# multiplication with open arrays

Discussion in 'Elementary Education' started by bella84, Dec 18, 2016.

1. ### czaczaMultitudinous

Joined:
Sep 30, 2001
Messages:
24,898
2,025

Dec 18, 2016

When you change the 'way you hold it' you are changing the arrangement of rows. Rows are horizontal. Columns are vertical

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

Right. But there is no mathematical reason (that I can find or that anyone has shared) that the number of rows must come first in the expression. It doesn't change the meaning, size, or shape of the actual object/array (box of chocolates, in this case).

Pashtun likes this.
3. ### czaczaMultitudinous

Joined:
Sep 30, 2001
Messages:
24,898
2,025

Dec 18, 2016

No one is saying they aren't equivalent. The value is the same ....commutative property of multiplication. But they don't represent the same situation

a2z likes this.
4. ### czaczaMultitudinous

Joined:
Sep 30, 2001
Messages:
24,898
2,025

Dec 18, 2016

The first factor always represents 'groups of'. And that translates, at least under CCSS which seems to be the impetus behind arrays, numberlines snd bat models, to rows of.

a2z likes this.

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

I don't see how they don't represent the same situation... That's what I'm asking for clarification on. If that's true, I'd like to understand why it's true - beyond because [someone] said so. Who says that rows must come first, before columns?

I wasn't aware that 'groups of' is defined as rows specifically in the case of an array. Where is that defined? My understanding is that groups could be formed in a variety of ways within an array, if no situational context is given. Isn't that why solving with the partial products works to find the value of the given array? I'm not arguing that 'groups of' comes first in other situations. It most definitely does when the context is given in a particular way.

6. ### a2zMaven

Joined:
Sep 16, 2010
Messages:
5,592
1,480

Dec 18, 2016

It would not matter for how much chocolate needs to be made to dip the chocolates or fill the molds. It would mater by the machine filling the box.

Here is a good article:
http://www.maa.org/external_archive/devlin/devlin_01_11.html

This explains how multiplication isn't repeated addition but is scaling. 2 rows each containing 10 chocolates each. The rows is the scaling factor and the column is the value being scaled. 2 x 10 = 20.

There are some other examples that might help you see why thinking of multiplication as scalability rather than repeated addition helps understand why the order is important.

After reading posts that were made while I was making mine, the scaling factor is the "groups of".

7. ### a2zMaven

Joined:
Sep 16, 2010
Messages:
5,592
1,480

Dec 18, 2016

Following on my previous post, if there was no concrete representation of the numbers and there are no units defined, both would be equal and equivalent. The concrete representation of the array sets the scalability factor.

8. ### msleepRookie

Joined:
Nov 1, 2011
Messages:
91
34

Dec 18, 2016

You do know that these are elementary school children? Good luck with teaching them multiplication is scaling.

9. ### czaczaMultitudinous

Joined:
Sep 30, 2001
Messages:
24,898
2,025

Dec 18, 2016

But they can and do learn rows of, groups of, jumps of....

a2z likes this.
10. ### a2zMaven

Joined:
Sep 16, 2010
Messages:
5,592
1,480

Dec 18, 2016

I am aware. And thanks. (I will ignore your insult.)

11. ### a2zMaven

Joined:
Sep 16, 2010
Messages:
5,592
1,480

Dec 18, 2016

Thanks for the examples of scalability, czacza! It seems scalability is a concept that young children can learn.

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

Thanks for sharing. To be clear, I was not suggesting that we teach multiplication strictly as repeated addition, but that's beside the point... This was an interesting article. That said, I really didn't get any more information to help me understand why some people are saying the rows must go before columns in an expression, assuming that no context is given indicating how the groups are to be formed (I completely agree and understand that the expression must be written with 'groups of' first when we are told how to contextualize the groups. My question is about when no context of groups is given. We are simply shown an array and left to group it as we see fit.).

