Okay, math people

[LAS in LA]

Would you help DH and I settle our friendly argument?

 

He says: 2x3 means 2+2+2 (Supported by our math dictionary)

 

She says: 2x3 means 3+3 (Supported by our math curriculum)

 

Is this a philosophical point that's up for debate among mathematicians? Do I have to make humble pie for dinner and revise what I taught DD this week? What do you think?

[theYoungerMrsWarde]

I believe it is both, not either or.

 

ETA 3+3 is one step less, in my way of thinking, than 2+2+2, so I think that would be easier to do mentally.

Edited September 7, 2011 by theYoungerMrsWarde

[dcjlkplus3]

Would you help DH and I settle our friendly argument?

 

He says: 2x3 means 2+2+2 (Supported by our math dictionary)

 

She says: 2x3 means 3+3 (Supported by our math curriculum)

 

Is this a philosophical point that's up for debate among mathematicians? Do I have to make humble pie for dinner and revise what I taught DD this week? What do you think?

 

it sound like you are both right.

he is doing 3 groups of 2 and you are doing 2 groups of 3. I think you should go out to dinner to celebrate both of you being right :D.

[Samiam]

Well, as order does not matter in multiplication, either can be right.

 

Just as order does not matter in addition.

 

So really, that would be the true math lesson in this conversation. That rule has a official name "Property of something or other". It is an important rule that will be built upon in higher maths. One would want to learn that concept early in the math process.

[obsidian]

It's the commutative rule of multiplication, I think.

[lorrainejmc]

Well, I would say that 2x3 is 2+2+2 and 3x2 is 3+3. Of course the answer is the same and I think it important for kids to know that it is the same, IYKWIM?

[LAS in LA]

Yes, we do get the same answer. The answer isn't exactly the focus. I guess I'm really asking: what picture do you make for 2x3? Is it three apples plus three apples, or two apples plus two apples plus two apples? DD definitely 'gets' the commutative property, but isn't the point of it that there are different ways of grouping that result in the same answer (meaning that the groups do look different and we express that difference by saying 2x3 or 3x2)?

[LAS in LA]

I think you should go out to dinner to celebrate both of you being right :D.

 

:iagree: Great idea!

[lorrainejmc]

Yes, we do get the same answer. The answer isn't exactly the focus. I guess I'm really asking: what picture do you make for 2x3? Is it three apples plus three apples, or two apples plus two apples plus two apples? DD definitely 'gets' the commutative property, but isn't the point of it that there are different ways of grouping that result in the same answer (meaning that the groups do look different and we express that difference by saying 2x3 or 3x2)?

 

 

What I was trying to say is that 2x3 is 2 counted 3 times and 3x2 is 3 counted 2 times. My dd did not get the commutative property. To group 2x3 I would make 3 groups of 2.

Edited September 7, 2011 by lorrainejmc

[regentrude]

This is a great example for the commutative property in multiplication.

2*3=3+3 (twice three)

3*2=2+2+2 (thrice two)

Now make an array of buttons: two rows with three buttons each, makes six buttons total.

Then make another array: three rows with two buttons each, makes six buttons.

You have visualized the commutative property.

[boscopup]

When I picture 2x3, I think of "two rows or three", which would be 3+3.

 

But since it doesn't matter, I don't correct my son if he does it backwards from what the math curriculum wants him to do. They're both correct.

[Aurelia]

I agree with the others that it's the same either way, but when I see 2x3, I read it as "two, three times" or 2+2+2.

 

Maybe you should post a poll.

[Mom22ns]

This is a great example for the commutative property in multiplication.

2*3=3+3 (twice three)

3*2=2+2+2 (thrice two)

Now make an array of buttons: two rows with three buttons each, makes six buttons total.

Then make another array: three rows with two buttons each, makes six buttons.

You have visualized the commutative property.

 

:iagree:

But use M&Ms not buttons! :lol:

 

It doesn't matter which way you think of it. Make sure they both understand that both are true and accurate representations and demonstrate the commutative property. Then eat the results.:D

[Heather in VA]

While 2+2+2 and 3+3 are the same answer, 2x3 only means 2 times of 3 so the real "translation" would be 3+3. Think of it this way... A picture representation of 2x3 would look like this

### ###

 

It's 2 groups of 3 which would also be 3+3 (3 twice)

 

A picture representation of 2+2+2 would look like this....

## ## ##

 

It's 3 groups of 2. (or 3x2).

 

While the answer and number of symbols are the same, they really aren't the same thing. (sorry I tried to use :) instead of # but the forum said I have a limit of 8 images.

