbers around!” I already hear this enough from people who don’t understand math very well. I can only imagine how terrible peoples understanding of math will be when you constantly give them rules and change them.

    I also agree with the first comment on the medium link:

    “However, (attempting to play devils advocate), I would say that the only downside to this method of teaching math is that it squashes ANY chance at creativity. Math is already a subject with little room for individual creativity, and this method eliminates any chance at that. It shows the children that unless they do the problem EXACTLY the way the teacher wants, they are wrong. What does this do to kids as they get older?How are they going to solve the worlds problems if they are limited to only solving problems the way their bosses see fit? As you mentioned, it does have benefits (for CS and matrices), but looking at the larger picture, how valuable is creativity in this day and age?”


  9. The boxes of chocolates analogy doesn’t really explain anything. 5 × 3 in such a context would solely pertain to the chocolates, not some new arbitrarily-introduced “sweethearts” value. Who’s to say that I didn’t build my own giant box and just got the boxes of chocolate to fill said box with to give to one sweetheart?

    Also, the commutative property is usually taught well before matrix arithmetic (at least in a sane education system); unless it is indeed the case that this is supposed to be some intro to matrices (which is not at all clear), the point is definitely pedantic anyway for what looks like elementary-school math homework.


  10. That makes no sense and it’s perfectly ok, because when dealing with real numbers both multiplication and addition are commutative operations. Sure, in some algebras they aren’t, but when dealing with real numbers, they are.


  11. Bottom line not all teachers remain smarter than the pupil. Hopefully the parent is at the school seeing that the grade is changed. Lol looking at this the theme song for are you smarter than a 5th grader popped in my head.


  12. I’d also like to bring up that 5 and 3 are just a bunch of 1’s added to each other.

    (1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1) = 15

    (1+1+1+1+1)+(1+1+1+1+1)+(1+1+1+1+1) = 15

    I’d like to bring up the argument that we are pretty much complaining about where to put parentheses.


  13. At first I read some of these comments and I could understand why you might interpret 3 x 5 as 5 three times, but then I realized that these are Arabic numbers! We read math and do it from right to left! Our primary schooling has given us the tools to get on with the learning but in fact or pronunciation is dyslexic.


  14. Like the example of sweethearts. I should say that I prefer 5 boxes of 3 candy each one than a set of 3 ones with 5 candy, because in the first case you have 5 sweethearts and not just 3 ones.

    Is it BTW some kind of modal logic or just pure biased calculation?


  15. Noah Hester I do believe that this is 2nd GRADE work, so I don’t think that it matters, but look at it, the array the child did was 4 across and 6 down. The correct answer in the teachers mind was 6 across 4 down.


  16. This whole exercise is why most of you will be poor all your lives.  For this way of teaching only teaches conformity and does not allow for independent thought. 

    Thus training an army of drones that cant think of new and inventive ways of doing things and instead tell everyone the same old song.  Do it our way or fail.  

    This is why our current Dewey/Rockfeller system is causing the failure of America.


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