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.
     
    Last edited: Dec 18, 2016
  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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