[NittanyJen]

Yes, we do get the same answer. The answer isn't exactly the focus. I guess I'm really asking: what picture do you make for 2x3? Is it three apples plus three apples, or two apples plus two apples plus two apples? DD definitely 'gets' the commutative property, but isn't the point of it that there are different ways of grouping that result in the same answer (meaning that the groups do look different and we express that difference by saying 2x3 or 3x2)?

 

 

You are sort of there; as I posted in another thread, the answer is the least important thing in mathematics. The logical process is what is being taught (hence the need for students to articulate in some form how they arriv at an answer).

 

2x3 really really really is either 2 rows of 3 items in a row or 3 rows of 2 items per row. From a purely mathematical standpoint there is no conceptual difference between 2x3 and 3x2. Or forming 3 bags of 2 cookies vs 2 bags of 3 cookies. If you do not use units when writing down your problem, there is no difference in the grouping based solely on how you wrote the number sentence. Teaching a child that there is one is mathematically incorrect (and confusing later when he learns that he learned it wrong).

[Parker Martin]

Mathematically it doesn't matter.

 

Grammatically you're saying "two times three." That's three two times. 3+3.

[Spy Car]

Mathematically it doesn't matter.

 

Grammatically you're saying "two times three." That's three two times. 3+3.

 

:iagree:

 

Bill

[Ellyndria]

Mathematically it doesn't matter.

 

Grammatically you're saying "two times three." That's three two times. 3+3.

 

:iagree:

 

 

I think the people who are reading it as 2+2+2 are putting a pause in there, such as, "two, times three." But, there is no pause, so it would be as it is in the quoted section.

[cottonmama]

2x3 really really really is either 2 rows of 3 items in a row or 3 rows of 2 items per row. From a purely mathematical standpoint there is no conceptual difference between 2x3 and 3x2. Or forming 3 bags of 2 cookies vs 2 bags of 3 cookies. If you do not use units when writing down your problem, there is no difference in the grouping based solely on how you wrote the number sentence. Teaching a child that there is one is mathematically incorrect (and confusing later when he learns that he learned it wrong).

 

This.

 

Multiplication is an abstract operation. I suppose you would want to choose one way or another when you define it, but that is just convention (i.e. two books may do it differently), because the two ways are equivalent -- not just equivalent in that they commute and so you get the same answer, but 100% the same thing. It's a different way of visualizing the same operation.

 

Understanding how different definitions can be describing exactly the same mathematical operation (or object) becomes extremely important in advanced math (i.e. late college for a math major). :)

 

ETA: I agree that our grammar implies one way of visualizing it, but I wouldn't insist on that way, especially to a child. Flexibility in how he visualizes it can only be a good thing.

Edited September 7, 2011 by cottonmama

[8filltheheart]

Would you help DH and I settle our friendly argument?

 

He says: 2x3 means 2+2+2 (Supported by our math dictionary)

 

She says: 2x3 means 3+3 (Supported by our math curriculum)

 

Is this a philosophical point that's up for debate among mathematicians? Do I have to make humble pie for dinner and revise what I taught DD this week? What do you think?

 

In mathematical language, in 2 x 3, 2 is the multiplicand and 3 is the multiplier. While it doesn't matter in the big picture b/c multiplication is commutative, it might be considered relevant in real world application. (how many groups of specific items do you need......it might make a difference if ordering a shipment if you have 3 containers of 2 tons or 2 containers of 3 tons if you 3 have trucks and their weight limit is 2 tons. :tongue_smilie:)

 

Anyway, the multiplicand is defined as the set w/the multiplier being the multiple of the set. So in 2 x 3, it technically means 2+2+2. Is it really important.....not really. You could simply call them both factors. :lol:

 

FWIW, here is a definition:

 

Multiplication. To multiply one number n ( a multiplicand ) by another m ( a multiplier ) means to repeat a multiplicand n as an addend m times. The result of multiplying is called a product. The operation of multiplication is written as: n x m or n · m . For example, 12 x 4 = 12 + 12 + 12 + 12 = 48. In our case 12 x 4 = 48 or 12 · 4 = 48. Here 12 is a multiplicand, 4 – a multiplier, 48 – a product. If a multiplicand n and a multiplier m are changed by places, their product is saved the same: 12 · 4 = 12 + 12 + 12 + 12 = 48 and 4 ·12 = 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 48. Therefore, a multiplicand and a multiplier are called usually factors or multipliers

Edited September 7, 2011 by 8FillTheHeart
eta: definition

[LAS in LA]

Mathematically it doesn't matter.

 

Grammatically you're saying "two times three." That's three two times. 3+3.

