There’s an image floating round of a student being marked wrong for writing “5+5+5” as the “repeated addition strategy” to solve “5x3”. The correct answer is apparently “3+3+3+3+3”. “5+5+5” would be the correct answer if it was asking about “3x5”
I’m not actually interested in whether the teacher was “right” (based on how the subject is taught). I’m interested in whether mathematically it’s wrong to say that 5x3 is “5+5+5”.
I think of 5x3 as being “five three times”, though of course I remember being taught to think of it as “five groups of three”. I always assumed though that that was simply a way of explaining it to kids. But in my mind both answers are correct and 5x3 and 3x5 are the same thing.
It’s called the commutative property, and is a pretty vital concept in basic algebra (and trig, and likely higher level math, but that’s what I’m familiar with).
But just as a literal matter, does “5 X 3” mean to communicate “five groups of three” or “three groups of five”?
That “5 X 3 = 3 X 5” is a different issue.
When you write those symbols in that specific order, Are you saying that “five groups of three equals three groups of five” or are you saying that “three groups of five equals five groups of three”?
I’m interested in whether mathematically it’s wrong to say that 5x3 is “5+5+5”.
Yes, it is wrong to say that 5X3 is 5+5+5.
Think of it as five times the number three. That is obviously 3+3+3+3+3.
You can’t forget that 3x5 and 5x3 are different expressions with different meanings. The fact that 3x5 and 5x3 are mathematically equivalent is a different issue.
Why should I think of it that way? Why not five, three times. “X[number]” after a listed item is a common way to indicate multiples of that item. What objective basis is there for your reading of the mathematical expression, versus this one?
Seems like the teacher (or test designer) is trying to make an incredibly stupid point. By the same logic, how would you show that 0*3=0? Leave a blank space?
Or, lets say you are required to calculate 5 apples * 3. How do you add 3 to itself 5 apples times?
Well, historically speaking, the first number is called the multiplicand, and it is considered the number that you are starting out with. The second number is called the multiplier, and it is the number that expresses what you are doing to the first number. So, 3 × 5 means 3 + 3 + 3 + 3 + 3, and 5 × 3 is 5 + 5 + 5.
It’s similar with addition, with the first number being called the augend and the second the addend.
None of these terms are common, of course, since both addition and multiplication are commutative, but this is the opposite of what the OP says is correct.
This definition of multiplication only works for positive integers (or natural numbers). You need new definitions of multiplication for negative integers, for rational numbers, for real numbers, and for complex numbers.
This is something that people who want to complain about the Common Core have been passing around. In fact, it has nothing to do with the Common Core. Opponents of the Common Core have been deliberately creating random bits of silliness to show what is supposedly wrong with it. Nothing like this is mentioned in the Common Core. Here’s an explanation of it:
As to whether any teacher actually, on their own, ever wrote a test like the one mentioned, well, who knows? Lots of teachers are pretty vague about the subjects they teach. Perhaps some have been telling their students that 5 x 3 means 3 + 3 + 3 + 3 + 3, and they will be marked wrong for saying that it means 5 + 5 + 5. That’s a pretty useless arbitrary distinction. It wouldn’t be the first time a teacher has tried to enforce some ridiculous arbitrary way of answering questions on tests.
I have strange bugs that each have 3 legs and there are 5 of them. Clearly that is 3+3+3+3+3
But is that:
3 legs/bug x 5 bugs or 5 bugs x 3 legs/bug? Which sentence applies and which one does not? Or are both correct?
Both are correct. And 3x5 can also be describing both 3 legs a bug with 5 bugs or 5 legs per bug with 3 bugs. And realizing that guides you to know that you can just count by 5s if you need to, rather than 3s, and by 25s, 50s or 100s instead of 3s too.
Yes something we should have all learned in second grade and that is part of common core standards for second grade.
Sure. But non-commutative multiplication won’t show up until you start with matrices, and that will be in high school at first. Why bother with the order of factors before that?
They are the same, and there is no one definition of what 3x5 means. The multiplicand and multiplier are no more–they are just both factors.
