Update: This post, originally titled 6 x 5 and 5 x 6 are NOT the Same! has generated the most amazing conversations! I have left the post as written, but I just felt that I needed to change the title. Enjoy the dialogue! 🙂
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Greetings from the long lost Math Coach! I am now officially a math interventionist and lovin’ life! My 4th graders have been representing multiplication with square tiles and writing equations to match pictures of arrays (like the ones shown above). These investigations have uncovered a misconception in their thinking that I wanted to share with you tonight. While 6 x 5 and 5 x 6 have the same product, they are not the same thing. The multiplication sign really means “groups of”, so I encourage students to read 6 x 5 as 6 groups of 5. This practice helps students make sense of multiplication.
I have two soapboxes to stand on tonight, so please humor me. First up is my CRA soapbox. Making the switch from additive thinking (6 + 5) to multiplicative thinking (6 x 5) is a huge transition for students, and it must begin with LOTS of concrete (hands-on) experiences. Over time, you need to overlap the concrete, representational, and abstract stages of learning to help students smoothly bridge the distance between concrete and abstract. Look, for example, at the cards below. After selecting these two cards, students can use counters to make 3 groups of 5 (concrete), draw 3 circles with 5 stars in each circle (representational), and write the equation 3 x 5 = 15 (abstract). All three stages of learning in one activity.
Next soapbox. As teachers, let’s be sure we are using precise language. If you are describing a multiplication equation, be sure that you are calling the numbers being multiplied factors and the answer to the multiplication problem the product. I’ve also found that students are easily confused about rows and columns. A simple anchor chart showing this vocabulary is a great reminder for both you and the students.
Congrats on your math interventionist position! They are blessed to have you.
I am so grateful for this post. It was an eye-opener to discover that several of my college students had no idea that 3 x 5 and 5 x 3, although having the same product, look differently. I do believe we too often jump to teaching the Commutative Property before allowing our students to delve into the concepts more deeply. A CRA approach really helps.
Thanks, Margie! I always appreciate your comments from the perspective of one who works with preservice teachers.
Hi Donna,
While I rarely comment on blogs, this issue has tremendous potential to undo deep understanding of commutative property. So, here goes: From the Common Core progressions document- http://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf , page 25- “The top row of Table 3 shows the usual order of writing multiplications of Equal Groups in the United States. The equation 3×6 means how many are in 3 groups of 6 things each: three sixes. But in many other countries the equation 3 x6 means how many are 3 things taken 6 times (6 groups of 3 things each): six threes. Some students bring this interpretation of multiplication equations into the classroom. So it is useful to discuss the different interpretations and allow students to use whichever is used in their home. This is a kind of linguistic commutativity that precedes the reasoning discussed above arising from rotating an array. These two sources of commutativity can be related when the rotation discussion occurs.”
Also, the description of the convention in the standard MCC3.OA.1: “Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7” is part of an “e.g.,” to be used as an example of one way in which the standard might be applied. The standard itself says interpret the product. As long as the student can do this and explain their thinking, they’ve met the standard, at least in my book. If, instead, the standard included an i.e., then I’d be much more inclined to expect from students exactly whatever followed the i.e.
So, it all comes down to classroom discussion and sense-making about the expression. Some students might say and see 5 taken 7 times, while another might say and see 5 groups of 7. Both uses are legitimate and the defense for one use over another is dependent upon a context and would be explored in classroom discussion. Bill McCallum has his say about this issue, here: http://commoncoretools.me/forums/topic/3-oa-a/
Let’s not add to the list of rules which aren’t really rules for kids and teachers. http://www.nctm.org/publications/blog/blog.aspx?id=42623
Respectfully,
Turtle Toms
Turtle, I am so pleased to have sparked such a lively debate! Before I address your other issues, let me state that this lesson, written by Marilyn Burns for the Do the Math program, was all about the commutative property. The point was that while the factors were the same and the products were the same, the models look different.
