multiplication with open arrays

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

  1. bella84

    bella84 Aficionado

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    Question...

    If you had a drawing of an open array with 4 rows and 6 columns, what would you accept as an appropriate equation to represent that array?

    a. 4x6=24
    b. 6x4=24
    c. both a and b

    Basically, my question is: Does it matter if the number indicating the rows (or columns) comes first in the equation, or is either one acceptable, based on the commutative property of multiplication and congruency?
     
    Last edited: Dec 18, 2016
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  3. Pashtun

    Pashtun Fanatic

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    They are both acceptable.
     
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  4. bella84

    bella84 Aficionado

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    That was my understanding, too, but I have colleagues who disagree. We were deciding on how to grade an assessment that we'll be giving soon, and two of my colleagues said that they would only accept the answer with [rows] x [columns] rather than the reverse of [columns] x [rows]. It's been irking me, because I think that's wrong... and it's not the way I've been teaching my class. I think I'm just going to accept both answers, even if they don't. Thanks!
     
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  5. czacza

    czacza Multitudinous

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    Nope.

    It's 4 rows of 6. 4x6.
    6x4 is a different array. Comparing the two arrays is a teaching point when it comes to commutative property of multiplication.

    Same thing with groups, bar model, using a numberline to show multiplication.
    The order of the factors in a multiplication problem has meaning.
     
  6. Missy

    Missy Aficionado

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    We teach rows times columns.
     
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  7. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
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  8. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
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  9. bella84

    bella84 Aficionado

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    I agree with the order of the factors having meaning in terms of groups and when representing situational problems, such as the context described in a word problem. However, I just don't understand why it matters in an array. What mathematical "rule" states that an array must be represented as rows times columns?

    It seems to me that it's an arbitrary rule we use in a classroom setting to make sure that all teachers and students are on the same page. It makes things simple. But, practically speaking, it doesn't matter if I say that I describe a pan of muffins as 4x6 or 6x4. I still have 24 muffins. It also doesn't matter if I say that my living room is 10x12 or 12x10. The square footage is the same. If I go to the hardware store and buy a 2x4, then turn it 90 degrees, it's still a 2x4.

    I don't claim to be "right" here. Clearly, based on the replies here and my discussion with my colleagues, not all teachers are seeing eye-to-eye here. I'm just wondering if someone can provide me with a "rule" that states that an array must be represented one way or the other. Where does it state that rows always come first in the equation?
     
  10. bella84

    bella84 Aficionado

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    How is the meaning different? It's a rectangle... If I have a box of chocolates that comes with 2 rows of 10 chocolates, it's the same box no matter which way I hold it. The box doesn't change from a 2x10 to a 10x2 just because I pass it across the table and it gets turned in the process.

    Again, not trying to argue, and I see your point about the state test. Clearly it makes things easier in the education setting if we can all agree on which should come first. I'm just wondering if there is a definitive answer to this based on mathematical or other rules that would also apply to practical real-life situations.
     
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  11. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
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  12. bella84

    bella84 Aficionado

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    Ok, I understand what you're getting at here, but I also think that's a little far removed from teaching about the practical context of an array (boxes, seats on a carpet, muffin tins, etc.). Matrix multiplication is nowhere on the radar at this time. They are simply learning about the concept of multiplication and how to multiply with partial products.

    I don't know that this offers a definitive answer either, but it adds to the confusion that these problems on the NCTM website show models of arrays representing equations as [columns] x [rows]. For example, they show an array with 6 rows and 14 columns, and the equation states 14x6. This is the NCTM... I would expect them to have it right. Since they don't have it as [rows] x [columns], that leads me to believe that it can be represented either way... but I'm still open to having someone prove otherwise. NCTM arrays link: https://illuminations.nctm.org/Lesson.aspx?id=3210
     
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  13. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
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  14. bella84

    bella84 Aficionado

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    Lol. Excellent. Let me know if you find anything! :)
     
  15. a2z

    a2z Maven

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    No. They are learning the foundations of math. So, what comes later and how you set up for those topics is critical. That is one of the reasons we fail so terribly in math in this country. We focus on the illusion of application rather than the true understanding.
     
  16. bella84

    bella84 Aficionado

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    Thanks, Leaborb, for sharing this link. This really gets to the heart of what I'm trying to say here. We can make up some arbitrary rule because a) it makes it easier for us to be on the same page, or b) it sets them up for matrices in high school. But, IMO, to truly understand the concept, they need to grapple with the numbers in a real-world context. I, personally, don't think either of those reasons are good enough to tell students that they have to write an equation or draw an array one way or the other.

    I particularly liked this part of the article: "Can you see that, by determining ahead of time what a given problem represents that we discourage students from considering what it means to use a mathematical tool such as an array (#5)? Rather than thinking ‘okay so what does my 3 represent in my drawing and how does the array help me make sense of that’ students will be thinking ‘okay the teacher said this is always the rows – I don’t know why but it just always is’. Do you see also how this directly affects the student’s sense of purpose and therefore impacts their ability to construct a viable argument for their drawing?