The only part of the article where the author mentioned rectangles was when he said this:
"What about when you use multiplication to compute the area of a rectangle? If the rectangle is pi inches by e inches (where e is the base for natural logarithms), then its area is

[pi in] x [e in] = pi.e sq.in. or, in approximate numerical terms, 3.14in x 2.72in = 8.54sq.in. Again, notice the units. (Note too that repeated addition is not going to get you very far with this example.)"

When I look at the CCSS (http://www.corestandards.org/Math/Content/mathematics-glossary/Table-2/), I see that their examples are all given as groups of rows, but I am wondering why we can't form 'groups of' columns.

See this example from CCSS: "There are 3 rows of apples with 6 apples in each row." Assuming that I was not told to see it that way but was only given a picture of the array, why can't I see the same array as 6 columns with 3 apples in each column?

And, furthermore, if we are talking about this example from CCSS: "What is the area of a 3 cm by 6 cm rectangle?", why can't I draw that rectangle vertically and you draw it horizontally? The area would be the same either way, and shouldn't either expression - 3cm x 6cm or 6cm x 3cm - represent both?

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

So, as I noted in my post right above this one, can we not also do "columns of"?

14. ### Leaborb192Enthusiast

Joined:
Jun 18, 2016
Messages:
2,405
1,181

Dec 18, 2016

,

Last edited: Feb 13, 2019
czacza likes this.
15. ### a2zMaven

Joined:
Sep 16, 2010
Messages:
5,592
1,480

Dec 18, 2016

It is a mathematical convention, not a rule. The convention is to be used in context and not as a pure number. Why was it decided that the row or the group comes first? Well, like any convention, someone decided that's how the convention would be noted. Why is the exponent put in the upper right hand corner of a number rather than in the upper left so you know something is going to be happening to that number? Someone decided it would be put there.

So, when assessing 2 x 10 or 10 x 2 you are assessing a convention being used not the equivalency of the equation or the equality of the equation.

Will the convention cause problems later on? Who knows.

czacza likes this.

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

We teach the commutative property right along with the "groups of" lessons at my school, before we even get to arrays. We just continue it once we get to arrays. When we get to arrays, we don't teach it as "rows of". We teach it in the context of the chocolate box I noted earlier (i.e. How many chocolates can this box with X number of columns and X number of rows hold?), so that they can build an understanding of equivalency and congruency. Next, we will learn area and perimeter.

So, it sounds like you are saying that we teach it as [rows] x [columns] because "it's probably just consistent to keep it that way" - or because it's easier on us as teachers. Again, I'm not arguing that it's not simpler to keep it that way. It is. But, I don't think it's mathematically correct to teach it that way.

Here is a message board where some other educators were discussing this same idea: https://ccgpsmathematicsk-5.wikispaces.com/share/view/60605538. This thought, at the end, really stuck out to me: "We never want to teach something which is mathematically incorrect, with the expectation that some teacher in the child's future will correct the misconception we've created. That teaching rationale is one of the greatest contributors leading to students thinking math is an arbitrary bunch of rules which don't make sense.”

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

Fair enough, and that's what I have been trying to determine all along. Now, if it's a convention, it sure doesn't seem to be one that is used consistently. And, given that so many educators, mathematicians, and people in the general public see it differently from one another, I'm not sure that it's a convention that we should be requiring our students to follow. If their work is mathematically correct and represents given situations correctly, then I'm not going to count it "wrong" just because it isn't conventional.

18. ### Leaborb192Enthusiast

Joined:
Jun 18, 2016
Messages:
2,405
1,181

Dec 18, 2016

,

Last edited: Feb 13, 2019

Joined:
Jul 20, 2012
Messages:
3,163
1,159

Dec 18, 2016

Ask him. I'd love to know what he says.

20. ### PashtunFanatic

Joined:
Jun 17, 2013
Messages:
2,985
435

Dec 18, 2016

Can someone give me an example of a problem where an array of 4x6 would be wrong without a context and not for assessing just the convention?

bella84 likes this.