 

Maybe this is our problem. He's a scientist. I'm a language person. :001_smile:

[PeterPan]

Just to further confuse the issue, wikipedia points out:

There are differences amongst educationalists as to which number should normally be considered as the number of copies and whether multiplication should even be introduced as repeated addition.[1] http://en.wikipedia.org/wiki/Multiplication

 

RightStart math used an array on the abacus to show it. So you'd make two rows of 3 beads, write the equation, then turn 90 degrees and write the other equation. As for which goes first, it's one of those unmemorable little things. I guess you could do a survey of how different curricula teach it and tally it up.

[stripe]

Well, as order does not matter in multiplication, either can be right.

 

 

Well, technically, multiplication is commutative over real numbers, and complex numbers, not in every case of multiplication. Matrices for example. Or quaternions.

 

I think there actually is some dispute about n x m. I've seen it in different books explained different ways -- some say, n groups of m, others say m groups of n.

[Homemama2]

I guess I'm really asking: what picture do you make for 2x3?

 

Well, I have 2 different math books that show the picture two different ways. One would say 2x3 is 2 taken three times. So 2+2+2. The other book teaches that 2x3 is 2 groups of 3, or 3+3. Even the math books can't agree. ;) When we were using both math programs, we just used objects and manipulated them to show that these are essentially the same thing.

[Mrs Twain]

In mathematical language, in 2 x 3, 2 is the multiplicand and 3 is the multiplier. While it doesn't matter in the big picture b/c multiplication is commutative, it might be considered relevant in real world application. (how many groups of specific items do you need......it might make a difference if ordering a shipment if you have 3 containers of 2 tons or 2 containers of 3 tons if you 3 have trucks and their weight limit is 2 tons. :tongue_smilie:)

 

Anyway, the multiplicand is defined as the set w/the multiplier being the multiple of the set. So in 2 x 3, it technically means 2+2+2. Is it really important.....not really. You could simply call them both factors. :lol:

 

FWIW, here is a definition:

 

Thank you for this. My husband and I had the same discussion, my husband disagreeing with our math book. I think my husband was actually correct. He will be glad to hear that. :001_smile:

[Haiku]

 

He says: 2x3 means 2+2+2 (Supported by our math dictionary)

 

She says: 2x3 means 3+3 (Supported by our math curriculum)

 

 

To me, 2x3 is 2+2+2 because it's saying "two, three times." 3x2 would be 3+3 (three, two times). But I don't think it really matters.

 

Tara

[chepyl]

:001_smile::001_smile::001_smile:

:001_smile::001_smile::001_smile:

 

3x2 is 3 across and 2 down, therefore, it is 2 groups of 3.

2x3 is 2 down and 3 across, therefore, it is 3 groups of 2.

[ChandlerMom]

The "2 groups of 3" versus "3 groups of 2" is an artifice to TEACH math, but has nothing to DO with math. Math is a concise language and one expression, like 2x3 can mean many things.

 

How would you figure the cost of 2 pounds of apples that where $3 per pound or 3lbs at $2 each (mmmm, honey crisps)? Is 2x3 the momentum of a 2 lb rock going 3 ft/s or a 3lb rock going 2 ft/s ? Ask dh as a scientist, if he saw the numbers and result if he'd be confident to *assume* which was which. Sure, the equation is normally I=m*v, but maybe the engineer figured out the v first. LOL Unless units were shown, you wouldn't assume.

 

If you look in the arithmetic section of Capt 1 of AoPS pre algebra, they would say they are the same thing since a figure made of 2 rows by 3 columns can be rotated to be the same as a figure of 3 rows of 2 columns. Yea, that is used to introduce commutative property, but who says you multiply rows by columns, not columns by rows?

[kristinannie]

I think either one is correct. I tend to think 2+2+2, but 3+3 is easier so that might be smarter! :lol:

[ChandlerMom]

:001_smile::001_smile::001_smile:

:001_smile::001_smile::001_smile:

 

3x2 is 3 across and 2 down, therefore, it is 2 groups of 3.

2x3 is 2 down and 3 across, therefore, it is 3 groups of 2.

 

Depends on which language you are programming in! :lol: Some store (2,3) others (3,2). Bane of my existence....

 

THe order is customary in a setting, not MATH itself.

[lejardin]

Would you help DH and I settle our friendly argument?

 

He says: 2x3 means 2+2+2 (Supported by our math dictionary)

 

She says: 2x3 means 3+3 (Supported by our math curriculum)

 

Is this a philosophical point that's up for debate among mathematicians? Do I have to make humble pie for dinner and revise what I taught DD this week? What do you think?

 

 

Over here -

He says: 2x3 means 3+3

She says: 2x3 means 2+2+2

LOL Just the oppposite of your DH and you!