    This then impacts Practice Standard #3 Construct Viable Arguments and Critique the Reasoning of Others. As we teach students to construct arguments we must remember that ‘because the teacher said so’ is not a viable argument. On the other hand ‘because in my drawing you can see here that the 3 represents my columns and the 4 represents my rows’ is a viable argument that connects the students thinking to the numeric expression with which they are working. This argument (3 for my columns and 4 for my rows) is only ‘wrong’ in the context of the teachers arbitrary ‘rule’. It is on the money with how we want students to be engaging with their models/drawings/mathematical tools and how we want them to be able to put into words what they represent as they learn to construct viable arguments."

    And this part: "They did, however, mention that there is one decent argument for this position. That is that when we get to Algebra there is a formal understanding that the first numeral/adjective/factor in a multiplication expression does represent the row in a matrix and the second numeral/adjective/factor represents the column. But this is still not reason enough to make this a ‘rule’ in an elementary class. Let’s refer to Practice Standard #4 again. In the explanation for this standard it says “Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.” The key here is that they are not to be generalized or made formal until later grades. When you formalize ideas into rules too early, you detract from students’ abilities to have discussions and describe their thinking. You also injure their ability to think about the difference between an abstract expression (3 x4), a tool to represent it’s meaning (array – rows and columns), and the actual thing you might want it to represent (a patio).

    We want to have these conversations with young students and commanding a rule-from-on-high hurts your efforts to have good math talk. Saying to students “when you are in high school you will see that mathematicians have an understood habit of having the first product represent the rows, but here we are focusing on whether you can describe your model and connect it to the numerals that represent it” is very different than saying “the first factor is always the row and the second factor is always the column”."
     
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  17. bella84

    bella84 Aficionado

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    I think letting them grapple with it in a real-world context does set them up for the topics that come later. Just telling them that they have to put factors in a certain order for no apparent reason seems like it would be the opposite of building a true understanding.
     
    Last edited: Dec 18, 2016
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  18. msleep

    msleep Rookie

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    What happened to common sense? No wonder students hate math. This just teaches students to follow some arbitrary rule just because the teacher says so.
     
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  19. a2z

    a2z Maven

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    Equality and equivalence are two different things.
     
  20. bella84

    bella84 Aficionado

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    Yes, I understand that, but I'm not sure how that is relevant in this case. Going back to my example a few posts above, if a box of chocolates has 2 rows of 10 chocolates, it doesn't matter if the box is turned horizontally or vertically. It holds 20 chocolates either way. A 2x10 box is equivalent to a 10x2 box, in that both hold 20 chocolates. Why does it matter which way the expression is written? Both boxes are congruent and equivalent.
     
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  21. bella84

    bella84 Aficionado

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    To build on this, I'm going to add that...

    Saying that I have 2 bags holding 10 chocolates each (2x10) is NOT equivalent to saying that I have 10 bags holding 2 chocolates each (10x2). Yet, they are equal.

    In the case of the box of chocolates, however, whether I represent the box that has 2 rows of 10 chocolates as 2x10 or 10x2 does not matter. Either way, it is equal, equivalent, and congruent to a box that is represented by the reverse of either expression.

    My point is that the context and application of the concept does matter. Arbitrary rules don't help students to reason with or understand a concept; rather, they only limit their understanding.
     
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  22. czacza

    czacza Multitudinous

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    When you change the 'way you hold it' you are changing the arrangement of rows. Rows are horizontal. Columns are vertical
     
  23. bella84

    bella84 Aficionado

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    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).
     
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  24. czacza

    czacza Multitudinous

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    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
     
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  25. czacza

    czacza Multitudinous

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    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.
     
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  26. bella84

    bella84 Aficionado

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    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.
     
  27. a2z

    a2z Maven

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    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".
     
  28. a2z

    a2z Maven

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    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.
     
  29. msleep

    msleep Rookie

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

    czacza Multitudinous

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    But they can and do learn rows of, groups of, jumps of....
     
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  31. a2z

    a2z Maven

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    I am aware. And thanks. (I will ignore your insult.)
     
  32. a2z

    a2z Maven

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    Thanks for the examples of scalability, czacza! It seems scalability is a concept that young children can learn.
     
  33. bella84

    bella84 Aficionado

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    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?
     
  34. bella84

    bella84 Aficionado

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    So, as I noted in my post right above this one, can we not also do "columns of"?
     
  35. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
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  36. a2z

    a2z Maven

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    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.
     
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  37. bella84

    bella84 Aficionado

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    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.”
     
  38. bella84

    bella84 Aficionado

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    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.
     
  39. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
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  40. bella84

    bella84 Aficionado

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

    Pashtun Fanatic

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    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?
     
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