As for standard MCC3.OA.1, I think it pretty clearly states exactly my point–5 x 7 is 5 groups of 7 objects each. While it says “interpret” and uses e.g., I think that if they wanted it to be viewed either way, they would have included both examples.
Finally, when it comes to questions about math, I always consult Van de Walle. In his 2006 edition of Teaching Student-Centered Mathematics he states, “Students should draw or build arrays and use them to demonstrate why each array represents two different multiplications with the same product.”
If you follow my blog, you know that I am not about tricks or rules. We may have to just agree to disagree on this one.
I hate to even follow up after Turtle Tom’s post, but I agree with what was stated. I think it is important to speak the same language for modeling purposes. For example, if I want my students to model 3 x 5 in the same manner as me, I would tell them to show 3 rows of 5. However, if my goal is for students to develop an understanding of number relationships and patterns, I wouldn’t want to tell them they are not the same. The Commutative Property of Multiplication proves that they are. However, the model of each equations will look different. It is my belief that stating they are not the same undermines the Commutative Property and disconnections the relationship based on properties.
Thanks for chiming in on the debate, Jenise! Here’s a definition I found for the Commutative Property, which I’m sure is pretty standard:
A property of real numbers that states that the product of two factors is unaffected by the order in which they are multiplied; i.e., the product remains the same. Examples: 3 * 5 = 5 * 3 or 5 * x = x * 5.
I don’t see how my example undermines this at all, in fact it supports it! While the models clearly look different, the product is unaffected.
Wow it is supported! I guess the writer of that definition should remove the equal sign, which Van de Walle says indicates equivalence. I’ve also found this site which refers to the equal sign as “the same as” http://mathcoachscorner.blogspot.com/2012/11/the-meaning-of-equal-sign.html. I’m loving this math discourse! Thanks Donna!
Hi Donna,
I completely agree that students must be able to differentiate the number of groups and the number in each group when identifying the factors in a multiplication expression or equation; however, I have to disagree on your explanation of using an array to model multiplication.
I recently took a course at Northeast Georgia RESA called Teaching Algebra Through Operations and Algebraic Thinking with Dr. Stephanie Smith, a leader in CGI instruction throughout the nation. This course taught me how to guide my students, mostly through CGI, toward thinking algebraically. We discussed the different levels of work that the students might use, including direct modeling, counting strategies and invented algorithms that will eventually lead students toward using more traditional algorithms. One thing I that I distinctly recall from Dr. Smith’s lectures is that arrays are commutative and that it DOES NOT matter which factor is in the rows and columns of the array. The array can be rotated in either way and the student can still possibly see and explain the number of groups and the number in each group that the array is modeling.
Thanks for all you do. Your blog is a tremendous help in making me a better math teacher!
So in working with my kiddos today, a student described an array with 3 rows and 4 columns as 4 x 3. So I rotated it and asked how she would describe it after being rotated, and she said 3 x 4. I was okay with that. She recognized that switching the factors, while resulting in the same product, changes the way the array looks. Just like 3 bags with 4 pieces of candy each is contextually different than 4 bags with 3 pieces of candy each.
Hi Donna…I figured I’d chime in for what it’s worth.
Is it wrong if a student writes the equation 5 * 3 = 15 for their model and explains it represents 5 columns of 3? I’m hoping not.
In frequenting your blog, I know how you feel about tricks and gimmicks as a means to understand the math. You’re also extremely cautious about over-generalizing concepts because of the misconceptions that teachers and students can develop. I believe by stating that rows MUST ALWAYS come first contradicts the ownership of mathematical understanding we all strive for.
The important thing: 5 * 3 has to be 5 rows of 3 is an implied understanding, when most teachers believe it’s explicit. I couldn’t find where Van de Walle states that rows MUST come first. His models and pictures imply this understanding, however it’s not explicitly stated for all of the aforementioned reasons and previous comments… which are extremely thought provoking. They stress the importance of conceptual understanding, teaching in a context, and the integration of the SMPs.