 

The 2x's multiplication table is written

2x1=2

2x2=4

2x3=6

2x4=8 and so on

To my way of thinking (and how I was obviously taught)

it means you are adding groups of 2 multiple times. So 2x3=6 or 2+2+2=6

 

When looking at 3x2=6 from the 3x's table

it means 3+3 because you are adding a group of 3 two times.

 

So, I agree with your DH. It doesn't mean you're wrong and he's right, just different ways of looking at it.

(Although I do believe I'm right :tongue_smilie::lol: )

However, I agree that the two of you should go out for dinner!:D

[LAS in LA]

Thanks for the replies everyone! (For the record, we did not go out to dinner. *sigh*)

[smillard00]

I'm going to throw in one more opinion from an elementary math teacher point of view. You can argue the technicalities and definitions all day and will find experts in both camps, but when learning how to multiply, 2 x 3 is the same as 3 x 2...two groups of three or three groups of two...whatever. Multiplication is commutative. I think there is a danger in making the differentiation between two groups of three vs. three groups of two because it gives the child the impression that the outcome is different which adds a layer of burden to learning multiplication facts. You want them to think 2 x 3 = 3 x 2 so they will make that connection in their facts. So, for instance, if they see 7 x 8 and feel shaky on their 7 facts, they can make that switch in their mind because they know the 8 facts better. If 8 x 7 is 56, then that must be the same for 7 x 8. When they get into larger number multiplication, you want them to feel free to arrange the numbers in the problem so the easier number is on bottom for efficiency in working the problem. I think having them try to remember if 2 x 3 is two groups of three or three groups of two could cause more confusion and complication than are necessary.

[Greenmama2]

it sound like you are both right.

he is doing 3 groups of 2 and you are doing 2 groups of 3. I think you should go out to dinner to celebrate both of you being right :D.

:iagree: Both correct.

[cottonmama]

Well, this all made for great dinnertime conversation between me and my husband! And after all was said and done, he has persuaded me that there is one correct interpretation of 2x3. Not that it matters for an elementary student, but this thread is already pretty pedantic, so I'm going to dig that hole a little deeper, lol.

 

First of all, in advanced math, we are not always multiplying an integer by an integer. As some folks in this thread have pointed out, sometimes we are multiplying a matrix by a matrix, but there are other things that we can multiply that are also not commutative. Commutativity in multiplication is a special property -- not everything has it. And the thing is that in non-commutative multiplication, mathemeticians do talk about "left multiplication" and "right multiplication", and by convention, we do left multiplication. (applying the thing on the left to the thing on the right.)

 

Also, in general, given any mathematical object Y and any integer n, mathemeticians define nxY as Y taken n times, i.e. Y+Y+...+Y n times. So in our example, 2x3 is 3+3. We use this convention in algebra when we say 3x instead of x3 to represent 3 times the variable x.

 

Hope that makes sense... I'm typing while holding a baby and making breakfast for my big girl, so not all my attention is here, lol.

Edited September 8, 2011 by cottonmama

[2squared]

:001_smile::001_smile::001_smile:

:001_smile::001_smile::001_smile:

 

3x2 is 3 across and 2 down, therefore, it is 2 groups of 3.

2x3 is 2 down and 3 across, therefore, it is 3 groups of 2.

 

:iagree:

 

This is exsctly how I present it, and I have my children tell me what they see.

[Haiku]

I'm language-oriented. As I pointed out earlier, I see 2x3 as 2, 3 times.

 

My husband is mathy. I asked him about this, and he said it's 3+3, because that is the most efficient way to get the answer.

 

:D

 

Tara

[MrsMe]

It depends...BJU teaches it as the first number is the number of sets and the second number is the amount in each set. But frankly, they're both right. I agree. Have dinner to celebrate you're both right.

[freerange]

I'm sorry. I have failed you. I hunted really hard, but could not find a picture of a mathematician in a kilt.:001_huh:

[NittanyJen]

I'm sorry. I have failed you. I hunted really hard, but could not find a picture of a mathematician in a kilt.:001_huh:

 

My brother-in-law is a kilt-wearing engineer. Is that close enough? :lol:

[freerange]

My brother-in-law is a kilt-wearing engineer. Is that close enough? :lol:

I couldn't say without seeing a photo....:D

[stripe]

I found one. Sort of.

 

http://diamondhacks.blogspot.com/2009/10/scottish-mathematician-retroactively.html

[Carpe]

I couldn't say without seeing a photo....:D

 

:iagree:

[catholicmommy]

I often like to skip to the last page of posts to see what the general consensus was on the original question, and I have to say....

 

you guys crack me up :001_smile: I guess I'll have to check back a few pages to find the math answer :lol::lol: