Is 3x5 the same as 5x3?

Factual Questions
21

There’s nothing new here. Over 50 years ago I was learning new math in elementary school. It was the same as old math except for these little games that were supposed to teach us something besides memorizing tables. I didn’t notice any greater ability to understand math in the kids I went to school with. 20 years later my friend’s kid was in elementary school and they had another set of new math skills kids were supposed to learn with no greater effect. 10 years after that my own kids were in school and they had another twist on it, with no better results. Now there’s another round and I predict the same results. Each time there are parents complaining because they don’t understand the assignments, the same set of kids who can’t get the right answer on the test because they don’t care, and most of the kids will learn their arithmetic just like the rest of us because it isn’t all that difficult and the new concepts aren’t teaching them anything they wouldn’t figure out over time.

So now we teach them there’s a difference between 3 sets of 5 and 5 sets of 3, a difference that makes no difference to them, or in terms of any math they’ll be doing anytime soon makes no difference at all. The kids who figure out the basic rule, give the teacher the answer they want, those kids will get it right on the test without caring why, and the others will get it wrong just as they would no matter what the question was. If there’s a new teaching technique needed it’s not in new math skills.

There is great irony here for me. When it came to new math and understanding basic arithmetic in elementary school I excelled. Later in school when math became more complex I lost interest and now I’m about as dumb as you can get in the field of math. Thank Og for computers.

22

I think the correct way to say this is that 3x5 and 5x3 are the same value.

23

Or maybe they actually understand the underlying concept.

You can learn that multiplication is commutative without being taught this by a teacher - it’s a fundamental property of numbers. A 3-year-old kid may notice that a 3x5 rectangle of square blocks is exactly the same thing as a 5x3 rectangle.

24

This is the crux of the issue right here. The OP’s question can only be answered definitively by citing a single, commonly accepted definition for what an expression like “3x5” means; but there is no such thing.

Some posters in this thread have been assuming that multiplication denotes repeated addition; but that notion is controversial: see Is Multiplication Repeated Addition? or What Exactly Is Multiplication?

25

5x3 to me means = 5 multiplied by 3 = 5+5+5
example: You get $5 three times. That’s $5+$5+$5

But then again there is: the five times 3
example: You get 5 times 3 apples. That’s 3 apples+3 apples+3 apples+3 apples+3 apples

26

Well, since I do a lot of printing, 3x5 and 5x3 are (usually) different, so I visualize 3x5 as 3 width and five height, so I’d’ve been marked wrong on that test, too. (The test I saw was 4 x 6, and the array drawn is exactly 4 units across by 6 units high.) Mathematically, I never remember being taught that 3 x 5 is specifically three groups of five as opposed to three five times.

27

That’s a language issue though, not a mathematical one. “5x3” and “five times three” are not quite the same: the first one is understood by people the world over, the second one not so much. In other languages the word in the middle is something that can’t be translated as “times” for any context other than multiplication.

28

Yes and yes. Another successful “encounter”…

29

Wait, the wrong reason? Are you saying multiplication is NOT commutative? Because if it is, this student’s reasoning is perfectly valid.

Shouldn’t they should be congratulated for figuring things out correctly by themselves? Obviously, if they reason that subtraction is commutative, that’s incorrect, but that’s not what they’re doing here; they’re in fact right. And, in fact, they are showing a deep understanding of what multiplication is, if they can work out properties like commutativity.

The teacher did a bad job grading this. Not saying they’re completely incompetent; we all have bad days and make occasional mistakes. But it shouldn’t have been marked wrong.

Well, a good teacher would congratulate the student on figuring out a rule of geometry (assuming the student got the rule right; if not maybe discuss what the correct rule is, and why), and then – if the goal is really protractor practice – revise the homework for next time so no students can take that shortcut [Or, explicitly say to measure every angle with the protractor, then ask them if they notice anything about the total of angles…]
No doubt, when young Gauss found a shortcut for adding up numbers, he should have been told he didn’t understand math, and been steered into ditch digging or something as a career.

30

OOOOO
OOOOO
OOOOO

There are 3x5 = 5x3 O’s there. Three rows of 5, 5 columns of 3 , whatever.

31

Ultimately this is a semantic issue, not a mathematical one, and the teacher ought to be on the ball enough to realize it, and not ding the kid for his language skills instead of his math skills.

32

This.

Trying to imply that 3x5 =/= 5x3 is fundamentally incorrect is, in itself, non-mathematical.

33

Yes it is. 3X5 doesn’t mean anything in particular at all. It could be a part number. But I’m sure the kids were instructed on how the question should be answered on the test. Everybody can get off the math aspects of this and look at the skill that isn’t being developed, how to follow instructions.

34

You could interpret the question as “3 groups of 5” or “5 groups of 3” ,either way its 15. I have a B.S. in chemistry with a strong math background. I fail to see how the student was wrong. Maybe I’m dumb, who knows. I just hope the “new math” goes the way of the “new coke” (I’m dating myself there…)

35

Here’s one thing to consider:

In algebra, whenever a numerical constant is multiplied by a variable, the convention is that the numerical coefficient is always written first. So, for example, you’d see expressions like 3x or 5y or 12x², but never x3 or y5 or x²12, even though those would have the same value.

If you were solving a problem that asked “How many $3 tickets were sold?” you might use the expression 3x. If you were solving a problem that asked “How much does a ticket cost if 350 of them were sold?” you’d use 350x.

This strikes me as evidence that there is not any particular significance to the order in which the factors are written in multiplication (though it could also be interpreted to mean that any such significance has been obscured for the sake of notational convenience).

36

I’m guessing neither of you read my entire thread. I specifically gave examples that showed that the child didn’t correctly come up with a quicker way to answer the problem, but just stumbled upon a shortcut that they didn’t understand.

In one example the child just happened to figure out that they could take 3x5 and work it as 5x3 (and more likely just did it backwards by accident), however, when that child tried to rearrange subtraction problems, it didn’t give the same answer.

In the second example I explained that the student found that if you add the interior angles of a triangle it came out to 180 degrees, however when they then continued to apply apply this shortcut to other shapes.

Yes, these shortcuts are valid, but they clearly weren’t ready to use them and probably should have just stuck to their actual homework. Learn how to do multiplication properly so when you come across a problem where you can’t rearrange it, you can still do it. Learn how to use a protractor so when you’re not measure triangles or squares you can still find the answer. Math, in general, isn’t the place to skip right to the shortcuts. At least not with the basic building blocks of arithmetic. And besides, maybe you should be able to do both 3x5 and 5x3.

Lastly, just because you find a different way to do the problem doesn’t mean you get to not do your homework the way you’re being asked to do it. Not only are you not learning what the teacher is attempting to teach that day (and thus falling behind) you may also be doing yourself a disservice by learning something incorrectly.

PS, for all the people that say ‘hey, the kid found a great way to make multiplication easier, he should be rewarded for making it easier on himself’. What if he just refused to multiply and only added? Would that be okay? I mean, if he finds that easier, why learn how to multiply?
And, FTR, I’m not a fan of the ‘you did it backwards, it’s wrong’, but I do believe you should understand each level of math before going to the next. IOW, you shouldn’t be using the commutative property because you just happened to find it.

37

I’m not even sure if the student was using the commutative property. If I was asked the same question I would have given the same answer, since I mentally read it as <5>, <times 3>. If there’s some kind of universal agreement on how to read such expression I missed it all the way to my engineering degree.

38

I have a recipe file. Most of the cards are 5x3, but a few of them, with longer recipes are 3x5.
They all come in the same pack.

39

Damn, augend. How could I have gone my whole life without knowing that word, and that it means something quite specific?

Actually, more seriously, I can see myself being taught, or demanding to know, the equivalents of “augen” and “added” as concepts before learning how to add: it has seriously impeded learning new general technical and intellectual skills to this day or trying to figure out human relations problems, and it can be terribly hampering. (Psychiatrists and therapists, for example, use terms at that level, but most of their patients just have to learn how to get along.)

Which is why, luckily, I never learned those two words, and learned how to add.

40

When the goal is for students to understand something, it is always wrong to stop a student from proceeding to do something the way the student already understands it unless the way the student already understands it is fundamentally wrong.

The usual culprit in the operation is that the teacher assumes that all the students lack any understanding until that understanding is conveyed to them via the lesson. That’s arrogance, and dangerous arrogance at that.

A lot has been said about the need for education to be flexible. Flexibility starts with suspending the notion that the directional flow of learning is in one direction (from the school to the student) and that the student’s role is a passive one, soaking up correct information and correct methodologies and correct explanations like a paper towel soaking up a coffee spill.

The student who writes 5+5+5 instead of 3+3+3+3+3 is probably aware, either from untaught experience or from being specifically taught “this is the commutative property of multiplication”, that it doesn’t freaking matter which way you do it. It is morally wrong, not merely factually wrong, to unteach that in order to teach whatever concept the teacher’s lesson plan was designed to convey.