It would be much easier to say that rows should always come first and be done with it, but that wouldn’t sit well with any of us. The context and student explanation should always supersede over-generalizations in mathematics which I believe is the case with arrays.
All the best,
Graham
Graham, thank you for sharing your thoughts! You’re absolutely right–I’m not a fan of “always” when it comes to math. 🙂
This is interesting because this identical conversation always arises in our math class when we start working in Investigations with properties of numbers. To begin the unit, students are asked to find the number of tiles that could make a rectangle 2 tiles wide, 5 tiles wide, etc leading to generalizations about the types of numbers that will always make a “2 by something” or a “5 by something.” That piece goes relatively smoothly and then students come upon the question that asks them to find the number of tiles that would make only one rectangle….hmmm….now we need to decide is a 1 x 17 the same as a 17 x 1? At this point the conversation always arises, and it matters. Before defining prime number as a number with two factors, one and itself, that question is completely relevant and I look forward to it. I feel in that moment, In terms of determining factors and finding area they are, in fact, the same. I love this conversation every time and we never come to a conclusion to “are they the same.” We make cases when they are not the same (typically context dictates this for me) and when they are the same (like the example above).
I am not saying I am right or wrong in this situation, just reflecting on recent interactions with my students. I think having them talk and explore this, without giving it a right or wrong feeling, students naturally come to the commutative property. They will typically start the sentence with, “The factors are the same, and the answer is the same, but I think it is different.” or “Everything is the same, the order doesn’t matter.”
Great, explain it.
Thank you for such a great discussion!
Kristin
It’s been an awesome discussion, Kristin! Letting students discover prime and composite numbers is one of my favorite 5th grade math lessons. Thanks for adding your thoughts.
I was working with a group today and we discussed this!!! Awesome!! I love anchor charts.
Anchor charts are such a great visual aid for students, Terri!
I have really enjoyed reading this post and all of the comments. I love a lively mathematical discussion. I can tell we are all math nerds on here! Gotta love it!!! I just wanted to share my thoughts based on what I’ve read in some of the posts.
I have to share in the nerdism.
Part 1 of my response…It wouldn’t all fit with one post :-(.
It is true conventionally that you typically see rows by columns (or rows x columns) written to describe matrices or arrays (dating back to 1851 introduced by James Joseph Sylvester). Conventions suggest that you generally refer to the rows first. Mathematics makes so much more sense in context, however. Whenever you attempt to justify and explain mathematics out of context, the discussion can become convoluted. I agree with Graham’s comment that “the context and student explanation should always supersede over-generalizations in mathematics” provided that the context and explanation make sense mathematically.
In my explanation, I will use a geometry context because that is how I see this problem. It’s all about orientation. If you were to view these two arrays as geometric figures or even take it a step further and plot them on a Cartesian plane, you can prove their congruence using basic geometric principles. Taking the first figure and rotating it -90 degrees (or 270°) produces the next figure. The shape, size, and area are maintained, so the two figures are considered congruent (or loosely referred to as equivalent, equal, or the same). In Euclidean Geometry, polygons are congruent if they are equal in all respects: same number of sides, all corresponding sides are the same length, and all corresponding interior angles are the same measure. I don’t think anyone would argue with that if it were a full 360 degree rotation. It’s easier to see it that way. If applying an isometry on a figure yields the same shape and size (and hence, area, perimeter and corresponding diagonals are the same), we can consider those two congruent. That definitely applies to the example provided. It’s not about them not being the same because mathematically they are; it’s more about their orientation being different. If I stand up and then later lie down, I’m still the same me; my orientation is just different. I won’t quote anyone here except myself, but feel free to verify my logic using any Geometry textbook or internet source.