There is an example in Common Core that would seem to indicate that 5x3=5+5+5, but it is given as exactly that, an example. 5x3=3+3+3+3+3 just as well.
And there is problem with teachers having to suddenly teach a new curriculum that they don’t understand, and this is indeed caused by Common Core. It encouraged a lack of local control on mathematics curricula and allowed a new market that was explicitly Common Core friendly. School administrators, looking for what was cheaper, have settled on a method that is commonly called “common core,” even if it is not actually mandated by that program.
These systems aren’t bad, per se. But it’s not uncommon for a teacher to not understand them any more than the parents who decry them. This will hopefully be a short-term problem.
I know I’ve argued the other way (for other reasons) in CC threads, but this really is the case. If a student that hasn’t been taught the commutative property has 5x3 and right 5+5+5, they got the right answer for the wrong reason. That is the whole basis for ‘show your work’. The just happened to get lucky on that one.
If they haven’t been taught the commutative property they don’t know where it applies. What happens when they have (remember, these are 1-4th (ish) graders) 10-7 and decide that 7-10 would be easier, ya know, because you can just rearrange equations (I’m sorry, math sentences).
First, let’s learn how to do the problem correctly, understand why it works so we can apply properly to things other than the exact scenario in which is was brought up in class. Once the students have at least a somewhat firm grip on that, then move on to tricks to make it easier, like learning the commutative property. If the kid just stumbles on to it by accident, they’re not going to understand it.
To take another example, imagine they’re working with shapes (middle school geometry) and trying to figure out missing angles in a triangles and squares using a protractor. A problem shows a triangle with a 90 degree angle a 30 degree angle and says ‘use your protractor to find the last angle’. If a kid happens to notice after the first few triangles it always adds up to 180, is it okay if they just do that for the rest of it homework.
No, they’re not getting practice with their protractor, but more appropriate to what I was saying, they may take this new found ‘knowledge’ and apply it to the squares.
I should mention that I can never remember which was it’s ‘supposed’ to go and, honestly, I can’t remember a time after those first few years of math (including high school and college math) where it really mattered which way I put it, but I agree that the students should at least learn the proper way. If for no other reason it helps them to understand what multiplication is.
Don’t get me wrong, I think a lot of math (currently) it taught in an absolutely stupid way, but understanding that there is, in fact, a fundamental difference between 5+5+5 and 3+3+3+3+3 is a solid concept. Being able to manipulate numbers relies on understanding the very basics. Skipping steps won’t do you any favors at all. As they say, math builds on itself. A student (or any age) can miss a few days of lit or social studies and probably be okay, but when they miss a few days of math they fall behind very quickly.
Most of these examples seem to miss the point. The goal is for the students to understand what they are doing versus memorizing a list of facts. Having students rewrite multiplication as addition in a specific way is an interim step in their learning, not the final method for them to use in adulthood.
When I’ve helped my children or the children of friends with math, I always skim the section of their text that they are working from. I had one kid ask if I didn’t know how to do it. I explained to him that I wanted to help him with the techniques his class was working on, not skip to a technique I learned later in high school or college.
Calling them number sentences could be a good way to get a student to understand. If they understand the grammar concepts, talking about variable as roughly pronouns and using rules to eliminate options until you know which person “she” is could help.
This is something I have often heard; and I will persistently encounter it with this: when performing a simple division, like 27/11; do you really understand what you’re doing, or are you merely following a memorized algorithm? Do you even understand the full significance of the answer you arrived at?
Sure, it would be wonderful if we could start as children with the definition of natural numbers, and trace the footsteps of Peano and Landau all the way to the complex numbers and further; but the fact that Peano and Landau didn’t do their work until the 19th-20th centuries speaks, I think, something of the human mind. Heck, people even could do calculus and logarithms before they really knew why a(b+c)=ab+ac !
“Twice” means “two times” so “two times 5” is 5+5. (You wouldn’t write “twice five” as 2+2+2+2+2. That’s “two, five times” not “two times five.”)
But I decided this just now! Sixty years ago, our 2nd-grade teacher may or may not have mentioned the commutative law, but we all took it for granted and never looked back.
Perhaps I’d flunk 2nd-grade if I had to take it again.