I also compare it to when teachers suggest that a square rotated 45 degrees is now a diamond…huge misconception because it is still a square with just a different orientation (and we won’t even get into the fact that a diamond is not a mathematical shape). Even though they may not look the same, it doesn’t mean we rule out the two figures being the same quantitatively. If I go to Home Depot and ask for a 4×8 piece of plywood and then tell them that I got it wrong initially, that I would rather have a 8×4 piece of plywood; they may look at me a little strange (if all things are equal with the wood).
Let’s look at it from a science perspective. Take the earth as an example. If we look at the earth (3-D sphere) rotating on its axis, is it different or the same on its path of revolution? It’s the same earth, right, with just a different orientation?? Even though the earth has rotated causing it to be dark outside, that doesn’t mean the earth is a different earth, right?? It just means it has rotated away from the sun. Tomorrow it may be daylight, but it’s still the same earth with just a different orientation. The same principle applies here. If you take a 5X6 rectangle and rotate it in space 90 degrees clockwise, you still get the same rectangle, but with a different orientation.
***to be continued soon…
I got the biggest kick out of your Home Depot example! All of your examples, really. There can be no denying that the rectangles are indeed congruent.
Part 2 of my post since it all wouldn’t fit the first time :-(…
When you say something is “NOT the same” in mathematics or science, there are typically inferences that there is no quantitative equivalence or geometric congruence associated with the two expressions or geometric figures. In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value or that the expressions represent the same mathematical object. That is definitely the case in the two examples provided. 5 rows of 6 can be seen as 6 columns of 5, just like 6 rows of 5 can be seen as 5 columns of 6; it’s all in how you see it. If a student sees either of these from the model provided, they should be supported because they are seeing it mathematically correct. Both are correct, equivalent, equal, and the same mathematically (geometrically and quantitatively), so we should encourage students’ mathematical correctness and sound logic when they indicate the appropriate relationship.
The Commutative Property for Multiplication also indicates that a*b = b*a. As students begin discovering this and making sense of this property, they realize quickly that a*b is the same as or equal to or equivalent as b*a. The fact that there is an equal sign between the two expressions, we can infer that the two expressions are equivalent, the same, or equal. Because the product doesn’t change, then mathematically we consider them the same. Here is a great visual/demonstration of that point: http://demonstrations.wolfram.com/TheCommutativePropertyOfMultiplication/.
I think you probably meant to say that even though the figures look different or have a different geometric orientation, they are actually the same. The way you see it just depends on how you turn your head to look at it :-). That was probably way too much geometry in a discussion about multiplication, but I just love integrated mathematics. It helps us all make sense of the mathematics using various conceptual categories. Of course, third graders don’t need to formally prove congruence; however, it truly lays a solid foundation when they understand rotations and congruence without using the terms, which transfers nicely into secondary mathematics. This is a prime opportunity to lay that solid foundation for future studies. I hope this helps someone somewhere!
Keeping the math fire burning all around,
Lya
Thank you again, Dr. Snell, for adding such richness to this conversation. My blog post clearly lacked precision, and I appreciate that so many readers took the time to comment and bring clarity to the muddy waters. The math fire is, indeed, burning a little brighter tonight. 🙂
Great discussion!
The question I keep hearing is: Are 6 x 5 and 5 x 6 the same? Maybe we should ask a different question. When are 6 x 5 and 5 x 6 the same? What context would lead us to think of them as the same. What is another context where we would think of the two as different?
Instead of forcing students to think of 6 x 5 as always 6 groups of 5 or as 6 taken 5 times, let’s let them think and make sense of the mathematics.
I think the best part about this is that everyone here is thinking. Maybe the best thing we can do with this is get students involved in the conversation as Kristin mentioned above. When students are thinking, it’s a huge plus for everyone – especially students!
Mike
Great questions, Mike!
Hi Donna,
This is such a great collaborative sense-making discussion. I love Dr. Snell’s vertical view, and Mike’s gentle coaching approach. I’ll give it one more go, keeping it simpler this time around.
The equation doesn’t drive the representation, nor does the representation drive the equation. Context drives the representation. So, a representation of 3 bags with 4 pieces of candy won’t look like a representation of 4 bags with 3 pieces of candy, but either can be expressed mathematically as 3×4 or 4×3, because the expressions are equivalent.
And, Donna, I love your blog, and love the terrific ideas and resources you share. I’ve shared it with many others, and reference it often. You are providing such a great virtual comfort zone for teachers.
Thank you for all you do.
Best,
Turtle
Thank you for the kind comments, Turtle! I have so enjoyed this conversation. 🙂
This is probably going to sound extremely ignorant. I have read through all of these very thought-provoking comments, but I am still left to believe that the actual modeling of multiplication is completely different than relating it to the communtitive property which demonstrates that the order of how two numbers are multiplied does not affect the PRODUCT. When a student looks at an array that has 4 rows and 8 columns (and then turn the model 90 degrees), there would be an obvious difference in the arrangement of the array which (for all practical purposes) would be represented by 8 rows with 4 columns. The communtitive property demonstates that the product is the same….however the way the array is modeled/arranged is visibly different. And in representing multiplication on a number line…..(using 4×8 as an example) yes, students “could” represent this with jumping 8 numbers 4 times OR jumping 4 numbers 8 times would end on the same number…..however, students would see a visible difference in the process of getting there. With that being said, if the model looks different in how it is represented, then why would the written expression also look different to be consistent with our representation? And how does the MODELING of multiplication contradict the communtitive property when we are looking at two different situations.
Not ignorant at all! As you can tell by the comments, there are many subtleties and nuances to this seemingly simple question. Thanks for jumping in!
Modeling doesn’t contradict the commutative property. But stating that 5×6 and 6×5 aren’t the same does. The point here is that the expression 5×6 can be represented with either 5 groups of 6 or 6 groups of 5. The deciding factor (no pun intended!) for which way it is represented will always be context. Hope this helps! Again, thanks, Donna, for providing so many great resources.
Hi
I am. Wry interested in this post and enjoy the debate. However, I find it odd that there is such a difference culturally. I am currently studying. masters in Primary maths here in the UK and have been taught by several lecturers, backed up by several academic articles thus:
5×6 is 5, 6 times, ie 6 groups of 5. The idea that x means ‘groups of’ is a common misconception ( as interpreted here).
I agree, the array model demonstrates beautifully the commutative law. But for clarity, in Our school, I am insisting that rows and columns are circled so, t is clear which model we are looking at.
I hope that’s not too controversial!?
I completely agree. In addition, I teach the word “of” means “out of” when using fractions. 1/4, would be said, “I have 1 out of the 4” or “1 of 4” to make a whole. So there is a connection with the word “of” to division. For multiplication, 4 x 6 (as was said) I would say that we have 4, 6 times. Last comment, when using arrays, like coordinates in graphing you would say “x” first the “y”
Playing devil’s advocate:
I went to my doctor. He prescribed some pills and said he wanted me to take 3 pills a day for 14 days.
On the third day of taking the pills I woke up feeling very sick. I went back to the doctor, who was horrified I had been taking 14 pills a day.
“3 x 14” is definitely not the same as “14 x 3” he told me, albeit that the product is the same.
“What’s the difference?”, I asked.
“Don’t be daft”, he told me. That would be subtraction.
Try skip counting 3 tablets a day until you run out of what’s remaining.
“What’s remaining?”, I asked.
“Get real”, he told me, “I’m not teaching you division”
When I woke up the next morning, I was pondering the thought, to get real, and wondered if I had only imagined my experience. Oh dear, it was all too complex. What a quandary “i” found myself in. Time to try some herbal remedies instead, so I tried ginger root.
I’m going in circles with a third grade teacher who insists that all multiplication absolutely is repeated addition (no cross words, just recreational , and I’m open to learning.) Take something simple like -4×3. I can see adding -4 to itself 3 times, but how does one add 3 to itself -4 times? The teachers say that x means groups of. So maybe it matters?
Khan Academy has a GREAT video explaining this!